secrecy capacities of compound quantum wiretap channels and applications

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Secrecy capacities of compound quantum wiretap channels and applications

Holger Boche1 Minglai Cai1 Ning Cai2 and Christian Deppe1

1Lehrstuhl fur Theoretische Informationstechnik Technische Universitat Munchen Munchen Germanylowast

2The State Key Laboratory of Integrated Services Networks University of Xidian Xian Chinadagger

We determine the secrecy capacity of the compound channel with quantum wiretapper and channelstate information at the transmitter Moreover we derive a lower bound on the secrecy capacity ofthis channel without channel state information and determine the secrecy capacity of the compoundclassical-quantum wiretap channel with channel state information at the transmitter We use thisresult to derive a proof for a lower bound on the entanglement generating capacity of the compoundquantum channel We also derive a proof for the formula for entanglement generating capacity ofthe compound quantum channel with channel state information at the encoder which was givenin additional information (cf I Bjelakovic H Boche and J Notzel Proceedings of InternationalSymposium on Information Theory ISIT 1889-1893 Korea 2009)

Keywords Compound channel Wiretap channels Quantum channels Entanglement generation

I INTRODUCTION

Our goal is to analyze information transmission over aset of indexed channels which is called a compound chan-nel The indices are referred to as channel states Onlyone channel in this set is actually used for the informationtransmission but the users can not control which chan-nel in the set will be used The capacity of the classicalcompound channel was determined in [14]A compound channel with an eavesdropper is called a

compound wiretap channel We define a compound wire-tap channel as a family of pairs of channels (WtVt) t = 1 T with a common input alphabet and possiblydifferent output alphabets connecting a sender with tworeceivers a legal one and a wiretapper where t stands forthe state of the channel pair (WtVt) The legitimate re-ceiver accesses the output of the first channel Wt in thepair (WtVt) and the wiretapper observes the outputof the second part Vt in the pair (WtVt) respectivelywhen a state t governs the channel A code for the chan-nel conveys information to the legal receiver such thatthe wiretapperrsquos knowledge of the transmitted informa-tion can be kept arbitrarily small This is a generaliza-tion of Wynerrsquos classical wiretap channel [42] to a case ofmultiple channel states In [42] the author required thatthe wiretapper can not detect the message using a weaksecurity criterion (cf Remark 2) For the achievable se-crecy rate we use the worst-case interpretation ie it islimited by the secrecy rate when the destination has theworst channel stateWe deal with two communication scenarios In the first

one only the sender is informed about the index t or inother words he has CSI where CSI is an abbreviationfor ldquochannel state informationrdquo In the second one boththe sender and the receiver do not have any informationabout that index at all

lowast [boche minglaicai christiandeppe]tumdedagger cainingmailxidianeducn

The classical compound wiretap channels were intro-duced in [28] A lower bound on the classical secrecy ca-pacity was obtained under the condition that the senderdoes not have any knowledge about the CSI In [28] theauthors required that the receiverrsquos average error goes tozero and that the wiretapper is not able to detect themessage with respect to the same security criterion asin [42] The result of [28] was improved in [13] by us-ing a stronger condition for the limit of the legitimatereceiverrsquos error ie the maximal error should go to zeroas well as a stronger condition for the security criterion(cf Remark 2) Furthermore the secrecy capacity wasdetermined for the case in which the sender had knowl-edge about the CSI

In this paper we consider quantum channels A quan-tum channel can transmit both classical and quantuminformation We consider the capacity of quantum chan-nels carrying classical information This is equivalent toconsidering the capacity of classical-quantum channelswhere the classical-quantum channels are quantum chan-nels whose senderrsquos inputs are classical variables Theclassical capacity of quantum channels has been deter-mined in [23] [24] [34] and [35] In general there aretwo ways to represent a quantum channel with linear al-gebraic tools (cf eg Section VII) either as a sum ofseveral transformations or as a single unitary transfor-mation which explicitly includes the unobserved environ-ment We use the latter one for our result in the entan-glement generating capacity These two representationscan both be used to determine the entanglement gener-ating capacity for quantum channels but it is unknownif this holds for the entanglement generating capacity ofcompound quantum channels

We analyze two variants of compound wiretap quan-tum channels in this paper The first variant is calledthe classical compound channel with quantum wiretap-per In this channel model we assume that the wiretapchannels are quantum channels while the legal transmis-sion channels are classical channels The second variantis called the compound classical-quantum wiretap chan-nel In this channel model we assume that both fami-

lies of channels are quantum channels while the sendertransmits classical information

A quantum wiretap channel is described by a map Nwhich maps the set of density operators on a system GA

to the set of density operators on a composite systemGBZ Here GA is the the system of the sender GB

is the system observed by the legal receiver and GZ isthe system observed by the wiretapper We only allowpassive eavesdropping attacks ie the actions which thewiretapper performs on his system have no influence onthe legal receiverrsquos system For active eavesdropping at-tacks ie the actions which the wiretapper performs onhis system have influences on the legal receiverrsquos sys-tem cf please arbitrarily varying quantum channelswhich are generalizations of compound quantum chan-nels ([2] [10] [15] [1] and [17]) Following [19] wedefine a quantum wiretap channel with passive eaves-dropping attacks as a pair of channels (WV ) For everydensity operator ζ on GA we define W (ζ) = trB(N(ζ))and V (ζ) = trZ(N(ζ))

References[8] and [7] are two well-known examples forsecure quantum information transmission using quantumkey distributions Good one-shot results for quantumchannels with a wiretapper who is limited in his actionshave been obtained But our goal is to have a moregeneral theory for channel security in quantum informa-tion theory ie message transmission should be secureagainst every possible kind of eavesdropping Further-more we are interested in asymptotic behavior when wedeliver a large volume of messages by many channel usesTherefore we consider an alternate paradigm for the de-sign of quantum channel systems which is called embed-ded security Here we embed protocols with a guaran-teed security into the bottom layerof the model of com-munications systems which is the physical layer We usechannels for example fiber-optic cables for generatingand transmitting secure messages which generate securekeys or secure message transmissions

Since we allow every possible kind of eavesdroppingwe use the Holevo χ quantity as our security criterion(cf (13)) By [24] and [35] the wiretapper can never ob-tain more information asymptotically than the Holevo χquantity no matter which strategy the wiretapper usesAnother widely used security criterion is the variationaldistance between pApZ and pAZ Here pAZ is the jointprobability describing the senderrsquos random variable andthe wiretapperrsquos random variable pA is the marginalprobability describing the senderrsquos random variable andpZ is the marginal probability describing the wiretap-perrsquos random variable respectively The Holevo χ quan-tity using a strong condition for the security criterion(cf Remark 2) is stronger than the variational distancebetween pApZ and pAZ in the sense that if the Holevoχ quantity between pA and the wiretap channelrsquos out-put (using strong condition) goes to zero the variationaldistance between pApZ and pAZ goes to zero as well (cf

[21] and [16])

Our results are summarized as follows Under the con-dition that the sender has knowledge about the CSI thesecrecy capacity for these two channel models is derivedAdditionally when the sender does not have any knowl-edge about the CSI we determine the secrecy capacityof the compound classical-quantum wiretap channel andgive a lower bound for the secrecy capacity of the classicalcompound channel with quantum wiretapper

As an application of the above results we turn to thequestion ldquoWhat is the maximal amount of entanglementthat we can generate or transmit over a given compoundquantum channelrdquo For the sender and the receiver theobjective is to share a nearly maximally entangled stateon a (2nR times 2nR) dimensional Hilbert space by usinga large number n instances of the compound quantumchannel In [5] it is shown how to send a large amountof entangled quantum states through a noisy quantumchannel such that the channel does not modify the entan-glement However the study of entanglement generationallows a noisy quantum channel to modify the entan-glement as long as the transmitters can use a recoveryalgorithm to restore the entanglement The entangle-ment generating capacity of a quantum channel has beendetermined in [20] and [29] The entanglement generat-ing capacities of a compound quantum channel with andwithout CSI have been determined in [11] and [12] Inour paper we derive a lower bound on the entanglementgenerating capacity of the compound quantum channelby using an alternative technique to the method in [11]and [12] (cf Section VII) Furthermore we derive the en-tanglement generating capacity of the compound quan-tum channel with CSI at the encoder using an alternativetechnique

The main definitions are given in Section II

In Section III we present some known results for theclassical compound wiretap channel which are used forthe proof of the results in Section IV

In Section IV we discuss the classical compound chan-nel with a quantum wiretapper For the case when thesender has the full knowledge about the CSI we derivethe secrecy capacity For the case when the sender doesnot know the CSI we give a lower bound for the se-crecy capacity In this channel model the wiretapper usesclassical-quantum channels

In Section V we derive the secrecy capacity of thecompound classical-quantum wiretap channel with CSIIn this model both the receiver and the wiretapper useclassical quantum channels and the set of the channelstates may be finite or infinite

In Section VI we use the results of Section V to derivea lower bound on the entanglement generating capacityfor the compound quantum channel The entanglementgenerating capacity of the compound quantum channelwith CSI at the encoder is also derived

In Section VII we discuss our proof of the previoussection and remind the reader about the two ways to

2

represent a quantum channel with linear algebraic tools

II PRELIMINARIES

For a finite set B we denote the set of probabilitydistributions on B by P (B) Let ρ1 and ρ2 be Hermitianoperators on a finite-dimensional complex Hilbert spaceG We say ρ1 ge ρ2 and ρ2 le ρ1 if ρ1 minus ρ2 is positive-semidefinite For a finite-dimensional complex Hilbertspace G we denote the set of density operators on G by

S(G) = ρ isin L(G) ρ is Hermitian ρ ge 0G tr(ρ) = 1

where L(G) is the set of linear operators on G and 0G isthe null matrix on G Note that any operator in S(G) isboundedFor finite sets A and B we define a (discrete) classical

channel V A rarr P (B) A ni x rarr V(x) isin P (B) to be asystem characterized by a probability transition matrixV(middot|middot) For x isin A and y isin B V(y|x) expresses the proba-bility of the output symbol y when we send the symbol xthrough the channel The channel is said to be memory-less if the probability distribution of the output dependsonly on the input at that time and is conditionally inde-pendent of previous channel inputs and outputsLet n isin N For a finite set A we define An =

(a1 an) ai isin A foralli isin 1 n For a finite setA and a finite-dimensional complex Hilbert space H thespace which the vector v1 otimes middot middot middot otimes vn vi isin H foralli isin1 n span is defined by Hotimesn We also write an

and vn for the elements of An and Hotimesn respectively

Let n isin N For a discrete random variable X on afinite set A and a discrete random variable Y on a finiteset B we denote the Shannon entropy by H(X) and themutual information between X and Y by I(X Y ) (cf[40])For a probability distribution P on a finite set A a

conditional stochastic matrix Λ and a positive constantδ we denote the set of typical sequences by T n

Pδ and the

set of conditionally typical sequences by T nΛδ(x

n) (here

we use the strong condition) (cf [40])

For finite-dimensional complex Hilbert spaces G andGprime a quantum channel V S(G) rarr S(Gprime) S(G) niρ rarr V (ρ) isin S(Gprime) is represented by a completely posi-tive trace preserving map which accepts input quantumstates in S(G) and produces output quantum states inS(Gprime)

If the sender wants to transmit a classical messagem isin M to the receiver using a quantum channel hisencoding procedure will include a classical-to-quantumencoder M rarr S(G) to prepare a quantum message stateρ isin S(G) suitable as an input for the channel If thesenderrsquos encoding is restricted to transmitting an indexedfinite set of quantum states ρx x isin A sub S(G) then wecan consider the choice of the signal quantum states ρx tobe a component of the channel Thus we obtain a chan-nel with classical inputs x isin A and quantum outputs

σx = V (ρx) which we call a classical-quantum channelThis is a map V A rarr S(Gprime) X ni x rarr V(x) isin S(Gprime)which is represented by the set of |A| possible outputquantum states σx = V(x) = V (ρx) x isin A sub S(Gprime)meaning that each classical input of x isin A leads to adistinct quantum output σx isin S(Gprime)

Let n isin N Following [40] we define the n-th memo-ryless extension of the stochastic matrix V by V

n iefor xn = (x1 xn) isin An and yn = (y1 yn) isin BnVn(yn|xn) =

prodni=1 V(yi|xi) Following [40] we define

the n-th extension of quantum channel and classical-quantum channel as follows Associated with V and V

are the channel maps on an n-block V otimesn S(Gotimesn) rarrS(Gprimeotimesn

) and Votimesn An rarr S(Gprimeotimesn

) such that for anyρn = ρ1otimesmiddot middot middototimesρn isin S(Gotimesn) and any xn = (x1 xn) isinAn V otimesn(ρn) = V (ρ1) otimes middot middot middot otimes V (ρn) and V

otimesnt (xn) =

V(x1)otimes middot middot middototimes V(xn) respectively Although the outcomesof V otimesn can be written as n tuples we still regard them aselements of S(Gotimesn) because for the proof of our resultswe need tools which are defined on the space Gotimesn (cf(1)-(7))

Let A be a finite set and G be a finite-dimensionalcomplex Hilbert space For a quantum state ρ isin S(G)we denote the von Neumann entropy of ρ by

S(ρ) = minustr(ρ log ρ)

Let V A rarr S(G) be a classical-quantum channel Fol-lowing [3] for P isin P (A) the conditional entropy of thechannel for V with input distribution P is denoted by

S(V|P ) =sum

xisinA

P (x)S(V(x))

Remark 1 The following definition is a more generaldefinition of conditional entropy in quantum informa-tion theory Let P and Q be quantum systems Wedenote the Hilbert space of P and Q by GP and GQrespectively Let φPQ be a bipartite quantum state inS(GPQ) We denote S(P | Q)ρ = S(φPQ) minus S(φQ)Here φQ = trP(φ

PQ)

For quantum states ρ and σ isin S(G) we denote the fi-delity of ρ and σ by

F (ρ σ) = radicρradicσ21

where middot 1 stands for the trace normWe denote the identity operator on a space G by IG

and the identity superoperator on G by idG

Let A be a finite set and let G and Gprime be a finite-dimensional complex Hilbert spaces For a quantum stateρ isin S(G) and a quantum channel V S(G) rarr S(Gprime) thecoherent information is defined as

IC(ρ V ) = S(V (ρ))minus S ((IG otimes V )(|ψ〉〈ψ|))

where |ψ〉〈ψ| is an arbitrary purification of ρ in S(G) otimesS(G) Let Φ = ρx x isin A be a set of quantum states

3

labeled by elements of A For a probability distributionP on A the Holevo χ quantity is defined as

χ(P Φ) = S

(

sum

xisinA

P (x)ρx

)

minussum

xisinA

P (x)S (ρx)

Let n isin N let A be a finite set and G be a finite-dimensional complex Hilbert space For ρ isin S(G) andα gt 0 there exists an orthogonal subspace projector Πρα

commuting with ρn and satisfying

tr (ρnΠρα) ge 1minus d

4nα2 (1)

tr (Πρα) le 2nS(ρ)+Kdαradicn (2)

Πρα middot ρn middotΠρα le 2minusnS(ρ)+KdαradicnΠρα (3)

where d = dimH and K is a positive constantLet V Ararr S(G) be a classical-quantum channel For

P isin P (A) α gt 0 and xn isin An there exists an or-thogonal subspace projector ΠVα(x

n) commuting withVotimesn(xn) and satisfying

tr(

Votimesn(xn)ΠVα(x

n))

ge 1minus ad

4nα2 (4)

tr (ΠVα(xn)) le 2nS(V|P )+Kadα

radicn (5)

ΠVα(xn) middot Votimesn(xn) middot ΠVα(x

n)

le 2minusnS(V|P )+KadαradicnΠVα(x

n) (6)

where a = |A| and K is a positive constant (cf [40])For the classical-quantum channel V Ararr S(G) every

probability distribution P on A defines a quantum statePV on S(G) which is the resulting quantum state atthe output of V when the input is sent according to P Thus for αprime gt 0 we can define an orthogonal subspaceprojector ΠPVαprime

radica which fulfills (1) (2) and (3) (here we

set ρ = PV and α = αprimeradica) Furthermore for ΠPVαprimeradica

we have the following inequality

tr(

Votimesn(xn) middotΠPVα

radica

)

ge 1minus ad

4nα2 (7)

Let A B and C be finite sets H H prime and H primeprime becomplex Hilbert spaces and P and Q be quantum sys-tems We denote the Hilbert space of P and Q by HP

and HQ respectively Let θ = 1 T be a finite setFor every t isin θ let

Wt be a classical channel Ararr P (B)Vt be a classical channel Ararr P (C)Vt be a classical-quantum channel Ararr S(H)Wt be a quantum channel S(H prime) rarr S(H primeprime)

Vt be a quantum channel S(H prime) rarr S(H)Nt be a quantum channel S(HP) rarr S(HQ)

We call the set of the classical channel pairs (WtVt)tisinθ

a (classical) compound wiretap channel When thechannel state is t and the sender inputs x isin A into thechannel the receiver receives the output y isin B withprobability Wt(y|x) while the wiretapper receives theoutput z isin Z with probability Vt(z|x)We call the set of the classical channel and classical-

quantum channel pairs (Wt Vt)tisinθ a compound chan-nel with quantum wiretapper When the channelstate is t and the sender inputs x isin A into the channelthe receiver receives the output y isin B with probabilityWt(y|x) while the wiretapper receives an output quan-tum state Votimesn

t (x) isin S(H)

We call the set of the quantum channel pairs(Wt Vt)tisinθ a quantum compound wiretap chan-nel When the channel state is t and the sender inputsa quantum state ρ isin S(H prime) into the channel the re-ceiver receives an output quantum state Wt(ρ) isin S(H primeprime)while the wiretapper receives an output quantum stateVt(ρ) isin S(H)

We call the set of the quantum channel (Nt)tisinθ aquantum compound channel When the channelstate is t and the sender inputs a quantum state ρP isinS(HP) into the channel the receiver receives an outputquantum state Nt(ρ

P) isin S(HQ)

We distinguish two different scenarios according to thesenderrsquos knowledge of the channel state

bull the sender has the CSI ie he knows which t thechannel state actually is

bull the sender does not have any CSI

In both cases we assume that the receiver does not haveany CSI but the wiretapper always has the full knowl-edge of the CSI Of course we also have the case whereboth the sender and the receiver have the CSI but thiscase is equivalent to the case when we only have one pairof channels (Wt Vt) instead of a family of pairs of chan-nels (Wt Vt) t = 1 T

An (n Jn) code for the classical compound wiretapchannel (WtVt)tisinθ consists of a stochastic encoder E 1 Jn rarr P (An) specified by a matrix of conditionalprobabilities E(middot|middot) and a collection of mutually disjointsets Dj sub Bn j isin 1 Jn (decoding sets)

If the sender has the CSI then instead of using a sin-gle code for all channel states we may use the followingstrategy For every t isin θ the sender and the receiverbuild an (n Jn) code (Et Dj j = 1 Jn) such

that all codes in

(Et Dj j = 1 Jn) t isin θ

share the same decoding sets Dj j = 1 Jn whichdo not depend on t to transform the message

A non-negative number R is an achievable secrecy rate

4

for the classical compound wiretap channel (WtVt) hav-ing CSI at the encoder if for every positive ε δ everyt isin θ and a sufficiently large n there is an (n Jn) code(Et Dj j = 1 Jn) such that 1

n log Jn ge R minus δand

maxtisinθ

maxjisin1Jn

sum

xnisinAn

Et(xn|j)Wn

t (Dcj |xn) le ε (8)

maxtisinθ

I(XuniKnt ) le ε (9)

where Xuni is a random variable uniformly distributedon 1 Jn Kn

t are the resulting random variables atthe output of wiretap channels V

nt Here we denote the

complement of a set Ξ by Ξc

Remark 2 A weaker and widely used security crite-rion eg in [28] (also cf [42] for wiretap channelrsquossecurity criterion) is obtained if we replace (9) withmaxtisinθ

1nI(XuniK

nt ) le ε In this paper we will follow

[13] and use (9)

A non-negative number R is an achievable secrecyrate for the classical compound wiretap channel (WtVt)having no CSI at the encoder if for every positive εδ and a sufficiently large n there is an (n Jn) code(E Dj j = 1 Jn) such that 1


maxtisinθ

maxjisin1Jn

sum

xnisinAn

E(xn|j)Wnt (D

cj |xn) le ε (10)

maxtisinθ


An (n Jn) code for the compound channel with quan-tum wiretapper (Wt Vt)tisinθ consists of a stochastic en-coder E 1 Jn rarr P (An) and a collection of mutu-ally disjoint sets Dj sub Bn j isin 1 Jn (decodingsets)

A non-negative number R is an achievable secrecyrate for the compound channel with quantum wiretap-per (Wt Vt)tisinθ having CSI at the encoder if for everypositive ε δ every t isin θ and a sufficiently large n thereis an (n Jn) code (Et Dj j = 1 Jn) such that1n log Jn ge Rminus δ and

maxtisinθ

maxjisin1Jn

sum

xnisinAn

Et(xn|j)Wn


maxtisinθ

χ(XuniZnt ) le ε (13)

Here Znt are the resulting quantum states at the output

of wiretap channels Vnt

A non-negative number R is an achievable secrecyrate for the compound channel with quantum wiretap-per (Wt Vt)tisinθ having no CSI at the encoder if for every

positive ε δ and a sufficiently large n there is an (n Jn)code (E Dj j = 1 Jn) such that 1

n log Jn ge Rminusδand

maxtisinθ

maxjisin1Jn

sum

xnisinAn

E(xn|j)Wnt (D

cj |xn) le ε (14)

maxtisinθ


An (n Jn) code carrying classical information for thecompound quantum wiretap channel (Wt Vt)tisinθ consistsof a family of quantum states w(j) j = 1 Jn subS(H primeotimesn

) and a collection of positive semi-definite oper-

ators Dj j isin 1 Jn on S(H primeprimeotimesn) which is a par-

tition of the identity iesumJn

j=1Dj = IHprimeprimeotimesn

A non-negative number R is an achievable secrecy ratewith classical input for the compound quantum wiretapchannel (Wt Vt)tisinθ having CSI at the encoder with av-erage error if for every positive ε δ every t isin θ anda sufficiently large n there is an (n Jn) code carryingclassical information (wt(j) j Dj j) such that1n log Jn ge Rminus δ and

maxtisinθ

1

Jn

Jnsum

j=1

tr(

(IHprimeprimeotimesn minusDj)Wotimesnt (wt(j))

)

le ε (16)

maxtisinθ


A non-negative number R is an achievable secrecy ratewith classical input for the compound quantum wiretapchannel (Wt Vt)tisinθ having no CSI at the encoder if forevery positive ε and δ and a sufficiently large n thereis an (n Jn) code carrying classical information (w(j) j Dj j) such that 1

n log Jn ge Rminus δ and

maxtisinθ

maxjisin1Jn

tr(

(IHprimeprimeotimesn minusDj)Wotimesnt (w(j))

)

le ε

(18)

maxtisinθ


Instead of ldquoachievable secrecy rate with classical inputfor the compound quantum wiretap channel rdquo we say Ris an achievable secrecy rate for the compound classical-quantum wiretap channel (Wt Vt)tisinθ

An (n Jn) code carrying quantum information forthe compound quantum channel

(

Notimesnt

)

tisinθconsists of a

Hilbert spaces HA such that dimHA = Jn and a gen-eral decoding quantum operation D ie a completelypositive trace-preserving map D S(HQn

) rarr S(HM)where HM is a Hilbert space such that dimHM = JnThe code can be used for entanglement generation inthe following way The sender prepares a pure bipar-tite quantum state |ψ〉APn

defined on HA otimesHPn

and

5

sends thePn portion of it through the channel Notimesnt The

receiver performs the general decoding quantum opera-tion on the channel output D S(HQn

) rarr S(HM) Thesender and the receiver share the resulting quantum state

ΩAMt = [IA otimes (D Notimesn

t )](

|ψ〉〈ψ|APn)

(20)

A non-negative number R is an achievable entangle-ment generating rate for the compound quantum channel(

Notimesnt

)

tisinθif for every positive ε δ and a sufficiently large

n there is an (n Jn) code carrying quantum information(

HA D)

such that 1n log Jn ge Rminus δ and

mintisinθ

F(

ΩAMt |ΦK〉〈ΦK |AM

)

ge 1minus ε (21)

where

|ΦK〉AM =

radic

1

Jn

Jnsum

j=1

|j〉A|j〉M

which is the standard maximally entangled state sharedby the sender and the receiver |j〉A and |j〉M areorthonormal bases for HA and HM respectively

The largest achievable secrecy rate is called the secrecycapacity The largest achievable entanglement generatingrate is called the entanglement generating capacity

III CLASSICAL COMPOUND WIRETAP

CHANNELS

In this section we present some known results for theclassical compound wiretap channel which are used forthe proof of the results in Section IVLet A B C θ and (WtVt)tisinθ be defined as in Sec-

tion II For every t isin θ we fix a probability distribution

pt on An Let pprimet(x

n) =

pnt (x

n)pnt (T n

ptδ) if xn isin T n

ptδ

0 else

and X(t) = X(t)jl jisin1Jnlisin1Lnt be a family of

random matrices whose entries are selected iid accord-ing to pprimet where Lnt is a natural number which will bespecified laterIt was shown in [13] that for any positive ω if we set

Jn = lfloor2n(mintisinθ(I(ptWt)minus 1nlogLntminusmicro)rfloor

where micro is a positive constant which does not depend onj t and can be arbitrarily small when ω goes to 0 thefollowing statement is valid There are such Dj j =1 Jn that for all t isin θ and for all Lnt isin N

Pr

maxjisin1Jn

Lntsum

l=1

1

LntW

nt (D

cj |X

(t)jl ) gt

radicT2minusnω2

leradicT2minusnω2 (22)

Since only the error of the legitimate receiver is analyzedfor the result (22) just the channels Wt but not those ofthe wiretapper are regarded For every j isin 1 Jnl isin 1 Lnt and t isin θ Wn

t (Dcj |X

(t)jl ) is a random

variable taking values in ]0 1[ which depends on X(t)jl

since we defined X(t)jl as a random variable with value in

AnIn view of (22) by choosing Lnt =

lfloor

2n[I(ptVt)+τ ]rfloor

forany positive constant τ the authors of [13] showed thatCSCSI the secrecy capacity of the compound wiretapchannel with CSI at the transmitter is given by

CSCSI ge mintisinθ

maxUrarrArarr(BK)t

(I(U Bt)minus I(U Kt)) (23)

where Bt are the resulting random variables at the out-put of legal receiver channels Kt are the resulting ran-dom variables at the output of wiretap channels Themaximum is taken over all random variables that satisfythe Markov chain relationships U rarr A rarr (BZ)t HereA rarr (BZ)t means A rarr Bt times Zt where A rarr Bt means

AWtminusminusrarr Bt and Ararr Zt means A

Vtminusrarr ZtBjelakovic Boche and Sommerfeld also proved in [13]

CSCSI ge mintisinθ

maxUrarrArarr(BK)t

(I(U Bt)minus I(U Kt))

Together with this inequality and (23) we have

CSCSI = mintisinθ

maxUrarrArarr(BK)t


Analogously in the case without CSI the idea is simi-lar to the case with CSI Fix a probability distribution p

on An Let pprime(xn) =

pn(xn)pn(T n

pδ) if xn isin T n

pδ

0 else

andXn = Xjljisin1Jnlisin1Ln where Ln a natu-ral number will be specified later be a family of randommatrices whose components are selected iid accordingto pprimeFor any ω gt 0 we define

Jn = lfloor2n(mintisinθ(I(pWt)minus 1nlogLnminusmicro)rfloor

where micro is a positive constant which does not depend onj and t and can be arbitrarily small when ω goes to 0There are Dj j = 1 Jn such that for all t isin θ andfor all Ln isin N

Pr

(

maxjisin1Jn

Lnsum

l=1

1

LnW

nt (D

cj |Xjl) gt

radicT 2minusnω2

)


In view of (25) by choosing Ln =lfloor

2n[maxt I(ptVt)+τ4 ]rfloor

where τ is a positive constant the authors of [13] showedthat CS the secrecy capacity of the compound wiretapchannel without CSI at the transmitter is lower boundedas follows

CS ge maxUrarrArarr(BK)t

(mintisinθ

I(U Bt)minusmaxtisinθ

I(U Kt)) (26)

6

IV COMPOUND CHANNELS WITH

QUANTUM WIRETAPPER

In this section we discuss the classical compound chan-nel with a quantum wiretapper For the case when thesender has the full knowledge about the CSI we derivethe secrecy capacity For the case when the sender doesnot know the CSI we give a lower bound for the se-crecy capacity In this channel model the wiretapperuses classical-quantum channelsLet A B H θ and (Wt Vt)tisinθ be defined as in Section

II

Theorem 1 The secrecy capacity of the compound chan-nel with quantum wiretapper (Wt Vt)tisinθ in the case withCSI at the transmitter CSCSI is given by

CSCSI = mintisinθ

maxUrarrArarr(BZ)t

(I(U Bt)minuslim supnrarrinfin

1

nχ(U Zn

t ))

(27)Respectively in the case without CSI the secrecy capac-ity of the compound channel with quantum wiretapper(Wt Vt)tisinθ CS is lower bounded as follows

CS ge maxUrarrArarr(BZ)t

(mintisinθ

I(U Bt)minusmaxtχ(U Zt)) (28)

where Bt are the resulting random variables at the outputof legal receiver channels and Zt are the resulting ran-dom quantum states at the output of wiretap channels

Remark 3 We have only the multi-letter formulas (27)and (28) since we do not have a single-letter formulaeven for a quantum channel which is neither compoundnor has wiretappers

Proof 1) Lower bound for case with CSIFor every t isin θ fix a probability distribution pt on A

nLet


where Lnt is a natural number that will be specified

below and micro is defined as in Section III Let pprimet X(t)

and Dj be defined as in the classical case Then (22)still holds since the sender transmits through a classicalchannel to the legitimate receiver

Let

Qt(xn) = ΠptVtα

radicaΠVtα(x

n)middotVotimesnt (xn)middotΠVtα(x

n)ΠptVtαradica

where α will be defined later

Lemma 1 (Tender Operator cf [41] and [32])Let ρ be a quantum state and X be a positive operatorwith X le I and 1minus tr(ρX) le λ le 1 Then

ρminusradicXρ

radicX le

radic2λ (29)

Tender Operator was first introduced in [41] where it

has been shown that ρ minusradicXρ

radicX le

radic8λ In [32]

the result of [41] has been improved and (29) has beenproved

In view of the fact that ΠptVtαradica and ΠVtα(x

n) areboth projection matrices by (1) (7) and Lemma 1 forany t and xn it holds that

Qt(xn)minus V

otimesnt (xn) le

radic

2(ad+ d)

nα2 (30)

We set Θt =sum

xnisinT nptδ

ppriment (xn)Qt(x

n) For given zn

and t 〈zn|Θt|zn〉 is the expected value of 〈zn|Qt(xn)|zn〉

under the condition xn isin T nptδ

Lemma 2 (Covering Lemma cf [3]) Let V be a fi-nite dimensional Hilbert space Let E sub S(V) be a col-lection of density operators such that σ le micro middot IV for allσ isin E and let p be a probability distribution on on EFor any positive λ we define a sequence of iid ran-dom variables X1 XL taking values in E such that

for all σ isin E we have p(σ) = Pr

Xi = Πprimeρλ middot σ middotΠprime

ρλ

where ρ =sum

σisinE p(σ)σ and Πprimeρλ is the projector onto

the subspace spanned by the eigenvectors of ρ whose cor-responding eigenvalues are greater than λ

dimV For anyǫ isin]0 1[ the following inequality holds

Pr

(

Lminus1Lsum

i=1

Xi minusΠprimeρλ middot ρ middot Πprime

ρλ gt ǫ

)

le 2 middot (dimV)exp(

minusL ǫ2λ

2 ln2(dimV)micro

)

(31)

Let V be the range space of ΠptVtαradica By (2) we have

dimV le 2nS(pt)+Kdαradican

Furthermore for all xn holds

Qt(xn)

= ΠptVtαradicaΠVtα(x

n) middot Votimesnt (xn) middotΠVtα(x

n)ΠptVtαradica

le 2minusn(S(Vt|pt)+Kadαradicn)ΠptVtα

radicaΠVtα(x

n)ΠptVtαradica

le 2minusnmiddotS(Vt|pt)+Kadαradicn middot ΠptVtα

radica

le 2minusnmiddotS(Vt|pt)+Kadαradicn middot IV (32)

The first inequality follows from (6) The second inequal-ity holds because ΠVtα and ΠptVtα

radica are projection ma-

trices The third inequality holds because ΠptVtαradica is a

projection matrix onto V Let λ = ǫ By applying Lemma 2 where we set micro =

2minusnmiddotS(Vt|pt)+Kadαradicn in (31) in view of (32) if n is large

enough we have

Pr

Lntsum

l=1

1

LntQt(Xjl)minusΘt gt ǫ

7

le 2n(S(pt)+Kdαradican) (33)

middot exp(

minusLntǫ2

2 ln 2λ middot 2n(S(Vt|pt)minusS(pt))+Kdα

radicn(

radicaminus1)

)

= 2n(S(pt)+Kdαradican)

middot exp(

minusLntǫ2

2 ln 2λ middot 2n(minusχ(ptZt))+Kdα

radicn(

radicaminus1)

)

le exp(

minusLnt middot 2minusn(χ(ptZt)+ζ))

(34)

where ζ is some suitable positive constant which doesnot depend on j t and can be arbitrarily small when ǫis close to 0 The equality in the last line holds since

S(pt)minus S(Vt|pt)

= S

sum

j

pt(j)sum

l

1

LntVotimesnt (X

(t)jl )

minussum

j

pt(j)S

(

sum

l

1

LntVotimesnt (X

(t)jl )

)

= χ(ptZt)

Let Lnt = lceil2n(χ(ptZt)+2ζ)rceil and n be large enoughthen by (34) for all j it holds that

Pr

Lntsum

l=1

1

LntQt(X

(t)jl )minusΘt gt ǫ

le exp(minus2nζ) (35)

and

Pr

Lntsum

l=1

1

LntQt(X

(t)jl )minusΘt le ǫ forallt forallj

= 1minus Pr

⋃

t

⋃

j

Lntsum

l=1

1

LntQt(X


ge 1minus TJnexp(minus2nζ)

ge 1minus T 2n(mintisinθ(I(ptWt)minus 1nlogLnt)exp(minus2nζ)

ge 1minus 2minusnυ (36)

where υ is some suitable positive constant which does notdepend on j and t

Remark 4 Since exp(minus2nζ) converges to zero double ex-ponentially quickly the inequality (36) remains true evenif T depends on n and is exponentially large over n iewe can still achieve an exponentially small error

From (22) and (36) it follows For any ǫ gt 0 if n is

large enough then the event

⋂

t

maxjisin1Jn

Lntsum

l=1

1

LntW

nt (D

cj(X )|X(t)

jl ) le ǫ

cap

Lntsum

l=1

1

LntQt(X


has a positive probability This means that we can find

a realization x(t)jl of X

(t)jl with a positive probability such

that for all t isin θ and j isin 1 Jn we have

Lntsum

l=1

1

LntW

nt (D

cj |x

(t)jl ) le ǫ (37)

and

Lntsum

l=1

1

LntQt(x

(t)jl )minusΘt le ǫ (38)

For an arbitrary γ gt 0 let

R = mintisinθ

maxUrarrArarr(BZ)t

(I(U Bt)minus lim supnrarrinfin

1

nχ(U Zn

t ))minusγ

Choose micro lt 12γ then for every t isin θ there is an (n Jn)

code(

(x(t)jl )j=1Jnl=1Lnt

Dj j = 1 Jn)

such that

1

nlog Jn ge R (39)

limnrarrinfin

maxtisinθ

maxjisin1Jn

Lntsum

l=1

1

LntW

nt (D

cj |x

(t)jl ) = 0 (40)

Choose a suitable α in (30) such that for all j it

holds Votimesnt (x

(t)jl ) minus Qt(x

(t)jl ) lt ǫ For any given jprime isin

1 Jn (30) and (38) yield

Lntsum

l=1

1

LntVotimesnt (x

(t)jprimel)minusΘt

le Lntsum

l=1

1

LntVotimesnt (x

(t)jprimel)minus

Lntsum

l=1

1

LntQt(x

(t)jprimel)

+ Lntsum

l=1

1

LntQt(x

(t)jprimel)minusΘt

leLntsum

l=1

1

LntVotimesn

t (x(t)jprimel)minusQt(x

(t)jprimel)

+ L

(t)ntsum

l=1

1

LntQt(x

(t)jprimel)minusΘt

le 2ǫ (41)

and sumJn

j=11Jn

sumLnt

l=11

LntVotimesnt (x

(t)jl )minusΘt le ǫ

8

Lemma 3 (Fannes-Audenaert Ineq cf [22] [4])Let Φ and Ψ be two quantum states in a d-dimensionalcomplex Hilbert space and ΦminusΨ le micro lt 1

e then

|S(Φ)minusS(Ψ)| le micro log(dminus1)minusmicro log microminus(1minusmicro) log(1minusmicro) (42)

The Fannes Inequality was first introduced in [22]where it has been shown that |S(X)minus S(Y)| le micro log dminusmicro logmicro In [4] the result of [22] has been improved and(42) has been proved

By Lemma 3 and the inequality (41) for a uniformlydistributed distributed random variable Xuni with valuein 1 Jn we have

χ(XuniZnt )

= S

Jnsum

j=1

1

Jn

Lntsum

l=1

1

LntVotimesnt (x

(t)jl )

minusJnsum

j=1

1

JnS

Lntsum

l=1

1

LntVotimesnt (x

(t)jl )

le

∣

∣

∣

∣

∣

∣

S

Jnsum

j=1

1

Jn

Lntsum

l=1

1

LntVotimesnt (x

(t)jl )

minus S (Θt)

∣

∣

∣

∣

∣

∣

+

∣

∣

∣

∣

∣

∣

S(Θt)minusJnsum

j=1

1

JnS

Lntsum

l=1

1

LntVotimesnt (x

(t)jl )

∣

∣

∣

∣

∣

∣

le ǫ log(dminus 1)minus ǫ log ǫ minus (1minus ǫ) log(1minus ǫ)

+

∣

∣

∣

∣

∣

∣

Jnsum

j=1

1

Jn

S(Θt)minus S

Lntsum

l=1

1

LntVotimesnt (x

(t)jl )

∣

∣

∣

∣

∣

∣

le 3ǫ log(dminus 1)minus ǫ log ǫminus (1minus ǫ) log(1 minus ǫ)minus 2ǫ log 2ǫ (43)

By (43) for any positive λ if n is sufficiently large wehave

maxtisinθ

χ(XuniZnt ) le λ (44)

For every t isin θ we define an (n Jn) code(Et Dj j = 1 Jn) where Et is built such that

Pr(

Et(j) = x(t)jl

)

= 1Lnt

for l isin 1 Lnt Combin-

ing (40) and (44) we obtain

CSCSI ge mintisinθ

maxUrarrArarr(BZ)t


1

nχ(U Zn

t ))

(45)

Thus we have shown the ldquogerdquo part of (27)

2) Upper bound for case with CSILet (Cn) be a sequence of (n Jn) codes such that

maxtisinθ

maxjisin1Jn

sum

xnisinAn

E(xn|j)Wnt (D

cj |xn) = ǫ1n (46)

maxtisinθ

χ(J Znt ) = ǫ2n (47)

where limnrarrinfin ǫ1n = 0 and limnrarrinfin ǫ2n = 0 J denotesthe random variable which is uniformly distributed onthe message set 1 JnWe denote the security capacity of the wiretap channel

(Wt Vt) in the sense of [40] by C(Wt Vt) Choose tprime isin θsuch that C(Wtprime Vtprime) = mintisinθ C(Wt Vt)

We denote a new random variable by X with valuesin 1 Jn determined by the Markov chain Xuni rarrA rarr Btprime rarr X where the first transition is governed bythe senderrsquos encoding strategy the second by Wtprime andthe last by the legal receiverrsquos decoding strategy Thenwe have from the data processing inequality

log Jn = H(Xuni)

= I(Xuni X) +H(Xuni | X)

le I(Xuni Bntprime) +H(Xuni | X)

Using Fanorsquos inequality we have

H(Xuni | X) le 1 + ǫ1n log Jn

Thus log Jn le I(Xuni Bntprime)+1+ǫ1n log Jn Applying the

standard technique for single letter formula in classicalinformation theory we have

log Jn le nI(Xuni Btprime) + 1 + ǫ1n log Jn (48)

Thus for any ǫ gt 0 if n is sufficiently large 1n log Jn

can not be greater than

I(XuniBtprime) +1

n+

1

nǫ1n log Jn

le [I(XuniBtprime)minus1

nχ(XuniZ

ntprime)] +

ǫ1n

n+

1

nlog Jn +

ǫ2n

n


nχ(XuniZ

ntprime)] + ǫ (49)

We can not exceed the secrecy capacity of the worstwiretap channel since we have to guarantee that the legalreceiver can decode the message in the worst case (cf(13) and Section I) Thus we have

CSCSI le mintisinθ

maxUrarrArarr(BZ)t


1

nχ(U Zn

t ))

(50)Combining (50) and (45) we obtain (27)

3) Lower bound for case without CSIFix a probability distribution p on An Let

Jn = lfloor2mintisinθ(nI(pWt)minuslogLn)minusnmicrorfloor

where Ln is a natural defined as in Section III Let pprimeXn and Dj (25) still holds

For a positive α we define

Qt(xn) = ΠpVtα

radicaΠVtα(x


n)ΠpVtαradica

9

and Θt =sum

xnisinT npδpprimen(xn)Qt(x

n)

For any positive δ let Ln = lceil2nmaxt(χ(pZt)+δ)rceil and nbe large enough in the same way as our proof of (36)for the case with CSI at the encoder there is a positiveconstant υ so that

Pr

(

Lnsum

l=1

1

LnQt(X


)

ge 1minus 2minusnυ

(51)

For any positive ǫ we choose a suitable α by (25) and(51) there is a realization xjl of Xjl with a positive prob-ability such that For all t isin θ and all j isin 1 Jn wehave

Lnsum

l=1

1

LnW

nt (D

cj |xjl) le ǫ

Lnsum

l=1

1

LnQt(xjl)minusΘt le ǫ

For any γ gt 0 let

R = maxUrarrArarr(BZ)t

(

mintisinθ

I(U Bt)minusmaxtχ(U Zt)

)

minus γ

Then there is an (n Jn) code (E Dj j = 1 Jn)where E is so built that Pr (E(j) = xjl) =

1Lnt

for l isin1 Lnt such that lim infnrarrinfin

1n log Jn ge R and

limnrarrinfin

maxtisinθ

maxjisin1Jn

Lnsum

l=1

1

LnW

nt (D

cj |xjl)) = 0 (52)

In the same way as our proof of (44) for the case withCSI at the encoder

maxtisinθ

χ(XuniZnt ) le ǫ (53)

for any uniformly distributed distributed random vari-able Xuni with value in 1 JnCombining (52) and (53) we obtain


(mintisinθ


χ(U Zt))

V COMPOUND CLASSICAL-QUANTUM

WIRETAP CHANNEL

In this section we derive the secrecy capacity of thecompound classical-quantum wiretap channel with CSIIn this model both the receiver and the wiretapper useclassical quantum channels and the set of the channelstates may be finite or infiniteLet A H H prime H primeprime θ and (Wt Vt)tisinθ be defined as in

Section II

Theorem 2 The secrecy capacity of the compoundclassical-quantum wiretap channel in the case with CSIis given by

CCSI = limnrarrinfin

mintisinθ

maxPinpwt

1

n(χ(PinpB

nt )minus χ(PinpZ

nt ))

(54)where Bt are the resulting random quantum states at theoutput of legal receiver channels and Zt are the resultingrandom quantum states at the output of wiretap channelsThe maximum is taken over all probability distributionsPinp on the input quantum states wt

Assume that the senderrsquos encoding is restricted totransmitting an indexed finite set of orthogonal quantumstates ρx x isin A sub S(H primeotimesn) then the secrecy capac-ity of the compound classical-quantum wiretap channel inthe case with no CSI at the encoder is given by

CS = limnrarrinfin

maxUrarrArarr(BZ)t

1

n

(

mintisinθ

χ(U Bnt )

minusmaxtisinθ

χ(U Znt )

)

(55)

Proof At first we are going to prove (54) Our idea isto send the information in two parts First we send thechannel state information with finite blocks of finite bitswith a code C1 to the receiver and then depending on t

we send the message with a code C(t)2 in the second part

11) Sending channel state information with finite bitsWe do not require that the first part should be secure

against the wiretapper since we assume that the wire-tapper already has the full knowledge of the CSIBy ignoring the security against the wiretapper we

consider only the compound channel (Wt)tisinθ Let W =(Wt)t be an arbitrary compound-classical quantum chan-nel Then by [9] for each λ isin (0 1) the λ capacityC(Wλ) equals

C(Wλ) = maxPinpisinP (A)

mintχ(PinpWt) (56)

If maxPinpmint χ(PinpWt) gt 0 holds then the sender

can build a code C1 such that the CSI can be sentto the legal receiver with a block with length l le

log Tmint maxPinp

χ(PinpWt)minus ǫ If maxPinp

mint χ(PinpWt) =

0 holds we can not build a code C1 such that the CSIcan be sent to the legal receiver But this does not causeany problem since if maxPinp

mint χ(PinpWt) = 0 theright-hand side of (54) is zero

12) Message transformation when both the sender andthe legal receiver know CSIIf both the sender and the legal receiver have the full

knowledge of t then we only have to look at the singlewiretap channel (Wt Vt)In [19] and [20] it was shown that if n is sufficiently

large there exists an (n Jn) code for the quantum wire-tap channel (WV ) with

log Jn = maxPinpw

(χ(PinpBn)minus χ(PinpZ

n))minus ǫ (57)

10

for any positive ǫ and positive δ where B is the resultingrandom variable at the output of legal receiverrsquos channeland Z the output of the wiretap channelWhen the sender and the legal receiver both know t

they can build an (n Jnt) code C(t)2 where

log Jnt = maxPinpwt

(χ(PinpBnt )minus χ(PinpZ

nt ))minus ǫ (58)

Thus

CCSI ge limnrarrinfin

mintisinθ

maxPinpwt

1

n(χ(PinpB

nt )minus χ(PinpZ

nt ))

(59)

Remark 5 For the construction of the second part of ourcode we use random coding and request that the random-ization can be sent (cf [19]) However it was shownin [13] that the randomization could not always be sentif we require that we use one unique code which is se-cure against the wiretapper and suitable for every chan-nel state ie it does not depend on t This is not acounterexample to our results above neither to the con-

struction of C1 nor to the construction of C(t)2 because

of the following factsThe first part of our code does not need to be

secure For our second part the legal transmitterscan use the following strategy At first they builda code C1 = (E Dt t = 1 |θ|) and a code

C(t)2 = (E(t) D(t)

j j = 1 Jn) for every t isin θ

If the sender wants to send the CSI tprime isin θ and themessage j he encodes tprime with E and j with E(tprime)then he sends both parts together through the channelAfter receiving both parts the legal receiver decodesthe first part with Dt t and chooses the right

decoders D(tprime)j j isin

D(t)j j t isin θ

to decode the

second part With this strategy we can avoid usingone unique code which is suitable for every channel state

13) Upper bound for the case CSI at the encoderFor any ǫ gt 0 we choose tprime isin θ such that C(Wtprime Vtprime) le

inftisinθ C(Wt Vt) + ǫFrom [19] and [20] we know that the secrecy capacity

of the quantum wiretap channel (Wtprime Vtprime) can not begreater than

limnrarrinfin

maxPinpwtprime

1

n(χ(PinpB

ntprime)minus χ(PinpZ

ntprime))

Since we can not exceed the capacity of the worst wiretapchannel we have

CCSI le limnrarrinfin

mintisinθ

maxPinpwt

1

n(χ(PinpB

nt )minus χ(PinpZ

nt ))

(60)This together with (59) completes the proof of (54)

Remark 6 In [39] it was shown that if for a given t andany n isin N

χ(PinpBnt ) ge χ(PinpZ

nt )

holds for all Pinp isin P (A) and wt(j) j = 1 Jn subS(Hotimesn) then

limnrarrinfin

maxPinpwt

1

n(χ(PinpB

nt )minus χ(PinpZ

nt ))

= maxPinpwt

(χ(PinpBt)minus χ(PinpZt))

Thus if for every t isin θ and n isin N

I(Pinp Bnt ) ge I(PinpZ

nt )

holds for all Pinp isin P (A) and wt(j) j = 1 Jn subS(Hotimesn) we have

CCSI = mintisinθ

maxPinpwt


Now we are going to prove (55)


Jn = lfloor2mintisinθ χ(pBnt )minusmaxtisinθ χ(pZn

t )minus2nmicrorfloor

Ln = lceil2maxt χ(pZnt )+nmicrorceil

and let pprime and Xn = Xjl j l be defined as inthe classical case (cf Section III) Since Jn middot Ln le2mint χ(pB

nt )minusnmicro in [18] it was shown that if n is suffi-

ciently large there exist a collection of quantum statesρxn xn isin An sub S(H primeotimesn

) a collection of positive-semidefinite operators Dtxn t isin θ xn isin An anda positive constant β such that for any (t j l) isin θ times1 Jn times 1 Ln it holds

Pr[

tr(

Wnt (ρ

nXjl

)DtXjl

)

ge 1minus 2minusnβ]

gt 1minus 2minusnβ

(61)and for any realization xjl j l of Xjl j l it holdsthat

sum

tisinθ

Jnsum

j=1

Lnsum

l=1

Dtxjlle I

We define

Qt(ρxn) = ΠpVtαradicaΠVtα(x

n)middotV otimesnt (ρxn)middotΠVtα(x

n)ΠpVtαradica

and Θt =sum

xnisinT npδpprimen(xn)Qt(ρxn)

Choosing n sufficiently large in the same way as ourproof of (36) for the classical compound channel with

11

quantum wiretapper there is a positive constant υ suchthat

Pr

(

Lnsum

l=1

1

LnQt(ρX(t)

jl

)minus Θt le ǫ forallt forallj)

ge 1minus 2minusnυ

(62)

We choose a suitable α If n is sufficiently large we canfind a realization xjl of Xjl with a positive probabilitysuch that for all j isin 1 Jn we have

mintisinθ

tr(

Wnt (ρ

nxjl

)Dtxjl

)

ge 1minus 2minusnβ

and

maxtisinθ

Lnsum

l=1

1

LnQt(ρxjl

)minusΘt le ǫ

We define Dj =sum

tisinθ

sumLn

l=1Dtxjl then

sumJn

j=1Dj =sum

tisinθ

sumJn

j=1

sumLn

l=1Dtxjlle I Furthermore for all tprime isin θ

and lprime isin 1 Ln we have

tr(

Wntprime (ρ

nxjlprime

)Dj

)

=sum

tisinθ

Lnsum

l=1

tr(

Wntprime (ρ

nxjlprime

)Dtxjl

)

ge tr(

Wntprime (ρ

otimesnxjlprime

)Dtprimexjlprime

)

ge 1minus 2minusnβ

the inequality in the third line holds because for two pos-itive semi-definite matrices M1 and M2 we always havetr (M1M2) = tr

(radicM1M2

radicM1

)

ge 0

For any γ gt 0 let


1

n

[

mintisinθ

χ(pBnt )minusmax

tisinθχ(pZn

t )

]

minus γ

Then for any positive λ there is an (n Jn λ) code(

w(j) =sumLn

l=11Lnρnxjl

j = 1 Jn Dj j =

1 Jn)

such that lim infnrarrinfin1n log Jn ge R

maxtisinθ

maxjisin1Jn

tr(


)

le λ

(63)and in the same way as our proof of (44) for the classicalcompound channel with quantum wiretapper

maxtisinθ


for any uniformly distributed random variable Xuni withvalue in 1 JnCombining (63) and (64) we obtain

CS ge limnrarrinfin

maxUrarrArarr(BZ)t

1

n

(

mintisinθ

χ(U Bnt )minusmax

tisinθχ(U Zn

t )

)

(65)

22) Upper bound for case without CSI

Let (Cn) = (ρ(n)j j D(n)j j) be a sequence of

(n Jn λn) code such that

maxtisinθ

maxjisin1Jn

tr(

(IminusD(n)j )Wotimesn

t

(

ρ(n)j

))

le λn (66)

maxtisinθ

χ(XuniZnt ) = ǫ2n (67)

where limnrarrinfin λn = 0 and limnrarrinfin ǫ2n = 0 Xuni de-notes the random variable which is uniformly distributedon the message set 1 JnWe denote the classical capacity of the quantum chan-

nel Wt in the sense of [40] by C(Wt) Choose tprime isin θ such

that C(Wtprime) = mintisinθ C(Wt)It is known (cf Section IV 2) Upper bound for case

with CSI and [31]) that can not exceed χ(XuniBntprime) + ξ

for any constant ξ gt 0 Since the secrecy capacity of acompound wiretap channel can not exceed the capacityof the worst channel without wiretapper for any ǫ gt 0choose ξ = 1

2ǫ if n is large enough the secrecy rate of(Cn) can not be greater than

1

nχ(XuniB

ntprime) + ξ

= mintisinθ

1

nχ(XuniB

nt ) + ξ

le mintisinθ

1

nχ(XuniB

nt )minusmax

tisinθ

1

nχ(XuniZ

nt ) + ξ +

1

nǫ2n

le 1

n

(

mintisinθ

χ(XuniBnt )minusmax

tisinθχ(XuniZ

nt )

)

+ ǫ (68)

Thus

CS le limnrarrinfin

maxUrarrArarr(BZ)t

1

n

(

mintisinθ

χ(U Bnt )minusmax

tisinθχ(U Zn

t )

)


So far we assumed that |θ| the number of the chan-nels is finite therefore we can send the CSI with finitebits to the receiver in the case where the sender has CSINow we look at the case where |θ| can be arbitrary Weof course are not allowed to send the CSI with finite bitsif |θ| = infin but in this case we may use a ldquofinite approx-imationrdquo to obtain the following corollary

Corollary 1 For an arbitrary set θ we have

CSCSI = limnrarrinfin

inftisinθ

maxPinpwt

1

n(χ(PinpB

nt )minus χ(PinpZ

nt ))

(70)

Proof Let W S(H prime) rarr S(H primeprime) be a linear map thenlet

Wdiams = supnisinN

maxaisinS(CnotimesHprime)a1=1

(In otimesW )(a)1 (71)

12

It is known [33] that this norm is multiplicative ieW otimesW primediams = Wdiams middot W primediamsA τ -net in the space of the completely positive

trace-preserving maps S(H prime) rarr S(H primeprime) is a finite set(

W (k))K

k=1of completely positive trace-preserving maps

S(H prime) rarr S(H primeprime) with the property that for each com-pletely positive trace-preserving map W S(H prime) rarrS(H primeprime) there is at least one k isin 1 K with W minusW (k)diams lt τ

Lemma 4 (τminusnet [30]) Let H prime and H primeprime be finite-dimensional complex Hilbert spaces For any τ isin (0 1]

there is a τ-net of quantum-channels(

W (k))K

k=1in the

space of the completely positive trace preserving maps

S(H prime) rarr S(H primeprime) with K le ( 3τ )2dprime4

where dprime = dimH prime

If |θ| is arbitrary then for any ξ gt 0 let τ = ξminus log ξ By

Lemma 4 there exists a finite set θprime with |θprime| le ( 3τ )2dprime4

and τ -nets (Wtprime)tprimeisinθprime (Vtprime )tprimeisinθprime such that for every t isin θwe can find a tprime isin θprime with Wt minusWtprimediams le τ andVt minus Vtprimediams le τ For every tprime isin θprime the legal transmitters

build a code C(tprime)2 = wtprime Dtprimej j Since by [19] the

error probability of the code C(tprime)2 decreases exponentially

with its length there is an N = O(minus log ξ) such that forall tprimeprime isin θprime it holds

1

JN

JNsum

j=1

tr(

WotimesNtprimeprime (wtprimeprime(j))Dtprimeprimej

)

ge 1minus λminus ξ (72)

χ(XuniZNtprime ) le ξ (73)

Then if the sender obtains the channel state informa-tion ldquotrdquo he chooses a ldquotprimerdquo isin θprime such that Wt minusWtprimediams leτ and Vt minus Vtprimediams le τ He can send ldquotprimerdquo to the legal re-ceiver in the first part with finite bits and then they

build a code C(tprime)2 that fulfills (72) and (73) to transmit

the message

For every tprime and j let |ψtprime(j)〉〈ψtprime (j)| isin S(H primeotimesN otimesH primeotimesN ) be an arbitrary purification of the quantum

state wtprime(j) then tr[(

WotimesNt minusWotimesN

tprime

)

(wtprime(j))]

=

tr(

trHprimeotimesN

[

IotimesNHprime otimes (WotimesN

t minusWotimesNtprime ) (|ψtprime(j)〉〈ψtprime(j)|)

])

We have

tr∣

∣

(


tprime

)

(wtprime (j))∣

∣

= tr(

trHprimeotimesN

∣

∣IotimesNHprime otimes (WotimesN

t minusWotimesNtprime ) (|ψtprime(j)〉〈ψtprime (j)|)

∣

∣

)

= tr∣

∣IotimesNHprime otimes (Wotimesn


∣

∣

=∥

∥IotimesNHprime otimes (WotimesN


∥

∥

1

le WotimesNt minusWotimesN

tprime diams middot (|ψtprime(j)〉〈ψtprime (j)|)1le Nτ

The second equality follows from the definition oftrace The second inequality follows by the definitionof middot diams The third inequality follows from the facts

that (|ψtprime(j)〉〈ψtprime (j)|) 1 = 1 and∥

∥WotimesNt minusWotimesN

tprime

∥

∥

diams=

∥

∥

∥(Wt minusWtprime)otimesN∥

∥

∥

diams= N middotWt minusWtprimediams since middotdiams is mul-

tiplicative

It follows that

∣

∣

∣

∣

1

JN

JNsum

j=1

tr(

WotimesNt (wtprime(j))Dtprimej

)

minus 1

JN

JNsum

j=1

tr(

WotimesNtprime (wtprime(j))Dtprimej

)

∣

∣

∣

∣

le 1

JN

JNsum

j=1

∣

∣tr[(


tprime

)

(wtprime(j))Dtprimej

]∣

∣

le 1

JN

JNsum

j=1

tr∣

∣

(


tprime

)


∣

∣

le 1

JN

JNsum

j=1

tr∣

∣

(


tprime

)

(wtprime(j))∣

∣

le 1

JNJNNτ

= Nτ (74)

Nτ can be arbitrarily small when ξ is close to zerosince N = O(minus log ξ)Let Xuni be a random variable uniformly distributed

on 1 JN and ρ(j) j = 1 Jn be a set ofquantum states labeled by elements of 1 Jn Wehave

|χ(XuniVt)minus χ(XuniVtprime)|

le

∣

∣

∣

∣

∣

∣

S

JNsum

j=1

1

JNVt(ρ(j))

minus S

JNsum

j=1

1

JNVtprime(ρ(j))

∣

∣

∣

∣

∣

∣

+

∣

∣

∣

∣

∣

∣

JNsum

j=1

1

JNS (Vt(ρ(j))) minus

JNsum

j=1

1

JNS (Vtprime(ρ(j)))

∣

∣

∣

∣

∣

∣

le τ log(dminus 1)minus τ log τ minus (1minus τ) log(1minus τ) (75)

where d = dimH The inequality in the last line holdsby Lemma 3 and because Vt(ρ)minus Vtprime(ρ) le τ for allρ isin S(H) when Vt minus Vtprimediams le τ

By (74) and (75) we have

suptisinθ

1

JN

JNsum

j=1

tr(


)

ge 1minus λminus ξ minusNτ

χ(XuniZNt ) le ξ+τ log(dminus1)minusτ log τminus(1minusτ) log(1minusτ)

Since ξ + Nτ and τ log(d minus 1) can be arbitrarily smallwhen ξ is close to zero we have

suptisinθ

1

JN

JNsum

j=1

tr(


)

ge 1minus λ

13

suptisinθ

χ(XuniZNt ) le ǫ

The bits that the sender uses to transform the CSI arelarge but constant so it is still negligible compared tothe second part We obtain


inftisinθ

maxPinpwt

1

n(χ(PinpB

nt )minus χ(PinpZ

nt ))

(76)The proof of the converse is similar to those given in

the proof of Theorem 2 where we consider a worst tprime

Remark 7 In (54) and Corollary 1 we have only re-quired that the legal receiver can decode the correct mes-sage with a high probability if n is sufficiently large Wehave not specified how fast the error probability tends tozero when the code length goes to infinity If we analyzethe relation between the error probability ε and the codelength then we have the following factsIn the case of finite θ let ε1 denote the error probability

of the first part of the code (ie the legal receiver doesnot decode the correct CSI) and let ε2 denote the errorprobability of the second part of the code (ie the legalreceiver decodes the correct CSI but does not decode themessage) Since the length of the first part of the code isl middot log c middot cprime = O(log ε1) we have εminus1

1 is O(exp(l middot log c middotcprime)) = O(exp(n)) where n stands for the length of thefirst part of the code For the second part of the codeε2 decreased exponentially with the length of the secondpart as proven in [19] Thus the error probability ε =maxε1 ε2 decreases exponentially with the code lengthin the case of finite θIf θ is infinite let ε1 denote the error probabil-

ity of the first part of the code probability Here wehave to build two τ-nets for a suitable τ each con-

tains O((minus log ε1ε1

)minus2dprime4

) channels If we want to sendthe CSI of these τ-nets the length of first part l will

be O(minus2dprime4 middot log(ε1 log ε1)) which means here εminus11 will

be O(exp( n4dprime4 )) = O(exp(n)) Thus we can still achieve

that the error probability decreases exponentially with thecode length in case of infinite θ

VI ENTANGLEMENT GENERATION OVER

COMPOUND QUANTUM CHANNELS

The entanglement generating capacity of a given quan-tum channel describes the maximal amount of entangle-ment that we can generate or transmit over the chan-nel A code for the secure message transmission over aclassical-quantum wiretap channel can be used to build acode for the entanglement transmission over a quantumchannel (cf [20]) Our technique for entanglement gener-ation over compound quantum channels is similar to theproof of entanglement generating capacity over quantumchannels in [20] The difference between our techniqueand the proofs in [20] is that we have to consider thechannel uncertainty (cf the discussion in Section VII)

Let P Q HP HQ θ and(

Notimesnt

)

tisinθbe defined as in

Section II (ie we assume that θ is finite)

We denote dimHP by a and denote X = 1 aConsider the eigen-decomposition of ρP into the or-thonormal pure quantum state ensemble p(x) |φx〉P x isin X

sum

xisinXp(x)|φx〉〈φx|P = ρP

The distribution p defines a random variable X

Theorem 3 The entanglement generating capacity of(Nt)tisinθ is bounded as follows

A ge maxp

(

mintisinθ

χ(pQt)minusmaxtisinθ

χ(pEt)

)

(77)

where Qt stands for the quantum outputs that the receiverobserves at the channel state t and Et the quantum out-puts at the environment

(Theorem 3 is weaker than the result in [12] the reasonis that we use for our proof a different quantum channelrepresentation For details and the result in [12] cf Sec-tion VII)Proof Let ρP rarr UNt

ρPUlowastNt

be a unitary transforma-tion which represents Nt (cf Section VII) where UNt

is a linear operator S(HP) rarr S(HQE) and E is thequantum system of the environment Fix a ρP witheigen-decomposition

sum

xisinX p(x)|φx〉P〈φx|P If the chan-nel state is t the local output density matrix seen by thereceiver is

trE

(

sum

x

p(x)UNt|φx〉〈φx|PUlowast

Nt

)

and the local output density matrix seen by the environ-ment (which we interpret as the wiretapper) is

trQ

(

sum

x


Nt

)

Therefore (Nt)tisinθ defines a compound classical-quantum

wiretap channel (WNt VNt

)tisinθ where WNt HP rarr HQ

sum

xisinX p(x)|φx〉〈φx|P rarr trE(sum

x p(x)UNt|φx〉〈φx|PUlowast

Nt

)

and VNt HP rarr HQ

sum

xisinX p(x)|φx〉〈φx|P rarrtrE(sum


Nt

)

1) Building the encoder and the first part of the decodingoperatorLet

Jn = lceil2n[mint χ(XQt)minusmaxt χ(XEt)minus2δ]rceil

and

Ln = lceil2n(maxt χ(XEt)+δ)rceil

14

For the compound classical-quantum wiretap channel(WNt

VNt)tisinθ since

|(j l) j = 1 Jn l = 1 Ln|= Jn middot Ln le 2nmint[χ(XQt)minusδ]

if n is large enough by Theorem 2 and [18] the followingholds There is a collection of quantum states ρPn

xjl

j = 1 Jn l = 1 Ln sub S(HPn

) a collectionof positive-semidefinite operators Dtjl = Dtxjl

t isinθ j = 1 Jn l = 1 Ln a positive constant βand a quantum state ξE

n

t on HEn

such that

tr(

(DQn

txjlotimes IE

n

)UNtρP

n

xjlUlowastNt

)

ge 1minus 2minusnβ (78)

and

ωEn

jt minus ξEn

t 1 lt ǫ (79)

where ωEn

jt = 1Lnt

sumLnt

l=1 trQn

(

UNtρP

n

xjlUlowastNt

)

Now the quantum state ρPn

xjlmay be pure or

mixed Assume ρPn

xjlis a mixed quantum state

sumni=1 p

primejl(i)|κ

(i)xjl〉〈κ(i)

xjl |Pn

then

nsum

i=1

pprimejl(i)tr(

(DQn

txjlotimes IE

n

)UNt|κ(i)

xjl〉〈κ(i)

xjl|Pn

UlowastNt

)

tr

(

(DQn

txjlotimes IE

n

)UNt(

nsum

i=1

pprimejl(i)|κ(i)xjl

〉〈κ(i)xjl

|Pn

)UlowastNt

)

ge 1minus 2minusnβ

Thus for all i such that pprimejl(i) ge 2minusnβ

1minus2minusnβ it must hold

tr(

(DQn

txjlotimes IE

n

)UNt|κ(i)

xjl〉〈κ(i)

xjl|Pn

UlowastNt

)

ge 1minus 2minusnβ

If n is large enough then there is at least one ilj isin1 n such that pprimejl(ilj) ge 2minusnβ

1minus2minusnβ By Theorem 2

there is a ξEn

t on HEn

such that

1

Lnt

Lntsum

l=1

trQn

(

UNt|κ(ilj)

xjl 〉〈κ(ilj )xjl |Pn

UlowastNt

)

minus ξEn

t 1 lt ǫ

Thus(

|κ(ilj)xjl 〉〈κ(ilj )

xjl |Pn

j l DQn

txjl j l t

)

is a code with the same security rate as(

ρPn

xjl j l DQn

txjl j l t

)

Hence we may assume that ρPn

xjlis a pure quantum state

Assume ρPn

xjl= |κjl〉〈κjl|P

n

Let HM be a Jn-dimensional Hilbert space with an orthonormal basis

|j〉M j = 1 Jn HL be a Ln-dimensional Hilbertspace with an orthonormal basis |l〉L l = 1 Lntand Hθ be a |θ|-dimensional Hilbert space with an or-thonormal basis |t〉θ t isin θ Let |0〉M|0〉L|0〉θ be theancillas on HM HL and Hθ respectively that the re-ceiver adds We can (cf [31]) define a unitary matrixV QnMLθ on HQnMLθ such that for any given quantumstate ρQ

n isin S(HQn

) we have

V QnMLθ

(

ρQn otimes |0〉〈0|M otimes |0〉〈0|L otimes |0〉〈0|θ

)

(V QnMLθ)lowast

=sum

t

sum

j

sum

l

(

DQn

txjlρQ

n)

otimes |j〉〈j|M|l〉〈l|L|t〉〈t|θ

We denote

ψQnEnMLθjlt

=(

IEn otimes V QnMLθ

)

(

UN otimes IMLθ)

[

|κjl〉〈κjl|Pn

otimes |0〉〈0|M otimes |0〉〈0|L otimes |0〉〈0|θ]

(

UN otimes IMLθ)lowast

(

IEn otimes V QnMLθ

)lowast

in view of (78) we have

F(

trQnEn

(

ψQnEnMLθjlt

)

|j〉〈j|M otimes |l〉〈l|L otimes |t〉〈t|θ)

ge 1minus ǫ (80)

By Uhlmannrsquos theorem (cf eg [40]) we can find a|ζjlt〉Q

nEn

on HQnEn

such that

〈0|θ〈0|L〈0|M〈κjl|Pn (

UNtotimes IMLθ

)lowast

(

IEn otimes V QnMLθ

)lowast|ζjlt〉Q

nEn |j〉M|l〉L|t〉θ

= F

(

ψQnEnMLθjlt |ζjlt〉〈ζjlt|Q

nEn

otimes |j〉〈j|M otimes |l〉〈l|L otimes |t〉〈t|θ)

ge 1minus ǫ (81)

2) Building the seconder part of the decoding operatorWe define

|ajl〉PnMLθ = |κjl〉P

n |0〉M|0〉L|0〉θ

and

|bjlt〉PnMLθ =

(

UNtotimes IMLθ

)lowast (IE

n otimes V QnMLθ)lowast

|ζjlt〉QnEn |j〉M|l〉L|t〉θ

For every j l and t we have 〈ajl|bjlt〉PnMLθ ge 1minus ǫ

We define

|ajk〉PnMLθ =

1radicLn

Lnsum

l=1

eminus2πil kLn |ajl〉P

nMLθ

15

|bjkt〉PnMLθ =

1radicLn

Lnsum

l=1

eminus2πil kLn |bjlt〉P

nMLθ

and

|bjk〉PnMLθ =

1

|θ|

|θ|sum

t=1

|bjkt〉PnMLθ

For every j isin 1 Jn by (81) it holds

1

Ln

Lnsum

k=1

〈ajk|bjk〉PnMLθ

=1

|θ|1

Ln

|θ|sum

t=1

Lnsum

k=1

〈ajk|bjkt〉PnMLθ

=1

|θ|1

Ln

|θ|sum

t=1

Lnsum

l=1

〈ajl|bjlt〉PnMLθ

ge 1minus ǫ (82)

Hence there is at least one kj isin 1 Ln such thatfor every j we have

1minus ǫ

le eminusiskj 〈ajkj

|bjkj〉PnMLθ

=1

|θ|

|θ|sum

t=1

eminusiskj 〈ajkj

|bjkj t〉PnMLθ

for a suitable phase skj Since for all t it holds

∣

∣

∣eminusiskj 〈ajkj

|bjkj t〉PnMLθ

∣

∣

∣ le 1 we have

mintisinθ

∣

∣


|bjkj t〉PnMLθ

∣

∣

∣ ge 1minus |θ|ǫ

Therefore there is a suitable phase rkjsuch that for all

t isin θ

1minus |θ|ǫ

le∣

∣


|bjkj t〉PnMLθ

∣

∣

∣

= eminusirkj 〈ajkj

|bjkj t〉PnMLθ

= eminusirkj

1

Ln

(

Lnsum

l=1

eminus2πilkjLn 〈ajl|P

nMLθ

)

(

Lnsum

l=1

eminus2πilkjLn |bjlt〉P

nMLθ

)

(83)

For every t isin θ we set

|jt〉QnEnL =

radic

1

Ln

Lnsum

l=1

eminus2πi(l

kjLn

+rkj )|ζjlt〉QnEn otimes|l〉L

and

|ϑjt〉QnEnMLθ =

radic

1

Ln

Lnsum

l=1

eminus2πilkj

Ln

[

IEn otimes V QnMLθ

]

(UnN |κjl〉P

n

)|0〉M|0〉L|0〉θ

For all t isin θ and j isin 1 Jn it holds by (83)

F

(

|ϑjt〉〈ϑjt|QnEnMLθ

|jt〉〈jt|QnEnL otimes |j〉〈j|M otimes |t〉〈t|θ

)

=∣

∣

∣〈ϑjt|QnEnMLθ|jt〉Q

nEnL|j〉M |t〉θ∣

∣

∣

=1

Ln

(

Lnsum

l=1


nMLθ

)

(

Lnsum

l=1

eminus2πilkjLn e

minusirkj |bjlt〉PnMLθ

)

ge 1minus |θ|ǫ (84)

Furthermore since (79) holds there is a quantum stateξE

n

t which does not depend on j and l on HEn

such that

∥

∥

∥ξEn

t minus trQn

(

UNt|κjl〉〈κjl|P

n

UlowastNt

)∥

∥

∥

1le ǫ (85)

By monotonicity of fidelity for any l isin 1 Ln∥

∥

∥trQn

(


n

UlowastNt

)

minus trQn

(

|ζjlt〉〈ζjlt|QnEn)∥

∥

∥

1

le 2

[

1minus F

(

trQn

(


n

UlowastNt

)

trQn

(

|ζjlt〉〈ζjlt|QnEn)

)]12

le 2

[

1minus F

(


nEn

otimes |j〉〈j|M otimes |l〉〈l|L otimes |t〉〈t|θ)]

12

le 2radicǫ (86)

the first inequality holds because for two quantum states and η we have 1

2minus η1 leradic

1minus F ( η)2By (85) and (86)

∥

∥

∥trQnL

(

|jt〉〈jt|QnEnL

)

minus ξEn

t

∥

∥

∥

1

=

∥

∥

∥

∥

∥

1

Ln

Lnsum

l=1

trQn

(


minus ξEn

t

∥

∥

∥

∥

∥

1

le 1

Ln

Lnsum

l=1

∥

∥

∥

∥

trQn

(


n

UlowastNt

)

minus trQn

(


∥

∥

∥

∥

1

+∥

∥

∥ξEn

t minus trQn

(


n

UlowastNt

)∥

∥

∥

1

le 2radicǫ+ ǫ (87)

16

holds for all t isin θ and j isin 1 JnIn [36] (cf also [20]) it was shown that when (87) holds

for every t isin θ we can find a unitary operator UQnML(t)

such that if we set

χQnEnMLjjprimet =

(

UQnML(t) otimes IE

n)

(

|jt〉〈jt|QnEnL otimes |j〉〈jprime|M

)(

UQnML(t) otimes IE

n)lowast

then

F(

|ξt〉〈ξt|QnEnL otimes |j〉〈jprime|M χQnEnML

jjprimet

)

ge 1minus 4ǫminus 4radicǫ

(88)where |ξt〉Q

nEnL is chosen so that |ξt〉〈ξt|QnEnL is a pu-

rification of ξEn

t on HQnEnL

3) Defining the codeWe can now define our entanglement generating code

Let tprime be arbitrary in θ The sender prepares the quantumstate

1

Jn

1

Ln

Jnsum

j=1

Lnsum

l=1

eminus2πilkjLn |κjl〉P

n |j〉A

Jnsum

j=1

Lnsum

l=1

eminus2πilkjLn 〈j|A〈κjl|P

n

(89)

keeps the system A and sends the system Pn throughthe channel Notimesn

tprime ie the resulting quantum state is

1

Jn

1

Ln

(

IA otimes UnNtprime

)

[

Jnsum

j=1

Lnsum

l=1

eminus2πilkj

Ln |j〉A|κjl〉Pn

Jnsum

j=1

Lnsum

l=1

eminus2πilkj

Ln 〈κjl|Pn〈j|A

]

(

IA otimes UnNtprime

)lowast

=1

Jn

1

Ln

Jnsum

j=1

|j〉A(

Lnsum

l=1

eminus2πilkjLn Un

Ntprime|κjl〉P

n

)

Jnsum

j=1

(

Lnsum

l=1

eminus2πilkjLn 〈κjl|P

n

(UnNtprime

)lowast)

〈j|A

The receiver subsequently applies the decoding operator

τQn rarr trQnLθ

[

(

sum

tisinθ

UQnML(t) otimes |t〉〈t|θ

)

V QnMLθ

(

τQn otimes |0〉〈0|M otimes |0〉〈0|L otimes |0〉〈0|θ

)

V QnMLθlowast(

sum

tisinθ


)lowast]

(90)

to his outcome

31) The resulting quantum state after performing the de-coding operator

We define

ιAQnEnMLθtprime

=

(

sum

tisinθ

UQnML(t) otimes IAEn otimes |t〉〈t|θ

)

(V QnMLθ otimes IAEn

)

(

1

Jn

1

Ln

Jnsum

j=1

|j〉A(

Lnsum

l=1

eminus2πilkjLn Un

Ntprime|κjl〉P

n

)

Jnsum

j=1

(

Lnsum

l=1


n

(UnNtprime

)lowast)

〈j|A

otimes |0〉〈0|M otimes |0〉〈0|L otimes |0〉〈0|θ)


)lowast

(

sum

tisinθ


)lowast

=

(

sum

tisinθ


)

1

Jn(

Jnsum

j=1

|j〉A|ϑjtprime〉QnEnMLθ)(

Jnsum

j=1

〈ϑjtprime |QnEnMLθ〈j|A)

(

sum

tisinθ


)lowast

(91)

then the resulting quantum state after performing thedecoding operator is trQnEnLθ(ι

AQnEnMLθtprime )

32) The fidelity of 1Jn

sumJn

j=1

sumJn

jprime=1 χQnEnMLjjprimetprime otimes |j〉〈j|A otimes

|tprime〉〈tprime|θ and the actual quantum state

(

sum

tisinθ


)

(

sum

tisinθ

UQnML(t) otimes IE

n otimes |t〉〈t|θ)lowast

= IAEn otimessum

tisinθ

UQnML(t) (UQnML

(t) )lowast otimes |t〉〈t|θ

= IAQnEnMLθ

sum

tisinθ UQnML(t) otimes IE

n otimes |t〉〈t|θ is unitary

Because of this unitarity and by (84)

F

ιAQnEnMLθtprime

1

Jn

Jnsum

j=1

Jnsum

jprime=1

χQnEnMLjjprimetprime otimes |j〉〈jprime|A otimes |tprime〉〈tprime|θ

= F

(

1

Jn(

Jnsum

j=1


Jnsum

j=1


1

Jn(

Jnsum

j=1

|jtprime〉QnEnL otimes |j〉A otimes |j〉M)

17

(

Jnsum

j=1

〈j|M otimes 〈j|A otimes 〈jtprime |QnEnL)otimes |tprime〉〈tprime|θ

)

=1

Jn

∣

∣

∣

∣

( Jnsum

j=1

〈ϑjtprime |QnEnMLθ

)

( Jnsum

j=1

|jtprime〉QnEnL otimes |j〉M otimes |tprime〉θ

)∣

∣

∣

∣



sumJn

j=1

sumJn

jprime=1 χQnEnMLjjprimetprime otimes|j〉〈jprime|Aotimes

|tprime〉〈tprime|θ and the standard maximally entanglement stateBy (88) we have

F

(

1

Jn

Jnsum

j=1

Jnsum

jprime=1

χQnEnMLjjprimetprime otimes |tprime〉〈tprime|θ otimes |j〉〈jprime|A

1

Jn(

Jnsum

j=1

|ξtprime 〉QnEnL otimes |j〉A otimes |j〉M otimes |tprime〉θ)

(

Jnsum

j=1

〈ξtprime |QnEnL otimes 〈j|A otimes 〈j|M otimes 〈tprime|θ)

)

ge 1minus 4ǫminus 4radicǫ (93)

34) The fidelity of the actual quantum state and the stan-dard maximally entanglement stateSince for two quantum states and η it holds

1minus F ( η) le 1

2minus η1 le

radic

1minus F ( η)2

for three quantum states η and υ we have

F ( η)

ge 1minus 1

2minus η1

ge 1minus 1

2minus υ1 minus

1

2υ minus η1

ge 1minusradic

1minus F ( υ)2 minusradic

1minus F (υ η)2

Combining (92) and (93) for all tprime isin θ we have

F

(

trQnEnLθ(ιAQnEnMLθtprime )

(

Jnsum

j=1

|j〉A otimes |j〉M)(

Jnsum

j=1

〈j|A otimes 〈j|M)

)

ge F

(

ιAQnEnMLθtprime

1

Jn(

Jnsum

j=1

|ξtprime〉QnEnL otimes |j〉A otimes |j〉M otimes |tprime〉θ)

(

Jnsum

j=1


)

ge 1minusradic

2|θ|ǫ minus |θ|2ǫ2 minusradic

8radicǫminus 16ǫ2 minus 32ǫ

radicǫminus 8ǫ

(94)

ge 1minusradic

2|θ|radicǫminus

radic8 4radicǫ (95)

This means that if n is large enough then for anypositive δ and ǫ there is an (n

radic

2|θ|radicǫ +radic8 4radicǫ) code

with rate

mintχ(X Qt)minusmax

tχ(X Et)minus 2δ

Proposition 1 The entanglement generating capacity of(Nt)tisinθ with CSI at the encoder is

ACSI = limnrarrinfin

1

nmintisinθ

maxρisinS(H)Qn

IC(ρNtotimesn) (96)

Proof As the authors of [18] showed after receiving adummy code word as the first block the receiver also canhave CSI Then we have the case where both the senderand the receiver have CSI But this case is equivalent tothe case where we only have one channel (Nt) insteadof a family of channels (Nt) t = 1 |θ| and wemay assume it is the worst channel The bits that weuse to detect the CSI are large but constant so it is neg-ligible compared to the rest By [20] the entanglementgenerating capacity of the quantum channel Nt is

limnrarrinfin

1

nmax

ρisinS(H)QnIC(ρN

otimesnt )

The proof of the converse is similar to those given inthe proof of Theorem 2 where we consider a worst tprime

Proposition 2 The entanglement generating capacity of(Nt)tisinθ with feedback is bounded as follows

Afeed ge limnrarrinfin

1

nmintisinθ

maxρisinS(H)Qn


Proof As the authors of [18] showed the receiver can de-tect the channel state t correctly after receiving a dummyword as the first block Then he can send t back to thesender via feedback

Remark 8 Feedback can improve the channel capacity ofquantum channels in some cases (cf [27]) Thus it canbe possible that the lower bound in Proposition 2 is nottight For a one-way entanglement distillation protocolusing secret key cf [21]

18

VII FURTHER NOTES

In this section we will discuss the proof of our resultof the previous sectionLet P Q HP and HQ be defined as in Section II Let

N be a quantum channel S(HP) rarr S(HQ) In generalthere are two ways to represent a quantum channel ie a completely positive trace preserving map S(HP) rarrS(HQ) with linear algebraic tools

1 Operator Sum Decomposition (Kraus Representation)

N(ρ) =

Ksum

i=1

AiρAilowast (98)

where A1 AK (Kraus operators) are linear operatorsS(HP) rarr S(HQ) (cf[26] [6] and [31]) They satisfy

the completeness relationsumK

i=1 AilowastAi = IHP The rep-

resentation of a quantum channel N according to (98)is not unique Let A1 AK and B1 BKprime be twosets of Kraus operators (by appending zero operatorsto the shorter list of operation elements we may ensurethat K prime = K) Suppose A1 AK represents N thenB1 BK also represents N if and only if there exist aK timesK unitary matrix (uij)ij=1K such that for all i

we have Ai =sumK

j=1 uijBj (cf [31])

2 Isometric Extension (Stinespring Dilation)

N(ρ) = trE (UNρUlowastN ) (99)

where UN is a linear operator S(HP) rarr S(HQE) suchthat Ulowast

NUN = IHP and E is the quantum system ofthe environment (cf [37] [6] and also cf [38] for amore general Stinespring Dilation Theorem) HE can bechosen such that dimHE le (dimHP)2 The isometricextension of a quantum channel N according to (99) isnot unique either Let U and U prime be two linear operatorsS(HP) rarr S(HQE) Suppose U represents N then U prime

also represents N if and only if U and U prime are unitarilyequivalent

It is well known that we can reduce each ofthese two representations of the quantum chan-nel from the other one Let A1 AK be aset of Kraus operators which represents N Let|j〉E j = 1 K be an orthonormal system on

HE Then UN =sumK

j=1 Aj otimes |j〉E is an isometric exten-

sion which represents N since(

sumKj=1 Aj otimes |j〉E

)

ρ(

sumKk=1Ak otimes |k〉E

)lowast=

sumKj=1AjρAj

lowast and(


)lowast (sumK

k=1Ak otimes |k〉E)

=sumK

j=1 AjlowastAj

For the other way around every isometric extensionUN that represents N can be written in the form

UN =sumK

j=1Aj otimes |j〉E ie if the sender sends ρ

and if the environmentrsquos measurement gives |i〉E thereceiverrsquos outcome will be AiρAi

lowast Here A1 AK

is a set of Kraus operators which represents N and|j〉E j = 1 K is an orthonormal system on HEUsing either of both methods to represent a quantum

channel one can show that (cf [20]) the entanglementgenerating capacity of a quantum channel N is

A(N) = limnrarrinfin

1

nmax

ρisinS(H)QnIC(ρN

otimesn) (100)

The Kraus representation describes the dynamics ofthe principal system without having to explicitly considerproperties of the environment whose dynamics are oftenunimportant All that we need to know is the systemof the receiver alone this simplifies calculations In [25]an explicit construction of a quantum error correctioncode (both perfect and approximate information recov-ery) with the Kraus operators is given In the Stinespringdilation we have a natural interpretation of the system ofthe environment From the Stinespring dilation we canconclude that the receiver can detect almost all quantuminformation if and only if the channel releases almostno information to the environment In [36] an alterna-tive way to build a quantum error correction code (bothperfect and approximate information recovery) is givenusing this fact The disadvantage is that we suppose it issuboptimal for calculating the entanglement generatingcapacity of a compound quantum channel without CSIat the encoderIn [12] the entanglement generating capacity for the

compound quantum channel is determined using a quan-tum error correction code of [25] which is built by Krausoperators Their result is the following The entangle-ment generating capacity of a quantum wiretap channelN = (Nt)tisinθ is


1

nmax

ρisinS(H)Qnmintisinθ


This result is stronger than our result in Theorem 3 Thisis due to the fact that we use for our proof a quan-tum error correction code of [36] which is based uponthe Stinespring dilation If we use the Kraus operatorsto represent a compound quantum channel we have abipartite system and for calculating the entanglementgenerating capacity of a compound quantum channelwe can use the technique which is similar to the caseof a single quantum channel However if we use theStinespring dilation to represent a compound quantumchannel we have a tripartite system which includes thesender the receiver and in addition the environmentUnlike in the case of a single quantum channel for com-pound quantum channel we have to deal with uncertaintyat the environment If the sender knows the CSI thetransmitters can build an (n ǫ) code for entanglementgenerating with rate mint [χ(X Qt)minus χ(X Et)] minus δ =mintisinθ IC(ρNt) minus δ (Proposition 1) for any positive δand ǫ This result is optimal (cf [12]) But if thesender does not know the CSI he has to build an en-coding operator by considering every possible channel

19

state for the environment Therefore the maximal ratethat we can achieve is mint χ(X Qt) minus maxt χ(X Et)but not mintisinθ IC(ρNt) = mint [χ(X Qt)minus χ(X Et)]This is only a lower bound of the entanglement generat-ing capacity It is unknown if we can achieve the strongerresult (101) using the Stinespring dilation

ACKNOWLEDGMENT

Support by the Bundesministerium fur Bildung undForschung (BMBF) via grant 16BQ1050 and 16BQ1052

and the National Natural Science Foundation of Chinavia grant 61271174 is gratefully acknowledged We wouldlike to thank the reviewers for their valuable commentswhich helped us to improve our manuscript

[1] R Ahlswede Elimination of correlation in random codesfor arbitrarily varying channels Z Wahrscheinlichkeits-theorie und verw Geb Vol 44 159-185 1978

[2] R Ahlswede and V Blinovsky Classical capacity ofclassical-quantum arbitrarily varying channels IEEETrans Inform Theory Vol 53 No 2 526-533 2007

[3] R Ahlswede and A Winter Strong converse for identifi-cation via quantum channels IEEE Trans Inform The-ory Vol 48 No 3 569-579 2002 Addendum IEEETrans Inform Theory Vol 49 No 1 346 2003

[4] K M R Audenaert A sharp continuity estimate for thevon Neumann entropy J Phys A Math Theor Vol40 8127-8136 2007

[5] H Barnum E Knill M A Nielsen On Quantum Fideli-ties and Channel Capacities IEEE Trans Inform The-ory Vol 46 1317-1329 2000

[6] H Barnum M A Nielsen and B Schumacher Infor-mation transmission through a noisy quantum channelPhys Rev A Vol 57 4153 1998

[7] C H Bennett Quantum cryptography using any twonon-orthogonal states Physical Review Letters Vol 683121-3124 1992

[8] C H Bennett and G Brassard Quantum cryptographypublic key distribution and coin tossing Proceedings ofthe IEEE International Conference on Computers Sys-tems and Signal Processing Bangalore 175 1984

[9] I Bjelakovic and H Boche Classical capacities of av-eraged and compound quantum channels IEEE TransInform Theory Vol 57 No 7 3360-3374 2009

[10] I Bjelakovic H Boche G Janszligen and J Notzel Ar-bitrarily varying and compound classical-quantum chan-nels and a note on quantum zero-error capacities Infor-mation Theory Combinatorics and Search Theory inMemory of Rudolf Ahlswede H Aydinian F Cicaleseand C Deppe eds Vol7777 247-283 arXiv120963252012

[11] I Bjelakovic H Boche and J Notzel Entanglementtransmission and generation under channel uncertaintyuniversal quantum channel coding Communications inMathematical Physics Vol 292 No 1 55-97 2009

[12] I Bjelakovic H Boche and J Notzel Entanglementtransmission capacity of compound channels Proc ofInternational Symposium on Information Theory ISIT2009 1889-1893 Korea 2009

[13] I Bjelakovic H Boche and J Sommerfeld Capacity re-sults for compound wiretap channels Problems of Infor-

mation Transmission Vol 49 No 1 73-98 2013 originalRussian text Problemy Peredachi Informatsii Vol 49No 1 83-111 2011

[14] D Blackwell L Breiman and A J Thomasian Thecapacity of a class of channels Ann Math Stat Vol 30No 4 1229-1241 1959

[15] V Blinovsky and M Cai Classical-quantum arbitrar-ily varying wiretap channel Information Theory Com-binatorics and Search Theory in Memory of RudolfAhlswede H Aydinian F Cicalese and C Deppe edsVol7777 234-246 2013

[16] M Bloch and J N Laneman On the secrecy capacityof arbitrary wiretap channels Communication Controland Computing Forty-Sixth Annual Allerton ConferenceAllerton House UIUC USA 818-825 2008

[17] H Boche M Cai and C Deppe Classical-Quantum Arbitrarily Varying Wiretap ChannelmdashA Ca-pacity Formula with Ahlswede DichotomymdashResourcesarXiv13078007 2013

[18] M Cai and N Cai Channel state detecting code for com-pound quantum channel preprint

[19] N Cai A Winter and R W Yeung Quantum privacyand quantum wiretap channels Problems of InformationTransmission Vol 40 No 4 318-336 2004

[20] I Devetak The private classical information capacityand quantum information capacity of a quantum chan-nel IEEE Trans Inform Theory Vol 51 No 1 44-552005

[21] I Devetak and A Winter Distillation of secret key andentanglement from quantum states Proc R Soc A Vol461 207-235 2005

[22] M Fannes A continuity property of the entropy densityfor spin lattice systems Communications in Mathemati-cal Physics Vol 31 291-294 1973

[23] A Holevo Statistical problems in quantum physics Pro-ceedings of the second Japan-USSR Symposium on Prob-ability Theory ser Lecture Notes in Mathematics GMaruyama and J V Prokhorov Eds Vol 330 104-119Springer-Verlag Berlin 1973

[24] A Holevo The Capacity of the Quantum Channel withGeneral Signal States IEEE Trans on Inf Theory Vol44 No 1 269-273 1998

[25] R Klesse Approximate quantum error correction ran-dom codes and quantum channel capacity Phys Rev A75 062315 2007

20

[26] K Kraus States Effects and Operations SpringerBerlin 1983

[27] D Leung J Lim and P Shor On quantum capacity oferasure channel assisted by back classical communicationPhys Rev Lett Vol 103 No 24 240505 2009

[28] Y Liang G Kramer H Poor and S Shamai Com-pound wiretap channels EURASIP Journal on WirelessCommunications and Networking - Special issue on wire-less physical layer security archive Vol 2009 Article No5 2009

[29] S Lloyd Capacity of the noisy quantum channel Phys-ical Review A Vol 55 No 3 1613-1622 1997

[30] V D Milman and G Schechtman Asymptotic The-ory of Finite Dimensional Normed Spaces Lecture Notesin Mathematics 1200 Springer-Verlag corrected secondprinting Berlin UK 2001

[31] M Nielsen and I Chuang Quantum Computationand Quantum Information Cambridge University Press2000

[32] T Ogawa and H Nagaoka Making good codes forclassical-quantum channel coding via quantum hypoth-esis testing IEEE Trans Inform Theory Vol 53 No 62261-2266 2007

[33] V Paulsen Completely Bounded Maps and Operator Al-gebras Cambridge Studies in Advanced Mathematics 78Cambridge University Press Cambridge UK 2002

[34] B Schumacher and M A Nielsen Quantum data pro-cessing and error correction Phys Rev A Vol 54 26291996

[35] B Schumacher and M D Westmoreland Sending Clas-sical Information via Noisy Quantum Channels PhysRev A Vol 56 No 1 131-138 1997

[36] B Schumacher and M D Westmoreland Approximatequantum error correction Quant Inf Proc Vol 1 No8 5-12 2002

[37] P W Shor The quantum channel capacity and coherentinformation lecture notes MSRI Workshop on QuantumComputation 2002

[38] W F Stinespring Positive functions on C-algebrasProc Amer Math Soc Vol 6 211 1955

[39] S Watanabe Private and quantum capacities of morecapable and less noisy quantum channels Phys Rev A85 012326 2012

[40] M Wilde Quantum Information Theory CambridgeUniversity Press 2013

[41] A Winter Coding theorem and strong converse for quan-tum channels IEEE Trans Inform Theory Vol 45 No7 2481-2485 1999

[42] A D Wyner The wire-tap channel Bell System Tech-nical Journal Vol 54 No 8 1355-1387 1975

21

lies of channels are quantum channels while the sendertransmits classical information

A quantum wiretap channel is described by a map Nwhich maps the set of density operators on a system GA

to the set of density operators on a composite systemGBZ Here GA is the the system of the sender GB

is the system observed by the legal receiver and GZ isthe system observed by the wiretapper We only allowpassive eavesdropping attacks ie the actions which thewiretapper performs on his system have no influence onthe legal receiverrsquos system For active eavesdropping at-tacks ie the actions which the wiretapper performs onhis system have influences on the legal receiverrsquos sys-tem cf please arbitrarily varying quantum channelswhich are generalizations of compound quantum chan-nels ([2] [10] [15] [1] and [17]) Following [19] wedefine a quantum wiretap channel with passive eaves-dropping attacks as a pair of channels (WV ) For everydensity operator ζ on GA we define W (ζ) = trB(N(ζ))and V (ζ) = trZ(N(ζ))

References[8] and [7] are two well-known examples forsecure quantum information transmission using quantumkey distributions Good one-shot results for quantumchannels with a wiretapper who is limited in his actionshave been obtained But our goal is to have a moregeneral theory for channel security in quantum informa-tion theory ie message transmission should be secureagainst every possible kind of eavesdropping Further-more we are interested in asymptotic behavior when wedeliver a large volume of messages by many channel usesTherefore we consider an alternate paradigm for the de-sign of quantum channel systems which is called embed-ded security Here we embed protocols with a guaran-teed security into the bottom layerof the model of com-munications systems which is the physical layer We usechannels for example fiber-optic cables for generatingand transmitting secure messages which generate securekeys or secure message transmissions

Since we allow every possible kind of eavesdroppingwe use the Holevo χ quantity as our security criterion(cf (13)) By [24] and [35] the wiretapper can never ob-tain more information asymptotically than the Holevo χquantity no matter which strategy the wiretapper usesAnother widely used security criterion is the variationaldistance between pApZ and pAZ Here pAZ is the jointprobability describing the senderrsquos random variable andthe wiretapperrsquos random variable pA is the marginalprobability describing the senderrsquos random variable andpZ is the marginal probability describing the wiretap-perrsquos random variable respectively The Holevo χ quan-tity using a strong condition for the security criterion(cf Remark 2) is stronger than the variational distancebetween pApZ and pAZ in the sense that if the Holevoχ quantity between pA and the wiretap channelrsquos out-put (using strong condition) goes to zero the variationaldistance between pApZ and pAZ goes to zero as well (cf

[21] and [16])

Our results are summarized as follows Under the con-dition that the sender has knowledge about the CSI thesecrecy capacity for these two channel models is derivedAdditionally when the sender does not have any knowl-edge about the CSI we determine the secrecy capacityof the compound classical-quantum wiretap channel andgive a lower bound for the secrecy capacity of the classicalcompound channel with quantum wiretapper

As an application of the above results we turn to thequestion ldquoWhat is the maximal amount of entanglementthat we can generate or transmit over a given compoundquantum channelrdquo For the sender and the receiver theobjective is to share a nearly maximally entangled stateon a (2nR times 2nR) dimensional Hilbert space by usinga large number n instances of the compound quantumchannel In [5] it is shown how to send a large amountof entangled quantum states through a noisy quantumchannel such that the channel does not modify the entan-glement However the study of entanglement generationallows a noisy quantum channel to modify the entan-glement as long as the transmitters can use a recoveryalgorithm to restore the entanglement The entangle-ment generating capacity of a quantum channel has beendetermined in [20] and [29] The entanglement generat-ing capacities of a compound quantum channel with andwithout CSI have been determined in [11] and [12] Inour paper we derive a lower bound on the entanglementgenerating capacity of the compound quantum channelby using an alternative technique to the method in [11]and [12] (cf Section VII) Furthermore we derive the en-tanglement generating capacity of the compound quan-tum channel with CSI at the encoder using an alternativetechnique

The main definitions are given in Section II

In Section III we present some known results for theclassical compound wiretap channel which are used forthe proof of the results in Section IV

In Section IV we discuss the classical compound chan-nel with a quantum wiretapper For the case when thesender has the full knowledge about the CSI we derivethe secrecy capacity For the case when the sender doesnot know the CSI we give a lower bound for the se-crecy capacity In this channel model the wiretapper usesclassical-quantum channels

In Section V we derive the secrecy capacity of thecompound classical-quantum wiretap channel with CSIIn this model both the receiver and the wiretapper useclassical quantum channels and the set of the channelstates may be finite or infinite

In Section VI we use the results of Section V to derivea lower bound on the entanglement generating capacityfor the compound quantum channel The entanglementgenerating capacity of the compound quantum channelwith CSI at the encoder is also derived

In Section VII we discuss our proof of the previoussection and remind the reader about the two ways to

2


II PRELIMINARIES









Pδ and the


n) (here












otimesnt (xn) =





S(V|P ) =sum

xisinA

P (x)S(V(x))


PQ)








3


χ(P Φ) = S

(

sum

xisinA

P (x)ρx

)

minussum

xisinA

P (x)S (ρx)




4nα2 (1)






tr(

Votimesn(xn)ΠVα(x

n))

ge 1minus ad

4nα2 (4)


radicn (5)


n)


n) (6)






tr(


radica

)

ge 1minus ad

4nα2 (7)








t (x) isin S(H)



P) isin S(HQ)







that all codes in




4



maxtisinθ

maxjisin1Jn

sum

xnisinAn

Et(xn|j)Wn


maxtisinθ







1nI(XuniK


[13] and use (9)



maxtisinθ

maxjisin1Jn

sum

xnisinAn

E(xn|j)Wnt (D

cj |xn) le ε (10)

maxtisinθ




maxtisinθ

maxjisin1Jn

sum

xnisinAn

Et(xn|j)Wn


maxtisinθ







maxtisinθ

maxjisin1Jn

sum

xnisinAn

E(xn|j)Wnt (D

cj |xn) le ε (14)

maxtisinθ








maxtisinθ

1

Jn

Jnsum

j=1

tr(


)

le ε (16)

maxtisinθ




maxtisinθ

maxjisin1Jn

tr(


)

le ε

(18)

maxtisinθ




(

Notimesnt

)





and

5





t )](

|ψ〉〈ψ|APn)

(20)


Notimesnt

)



HA D)


mintisinθ

F(


)

ge 1minus ε (21)

where

|ΦK〉AM =

radic

1

Jn

Jnsum

j=1

|j〉A|j〉M




CHANNELS




n) =

pnt (x

n)pnt (T n


ptδ

0 else





Pr

maxjisin1Jn

Lntsum

l=1

1

LntW

nt (D

cj |X

(t)jl ) gt

radicT2minusnω2



t (Dcj |X

(t)jl ) is a random




lfloor



CSCSI ge mintisinθ

maxUrarrArarr(BK)t





CSCSI ge mintisinθ

maxUrarrArarr(BK)t



CSCSI = mintisinθ

maxUrarrArarr(BK)t




pn(xn)pn(T n

pδ) if xn isin T n

pδ

0 else




Pr

(

maxjisin1Jn

Lnsum

l=1

1

LnW

nt (D

cj |Xjl) gt

radicT 2minusnω2

)






(mintisinθ


I(U Kt)) (26)

6


QUANTUM WIRETAPPER


II


CSCSI = mintisinθ

maxUrarrArarr(BZ)t


1

nχ(U Zn

t ))



(mintisinθ





nLet





Let

Qt(xn) = ΠptVtα

radicaΠVtα(x


n)ΠptVtαradica



ρminusradicXρ

radicX le

radic2λ (29)



radicX le

radic8λ In [32]




Qt(xn)minus V

otimesnt (xn) le

radic

2(ad+ d)

nα2 (30)

We set Θt =sum

xnisinT nptδ

ppriment (xn)Qt(x

n) For given zn






ρλ

where ρ =sum




Pr

(

Lminus1Lsum

i=1


ρλ gt ǫ

)


minusL ǫ2λ

2 ln2(dimV)micro

)

(31)




Qt(xn)



n)ΠptVtαradica


radicaΠVtα(x

n)ΠptVtαradica


radica







enough we have

Pr

Lntsum

l=1

1


7


middot exp(

minusLntǫ2


radicn(

radicaminus1)

)


middot exp(

minusLntǫ2


radicn(

radicaminus1)

)

le exp(


(34)


S(pt)minus S(Vt|pt)

= S

sum

j

pt(j)sum

l

1

LntVotimesnt (X

(t)jl )

minussum

j

pt(j)S

(

sum

l

1

LntVotimesnt (X

(t)jl )

)

= χ(ptZt)


Pr

Lntsum

l=1

1

LntQt(X



and

Pr

Lntsum

l=1

1

LntQt(X


= 1minus Pr

⋃

t

⋃

j

Lntsum

l=1

1

LntQt(X









⋂

t

maxjisin1Jn

Lntsum

l=1

1

LntW

nt (D

cj(X )|X(t)

jl ) le ǫ

cap

Lntsum

l=1

1

LntQt(X






Lntsum

l=1

1

LntW

nt (D

cj |x

(t)jl ) le ǫ (37)

and

Lntsum

l=1

1

LntQt(x



R = mintisinθ

maxUrarrArarr(BZ)t


1

nχ(U Zn

t ))minusγ


code(


Dj j = 1 Jn)

such that

1

nlog Jn ge R (39)

limnrarrinfin

maxtisinθ

maxjisin1Jn

Lntsum

l=1

1

LntW

nt (D

cj |x

(t)jl ) = 0 (40)


holds Votimesnt (x

(t)jl ) minus Qt(x



Lntsum

l=1

1

LntVotimesnt (x

(t)jprimel)minusΘt

le Lntsum

l=1

1

LntVotimesnt (x

(t)jprimel)minus

Lntsum

l=1

1

LntQt(x

(t)jprimel)

+ Lntsum

l=1

1

LntQt(x

(t)jprimel)minusΘt

leLntsum

l=1

1

LntVotimesn


(t)jprimel)

+ L

(t)ntsum

l=1

1

LntQt(x

(t)jprimel)minusΘt

le 2ǫ (41)

and sumJn

j=11Jn

sumLnt

l=11

LntVotimesnt (x


8


e then




χ(XuniZnt )

= S

Jnsum

j=1

1

Jn

Lntsum

l=1

1

LntVotimesnt (x

(t)jl )

minusJnsum

j=1

1

JnS

Lntsum

l=1

1

LntVotimesnt (x

(t)jl )

le

∣

∣

∣

∣

∣

∣

S

Jnsum

j=1

1

Jn

Lntsum

l=1

1

LntVotimesnt (x

(t)jl )

minus S (Θt)

∣

∣

∣

∣

∣

∣

+

∣

∣

∣

∣

∣

∣

S(Θt)minusJnsum

j=1

1

JnS

Lntsum

l=1

1

LntVotimesnt (x

(t)jl )

∣

∣

∣

∣

∣

∣


+

∣

∣

∣

∣

∣

∣

Jnsum

j=1

1

Jn

S(Θt)minus S

Lntsum

l=1

1

LntVotimesnt (x

(t)jl )

∣

∣

∣

∣

∣

∣



maxtisinθ



Pr(

Et(j) = x(t)jl

)

= 1Lnt



CSCSI ge mintisinθ

maxUrarrArarr(BZ)t


1

nχ(U Zn

t ))

(45)



maxtisinθ

maxjisin1Jn

sum

xnisinAn

E(xn|j)Wnt (D

cj |xn) = ǫ1n (46)

maxtisinθ

χ(J Znt ) = ǫ2n (47)




log Jn = H(Xuni)










I(XuniBtprime) +1

n+

1

nǫ1n log Jn


nχ(XuniZ

ntprime)] +

ǫ1n

n+

1

nlog Jn +

ǫ2n

n


nχ(XuniZ

ntprime)] + ǫ (49)


CSCSI le mintisinθ

maxUrarrArarr(BZ)t


1

nχ(U Zn

t ))






Qt(xn) = ΠpVtα

radicaΠVtα(x


n)ΠpVtαradica

9

and Θt =sum


n)


Pr

(

Lnsum

l=1

1

LnQt(X


)

ge 1minus 2minusnυ

(51)


Lnsum

l=1

1

LnW

nt (D

cj |xjl) le ǫ

Lnsum

l=1

1


For any γ gt 0 let


(

mintisinθ


)

minus γ


1Lnt


1n log Jn ge R and

limnrarrinfin

maxtisinθ

maxjisin1Jn

Lnsum

l=1

1

LnW

nt (D

cj |xjl)) = 0 (52)


maxtisinθ




(mintisinθ


χ(U Zt))


WIRETAP CHANNEL


Section II



mintisinθ

maxPinpwt

1

n(χ(PinpB

nt )minus χ(PinpZ

nt ))



CS = limnrarrinfin

maxUrarrArarr(BZ)t

1

n

(

mintisinθ

χ(U Bnt )

minusmaxtisinθ

χ(U Znt )

)

(55)







mintχ(PinpWt) (56)



log Tmint maxPinp


mint χ(PinpWt) =






log Jn = maxPinpw


n))minus ǫ (57)

10



log Jnt = maxPinpwt


nt ))minus ǫ (58)

Thus


mintisinθ

maxPinpwt

1

n(χ(PinpB

nt )minus χ(PinpZ

nt ))

(59)





C(t)2 = (E(t) D(t)




D(t)j j t isin θ

to decode the





limnrarrinfin

maxPinpwtprime

1

n(χ(PinpB


ntprime))



mintisinθ

maxPinpwt

1

n(χ(PinpB

nt )minus χ(PinpZ

nt ))




nt )


limnrarrinfin

maxPinpwt

1

n(χ(PinpB

nt )minus χ(PinpZ

nt ))

= maxPinpwt




nt )


CCSI = mintisinθ

maxPinpwt











Pr[

tr(

Wnt (ρ

nXjl

)DtXjl

)


gt 1minus 2minusnβ


sum

tisinθ

Jnsum

j=1

Lnsum

l=1

Dtxjlle I

We define



n)ΠpVtαradica

and Θt =sum



11


Pr

(

Lnsum

l=1

1

LnQt(ρX(t)

jl


ge 1minus 2minusnυ

(62)


mintisinθ

tr(

Wnt (ρ

nxjl

)Dtxjl

)

ge 1minus 2minusnβ

and

maxtisinθ

Lnsum

l=1

1

LnQt(ρxjl

)minusΘt le ǫ

We define Dj =sum

tisinθ

sumLn

l=1Dtxjl then

sumJn

j=1Dj =sum

tisinθ

sumJn

j=1

sumLn



tr(

Wntprime (ρ

nxjlprime

)Dj

)

=sum

tisinθ

Lnsum

l=1

tr(

Wntprime (ρ

nxjlprime

)Dtxjl

)

ge tr(

Wntprime (ρ

otimesnxjlprime

)Dtprimexjlprime

)

ge 1minus 2minusnβ


(radicM1M2

radicM1

)

ge 0

For any γ gt 0 let


1

n

[

mintisinθ

χ(pBnt )minusmax

tisinθχ(pZn

t )

]

minus γ


w(j) =sumLn

l=11Lnρnxjl

j = 1 Jn Dj j =

1 Jn)


maxtisinθ

maxjisin1Jn

tr(


)

le λ


maxtisinθ



CS ge limnrarrinfin

maxUrarrArarr(BZ)t

1

n

(

mintisinθ

χ(U Bnt )minusmax

tisinθχ(U Zn

t )

)

(65)




maxtisinθ

maxjisin1Jn

tr(


t

(

ρ(n)j

))

le λn (66)

maxtisinθ








1

nχ(XuniB

ntprime) + ξ

= mintisinθ

1

nχ(XuniB

nt ) + ξ

le mintisinθ

1

nχ(XuniB

nt )minusmax

tisinθ

1

nχ(XuniZ

nt ) + ξ +

1

nǫ2n

le 1

n

(

mintisinθ


tisinθχ(XuniZ

nt )

)

+ ǫ (68)

Thus

CS le limnrarrinfin

maxUrarrArarr(BZ)t

1

n

(

mintisinθ

χ(U Bnt )minusmax

tisinθχ(U Zn

t )

)





inftisinθ

maxPinpwt

1

n(χ(PinpB

nt )minus χ(PinpZ

nt ))

(70)


Wdiams = supnisinN



12



W (k))K





W (k))K

k=1in the










1

JN

JNsum

j=1

tr(


)





the message




tprime

)

(wtprime(j))]

=

tr(

trHprimeotimesN

[



])

We have

tr∣

∣

(


tprime

)

(wtprime (j))∣

∣

= tr(

trHprimeotimesN

∣



∣

∣

)

= tr∣



∣

∣

=∥



∥

∥

1






tprime

∥

∥

diams=

∥

∥


∥

∥


tiplicative

It follows that

∣

∣

∣

∣

1

JN

JNsum

j=1

tr(


)

minus 1

JN

JNsum

j=1

tr(


)

∣

∣

∣

∣

le 1

JN

JNsum

j=1

∣

∣tr[(


tprime

)


]∣

∣

le 1

JN

JNsum

j=1

tr∣

∣

(


tprime

)


∣

∣

le 1

JN

JNsum

j=1

tr∣

∣

(


tprime

)

(wtprime(j))∣

∣

le 1

JNJNNτ

= Nτ (74)




le

∣

∣

∣

∣

∣

∣

S

JNsum

j=1

1

JNVt(ρ(j))

minus S

JNsum

j=1

1

JNVtprime(ρ(j))

∣

∣

∣

∣

∣

∣

+

∣

∣

∣

∣

∣

∣

JNsum

j=1

1


JNsum

j=1

1


∣

∣

∣

∣

∣

∣




suptisinθ

1

JN

JNsum

j=1

tr(


)




suptisinθ

1

JN

JNsum

j=1

tr(


)

ge 1minus λ

13

suptisinθ

χ(XuniZNt ) le ǫ



inftisinθ

maxPinpwt

1

n(χ(PinpB

nt )minus χ(PinpZ

nt ))








)minus2dprime4









Notimesnt

)




sum




A ge maxp

(

mintisinθ


χ(pEt)

)

(77)



ρPUlowastNt



sum


trE

(

sum

x


Nt

)


trQ

(

sum

x


Nt

)




sum



Nt

)

and VNt HP rarr HQ

sum



Nt

)



and


14


VNt)tisinθ since



xjl




n

t on HEn

such that

tr(

(DQn

txjlotimes IE

n

)UNtρP

n

xjlUlowastNt

)


and

ωEn

jt minus ξEn

t 1 lt ǫ (79)

where ωEn

jt = 1Lnt

sumLnt

l=1 trQn

(

UNtρP

n

xjlUlowastNt

)


xjlmay be pure or

mixed Assume ρPn


sumni=1 p

primejl(i)|κ

(i)xjl〉〈κ(i)

xjl |Pn

then

nsum

i=1

pprimejl(i)tr(

(DQn

txjlotimes IE

n

)UNt|κ(i)

xjl〉〈κ(i)

xjl|Pn

UlowastNt

)

tr

(

(DQn

txjlotimes IE

n

)UNt(

nsum

i=1


〉〈κ(i)xjl

|Pn

)UlowastNt

)

ge 1minus 2minusnβ



tr(

(DQn

txjlotimes IE

n

)UNt|κ(i)

xjl〉〈κ(i)

xjl|Pn

UlowastNt

)

ge 1minus 2minusnβ



there is a ξEn

t on HEn

such that

1

Lnt

Lntsum

l=1

trQn

(

UNt|κ(ilj)


UlowastNt

)

minus ξEn

t 1 lt ǫ

Thus(


xjl |Pn

j l DQn

txjl j l t

)


ρPn

xjl j l DQn

txjl j l t

)



Assume ρPn


n



n isin S(HQn

) we have

V QnMLθ

(


)

(V QnMLθ)lowast

=sum

t

sum

j

sum

l

(

DQn

txjlρQ

n)


We denote

ψQnEnMLθjlt

=(

IEn otimes V QnMLθ

)

(

UN otimes IMLθ)

[

|κjl〉〈κjl|Pn


(


(

IEn otimes V QnMLθ

)lowast


F(

trQnEn

(

ψQnEnMLθjlt

)


ge 1minus ǫ (80)


nEn

on HQnEn

such that

〈0|θ〈0|L〈0|M〈κjl|Pn (

UNtotimes IMLθ

)lowast

(

IEn otimes V QnMLθ

)lowast|ζjlt〉Q


= F

(


nEn


ge 1minus ǫ (81)



n |0〉M|0〉L|0〉θ

and

|bjlt〉PnMLθ =

(

UNtotimes IMLθ

)lowast (IE




We define

|ajk〉PnMLθ =

1radicLn

Lnsum

l=1


nMLθ

15

|bjkt〉PnMLθ =

1radicLn

Lnsum

l=1


nMLθ

and

|bjk〉PnMLθ =

1

|θ|

|θ|sum

t=1

|bjkt〉PnMLθ


1

Ln

Lnsum

k=1

〈ajk|bjk〉PnMLθ

=1

|θ|1

Ln

|θ|sum

t=1

Lnsum

k=1


=1

|θ|1

Ln

|θ|sum

t=1

Lnsum

l=1


ge 1minus ǫ (82)


1minus ǫ


|bjkj〉PnMLθ

=1

|θ|

|θ|sum

t=1

eminusiskj 〈ajkj

|bjkj t〉PnMLθ


∣

∣


|bjkj t〉PnMLθ

∣

∣

∣ le 1 we have

mintisinθ

∣

∣


|bjkj t〉PnMLθ

∣

∣



t isin θ

1minus |θ|ǫ

le∣

∣


|bjkj t〉PnMLθ

∣

∣

∣


|bjkj t〉PnMLθ

= eminusirkj

1

Ln

(

Lnsum

l=1


nMLθ

)

(

Lnsum

l=1


nMLθ

)

(83)


|jt〉QnEnL =

radic

1

Ln

Lnsum

l=1

eminus2πi(l

kjLn


and

|ϑjt〉QnEnMLθ =

radic

1

Ln

Lnsum

l=1

eminus2πilkj

Ln

[

IEn otimes V QnMLθ

]

(UnN |κjl〉P

n

)|0〉M|0〉L|0〉θ


F

(



)

=∣

∣



∣

∣

=1

Ln

(

Lnsum

l=1


nMLθ

)

(

Lnsum

l=1

eminus2πilkjLn e


)



n


such that

∥

∥

∥ξEn

t minus trQn

(


n

UlowastNt

)∥

∥

∥

1le ǫ (85)


∥

∥trQn

(


n

UlowastNt

)

minus trQn

(


∥

∥

1

le 2

[

1minus F

(

trQn

(


n

UlowastNt

)

trQn

(


)]12

le 2

[

1minus F

(


nEn


12

le 2radicǫ (86)


2minus η1 leradic


∥

∥

∥trQnL

(

|jt〉〈jt|QnEnL

)

minus ξEn

t

∥

∥

∥

1

=

∥

∥

∥

∥

∥

1

Ln

Lnsum

l=1

trQn

(


minus ξEn

t

∥

∥

∥

∥

∥

1

le 1

Ln

Lnsum

l=1

∥

∥

∥

∥

trQn

(


n

UlowastNt

)

minus trQn

(


∥

∥

∥

∥

1

+∥

∥

∥ξEn

t minus trQn

(


n

UlowastNt

)∥

∥

∥

1


16



such that if we set

χQnEnMLjjprimet =

(

UQnML(t) otimes IE

n)

(


)(

UQnML(t) otimes IE

n)lowast

then

F(


jjprimet

)


(88)where |ξt〉Q


rification of ξEn

t on HQnEnL



1

Jn

1

Ln

Jnsum

j=1

Lnsum

l=1


n |j〉A

Jnsum

j=1

Lnsum

l=1


n

(89)



1

Jn

1

Ln

(

IA otimes UnNtprime

)

[

Jnsum

j=1

Lnsum

l=1

eminus2πilkj

Ln |j〉A|κjl〉Pn

Jnsum

j=1

Lnsum

l=1

eminus2πilkj

Ln 〈κjl|Pn〈j|A

]

(

IA otimes UnNtprime

)lowast

=1

Jn

1

Ln

Jnsum

j=1

|j〉A(

Lnsum

l=1

eminus2πilkjLn Un

Ntprime|κjl〉P

n

)

Jnsum

j=1

(

Lnsum

l=1


n

(UnNtprime

)lowast)

〈j|A


τQn rarr trQnLθ

[

(

sum

tisinθ


)

V QnMLθ

(


)

V QnMLθlowast(

sum

tisinθ


)lowast]

(90)

to his outcome


We define

ιAQnEnMLθtprime

=

(

sum

tisinθ


)


)

(

1

Jn

1

Ln

Jnsum

j=1

|j〉A(

Lnsum

l=1

eminus2πilkjLn Un

Ntprime|κjl〉P

n

)

Jnsum

j=1

(

Lnsum

l=1


n

(UnNtprime

)lowast)

〈j|A



)lowast

(

sum

tisinθ


)lowast

=

(

sum

tisinθ


)

1

Jn(

Jnsum

j=1


Jnsum

j=1


(

sum

tisinθ


)lowast

(91)


AQnEnMLθtprime )


sumJn

j=1

sumJn



(

sum

tisinθ


)

(

sum

tisinθ

UQnML(t) otimes IE


= IAEn otimessum

tisinθ

UQnML(t) (UQnML


= IAQnEnMLθ

sum




F

ιAQnEnMLθtprime

1

Jn

Jnsum

j=1

Jnsum

jprime=1


= F

(

1

Jn(

Jnsum

j=1


Jnsum

j=1


1

Jn(

Jnsum

j=1


17

(

Jnsum

j=1


)

=1

Jn

∣

∣

∣

∣

( Jnsum

j=1


)

( Jnsum

j=1


)∣

∣

∣

∣



sumJn

j=1

sumJn



F

(

1

Jn

Jnsum

j=1

Jnsum

jprime=1


1

Jn(

Jnsum

j=1


(

Jnsum

j=1


)



1minus F ( η) le 1

2minus η1 le

radic

1minus F ( η)2


F ( η)

ge 1minus 1

2minus η1

ge 1minus 1

2minus υ1 minus

1

2υ minus η1

ge 1minusradic


1minus F (υ η)2


F

(


(

Jnsum

j=1


Jnsum

j=1


)

ge F

(

ιAQnEnMLθtprime

1

Jn(

Jnsum

j=1


(

Jnsum

j=1


)

ge 1minusradic



radicǫminus 8ǫ

(94)

ge 1minusradic

2|θ|radicǫminus



radic


with rate


tχ(X Et)minus 2δ



1

nmintisinθ

maxρisinS(H)Qn



limnrarrinfin

1

nmax

ρisinS(H)QnIC(ρN

otimesnt )




1

nmintisinθ

maxρisinS(H)Qn




18

VII FURTHER NOTES




N(ρ) =

Ksum

i=1

AiρAilowast (98)





we have Ai =sumK

j=1 uijBj (cf [31])







HE Then UN =sumK




)

ρ(


)lowast=

sumKj=1AjρAj

lowast and(


)lowast (sumK


=sumK

j=1 AjlowastAj


UN =sumK



lowast Here A1 AK




1

nmax

ρisinS(H)QnIC(ρN

otimesn) (100)




1

nmax




19


ACKNOWLEDGMENT





























20


















21


II PRELIMINARIES









Pδ and the


n) (here












otimesnt (xn) =





S(V|P ) =sum

xisinA

P (x)S(V(x))


PQ)








3


χ(P Φ) = S

(

sum

xisinA

P (x)ρx

)

minussum

xisinA

P (x)S (ρx)




4nα2 (1)






tr(

Votimesn(xn)ΠVα(x

n))

ge 1minus ad

4nα2 (4)


radicn (5)


n)


n) (6)






tr(


radica

)

ge 1minus ad

4nα2 (7)








t (x) isin S(H)



P) isin S(HQ)







that all codes in




4



maxtisinθ

maxjisin1Jn

sum

xnisinAn

Et(xn|j)Wn


maxtisinθ







1nI(XuniK


[13] and use (9)



maxtisinθ

maxjisin1Jn

sum

xnisinAn

E(xn|j)Wnt (D

cj |xn) le ε (10)

maxtisinθ




maxtisinθ

maxjisin1Jn

sum

xnisinAn

Et(xn|j)Wn


maxtisinθ







maxtisinθ

maxjisin1Jn

sum

xnisinAn

E(xn|j)Wnt (D

cj |xn) le ε (14)

maxtisinθ








maxtisinθ

1

Jn

Jnsum

j=1

tr(


)

le ε (16)

maxtisinθ




maxtisinθ

maxjisin1Jn

tr(


)

le ε

(18)

maxtisinθ




(

Notimesnt

)





and

5





t )](

|ψ〉〈ψ|APn)

(20)


Notimesnt

)



HA D)


mintisinθ

F(


)

ge 1minus ε (21)

where

|ΦK〉AM =

radic

1

Jn

Jnsum

j=1

|j〉A|j〉M




CHANNELS




n) =

pnt (x

n)pnt (T n


ptδ

0 else





Pr

maxjisin1Jn

Lntsum

l=1

1

LntW

nt (D

cj |X

(t)jl ) gt

radicT2minusnω2



t (Dcj |X

(t)jl ) is a random




lfloor



CSCSI ge mintisinθ

maxUrarrArarr(BK)t





CSCSI ge mintisinθ

maxUrarrArarr(BK)t



CSCSI = mintisinθ

maxUrarrArarr(BK)t




pn(xn)pn(T n

pδ) if xn isin T n

pδ

0 else




Pr

(

maxjisin1Jn

Lnsum

l=1

1

LnW

nt (D

cj |Xjl) gt

radicT 2minusnω2

)






(mintisinθ


I(U Kt)) (26)

6


QUANTUM WIRETAPPER


II


CSCSI = mintisinθ

maxUrarrArarr(BZ)t


1

nχ(U Zn

t ))



(mintisinθ





nLet





Let

Qt(xn) = ΠptVtα

radicaΠVtα(x


n)ΠptVtαradica



ρminusradicXρ

radicX le

radic2λ (29)



radicX le

radic8λ In [32]




Qt(xn)minus V

otimesnt (xn) le

radic

2(ad+ d)

nα2 (30)

We set Θt =sum

xnisinT nptδ

ppriment (xn)Qt(x

n) For given zn






ρλ

where ρ =sum




Pr

(

Lminus1Lsum

i=1


ρλ gt ǫ

)


minusL ǫ2λ

2 ln2(dimV)micro

)

(31)




Qt(xn)



n)ΠptVtαradica


radicaΠVtα(x

n)ΠptVtαradica


radica







enough we have

Pr

Lntsum

l=1

1


7


middot exp(

minusLntǫ2


radicn(

radicaminus1)

)


middot exp(

minusLntǫ2


radicn(

radicaminus1)

)

le exp(


(34)


S(pt)minus S(Vt|pt)

= S

sum

j

pt(j)sum

l

1

LntVotimesnt (X

(t)jl )

minussum

j

pt(j)S

(

sum

l

1

LntVotimesnt (X

(t)jl )

)

= χ(ptZt)


Pr

Lntsum

l=1

1

LntQt(X



and

Pr

Lntsum

l=1

1

LntQt(X


= 1minus Pr

⋃

t

⋃

j

Lntsum

l=1

1

LntQt(X









⋂

t

maxjisin1Jn

Lntsum

l=1

1

LntW

nt (D

cj(X )|X(t)

jl ) le ǫ

cap

Lntsum

l=1

1

LntQt(X






Lntsum

l=1

1

LntW

nt (D

cj |x

(t)jl ) le ǫ (37)

and

Lntsum

l=1

1

LntQt(x



R = mintisinθ

maxUrarrArarr(BZ)t


1

nχ(U Zn

t ))minusγ


code(


Dj j = 1 Jn)

such that

1

nlog Jn ge R (39)

limnrarrinfin

maxtisinθ

maxjisin1Jn

Lntsum

l=1

1

LntW

nt (D

cj |x

(t)jl ) = 0 (40)


holds Votimesnt (x

(t)jl ) minus Qt(x



Lntsum

l=1

1

LntVotimesnt (x

(t)jprimel)minusΘt

le Lntsum

l=1

1

LntVotimesnt (x

(t)jprimel)minus

Lntsum

l=1

1

LntQt(x

(t)jprimel)

+ Lntsum

l=1

1

LntQt(x

(t)jprimel)minusΘt

leLntsum

l=1

1

LntVotimesn


(t)jprimel)

+ L

(t)ntsum

l=1

1

LntQt(x

(t)jprimel)minusΘt

le 2ǫ (41)

and sumJn

j=11Jn

sumLnt

l=11

LntVotimesnt (x


8


e then




χ(XuniZnt )

= S

Jnsum

j=1

1

Jn

Lntsum

l=1

1

LntVotimesnt (x

(t)jl )

minusJnsum

j=1

1

JnS

Lntsum

l=1

1

LntVotimesnt (x

(t)jl )

le

∣

∣

∣

∣

∣

∣

S

Jnsum

j=1

1

Jn

Lntsum

l=1

1

LntVotimesnt (x

(t)jl )

minus S (Θt)

∣

∣

∣

∣

∣

∣

+

∣

∣

∣

∣

∣

∣

S(Θt)minusJnsum

j=1

1

JnS

Lntsum

l=1

1

LntVotimesnt (x

(t)jl )

∣

∣

∣

∣

∣

∣


+

∣

∣

∣

∣

∣

∣

Jnsum

j=1

1

Jn

S(Θt)minus S

Lntsum

l=1

1

LntVotimesnt (x

(t)jl )

∣

∣

∣

∣

∣

∣



maxtisinθ



Pr(

Et(j) = x(t)jl

)

= 1Lnt



CSCSI ge mintisinθ

maxUrarrArarr(BZ)t


1

nχ(U Zn

t ))

(45)



maxtisinθ

maxjisin1Jn

sum

xnisinAn

E(xn|j)Wnt (D

cj |xn) = ǫ1n (46)

maxtisinθ

χ(J Znt ) = ǫ2n (47)




log Jn = H(Xuni)










I(XuniBtprime) +1

n+

1

nǫ1n log Jn


nχ(XuniZ

ntprime)] +

ǫ1n

n+

1

nlog Jn +

ǫ2n

n


nχ(XuniZ

ntprime)] + ǫ (49)


CSCSI le mintisinθ

maxUrarrArarr(BZ)t


1

nχ(U Zn

t ))






Qt(xn) = ΠpVtα

radicaΠVtα(x


n)ΠpVtαradica

9

and Θt =sum


n)


Pr

(

Lnsum

l=1

1

LnQt(X


)

ge 1minus 2minusnυ

(51)


Lnsum

l=1

1

LnW

nt (D

cj |xjl) le ǫ

Lnsum

l=1

1


For any γ gt 0 let


(

mintisinθ


)

minus γ


1Lnt


1n log Jn ge R and

limnrarrinfin

maxtisinθ

maxjisin1Jn

Lnsum

l=1

1

LnW

nt (D

cj |xjl)) = 0 (52)


maxtisinθ




(mintisinθ


χ(U Zt))


WIRETAP CHANNEL


Section II



mintisinθ

maxPinpwt

1

n(χ(PinpB

nt )minus χ(PinpZ

nt ))



CS = limnrarrinfin

maxUrarrArarr(BZ)t

1

n

(

mintisinθ

χ(U Bnt )

minusmaxtisinθ

χ(U Znt )

)

(55)







mintχ(PinpWt) (56)



log Tmint maxPinp


mint χ(PinpWt) =






log Jn = maxPinpw


n))minus ǫ (57)

10



log Jnt = maxPinpwt


nt ))minus ǫ (58)

Thus


mintisinθ

maxPinpwt

1

n(χ(PinpB

nt )minus χ(PinpZ

nt ))

(59)





C(t)2 = (E(t) D(t)




D(t)j j t isin θ

to decode the





limnrarrinfin

maxPinpwtprime

1

n(χ(PinpB


ntprime))



mintisinθ

maxPinpwt

1

n(χ(PinpB

nt )minus χ(PinpZ

nt ))




nt )


limnrarrinfin

maxPinpwt

1

n(χ(PinpB

nt )minus χ(PinpZ

nt ))

= maxPinpwt




nt )


CCSI = mintisinθ

maxPinpwt











Pr[

tr(

Wnt (ρ

nXjl

)DtXjl

)


gt 1minus 2minusnβ


sum

tisinθ

Jnsum

j=1

Lnsum

l=1

Dtxjlle I

We define



n)ΠpVtαradica

and Θt =sum



11


Pr

(

Lnsum

l=1

1

LnQt(ρX(t)

jl


ge 1minus 2minusnυ

(62)


mintisinθ

tr(

Wnt (ρ

nxjl

)Dtxjl

)

ge 1minus 2minusnβ

and

maxtisinθ

Lnsum

l=1

1

LnQt(ρxjl

)minusΘt le ǫ

We define Dj =sum

tisinθ

sumLn

l=1Dtxjl then

sumJn

j=1Dj =sum

tisinθ

sumJn

j=1

sumLn



tr(

Wntprime (ρ

nxjlprime

)Dj

)

=sum

tisinθ

Lnsum

l=1

tr(

Wntprime (ρ

nxjlprime

)Dtxjl

)

ge tr(

Wntprime (ρ

otimesnxjlprime

)Dtprimexjlprime

)

ge 1minus 2minusnβ


(radicM1M2

radicM1

)

ge 0

For any γ gt 0 let


1

n

[

mintisinθ

χ(pBnt )minusmax

tisinθχ(pZn

t )

]

minus γ


w(j) =sumLn

l=11Lnρnxjl

j = 1 Jn Dj j =

1 Jn)


maxtisinθ

maxjisin1Jn

tr(


)

le λ


maxtisinθ



CS ge limnrarrinfin

maxUrarrArarr(BZ)t

1

n

(

mintisinθ

χ(U Bnt )minusmax

tisinθχ(U Zn

t )

)

(65)




maxtisinθ

maxjisin1Jn

tr(


t

(

ρ(n)j

))

le λn (66)

maxtisinθ








1

nχ(XuniB

ntprime) + ξ

= mintisinθ

1

nχ(XuniB

nt ) + ξ

le mintisinθ

1

nχ(XuniB

nt )minusmax

tisinθ

1

nχ(XuniZ

nt ) + ξ +

1

nǫ2n

le 1

n

(

mintisinθ


tisinθχ(XuniZ

nt )

)

+ ǫ (68)

Thus

CS le limnrarrinfin

maxUrarrArarr(BZ)t

1

n

(

mintisinθ

χ(U Bnt )minusmax

tisinθχ(U Zn

t )

)





inftisinθ

maxPinpwt

1

n(χ(PinpB

nt )minus χ(PinpZ

nt ))

(70)


Wdiams = supnisinN



12



W (k))K





W (k))K

k=1in the










1

JN

JNsum

j=1

tr(


)





the message




tprime

)

(wtprime(j))]

=

tr(

trHprimeotimesN

[



])

We have

tr∣

∣

(


tprime

)

(wtprime (j))∣

∣

= tr(

trHprimeotimesN

∣



∣

∣

)

= tr∣



∣

∣

=∥



∥

∥

1






tprime

∥

∥

diams=

∥

∥


∥

∥


tiplicative

It follows that

∣

∣

∣

∣

1

JN

JNsum

j=1

tr(


)

minus 1

JN

JNsum

j=1

tr(


)

∣

∣

∣

∣

le 1

JN

JNsum

j=1

∣

∣tr[(


tprime

)


]∣

∣

le 1

JN

JNsum

j=1

tr∣

∣

(


tprime

)


∣

∣

le 1

JN

JNsum

j=1

tr∣

∣

(


tprime

)

(wtprime(j))∣

∣

le 1

JNJNNτ

= Nτ (74)




le

∣

∣

∣

∣

∣

∣

S

JNsum

j=1

1

JNVt(ρ(j))

minus S

JNsum

j=1

1

JNVtprime(ρ(j))

∣

∣

∣

∣

∣

∣

+

∣

∣

∣

∣

∣

∣

JNsum

j=1

1


JNsum

j=1

1


∣

∣

∣

∣

∣

∣




suptisinθ

1

JN

JNsum

j=1

tr(


)




suptisinθ

1

JN

JNsum

j=1

tr(


)

ge 1minus λ

13

suptisinθ

χ(XuniZNt ) le ǫ



inftisinθ

maxPinpwt

1

n(χ(PinpB

nt )minus χ(PinpZ

nt ))








)minus2dprime4









Notimesnt

)




sum




A ge maxp

(

mintisinθ


χ(pEt)

)

(77)



ρPUlowastNt



sum


trE

(

sum

x


Nt

)


trQ

(

sum

x


Nt

)




sum



Nt

)

and VNt HP rarr HQ

sum



Nt

)



and


14


VNt)tisinθ since



xjl




n

t on HEn

such that

tr(

(DQn

txjlotimes IE

n

)UNtρP

n

xjlUlowastNt

)


and

ωEn

jt minus ξEn

t 1 lt ǫ (79)

where ωEn

jt = 1Lnt

sumLnt

l=1 trQn

(

UNtρP

n

xjlUlowastNt

)


xjlmay be pure or

mixed Assume ρPn


sumni=1 p

primejl(i)|κ

(i)xjl〉〈κ(i)

xjl |Pn

then

nsum

i=1

pprimejl(i)tr(

(DQn

txjlotimes IE

n

)UNt|κ(i)

xjl〉〈κ(i)

xjl|Pn

UlowastNt

)

tr

(

(DQn

txjlotimes IE

n

)UNt(

nsum

i=1


〉〈κ(i)xjl

|Pn

)UlowastNt

)

ge 1minus 2minusnβ



tr(

(DQn

txjlotimes IE

n

)UNt|κ(i)

xjl〉〈κ(i)

xjl|Pn

UlowastNt

)

ge 1minus 2minusnβ



there is a ξEn

t on HEn

such that

1

Lnt

Lntsum

l=1

trQn

(

UNt|κ(ilj)


UlowastNt

)

minus ξEn

t 1 lt ǫ

Thus(


xjl |Pn

j l DQn

txjl j l t

)


ρPn

xjl j l DQn

txjl j l t

)



Assume ρPn


n



n isin S(HQn

) we have

V QnMLθ

(


)

(V QnMLθ)lowast

=sum

t

sum

j

sum

l

(

DQn

txjlρQ

n)


We denote

ψQnEnMLθjlt

=(

IEn otimes V QnMLθ

)

(

UN otimes IMLθ)

[

|κjl〉〈κjl|Pn


(


(

IEn otimes V QnMLθ

)lowast


F(

trQnEn

(

ψQnEnMLθjlt

)


ge 1minus ǫ (80)


nEn

on HQnEn

such that

〈0|θ〈0|L〈0|M〈κjl|Pn (

UNtotimes IMLθ

)lowast

(

IEn otimes V QnMLθ

)lowast|ζjlt〉Q


= F

(


nEn


ge 1minus ǫ (81)



n |0〉M|0〉L|0〉θ

and

|bjlt〉PnMLθ =

(

UNtotimes IMLθ

)lowast (IE




We define

|ajk〉PnMLθ =

1radicLn

Lnsum

l=1


nMLθ

15

|bjkt〉PnMLθ =

1radicLn

Lnsum

l=1


nMLθ

and

|bjk〉PnMLθ =

1

|θ|

|θ|sum

t=1

|bjkt〉PnMLθ


1

Ln

Lnsum

k=1

〈ajk|bjk〉PnMLθ

=1

|θ|1

Ln

|θ|sum

t=1

Lnsum

k=1


=1

|θ|1

Ln

|θ|sum

t=1

Lnsum

l=1


ge 1minus ǫ (82)


1minus ǫ


|bjkj〉PnMLθ

=1

|θ|

|θ|sum

t=1

eminusiskj 〈ajkj

|bjkj t〉PnMLθ


∣

∣


|bjkj t〉PnMLθ

∣

∣

∣ le 1 we have

mintisinθ

∣

∣


|bjkj t〉PnMLθ

∣

∣



t isin θ

1minus |θ|ǫ

le∣

∣


|bjkj t〉PnMLθ

∣

∣

∣


|bjkj t〉PnMLθ

= eminusirkj

1

Ln

(

Lnsum

l=1


nMLθ

)

(

Lnsum

l=1


nMLθ

)

(83)


|jt〉QnEnL =

radic

1

Ln

Lnsum

l=1

eminus2πi(l

kjLn


and

|ϑjt〉QnEnMLθ =

radic

1

Ln

Lnsum

l=1

eminus2πilkj

Ln

[

IEn otimes V QnMLθ

]

(UnN |κjl〉P

n

)|0〉M|0〉L|0〉θ


F

(



)

=∣

∣



∣

∣

=1

Ln

(

Lnsum

l=1


nMLθ

)

(

Lnsum

l=1

eminus2πilkjLn e


)



n


such that

∥

∥

∥ξEn

t minus trQn

(


n

UlowastNt

)∥

∥

∥

1le ǫ (85)


∥

∥trQn

(


n

UlowastNt

)

minus trQn

(


∥

∥

1

le 2

[

1minus F

(

trQn

(


n

UlowastNt

)

trQn

(


)]12

le 2

[

1minus F

(


nEn


12

le 2radicǫ (86)


2minus η1 leradic


∥

∥

∥trQnL

(

|jt〉〈jt|QnEnL

)

minus ξEn

t

∥

∥

∥

1

=

∥

∥

∥

∥

∥

1

Ln

Lnsum

l=1

trQn

(


minus ξEn

t

∥

∥

∥

∥

∥

1

le 1

Ln

Lnsum

l=1

∥

∥

∥

∥

trQn

(


n

UlowastNt

)

minus trQn

(


∥

∥

∥

∥

1

+∥

∥

∥ξEn

t minus trQn

(


n

UlowastNt

)∥

∥

∥

1


16



such that if we set

χQnEnMLjjprimet =

(

UQnML(t) otimes IE

n)

(


)(

UQnML(t) otimes IE

n)lowast

then

F(


jjprimet

)


(88)where |ξt〉Q


rification of ξEn

t on HQnEnL



1

Jn

1

Ln

Jnsum

j=1

Lnsum

l=1


n |j〉A

Jnsum

j=1

Lnsum

l=1


n

(89)



1

Jn

1

Ln

(

IA otimes UnNtprime

)

[

Jnsum

j=1

Lnsum

l=1

eminus2πilkj

Ln |j〉A|κjl〉Pn

Jnsum

j=1

Lnsum

l=1

eminus2πilkj

Ln 〈κjl|Pn〈j|A

]

(

IA otimes UnNtprime

)lowast

=1

Jn

1

Ln

Jnsum

j=1

|j〉A(

Lnsum

l=1

eminus2πilkjLn Un

Ntprime|κjl〉P

n

)

Jnsum

j=1

(

Lnsum

l=1


n

(UnNtprime

)lowast)

〈j|A


τQn rarr trQnLθ

[

(

sum

tisinθ


)

V QnMLθ

(


)

V QnMLθlowast(

sum

tisinθ


)lowast]

(90)

to his outcome


We define

ιAQnEnMLθtprime

=

(

sum

tisinθ


)


)

(

1

Jn

1

Ln

Jnsum

j=1

|j〉A(

Lnsum

l=1

eminus2πilkjLn Un

Ntprime|κjl〉P

n

)

Jnsum

j=1

(

Lnsum

l=1


n

(UnNtprime

)lowast)

〈j|A



)lowast

(

sum

tisinθ


)lowast

=

(

sum

tisinθ


)

1

Jn(

Jnsum

j=1


Jnsum

j=1


(

sum

tisinθ


)lowast

(91)


AQnEnMLθtprime )


sumJn

j=1

sumJn



(

sum

tisinθ


)

(

sum

tisinθ

UQnML(t) otimes IE


= IAEn otimessum

tisinθ

UQnML(t) (UQnML


= IAQnEnMLθ

sum




F

ιAQnEnMLθtprime

1

Jn

Jnsum

j=1

Jnsum

jprime=1


= F

(

1

Jn(

Jnsum

j=1


Jnsum

j=1


1

Jn(

Jnsum

j=1


17

(

Jnsum

j=1


)

=1

Jn

∣

∣

∣

∣

( Jnsum

j=1


)

( Jnsum

j=1


)∣

∣

∣

∣



sumJn

j=1

sumJn



F

(

1

Jn

Jnsum

j=1

Jnsum

jprime=1


1

Jn(

Jnsum

j=1


(

Jnsum

j=1


)



1minus F ( η) le 1

2minus η1 le

radic

1minus F ( η)2


F ( η)

ge 1minus 1

2minus η1

ge 1minus 1

2minus υ1 minus

1

2υ minus η1

ge 1minusradic


1minus F (υ η)2


F

(


(

Jnsum

j=1


Jnsum

j=1


)

ge F

(

ιAQnEnMLθtprime

1

Jn(

Jnsum

j=1


(

Jnsum

j=1


)

ge 1minusradic



radicǫminus 8ǫ

(94)

ge 1minusradic

2|θ|radicǫminus



radic


with rate


tχ(X Et)minus 2δ



1

nmintisinθ

maxρisinS(H)Qn



limnrarrinfin

1

nmax

ρisinS(H)QnIC(ρN

otimesnt )




1

nmintisinθ

maxρisinS(H)Qn




18

VII FURTHER NOTES




N(ρ) =

Ksum

i=1

AiρAilowast (98)





we have Ai =sumK

j=1 uijBj (cf [31])







HE Then UN =sumK




)

ρ(


)lowast=

sumKj=1AjρAj

lowast and(


)lowast (sumK


=sumK

j=1 AjlowastAj


UN =sumK



lowast Here A1 AK




1

nmax

ρisinS(H)QnIC(ρN

otimesn) (100)




1

nmax




19


ACKNOWLEDGMENT





























20


















21


χ(P Φ) = S

(

sum

xisinA

P (x)ρx

)

minussum

xisinA

P (x)S (ρx)




4nα2 (1)






tr(

Votimesn(xn)ΠVα(x

n))

ge 1minus ad

4nα2 (4)


radicn (5)


n)


n) (6)






tr(


radica

)

ge 1minus ad

4nα2 (7)








t (x) isin S(H)



P) isin S(HQ)







that all codes in




4



maxtisinθ

maxjisin1Jn

sum

xnisinAn

Et(xn|j)Wn


maxtisinθ







1nI(XuniK


[13] and use (9)



maxtisinθ

maxjisin1Jn

sum

xnisinAn

E(xn|j)Wnt (D

cj |xn) le ε (10)

maxtisinθ




maxtisinθ

maxjisin1Jn

sum

xnisinAn

Et(xn|j)Wn


maxtisinθ







maxtisinθ

maxjisin1Jn

sum

xnisinAn

E(xn|j)Wnt (D

cj |xn) le ε (14)

maxtisinθ








maxtisinθ

1

Jn

Jnsum

j=1

tr(


)

le ε (16)

maxtisinθ




maxtisinθ

maxjisin1Jn

tr(


)

le ε

(18)

maxtisinθ




(

Notimesnt

)





and

5





t )](

|ψ〉〈ψ|APn)

(20)


Notimesnt

)



HA D)


mintisinθ

F(


)

ge 1minus ε (21)

where

|ΦK〉AM =

radic

1

Jn

Jnsum

j=1

|j〉A|j〉M




CHANNELS




n) =

pnt (x

n)pnt (T n


ptδ

0 else





Pr

maxjisin1Jn

Lntsum

l=1

1

LntW

nt (D

cj |X

(t)jl ) gt

radicT2minusnω2



t (Dcj |X

(t)jl ) is a random




lfloor



CSCSI ge mintisinθ

maxUrarrArarr(BK)t





CSCSI ge mintisinθ

maxUrarrArarr(BK)t



CSCSI = mintisinθ

maxUrarrArarr(BK)t




pn(xn)pn(T n

pδ) if xn isin T n

pδ

0 else




Pr

(

maxjisin1Jn

Lnsum

l=1

1

LnW

nt (D

cj |Xjl) gt

radicT 2minusnω2

)






(mintisinθ


I(U Kt)) (26)

6


QUANTUM WIRETAPPER


II


CSCSI = mintisinθ

maxUrarrArarr(BZ)t


1

nχ(U Zn

t ))



(mintisinθ





nLet





Let

Qt(xn) = ΠptVtα

radicaΠVtα(x


n)ΠptVtαradica



ρminusradicXρ

radicX le

radic2λ (29)



radicX le

radic8λ In [32]




Qt(xn)minus V

otimesnt (xn) le

radic

2(ad+ d)

nα2 (30)

We set Θt =sum

xnisinT nptδ

ppriment (xn)Qt(x

n) For given zn






ρλ

where ρ =sum




Pr

(

Lminus1Lsum

i=1


ρλ gt ǫ

)


minusL ǫ2λ

2 ln2(dimV)micro

)

(31)




Qt(xn)



n)ΠptVtαradica


radicaΠVtα(x

n)ΠptVtαradica


radica







enough we have

Pr

Lntsum

l=1

1


7


middot exp(

minusLntǫ2


radicn(

radicaminus1)

)


middot exp(

minusLntǫ2


radicn(

radicaminus1)

)

le exp(


(34)


S(pt)minus S(Vt|pt)

= S

sum

j

pt(j)sum

l

1

LntVotimesnt (X

(t)jl )

minussum

j

pt(j)S

(

sum

l

1

LntVotimesnt (X

(t)jl )

)

= χ(ptZt)


Pr

Lntsum

l=1

1

LntQt(X



and

Pr

Lntsum

l=1

1

LntQt(X


= 1minus Pr

⋃

t

⋃

j

Lntsum

l=1

1

LntQt(X









⋂

t

maxjisin1Jn

Lntsum

l=1

1

LntW

nt (D

cj(X )|X(t)

jl ) le ǫ

cap

Lntsum

l=1

1

LntQt(X






Lntsum

l=1

1

LntW

nt (D

cj |x

(t)jl ) le ǫ (37)

and

Lntsum

l=1

1

LntQt(x



R = mintisinθ

maxUrarrArarr(BZ)t


1

nχ(U Zn

t ))minusγ


code(


Dj j = 1 Jn)

such that

1

nlog Jn ge R (39)

limnrarrinfin

maxtisinθ

maxjisin1Jn

Lntsum

l=1

1

LntW

nt (D

cj |x

(t)jl ) = 0 (40)


holds Votimesnt (x

(t)jl ) minus Qt(x



Lntsum

l=1

1

LntVotimesnt (x

(t)jprimel)minusΘt

le Lntsum

l=1

1

LntVotimesnt (x

(t)jprimel)minus

Lntsum

l=1

1

LntQt(x

(t)jprimel)

+ Lntsum

l=1

1

LntQt(x

(t)jprimel)minusΘt

leLntsum

l=1

1

LntVotimesn


(t)jprimel)

+ L

(t)ntsum

l=1

1

LntQt(x

(t)jprimel)minusΘt

le 2ǫ (41)

and sumJn

j=11Jn

sumLnt

l=11

LntVotimesnt (x


8


e then




χ(XuniZnt )

= S

Jnsum

j=1

1

Jn

Lntsum

l=1

1

LntVotimesnt (x

(t)jl )

minusJnsum

j=1

1

JnS

Lntsum

l=1

1

LntVotimesnt (x

(t)jl )

le

∣

∣

∣

∣

∣

∣

S

Jnsum

j=1

1

Jn

Lntsum

l=1

1

LntVotimesnt (x

(t)jl )

minus S (Θt)

∣

∣

∣

∣

∣

∣

+

∣

∣

∣

∣

∣

∣

S(Θt)minusJnsum

j=1

1

JnS

Lntsum

l=1

1

LntVotimesnt (x

(t)jl )

∣

∣

∣

∣

∣

∣


+

∣

∣

∣

∣

∣

∣

Jnsum

j=1

1

Jn

S(Θt)minus S

Lntsum

l=1

1

LntVotimesnt (x

(t)jl )

∣

∣

∣

∣

∣

∣



maxtisinθ



Pr(

Et(j) = x(t)jl

)

= 1Lnt



CSCSI ge mintisinθ

maxUrarrArarr(BZ)t


1

nχ(U Zn

t ))

(45)



maxtisinθ

maxjisin1Jn

sum

xnisinAn

E(xn|j)Wnt (D

cj |xn) = ǫ1n (46)

maxtisinθ

χ(J Znt ) = ǫ2n (47)




log Jn = H(Xuni)










I(XuniBtprime) +1

n+

1

nǫ1n log Jn


nχ(XuniZ

ntprime)] +

ǫ1n

n+

1

nlog Jn +

ǫ2n

n


nχ(XuniZ

ntprime)] + ǫ (49)


CSCSI le mintisinθ

maxUrarrArarr(BZ)t


1

nχ(U Zn

t ))






Qt(xn) = ΠpVtα

radicaΠVtα(x


n)ΠpVtαradica

9

and Θt =sum


n)


Pr

(

Lnsum

l=1

1

LnQt(X


)

ge 1minus 2minusnυ

(51)


Lnsum

l=1

1

LnW

nt (D

cj |xjl) le ǫ

Lnsum

l=1

1


For any γ gt 0 let


(

mintisinθ


)

minus γ


1Lnt


1n log Jn ge R and

limnrarrinfin

maxtisinθ

maxjisin1Jn

Lnsum

l=1

1

LnW

nt (D

cj |xjl)) = 0 (52)


maxtisinθ




(mintisinθ


χ(U Zt))


WIRETAP CHANNEL


Section II



mintisinθ

maxPinpwt

1

n(χ(PinpB

nt )minus χ(PinpZ

nt ))



CS = limnrarrinfin

maxUrarrArarr(BZ)t

1

n

(

mintisinθ

χ(U Bnt )

minusmaxtisinθ

χ(U Znt )

)

(55)







mintχ(PinpWt) (56)



log Tmint maxPinp


mint χ(PinpWt) =






log Jn = maxPinpw


n))minus ǫ (57)

10



log Jnt = maxPinpwt


nt ))minus ǫ (58)

Thus


mintisinθ

maxPinpwt

1

n(χ(PinpB

nt )minus χ(PinpZ

nt ))

(59)





C(t)2 = (E(t) D(t)




D(t)j j t isin θ

to decode the





limnrarrinfin

maxPinpwtprime

1

n(χ(PinpB


ntprime))



mintisinθ

maxPinpwt

1

n(χ(PinpB

nt )minus χ(PinpZ

nt ))




nt )


limnrarrinfin

maxPinpwt

1

n(χ(PinpB

nt )minus χ(PinpZ

nt ))

= maxPinpwt




nt )


CCSI = mintisinθ

maxPinpwt











Pr[

tr(

Wnt (ρ

nXjl

)DtXjl

)


gt 1minus 2minusnβ


sum

tisinθ

Jnsum

j=1

Lnsum

l=1

Dtxjlle I

We define



n)ΠpVtαradica

and Θt =sum



11


Pr

(

Lnsum

l=1

1

LnQt(ρX(t)

jl


ge 1minus 2minusnυ

(62)


mintisinθ

tr(

Wnt (ρ

nxjl

)Dtxjl

)

ge 1minus 2minusnβ

and

maxtisinθ

Lnsum

l=1

1

LnQt(ρxjl

)minusΘt le ǫ

We define Dj =sum

tisinθ

sumLn

l=1Dtxjl then

sumJn

j=1Dj =sum

tisinθ

sumJn

j=1

sumLn



tr(

Wntprime (ρ

nxjlprime

)Dj

)

=sum

tisinθ

Lnsum

l=1

tr(

Wntprime (ρ

nxjlprime

)Dtxjl

)

ge tr(

Wntprime (ρ

otimesnxjlprime

)Dtprimexjlprime

)

ge 1minus 2minusnβ


(radicM1M2

radicM1

)

ge 0

For any γ gt 0 let


1

n

[

mintisinθ

χ(pBnt )minusmax

tisinθχ(pZn

t )

]

minus γ


w(j) =sumLn

l=11Lnρnxjl

j = 1 Jn Dj j =

1 Jn)


maxtisinθ

maxjisin1Jn

tr(


)

le λ


maxtisinθ



CS ge limnrarrinfin

maxUrarrArarr(BZ)t

1

n

(

mintisinθ

χ(U Bnt )minusmax

tisinθχ(U Zn

t )

)

(65)




maxtisinθ

maxjisin1Jn

tr(


t

(

ρ(n)j

))

le λn (66)

maxtisinθ








1

nχ(XuniB

ntprime) + ξ

= mintisinθ

1

nχ(XuniB

nt ) + ξ

le mintisinθ

1

nχ(XuniB

nt )minusmax

tisinθ

1

nχ(XuniZ

nt ) + ξ +

1

nǫ2n

le 1

n

(

mintisinθ


tisinθχ(XuniZ

nt )

)

+ ǫ (68)

Thus

CS le limnrarrinfin

maxUrarrArarr(BZ)t

1

n

(

mintisinθ

χ(U Bnt )minusmax

tisinθχ(U Zn

t )

)





inftisinθ

maxPinpwt

1

n(χ(PinpB

nt )minus χ(PinpZ

nt ))

(70)


Wdiams = supnisinN



12



W (k))K





W (k))K

k=1in the










1

JN

JNsum

j=1

tr(


)





the message




tprime

)

(wtprime(j))]

=

tr(

trHprimeotimesN

[



])

We have

tr∣

∣

(


tprime

)

(wtprime (j))∣

∣

= tr(

trHprimeotimesN

∣



∣

∣

)

= tr∣



∣

∣

=∥



∥

∥

1






tprime

∥

∥

diams=

∥

∥


∥

∥


tiplicative

It follows that

∣

∣

∣

∣

1

JN

JNsum

j=1

tr(


)

minus 1

JN

JNsum

j=1

tr(


)

∣

∣

∣

∣

le 1

JN

JNsum

j=1

∣

∣tr[(


tprime

)


]∣

∣

le 1

JN

JNsum

j=1

tr∣

∣

(


tprime

)


∣

∣

le 1

JN

JNsum

j=1

tr∣

∣

(


tprime

)

(wtprime(j))∣

∣

le 1

JNJNNτ

= Nτ (74)




le

∣

∣

∣

∣

∣

∣

S

JNsum

j=1

1

JNVt(ρ(j))

minus S

JNsum

j=1

1

JNVtprime(ρ(j))

∣

∣

∣

∣

∣

∣

+

∣

∣

∣

∣

∣

∣

JNsum

j=1

1


JNsum

j=1

1


∣

∣

∣

∣

∣

∣




suptisinθ

1

JN

JNsum

j=1

tr(


)




suptisinθ

1

JN

JNsum

j=1

tr(


)

ge 1minus λ

13

suptisinθ

χ(XuniZNt ) le ǫ



inftisinθ

maxPinpwt

1

n(χ(PinpB

nt )minus χ(PinpZ

nt ))








)minus2dprime4









Notimesnt

)




sum




A ge maxp

(

mintisinθ


χ(pEt)

)

(77)



ρPUlowastNt



sum


trE

(

sum

x


Nt

)


trQ

(

sum

x


Nt

)




sum



Nt

)

and VNt HP rarr HQ

sum



Nt

)



and


14


VNt)tisinθ since



xjl




n

t on HEn

such that

tr(

(DQn

txjlotimes IE

n

)UNtρP

n

xjlUlowastNt

)


and

ωEn

jt minus ξEn

t 1 lt ǫ (79)

where ωEn

jt = 1Lnt

sumLnt

l=1 trQn

(

UNtρP

n

xjlUlowastNt

)


xjlmay be pure or

mixed Assume ρPn


sumni=1 p

primejl(i)|κ

(i)xjl〉〈κ(i)

xjl |Pn

then

nsum

i=1

pprimejl(i)tr(

(DQn

txjlotimes IE

n

)UNt|κ(i)

xjl〉〈κ(i)

xjl|Pn

UlowastNt

)

tr

(

(DQn

txjlotimes IE

n

)UNt(

nsum

i=1


〉〈κ(i)xjl

|Pn

)UlowastNt

)

ge 1minus 2minusnβ



tr(

(DQn

txjlotimes IE

n

)UNt|κ(i)

xjl〉〈κ(i)

xjl|Pn

UlowastNt

)

ge 1minus 2minusnβ



there is a ξEn

t on HEn

such that

1

Lnt

Lntsum

l=1

trQn

(

UNt|κ(ilj)


UlowastNt

)

minus ξEn

t 1 lt ǫ

Thus(


xjl |Pn

j l DQn

txjl j l t

)


ρPn

xjl j l DQn

txjl j l t

)



Assume ρPn


n



n isin S(HQn

) we have

V QnMLθ

(


)

(V QnMLθ)lowast

=sum

t

sum

j

sum

l

(

DQn

txjlρQ

n)


We denote

ψQnEnMLθjlt

=(

IEn otimes V QnMLθ

)

(

UN otimes IMLθ)

[

|κjl〉〈κjl|Pn


(


(

IEn otimes V QnMLθ

)lowast


F(

trQnEn

(

ψQnEnMLθjlt

)


ge 1minus ǫ (80)


nEn

on HQnEn

such that

〈0|θ〈0|L〈0|M〈κjl|Pn (

UNtotimes IMLθ

)lowast

(

IEn otimes V QnMLθ

)lowast|ζjlt〉Q


= F

(


nEn


ge 1minus ǫ (81)



n |0〉M|0〉L|0〉θ

and

|bjlt〉PnMLθ =

(

UNtotimes IMLθ

)lowast (IE




We define

|ajk〉PnMLθ =

1radicLn

Lnsum

l=1


nMLθ

15

|bjkt〉PnMLθ =

1radicLn

Lnsum

l=1


nMLθ

and

|bjk〉PnMLθ =

1

|θ|

|θ|sum

t=1

|bjkt〉PnMLθ


1

Ln

Lnsum

k=1

〈ajk|bjk〉PnMLθ

=1

|θ|1

Ln

|θ|sum

t=1

Lnsum

k=1


=1

|θ|1

Ln

|θ|sum

t=1

Lnsum

l=1


ge 1minus ǫ (82)


1minus ǫ


|bjkj〉PnMLθ

=1

|θ|

|θ|sum

t=1

eminusiskj 〈ajkj

|bjkj t〉PnMLθ


∣

∣


|bjkj t〉PnMLθ

∣

∣

∣ le 1 we have

mintisinθ

∣

∣


|bjkj t〉PnMLθ

∣

∣



t isin θ

1minus |θ|ǫ

le∣

∣


|bjkj t〉PnMLθ

∣

∣

∣


|bjkj t〉PnMLθ

= eminusirkj

1

Ln

(

Lnsum

l=1


nMLθ

)

(

Lnsum

l=1


nMLθ

)

(83)


|jt〉QnEnL =

radic

1

Ln

Lnsum

l=1

eminus2πi(l

kjLn


and

|ϑjt〉QnEnMLθ =

radic

1

Ln

Lnsum

l=1

eminus2πilkj

Ln

[

IEn otimes V QnMLθ

]

(UnN |κjl〉P

n

)|0〉M|0〉L|0〉θ


F

(



)

=∣

∣



∣

∣

=1

Ln

(

Lnsum

l=1


nMLθ

)

(

Lnsum

l=1

eminus2πilkjLn e


)



n


such that

∥

∥

∥ξEn

t minus trQn

(


n

UlowastNt

)∥

∥

∥

1le ǫ (85)


∥

∥trQn

(


n

UlowastNt

)

minus trQn

(


∥

∥

1

le 2

[

1minus F

(

trQn

(


n

UlowastNt

)

trQn

(


)]12

le 2

[

1minus F

(


nEn


12

le 2radicǫ (86)


2minus η1 leradic


∥

∥

∥trQnL

(

|jt〉〈jt|QnEnL

)

minus ξEn

t

∥

∥

∥

1

=

∥

∥

∥

∥

∥

1

Ln

Lnsum

l=1

trQn

(


minus ξEn

t

∥

∥

∥

∥

∥

1

le 1

Ln

Lnsum

l=1

∥

∥

∥

∥

trQn

(


n

UlowastNt

)

minus trQn

(


∥

∥

∥

∥

1

+∥

∥

∥ξEn

t minus trQn

(


n

UlowastNt

)∥

∥

∥

1


16



such that if we set

χQnEnMLjjprimet =

(

UQnML(t) otimes IE

n)

(


)(

UQnML(t) otimes IE

n)lowast

then

F(


jjprimet

)


(88)where |ξt〉Q


rification of ξEn

t on HQnEnL



1

Jn

1

Ln

Jnsum

j=1

Lnsum

l=1


n |j〉A

Jnsum

j=1

Lnsum

l=1


n

(89)



1

Jn

1

Ln

(

IA otimes UnNtprime

)

[

Jnsum

j=1

Lnsum

l=1

eminus2πilkj

Ln |j〉A|κjl〉Pn

Jnsum

j=1

Lnsum

l=1

eminus2πilkj

Ln 〈κjl|Pn〈j|A

]

(

IA otimes UnNtprime

)lowast

=1

Jn

1

Ln

Jnsum

j=1

|j〉A(

Lnsum

l=1

eminus2πilkjLn Un

Ntprime|κjl〉P

n

)

Jnsum

j=1

(

Lnsum

l=1


n

(UnNtprime

)lowast)

〈j|A


τQn rarr trQnLθ

[

(

sum

tisinθ


)

V QnMLθ

(


)

V QnMLθlowast(

sum

tisinθ


)lowast]

(90)

to his outcome


We define

ιAQnEnMLθtprime

=

(

sum

tisinθ


)


)

(

1

Jn

1

Ln

Jnsum

j=1

|j〉A(

Lnsum

l=1

eminus2πilkjLn Un

Ntprime|κjl〉P

n

)

Jnsum

j=1

(

Lnsum

l=1


n

(UnNtprime

)lowast)

〈j|A



)lowast

(

sum

tisinθ


)lowast

=

(

sum

tisinθ


)

1

Jn(

Jnsum

j=1


Jnsum

j=1


(

sum

tisinθ


)lowast

(91)


AQnEnMLθtprime )


sumJn

j=1

sumJn



(

sum

tisinθ


)

(

sum

tisinθ

UQnML(t) otimes IE


= IAEn otimessum

tisinθ

UQnML(t) (UQnML


= IAQnEnMLθ

sum




F

ιAQnEnMLθtprime

1

Jn

Jnsum

j=1

Jnsum

jprime=1


= F

(

1

Jn(

Jnsum

j=1


Jnsum

j=1


1

Jn(

Jnsum

j=1


17

(

Jnsum

j=1


)

=1

Jn

∣

∣

∣

∣

( Jnsum

j=1


)

( Jnsum

j=1


)∣

∣

∣

∣



sumJn

j=1

sumJn



F

(

1

Jn

Jnsum

j=1

Jnsum

jprime=1


1

Jn(

Jnsum

j=1


(

Jnsum

j=1


)



1minus F ( η) le 1

2minus η1 le

radic

1minus F ( η)2


F ( η)

ge 1minus 1

2minus η1

ge 1minus 1

2minus υ1 minus

1

2υ minus η1

ge 1minusradic


1minus F (υ η)2


F

(


(

Jnsum

j=1


Jnsum

j=1


)

ge F

(

ιAQnEnMLθtprime

1

Jn(

Jnsum

j=1


(

Jnsum

j=1


)

ge 1minusradic



radicǫminus 8ǫ

(94)

ge 1minusradic

2|θ|radicǫminus



radic


with rate


tχ(X Et)minus 2δ



1

nmintisinθ

maxρisinS(H)Qn



limnrarrinfin

1

nmax

ρisinS(H)QnIC(ρN

otimesnt )




1

nmintisinθ

maxρisinS(H)Qn




18

VII FURTHER NOTES




N(ρ) =

Ksum

i=1

AiρAilowast (98)





we have Ai =sumK

j=1 uijBj (cf [31])







HE Then UN =sumK




)

ρ(


)lowast=

sumKj=1AjρAj

lowast and(


)lowast (sumK


=sumK

j=1 AjlowastAj


UN =sumK



lowast Here A1 AK




1

nmax

ρisinS(H)QnIC(ρN

otimesn) (100)




1

nmax




19


ACKNOWLEDGMENT





























20


















21



maxtisinθ

maxjisin1Jn

sum

xnisinAn

Et(xn|j)Wn


maxtisinθ







1nI(XuniK


[13] and use (9)



maxtisinθ

maxjisin1Jn

sum

xnisinAn

E(xn|j)Wnt (D

cj |xn) le ε (10)

maxtisinθ




maxtisinθ

maxjisin1Jn

sum

xnisinAn

Et(xn|j)Wn


maxtisinθ







maxtisinθ

maxjisin1Jn

sum

xnisinAn

E(xn|j)Wnt (D

cj |xn) le ε (14)

maxtisinθ








maxtisinθ

1

Jn

Jnsum

j=1

tr(


)

le ε (16)

maxtisinθ




maxtisinθ

maxjisin1Jn

tr(


)

le ε

(18)

maxtisinθ




(

Notimesnt

)





and

5





t )](

|ψ〉〈ψ|APn)

(20)


Notimesnt

)



HA D)


mintisinθ

F(


)

ge 1minus ε (21)

where

|ΦK〉AM =

radic

1

Jn

Jnsum

j=1

|j〉A|j〉M




CHANNELS




n) =

pnt (x

n)pnt (T n


ptδ

0 else





Pr

maxjisin1Jn

Lntsum

l=1

1

LntW

nt (D

cj |X

(t)jl ) gt

radicT2minusnω2



t (Dcj |X

(t)jl ) is a random




lfloor



CSCSI ge mintisinθ

maxUrarrArarr(BK)t





CSCSI ge mintisinθ

maxUrarrArarr(BK)t



CSCSI = mintisinθ

maxUrarrArarr(BK)t




pn(xn)pn(T n

pδ) if xn isin T n

pδ

0 else




Pr

(

maxjisin1Jn

Lnsum

l=1

1

LnW

nt (D

cj |Xjl) gt

radicT 2minusnω2

)






(mintisinθ


I(U Kt)) (26)

6


QUANTUM WIRETAPPER


II


CSCSI = mintisinθ

maxUrarrArarr(BZ)t


1

nχ(U Zn

t ))



(mintisinθ





nLet





Let

Qt(xn) = ΠptVtα

radicaΠVtα(x


n)ΠptVtαradica



ρminusradicXρ

radicX le

radic2λ (29)



radicX le

radic8λ In [32]




Qt(xn)minus V

otimesnt (xn) le

radic

2(ad+ d)

nα2 (30)

We set Θt =sum

xnisinT nptδ

ppriment (xn)Qt(x

n) For given zn






ρλ

where ρ =sum




Pr

(

Lminus1Lsum

i=1


ρλ gt ǫ

)


minusL ǫ2λ

2 ln2(dimV)micro

)

(31)




Qt(xn)



n)ΠptVtαradica


radicaΠVtα(x

n)ΠptVtαradica


radica







enough we have

Pr

Lntsum

l=1

1


7


middot exp(

minusLntǫ2


radicn(

radicaminus1)

)


middot exp(

minusLntǫ2


radicn(

radicaminus1)

)

le exp(


(34)


S(pt)minus S(Vt|pt)

= S

sum

j

pt(j)sum

l

1

LntVotimesnt (X

(t)jl )

minussum

j

pt(j)S

(

sum

l

1

LntVotimesnt (X

(t)jl )

)

= χ(ptZt)


Pr

Lntsum

l=1

1

LntQt(X



and

Pr

Lntsum

l=1

1

LntQt(X


= 1minus Pr

⋃

t

⋃

j

Lntsum

l=1

1

LntQt(X









⋂

t

maxjisin1Jn

Lntsum

l=1

1

LntW

nt (D

cj(X )|X(t)

jl ) le ǫ

cap

Lntsum

l=1

1

LntQt(X






Lntsum

l=1

1

LntW

nt (D

cj |x

(t)jl ) le ǫ (37)

and

Lntsum

l=1

1

LntQt(x



R = mintisinθ

maxUrarrArarr(BZ)t


1

nχ(U Zn

t ))minusγ


code(


Dj j = 1 Jn)

such that

1

nlog Jn ge R (39)

limnrarrinfin

maxtisinθ

maxjisin1Jn

Lntsum

l=1

1

LntW

nt (D

cj |x

(t)jl ) = 0 (40)


holds Votimesnt (x

(t)jl ) minus Qt(x



Lntsum

l=1

1

LntVotimesnt (x

(t)jprimel)minusΘt

le Lntsum

l=1

1

LntVotimesnt (x

(t)jprimel)minus

Lntsum

l=1

1

LntQt(x

(t)jprimel)

+ Lntsum

l=1

1

LntQt(x

(t)jprimel)minusΘt

leLntsum

l=1

1

LntVotimesn


(t)jprimel)

+ L

(t)ntsum

l=1

1

LntQt(x

(t)jprimel)minusΘt

le 2ǫ (41)

and sumJn

j=11Jn

sumLnt

l=11

LntVotimesnt (x


8


e then




χ(XuniZnt )

= S

Jnsum

j=1

1

Jn

Lntsum

l=1

1

LntVotimesnt (x

(t)jl )

minusJnsum

j=1

1

JnS

Lntsum

l=1

1

LntVotimesnt (x

(t)jl )

le

∣

∣

∣

∣

∣

∣

S

Jnsum

j=1

1

Jn

Lntsum

l=1

1

LntVotimesnt (x

(t)jl )

minus S (Θt)

∣

∣

∣

∣

∣

∣

+

∣

∣

∣

∣

∣

∣

S(Θt)minusJnsum

j=1

1

JnS

Lntsum

l=1

1

LntVotimesnt (x

(t)jl )

∣

∣

∣

∣

∣

∣


+

∣

∣

∣

∣

∣

∣

Jnsum

j=1

1

Jn

S(Θt)minus S

Lntsum

l=1

1

LntVotimesnt (x

(t)jl )

∣

∣

∣

∣

∣

∣



maxtisinθ



Pr(

Et(j) = x(t)jl

)

= 1Lnt



CSCSI ge mintisinθ

maxUrarrArarr(BZ)t


1

nχ(U Zn

t ))

(45)



maxtisinθ

maxjisin1Jn

sum

xnisinAn

E(xn|j)Wnt (D

cj |xn) = ǫ1n (46)

maxtisinθ

χ(J Znt ) = ǫ2n (47)




log Jn = H(Xuni)










I(XuniBtprime) +1

n+

1

nǫ1n log Jn


nχ(XuniZ

ntprime)] +

ǫ1n

n+

1

nlog Jn +

ǫ2n

n


nχ(XuniZ

ntprime)] + ǫ (49)


CSCSI le mintisinθ

maxUrarrArarr(BZ)t


1

nχ(U Zn

t ))






Qt(xn) = ΠpVtα

radicaΠVtα(x


n)ΠpVtαradica

9

and Θt =sum


n)


Pr

(

Lnsum

l=1

1

LnQt(X


)

ge 1minus 2minusnυ

(51)


Lnsum

l=1

1

LnW

nt (D

cj |xjl) le ǫ

Lnsum

l=1

1


For any γ gt 0 let


(

mintisinθ


)

minus γ


1Lnt


1n log Jn ge R and

limnrarrinfin

maxtisinθ

maxjisin1Jn

Lnsum

l=1

1

LnW

nt (D

cj |xjl)) = 0 (52)


maxtisinθ




(mintisinθ


χ(U Zt))


WIRETAP CHANNEL


Section II



mintisinθ

maxPinpwt

1

n(χ(PinpB

nt )minus χ(PinpZ

nt ))



CS = limnrarrinfin

maxUrarrArarr(BZ)t

1

n

(

mintisinθ

χ(U Bnt )

minusmaxtisinθ

χ(U Znt )

)

(55)







mintχ(PinpWt) (56)



log Tmint maxPinp


mint χ(PinpWt) =






log Jn = maxPinpw


n))minus ǫ (57)

10



log Jnt = maxPinpwt


nt ))minus ǫ (58)

Thus


mintisinθ

maxPinpwt

1

n(χ(PinpB

nt )minus χ(PinpZ

nt ))

(59)





C(t)2 = (E(t) D(t)




D(t)j j t isin θ

to decode the





limnrarrinfin

maxPinpwtprime

1

n(χ(PinpB


ntprime))



mintisinθ

maxPinpwt

1

n(χ(PinpB

nt )minus χ(PinpZ

nt ))




nt )


limnrarrinfin

maxPinpwt

1

n(χ(PinpB

nt )minus χ(PinpZ

nt ))

= maxPinpwt




nt )


CCSI = mintisinθ

maxPinpwt











Pr[

tr(

Wnt (ρ

nXjl

)DtXjl

)


gt 1minus 2minusnβ


sum

tisinθ

Jnsum

j=1

Lnsum

l=1

Dtxjlle I

We define



n)ΠpVtαradica

and Θt =sum



11


Pr

(

Lnsum

l=1

1

LnQt(ρX(t)

jl


ge 1minus 2minusnυ

(62)


mintisinθ

tr(

Wnt (ρ

nxjl

)Dtxjl

)

ge 1minus 2minusnβ

and

maxtisinθ

Lnsum

l=1

1

LnQt(ρxjl

)minusΘt le ǫ

We define Dj =sum

tisinθ

sumLn

l=1Dtxjl then

sumJn

j=1Dj =sum

tisinθ

sumJn

j=1

sumLn



tr(

Wntprime (ρ

nxjlprime

)Dj

)

=sum

tisinθ

Lnsum

l=1

tr(

Wntprime (ρ

nxjlprime

)Dtxjl

)

ge tr(

Wntprime (ρ

otimesnxjlprime

)Dtprimexjlprime

)

ge 1minus 2minusnβ


(radicM1M2

radicM1

)

ge 0

For any γ gt 0 let


1

n

[

mintisinθ

χ(pBnt )minusmax

tisinθχ(pZn

t )

]

minus γ


w(j) =sumLn

l=11Lnρnxjl

j = 1 Jn Dj j =

1 Jn)


maxtisinθ

maxjisin1Jn

tr(


)

le λ


maxtisinθ



CS ge limnrarrinfin

maxUrarrArarr(BZ)t

1

n

(

mintisinθ

χ(U Bnt )minusmax

tisinθχ(U Zn

t )

)

(65)




maxtisinθ

maxjisin1Jn

tr(


t

(

ρ(n)j

))

le λn (66)

maxtisinθ








1

nχ(XuniB

ntprime) + ξ

= mintisinθ

1

nχ(XuniB

nt ) + ξ

le mintisinθ

1

nχ(XuniB

nt )minusmax

tisinθ

1

nχ(XuniZ

nt ) + ξ +

1

nǫ2n

le 1

n

(

mintisinθ


tisinθχ(XuniZ

nt )

)

+ ǫ (68)

Thus

CS le limnrarrinfin

maxUrarrArarr(BZ)t

1

n

(

mintisinθ

χ(U Bnt )minusmax

tisinθχ(U Zn

t )

)





inftisinθ

maxPinpwt

1

n(χ(PinpB

nt )minus χ(PinpZ

nt ))

(70)


Wdiams = supnisinN



12



W (k))K





W (k))K

k=1in the










1

JN

JNsum

j=1

tr(


)





the message




tprime

)

(wtprime(j))]

=

tr(

trHprimeotimesN

[



])

We have

tr∣

∣

(


tprime

)

(wtprime (j))∣

∣

= tr(

trHprimeotimesN

∣



∣

∣

)

= tr∣



∣

∣

=∥



∥

∥

1






tprime

∥

∥

diams=

∥

∥


∥

∥


tiplicative

It follows that

∣

∣

∣

∣

1

JN

JNsum

j=1

tr(


)

minus 1

JN

JNsum

j=1

tr(


)

∣

∣

∣

∣

le 1

JN

JNsum

j=1

∣

∣tr[(


tprime

)


]∣

∣

le 1

JN

JNsum

j=1

tr∣

∣

(


tprime

)


∣

∣

le 1

JN

JNsum

j=1

tr∣

∣

(


tprime

)

(wtprime(j))∣

∣

le 1

JNJNNτ

= Nτ (74)




le

∣

∣

∣

∣

∣

∣

S

JNsum

j=1

1

JNVt(ρ(j))

minus S

JNsum

j=1

1

JNVtprime(ρ(j))

∣

∣

∣

∣

∣

∣

+

∣

∣

∣

∣

∣

∣

JNsum

j=1

1


JNsum

j=1

1


∣

∣

∣

∣

∣

∣




suptisinθ

1

JN

JNsum

j=1

tr(


)




suptisinθ

1

JN

JNsum

j=1

tr(


)

ge 1minus λ

13

suptisinθ

χ(XuniZNt ) le ǫ



inftisinθ

maxPinpwt

1

n(χ(PinpB

nt )minus χ(PinpZ

nt ))








)minus2dprime4









Notimesnt

)




sum




A ge maxp

(

mintisinθ


χ(pEt)

)

(77)



ρPUlowastNt



sum


trE

(

sum

x


Nt

)


trQ

(

sum

x


Nt

)




sum



Nt

)

and VNt HP rarr HQ

sum



Nt

)



and


14


VNt)tisinθ since



xjl




n

t on HEn

such that

tr(

(DQn

txjlotimes IE

n

)UNtρP

n

xjlUlowastNt

)


and

ωEn

jt minus ξEn

t 1 lt ǫ (79)

where ωEn

jt = 1Lnt

sumLnt

l=1 trQn

(

UNtρP

n

xjlUlowastNt

)


xjlmay be pure or

mixed Assume ρPn


sumni=1 p

primejl(i)|κ

(i)xjl〉〈κ(i)

xjl |Pn

then

nsum

i=1

pprimejl(i)tr(

(DQn

txjlotimes IE

n

)UNt|κ(i)

xjl〉〈κ(i)

xjl|Pn

UlowastNt

)

tr

(

(DQn

txjlotimes IE

n

)UNt(

nsum

i=1


〉〈κ(i)xjl

|Pn

)UlowastNt

)

ge 1minus 2minusnβ



tr(

(DQn

txjlotimes IE

n

)UNt|κ(i)

xjl〉〈κ(i)

xjl|Pn

UlowastNt

)

ge 1minus 2minusnβ



there is a ξEn

t on HEn

such that

1

Lnt

Lntsum

l=1

trQn

(

UNt|κ(ilj)


UlowastNt

)

minus ξEn

t 1 lt ǫ

Thus(


xjl |Pn

j l DQn

txjl j l t

)


ρPn

xjl j l DQn

txjl j l t

)



Assume ρPn


n



n isin S(HQn

) we have

V QnMLθ

(


)

(V QnMLθ)lowast

=sum

t

sum

j

sum

l

(

DQn

txjlρQ

n)


We denote

ψQnEnMLθjlt

=(

IEn otimes V QnMLθ

)

(

UN otimes IMLθ)

[

|κjl〉〈κjl|Pn


(


(

IEn otimes V QnMLθ

)lowast


F(

trQnEn

(

ψQnEnMLθjlt

)


ge 1minus ǫ (80)


nEn

on HQnEn

such that

〈0|θ〈0|L〈0|M〈κjl|Pn (

UNtotimes IMLθ

)lowast

(

IEn otimes V QnMLθ

)lowast|ζjlt〉Q


= F

(


nEn


ge 1minus ǫ (81)



n |0〉M|0〉L|0〉θ

and

|bjlt〉PnMLθ =

(

UNtotimes IMLθ

)lowast (IE




We define

|ajk〉PnMLθ =

1radicLn

Lnsum

l=1


nMLθ

15

|bjkt〉PnMLθ =

1radicLn

Lnsum

l=1


nMLθ

and

|bjk〉PnMLθ =

1

|θ|

|θ|sum

t=1

|bjkt〉PnMLθ


1

Ln

Lnsum

k=1

〈ajk|bjk〉PnMLθ

=1

|θ|1

Ln

|θ|sum

t=1

Lnsum

k=1


=1

|θ|1

Ln

|θ|sum

t=1

Lnsum

l=1


ge 1minus ǫ (82)


1minus ǫ


|bjkj〉PnMLθ

=1

|θ|

|θ|sum

t=1

eminusiskj 〈ajkj

|bjkj t〉PnMLθ


∣

∣


|bjkj t〉PnMLθ

∣

∣

∣ le 1 we have

mintisinθ

∣

∣


|bjkj t〉PnMLθ

∣

∣



t isin θ

1minus |θ|ǫ

le∣

∣


|bjkj t〉PnMLθ

∣

∣

∣


|bjkj t〉PnMLθ

= eminusirkj

1

Ln

(

Lnsum

l=1


nMLθ

)

(

Lnsum

l=1


nMLθ

)

(83)


|jt〉QnEnL =

radic

1

Ln

Lnsum

l=1

eminus2πi(l

kjLn


and

|ϑjt〉QnEnMLθ =

radic

1

Ln

Lnsum

l=1

eminus2πilkj

Ln

[

IEn otimes V QnMLθ

]

(UnN |κjl〉P

n

)|0〉M|0〉L|0〉θ


F

(



)

=∣

∣



∣

∣

=1

Ln

(

Lnsum

l=1


nMLθ

)

(

Lnsum

l=1

eminus2πilkjLn e


)



n


such that

∥

∥

∥ξEn

t minus trQn

(


n

UlowastNt

)∥

∥

∥

1le ǫ (85)


∥

∥trQn

(


n

UlowastNt

)

minus trQn

(


∥

∥

1

le 2

[

1minus F

(

trQn

(


n

UlowastNt

)

trQn

(


)]12

le 2

[

1minus F

(


nEn


12

le 2radicǫ (86)


2minus η1 leradic


∥

∥

∥trQnL

(

|jt〉〈jt|QnEnL

)

minus ξEn

t

∥

∥

∥

1

=

∥

∥

∥

∥

∥

1

Ln

Lnsum

l=1

trQn

(


minus ξEn

t

∥

∥

∥

∥

∥

1

le 1

Ln

Lnsum

l=1

∥

∥

∥

∥

trQn

(


n

UlowastNt

)

minus trQn

(


∥

∥

∥

∥

1

+∥

∥

∥ξEn

t minus trQn

(


n

UlowastNt

)∥

∥

∥

1


16



such that if we set

χQnEnMLjjprimet =

(

UQnML(t) otimes IE

n)

(


)(

UQnML(t) otimes IE

n)lowast

then

F(


jjprimet

)


(88)where |ξt〉Q


rification of ξEn

t on HQnEnL



1

Jn

1

Ln

Jnsum

j=1

Lnsum

l=1


n |j〉A

Jnsum

j=1

Lnsum

l=1


n

(89)



1

Jn

1

Ln

(

IA otimes UnNtprime

)

[

Jnsum

j=1

Lnsum

l=1

eminus2πilkj

Ln |j〉A|κjl〉Pn

Jnsum

j=1

Lnsum

l=1

eminus2πilkj

Ln 〈κjl|Pn〈j|A

]

(

IA otimes UnNtprime

)lowast

=1

Jn

1

Ln

Jnsum

j=1

|j〉A(

Lnsum

l=1

eminus2πilkjLn Un

Ntprime|κjl〉P

n

)

Jnsum

j=1

(

Lnsum

l=1


n

(UnNtprime

)lowast)

〈j|A


τQn rarr trQnLθ

[

(

sum

tisinθ


)

V QnMLθ

(


)

V QnMLθlowast(

sum

tisinθ


)lowast]

(90)

to his outcome


We define

ιAQnEnMLθtprime

=

(

sum

tisinθ


)


)

(

1

Jn

1

Ln

Jnsum

j=1

|j〉A(

Lnsum

l=1

eminus2πilkjLn Un

Ntprime|κjl〉P

n

)

Jnsum

j=1

(

Lnsum

l=1


n

(UnNtprime

)lowast)

〈j|A



)lowast

(

sum

tisinθ


)lowast

=

(

sum

tisinθ


)

1

Jn(

Jnsum

j=1


Jnsum

j=1


(

sum

tisinθ


)lowast

(91)


AQnEnMLθtprime )


sumJn

j=1

sumJn



(

sum

tisinθ


)

(

sum

tisinθ

UQnML(t) otimes IE


= IAEn otimessum

tisinθ

UQnML(t) (UQnML


= IAQnEnMLθ

sum




F

ιAQnEnMLθtprime

1

Jn

Jnsum

j=1

Jnsum

jprime=1


= F

(

1

Jn(

Jnsum

j=1


Jnsum

j=1


1

Jn(

Jnsum

j=1


17

(

Jnsum

j=1


)

=1

Jn

∣

∣

∣

∣

( Jnsum

j=1


)

( Jnsum

j=1


)∣

∣

∣

∣



sumJn

j=1

sumJn



F

(

1

Jn

Jnsum

j=1

Jnsum

jprime=1


1

Jn(

Jnsum

j=1


(

Jnsum

j=1


)



1minus F ( η) le 1

2minus η1 le

radic

1minus F ( η)2


F ( η)

ge 1minus 1

2minus η1

ge 1minus 1

2minus υ1 minus

1

2υ minus η1

ge 1minusradic


1minus F (υ η)2


F

(


(

Jnsum

j=1


Jnsum

j=1


)

ge F

(

ιAQnEnMLθtprime

1

Jn(

Jnsum

j=1


(

Jnsum

j=1


)

ge 1minusradic



radicǫminus 8ǫ

(94)

ge 1minusradic

2|θ|radicǫminus



radic


with rate


tχ(X Et)minus 2δ



1

nmintisinθ

maxρisinS(H)Qn



limnrarrinfin

1

nmax

ρisinS(H)QnIC(ρN

otimesnt )




1

nmintisinθ

maxρisinS(H)Qn




18

VII FURTHER NOTES




N(ρ) =

Ksum

i=1

AiρAilowast (98)





we have Ai =sumK

j=1 uijBj (cf [31])







HE Then UN =sumK




)

ρ(


)lowast=

sumKj=1AjρAj

lowast and(


)lowast (sumK


=sumK

j=1 AjlowastAj


UN =sumK



lowast Here A1 AK




1

nmax

ρisinS(H)QnIC(ρN

otimesn) (100)




1

nmax




19


ACKNOWLEDGMENT





























20


















21





t )](

|ψ〉〈ψ|APn)

(20)


Notimesnt

)



HA D)


mintisinθ

F(


)

ge 1minus ε (21)

where

|ΦK〉AM =

radic

1

Jn

Jnsum

j=1

|j〉A|j〉M




CHANNELS




n) =

pnt (x

n)pnt (T n


ptδ

0 else





Pr

maxjisin1Jn

Lntsum

l=1

1

LntW

nt (D

cj |X

(t)jl ) gt

radicT2minusnω2



t (Dcj |X

(t)jl ) is a random




lfloor



CSCSI ge mintisinθ

maxUrarrArarr(BK)t





CSCSI ge mintisinθ

maxUrarrArarr(BK)t



CSCSI = mintisinθ

maxUrarrArarr(BK)t




pn(xn)pn(T n

pδ) if xn isin T n

pδ

0 else




Pr

(

maxjisin1Jn

Lnsum

l=1

1

LnW

nt (D

cj |Xjl) gt

radicT 2minusnω2

)






(mintisinθ


I(U Kt)) (26)

6


QUANTUM WIRETAPPER


II


CSCSI = mintisinθ

maxUrarrArarr(BZ)t


1

nχ(U Zn

t ))



(mintisinθ





nLet





Let

Qt(xn) = ΠptVtα

radicaΠVtα(x


n)ΠptVtαradica



ρminusradicXρ

radicX le

radic2λ (29)



radicX le

radic8λ In [32]




Qt(xn)minus V

otimesnt (xn) le

radic

2(ad+ d)

nα2 (30)

We set Θt =sum

xnisinT nptδ

ppriment (xn)Qt(x

n) For given zn






ρλ

where ρ =sum




Pr

(

Lminus1Lsum

i=1


ρλ gt ǫ

)


minusL ǫ2λ

2 ln2(dimV)micro

)

(31)




Qt(xn)



n)ΠptVtαradica


radicaΠVtα(x

n)ΠptVtαradica


radica







enough we have

Pr

Lntsum

l=1

1


7


middot exp(

minusLntǫ2


radicn(

radicaminus1)

)


middot exp(

minusLntǫ2


radicn(

radicaminus1)

)

le exp(


(34)


S(pt)minus S(Vt|pt)

= S

sum

j

pt(j)sum

l

1

LntVotimesnt (X

(t)jl )

minussum

j

pt(j)S

(

sum

l

1

LntVotimesnt (X

(t)jl )

)

= χ(ptZt)


Pr

Lntsum

l=1

1

LntQt(X



and

Pr

Lntsum

l=1

1

LntQt(X


= 1minus Pr

⋃

t

⋃

j

Lntsum

l=1

1

LntQt(X









⋂

t

maxjisin1Jn

Lntsum

l=1

1

LntW

nt (D

cj(X )|X(t)

jl ) le ǫ

cap

Lntsum

l=1

1

LntQt(X






Lntsum

l=1

1

LntW

nt (D

cj |x

(t)jl ) le ǫ (37)

and

Lntsum

l=1

1

LntQt(x



R = mintisinθ

maxUrarrArarr(BZ)t


1

nχ(U Zn

t ))minusγ


code(


Dj j = 1 Jn)

such that

1

nlog Jn ge R (39)

limnrarrinfin

maxtisinθ

maxjisin1Jn

Lntsum

l=1

1

LntW

nt (D

cj |x

(t)jl ) = 0 (40)


holds Votimesnt (x

(t)jl ) minus Qt(x



Lntsum

l=1

1

LntVotimesnt (x

(t)jprimel)minusΘt

le Lntsum

l=1

1

LntVotimesnt (x

(t)jprimel)minus

Lntsum

l=1

1

LntQt(x

(t)jprimel)

+ Lntsum

l=1

1

LntQt(x

(t)jprimel)minusΘt

leLntsum

l=1

1

LntVotimesn


(t)jprimel)

+ L

(t)ntsum

l=1

1

LntQt(x

(t)jprimel)minusΘt

le 2ǫ (41)

and sumJn

j=11Jn

sumLnt

l=11

LntVotimesnt (x


8


e then




χ(XuniZnt )

= S

Jnsum

j=1

1

Jn

Lntsum

l=1

1

LntVotimesnt (x

(t)jl )

minusJnsum

j=1

1

JnS

Lntsum

l=1

1

LntVotimesnt (x

(t)jl )

le

∣

∣

∣

∣

∣

∣

S

Jnsum

j=1

1

Jn

Lntsum

l=1

1

LntVotimesnt (x

(t)jl )

minus S (Θt)

∣

∣

∣

∣

∣

∣

+

∣

∣

∣

∣

∣

∣

S(Θt)minusJnsum

j=1

1

JnS

Lntsum

l=1

1

LntVotimesnt (x

(t)jl )

∣

∣

∣

∣

∣

∣


+

∣

∣

∣

∣

∣

∣

Jnsum

j=1

1

Jn

S(Θt)minus S

Lntsum

l=1

1

LntVotimesnt (x

(t)jl )

∣

∣

∣

∣

∣

∣



maxtisinθ



Pr(

Et(j) = x(t)jl

)

= 1Lnt



CSCSI ge mintisinθ

maxUrarrArarr(BZ)t


1

nχ(U Zn

t ))

(45)



maxtisinθ

maxjisin1Jn

sum

xnisinAn

E(xn|j)Wnt (D

cj |xn) = ǫ1n (46)

maxtisinθ

χ(J Znt ) = ǫ2n (47)




log Jn = H(Xuni)










I(XuniBtprime) +1

n+

1

nǫ1n log Jn


nχ(XuniZ

ntprime)] +

ǫ1n

n+

1

nlog Jn +

ǫ2n

n


nχ(XuniZ

ntprime)] + ǫ (49)


CSCSI le mintisinθ

maxUrarrArarr(BZ)t


1

nχ(U Zn

t ))






Qt(xn) = ΠpVtα

radicaΠVtα(x


n)ΠpVtαradica

9

and Θt =sum


n)


Pr

(

Lnsum

l=1

1

LnQt(X


)

ge 1minus 2minusnυ

(51)


Lnsum

l=1

1

LnW

nt (D

cj |xjl) le ǫ

Lnsum

l=1

1


For any γ gt 0 let


(

mintisinθ


)

minus γ


1Lnt


1n log Jn ge R and

limnrarrinfin

maxtisinθ

maxjisin1Jn

Lnsum

l=1

1

LnW

nt (D

cj |xjl)) = 0 (52)


maxtisinθ




(mintisinθ


χ(U Zt))


WIRETAP CHANNEL


Section II



mintisinθ

maxPinpwt

1

n(χ(PinpB

nt )minus χ(PinpZ

nt ))



CS = limnrarrinfin

maxUrarrArarr(BZ)t

1

n

(

mintisinθ

χ(U Bnt )

minusmaxtisinθ

χ(U Znt )

)

(55)







mintχ(PinpWt) (56)



log Tmint maxPinp


mint χ(PinpWt) =






log Jn = maxPinpw


n))minus ǫ (57)

10



log Jnt = maxPinpwt


nt ))minus ǫ (58)

Thus


mintisinθ

maxPinpwt

1

n(χ(PinpB

nt )minus χ(PinpZ

nt ))

(59)





C(t)2 = (E(t) D(t)




D(t)j j t isin θ

to decode the





limnrarrinfin

maxPinpwtprime

1

n(χ(PinpB


ntprime))



mintisinθ

maxPinpwt

1

n(χ(PinpB

nt )minus χ(PinpZ

nt ))




nt )


limnrarrinfin

maxPinpwt

1

n(χ(PinpB

nt )minus χ(PinpZ

nt ))

= maxPinpwt




nt )


CCSI = mintisinθ

maxPinpwt











Pr[

tr(

Wnt (ρ

nXjl

)DtXjl

)


gt 1minus 2minusnβ


sum

tisinθ

Jnsum

j=1

Lnsum

l=1

Dtxjlle I

We define



n)ΠpVtαradica

and Θt =sum



11


Pr

(

Lnsum

l=1

1

LnQt(ρX(t)

jl


ge 1minus 2minusnυ

(62)


mintisinθ

tr(

Wnt (ρ

nxjl

)Dtxjl

)

ge 1minus 2minusnβ

and

maxtisinθ

Lnsum

l=1

1

LnQt(ρxjl

)minusΘt le ǫ

We define Dj =sum

tisinθ

sumLn

l=1Dtxjl then

sumJn

j=1Dj =sum

tisinθ

sumJn

j=1

sumLn



tr(

Wntprime (ρ

nxjlprime

)Dj

)

=sum

tisinθ

Lnsum

l=1

tr(

Wntprime (ρ

nxjlprime

)Dtxjl

)

ge tr(

Wntprime (ρ

otimesnxjlprime

)Dtprimexjlprime

)

ge 1minus 2minusnβ


(radicM1M2

radicM1

)

ge 0

For any γ gt 0 let


1

n

[

mintisinθ

χ(pBnt )minusmax

tisinθχ(pZn

t )

]

minus γ


w(j) =sumLn

l=11Lnρnxjl

j = 1 Jn Dj j =

1 Jn)


maxtisinθ

maxjisin1Jn

tr(


)

le λ


maxtisinθ



CS ge limnrarrinfin

maxUrarrArarr(BZ)t

1

n

(

mintisinθ

χ(U Bnt )minusmax

tisinθχ(U Zn

t )

)

(65)




maxtisinθ

maxjisin1Jn

tr(


t

(

ρ(n)j

))

le λn (66)

maxtisinθ








1

nχ(XuniB

ntprime) + ξ

= mintisinθ

1

nχ(XuniB

nt ) + ξ

le mintisinθ

1

nχ(XuniB

nt )minusmax

tisinθ

1

nχ(XuniZ

nt ) + ξ +

1

nǫ2n

le 1

n

(

mintisinθ


tisinθχ(XuniZ

nt )

)

+ ǫ (68)

Thus

CS le limnrarrinfin

maxUrarrArarr(BZ)t

1

n

(

mintisinθ

χ(U Bnt )minusmax

tisinθχ(U Zn

t )

)





inftisinθ

maxPinpwt

1

n(χ(PinpB

nt )minus χ(PinpZ

nt ))

(70)


Wdiams = supnisinN



12



W (k))K





W (k))K

k=1in the










1

JN

JNsum

j=1

tr(


)





the message




tprime

)

(wtprime(j))]

=

tr(

trHprimeotimesN

[



])

We have

tr∣

∣

(


tprime

)

(wtprime (j))∣

∣

= tr(

trHprimeotimesN

∣



∣

∣

)

= tr∣



∣

∣

=∥



∥

∥

1






tprime

∥

∥

diams=

∥

∥


∥

∥


tiplicative

It follows that

∣

∣

∣

∣

1

JN

JNsum

j=1

tr(


)

minus 1

JN

JNsum

j=1

tr(


)

∣

∣

∣

∣

le 1

JN

JNsum

j=1

∣

∣tr[(


tprime

)


]∣

∣

le 1

JN

JNsum

j=1

tr∣

∣

(


tprime

)


∣

∣

le 1

JN

JNsum

j=1

tr∣

∣

(


tprime

)

(wtprime(j))∣

∣

le 1

JNJNNτ

= Nτ (74)




le

∣

∣

∣

∣

∣

∣

S

JNsum

j=1

1

JNVt(ρ(j))

minus S

JNsum

j=1

1

JNVtprime(ρ(j))

∣

∣

∣

∣

∣

∣

+

∣

∣

∣

∣

∣

∣

JNsum

j=1

1


JNsum

j=1

1


∣

∣

∣

∣

∣

∣




suptisinθ

1

JN

JNsum

j=1

tr(


)




suptisinθ

1

JN

JNsum

j=1

tr(


)

ge 1minus λ

13

suptisinθ

χ(XuniZNt ) le ǫ



inftisinθ

maxPinpwt

1

n(χ(PinpB

nt )minus χ(PinpZ

nt ))








)minus2dprime4









Notimesnt

)




sum




A ge maxp

(

mintisinθ


χ(pEt)

)

(77)



ρPUlowastNt



sum


trE

(

sum

x


Nt

)


trQ

(

sum

x


Nt

)




sum



Nt

)

and VNt HP rarr HQ

sum



Nt

)



and


14


VNt)tisinθ since



xjl




n

t on HEn

such that

tr(

(DQn

txjlotimes IE

n

)UNtρP

n

xjlUlowastNt

)


and

ωEn

jt minus ξEn

t 1 lt ǫ (79)

where ωEn

jt = 1Lnt

sumLnt

l=1 trQn

(

UNtρP

n

xjlUlowastNt

)


xjlmay be pure or

mixed Assume ρPn


sumni=1 p

primejl(i)|κ

(i)xjl〉〈κ(i)

xjl |Pn

then

nsum

i=1

pprimejl(i)tr(

(DQn

txjlotimes IE

n

)UNt|κ(i)

xjl〉〈κ(i)

xjl|Pn

UlowastNt

)

tr

(

(DQn

txjlotimes IE

n

)UNt(

nsum

i=1


〉〈κ(i)xjl

|Pn

)UlowastNt

)

ge 1minus 2minusnβ



tr(

(DQn

txjlotimes IE

n

)UNt|κ(i)

xjl〉〈κ(i)

xjl|Pn

UlowastNt

)

ge 1minus 2minusnβ



there is a ξEn

t on HEn

such that

1

Lnt

Lntsum

l=1

trQn

(

UNt|κ(ilj)


UlowastNt

)

minus ξEn

t 1 lt ǫ

Thus(


xjl |Pn

j l DQn

txjl j l t

)


ρPn

xjl j l DQn

txjl j l t

)



Assume ρPn


n



n isin S(HQn

) we have

V QnMLθ

(


)

(V QnMLθ)lowast

=sum

t

sum

j

sum

l

(

DQn

txjlρQ

n)


We denote

ψQnEnMLθjlt

=(

IEn otimes V QnMLθ

)

(

UN otimes IMLθ)

[

|κjl〉〈κjl|Pn


(


(

IEn otimes V QnMLθ

)lowast


F(

trQnEn

(

ψQnEnMLθjlt

)


ge 1minus ǫ (80)


nEn

on HQnEn

such that

〈0|θ〈0|L〈0|M〈κjl|Pn (

UNtotimes IMLθ

)lowast

(

IEn otimes V QnMLθ

)lowast|ζjlt〉Q


= F

(


nEn


ge 1minus ǫ (81)



n |0〉M|0〉L|0〉θ

and

|bjlt〉PnMLθ =

(

UNtotimes IMLθ

)lowast (IE




We define

|ajk〉PnMLθ =

1radicLn

Lnsum

l=1


nMLθ

15

|bjkt〉PnMLθ =

1radicLn

Lnsum

l=1


nMLθ

and

|bjk〉PnMLθ =

1

|θ|

|θ|sum

t=1

|bjkt〉PnMLθ


1

Ln

Lnsum

k=1

〈ajk|bjk〉PnMLθ

=1

|θ|1

Ln

|θ|sum

t=1

Lnsum

k=1


=1

|θ|1

Ln

|θ|sum

t=1

Lnsum

l=1


ge 1minus ǫ (82)


1minus ǫ


|bjkj〉PnMLθ

=1

|θ|

|θ|sum

t=1

eminusiskj 〈ajkj

|bjkj t〉PnMLθ


∣

∣


|bjkj t〉PnMLθ

∣

∣

∣ le 1 we have

mintisinθ

∣

∣


|bjkj t〉PnMLθ

∣

∣



t isin θ

1minus |θ|ǫ

le∣

∣


|bjkj t〉PnMLθ

∣

∣

∣


|bjkj t〉PnMLθ

= eminusirkj

1

Ln

(

Lnsum

l=1


nMLθ

)

(

Lnsum

l=1


nMLθ

)

(83)


|jt〉QnEnL =

radic

1

Ln

Lnsum

l=1

eminus2πi(l

kjLn


and

|ϑjt〉QnEnMLθ =

radic

1

Ln

Lnsum

l=1

eminus2πilkj

Ln

[

IEn otimes V QnMLθ

]

(UnN |κjl〉P

n

)|0〉M|0〉L|0〉θ


F

(



)

=∣

∣



∣

∣

=1

Ln

(

Lnsum

l=1


nMLθ

)

(

Lnsum

l=1

eminus2πilkjLn e


)



n


such that

∥

∥

∥ξEn

t minus trQn

(


n

UlowastNt

)∥

∥

∥

1le ǫ (85)


∥

∥trQn

(


n

UlowastNt

)

minus trQn

(


∥

∥

1

le 2

[

1minus F

(

trQn

(


n

UlowastNt

)

trQn

(


)]12

le 2

[

1minus F

(


nEn


12

le 2radicǫ (86)


2minus η1 leradic


∥

∥

∥trQnL

(

|jt〉〈jt|QnEnL

)

minus ξEn

t

∥

∥

∥

1

=

∥

∥

∥

∥

∥

1

Ln

Lnsum

l=1

trQn

(


minus ξEn

t

∥

∥

∥

∥

∥

1

le 1

Ln

Lnsum

l=1

∥

∥

∥

∥

trQn

(


n

UlowastNt

)

minus trQn

(


∥

∥

∥

∥

1

+∥

∥

∥ξEn

t minus trQn

(


n

UlowastNt

)∥

∥

∥

1


16



such that if we set

χQnEnMLjjprimet =

(

UQnML(t) otimes IE

n)

(


)(

UQnML(t) otimes IE

n)lowast

then

F(


jjprimet

)


(88)where |ξt〉Q


rification of ξEn

t on HQnEnL



1

Jn

1

Ln

Jnsum

j=1

Lnsum

l=1


n |j〉A

Jnsum

j=1

Lnsum

l=1


n

(89)



1

Jn

1

Ln

(

IA otimes UnNtprime

)

[

Jnsum

j=1

Lnsum

l=1

eminus2πilkj

Ln |j〉A|κjl〉Pn

Jnsum

j=1

Lnsum

l=1

eminus2πilkj

Ln 〈κjl|Pn〈j|A

]

(

IA otimes UnNtprime

)lowast

=1

Jn

1

Ln

Jnsum

j=1

|j〉A(

Lnsum

l=1

eminus2πilkjLn Un

Ntprime|κjl〉P

n

)

Jnsum

j=1

(

Lnsum

l=1


n

(UnNtprime

)lowast)

〈j|A


τQn rarr trQnLθ

[

(

sum

tisinθ


)

V QnMLθ

(


)

V QnMLθlowast(

sum

tisinθ


)lowast]

(90)

to his outcome


We define

ιAQnEnMLθtprime

=

(

sum

tisinθ


)


)

(

1

Jn

1

Ln

Jnsum

j=1

|j〉A(

Lnsum

l=1

eminus2πilkjLn Un

Ntprime|κjl〉P

n

)

Jnsum

j=1

(

Lnsum

l=1


n

(UnNtprime

)lowast)

〈j|A



)lowast

(

sum

tisinθ


)lowast

=

(

sum

tisinθ


)

1

Jn(

Jnsum

j=1


Jnsum

j=1


(

sum

tisinθ


)lowast

(91)


AQnEnMLθtprime )


sumJn

j=1

sumJn



(

sum

tisinθ


)

(

sum

tisinθ

UQnML(t) otimes IE


= IAEn otimessum

tisinθ

UQnML(t) (UQnML


= IAQnEnMLθ

sum




F

ιAQnEnMLθtprime

1

Jn

Jnsum

j=1

Jnsum

jprime=1


= F

(

1

Jn(

Jnsum

j=1


Jnsum

j=1


1

Jn(

Jnsum

j=1


17

(

Jnsum

j=1


)

=1

Jn

∣

∣

∣

∣

( Jnsum

j=1


)

( Jnsum

j=1


)∣

∣

∣

∣



sumJn

j=1

sumJn



F

(

1

Jn

Jnsum

j=1

Jnsum

jprime=1


1

Jn(

Jnsum

j=1


(

Jnsum

j=1


)



1minus F ( η) le 1

2minus η1 le

radic

1minus F ( η)2


F ( η)

ge 1minus 1

2minus η1

ge 1minus 1

2minus υ1 minus

1

2υ minus η1

ge 1minusradic


1minus F (υ η)2


F

(


(

Jnsum

j=1


Jnsum

j=1


)

ge F

(

ιAQnEnMLθtprime

1

Jn(

Jnsum

j=1


(

Jnsum

j=1


)

ge 1minusradic



radicǫminus 8ǫ

(94)

ge 1minusradic

2|θ|radicǫminus



radic


with rate


tχ(X Et)minus 2δ



1

nmintisinθ

maxρisinS(H)Qn



limnrarrinfin

1

nmax

ρisinS(H)QnIC(ρN

otimesnt )




1

nmintisinθ

maxρisinS(H)Qn




18

VII FURTHER NOTES




N(ρ) =

Ksum

i=1

AiρAilowast (98)





we have Ai =sumK

j=1 uijBj (cf [31])







HE Then UN =sumK




)

ρ(


)lowast=

sumKj=1AjρAj

lowast and(


)lowast (sumK


=sumK

j=1 AjlowastAj


UN =sumK



lowast Here A1 AK




1

nmax

ρisinS(H)QnIC(ρN

otimesn) (100)




1

nmax




19


ACKNOWLEDGMENT





























20


















21


QUANTUM WIRETAPPER


II


CSCSI = mintisinθ

maxUrarrArarr(BZ)t


1

nχ(U Zn

t ))



(mintisinθ





nLet





Let

Qt(xn) = ΠptVtα

radicaΠVtα(x


n)ΠptVtαradica



ρminusradicXρ

radicX le

radic2λ (29)



radicX le

radic8λ In [32]




Qt(xn)minus V

otimesnt (xn) le

radic

2(ad+ d)

nα2 (30)

We set Θt =sum

xnisinT nptδ

ppriment (xn)Qt(x

n) For given zn






ρλ

where ρ =sum




Pr

(

Lminus1Lsum

i=1


ρλ gt ǫ

)


minusL ǫ2λ

2 ln2(dimV)micro

)

(31)




Qt(xn)



n)ΠptVtαradica


radicaΠVtα(x

n)ΠptVtαradica


radica







enough we have

Pr

Lntsum

l=1

1


7


middot exp(

minusLntǫ2


radicn(

radicaminus1)

)


middot exp(

minusLntǫ2


radicn(

radicaminus1)

)

le exp(


(34)


S(pt)minus S(Vt|pt)

= S

sum

j

pt(j)sum

l

1

LntVotimesnt (X

(t)jl )

minussum

j

pt(j)S

(

sum

l

1

LntVotimesnt (X

(t)jl )

)

= χ(ptZt)


Pr

Lntsum

l=1

1

LntQt(X



and

Pr

Lntsum

l=1

1

LntQt(X


= 1minus Pr

⋃

t

⋃

j

Lntsum

l=1

1

LntQt(X









⋂

t

maxjisin1Jn

Lntsum

l=1

1

LntW

nt (D

cj(X )|X(t)

jl ) le ǫ

cap

Lntsum

l=1

1

LntQt(X






Lntsum

l=1

1

LntW

nt (D

cj |x

(t)jl ) le ǫ (37)

and

Lntsum

l=1

1

LntQt(x



R = mintisinθ

maxUrarrArarr(BZ)t


1

nχ(U Zn

t ))minusγ


code(


Dj j = 1 Jn)

such that

1

nlog Jn ge R (39)

limnrarrinfin

maxtisinθ

maxjisin1Jn

Lntsum

l=1

1

LntW

nt (D

cj |x

(t)jl ) = 0 (40)


holds Votimesnt (x

(t)jl ) minus Qt(x



Lntsum

l=1

1

LntVotimesnt (x

(t)jprimel)minusΘt

le Lntsum

l=1

1

LntVotimesnt (x

(t)jprimel)minus

Lntsum

l=1

1

LntQt(x

(t)jprimel)

+ Lntsum

l=1

1

LntQt(x

(t)jprimel)minusΘt

leLntsum

l=1

1

LntVotimesn


(t)jprimel)

+ L

(t)ntsum

l=1

1

LntQt(x

(t)jprimel)minusΘt

le 2ǫ (41)

and sumJn

j=11Jn

sumLnt

l=11

LntVotimesnt (x


8


e then




χ(XuniZnt )

= S

Jnsum

j=1

1

Jn

Lntsum

l=1

1

LntVotimesnt (x

(t)jl )

minusJnsum

j=1

1

JnS

Lntsum

l=1

1

LntVotimesnt (x

(t)jl )

le

∣

∣

∣

∣

∣

∣

S

Jnsum

j=1

1

Jn

Lntsum

l=1

1

LntVotimesnt (x

(t)jl )

minus S (Θt)

∣

∣

∣

∣

∣

∣

+

∣

∣

∣

∣

∣

∣

S(Θt)minusJnsum

j=1

1

JnS

Lntsum

l=1

1

LntVotimesnt (x

(t)jl )

∣

∣

∣

∣

∣

∣


+

∣

∣

∣

∣

∣

∣

Jnsum

j=1

1

Jn

S(Θt)minus S

Lntsum

l=1

1

LntVotimesnt (x

(t)jl )

∣

∣

∣

∣

∣

∣



maxtisinθ



Pr(

Et(j) = x(t)jl

)

= 1Lnt



CSCSI ge mintisinθ

maxUrarrArarr(BZ)t


1

nχ(U Zn

t ))

(45)



maxtisinθ

maxjisin1Jn

sum

xnisinAn

E(xn|j)Wnt (D

cj |xn) = ǫ1n (46)

maxtisinθ

χ(J Znt ) = ǫ2n (47)




log Jn = H(Xuni)










I(XuniBtprime) +1

n+

1

nǫ1n log Jn


nχ(XuniZ

ntprime)] +

ǫ1n

n+

1

nlog Jn +

ǫ2n

n


nχ(XuniZ

ntprime)] + ǫ (49)


CSCSI le mintisinθ

maxUrarrArarr(BZ)t


1

nχ(U Zn

t ))






Qt(xn) = ΠpVtα

radicaΠVtα(x


n)ΠpVtαradica

9

and Θt =sum


n)


Pr

(

Lnsum

l=1

1

LnQt(X


)

ge 1minus 2minusnυ

(51)


Lnsum

l=1

1

LnW

nt (D

cj |xjl) le ǫ

Lnsum

l=1

1


For any γ gt 0 let


(

mintisinθ


)

minus γ


1Lnt


1n log Jn ge R and

limnrarrinfin

maxtisinθ

maxjisin1Jn

Lnsum

l=1

1

LnW

nt (D

cj |xjl)) = 0 (52)


maxtisinθ




(mintisinθ


χ(U Zt))


WIRETAP CHANNEL


Section II



mintisinθ

maxPinpwt

1

n(χ(PinpB

nt )minus χ(PinpZ

nt ))



CS = limnrarrinfin

maxUrarrArarr(BZ)t

1

n

(

mintisinθ

χ(U Bnt )

minusmaxtisinθ

χ(U Znt )

)

(55)







mintχ(PinpWt) (56)



log Tmint maxPinp


mint χ(PinpWt) =






log Jn = maxPinpw


n))minus ǫ (57)

10



log Jnt = maxPinpwt


nt ))minus ǫ (58)

Thus


mintisinθ

maxPinpwt

1

n(χ(PinpB

nt )minus χ(PinpZ

nt ))

(59)





C(t)2 = (E(t) D(t)




D(t)j j t isin θ

to decode the





limnrarrinfin

maxPinpwtprime

1

n(χ(PinpB


ntprime))



mintisinθ

maxPinpwt

1

n(χ(PinpB

nt )minus χ(PinpZ

nt ))




nt )


limnrarrinfin

maxPinpwt

1

n(χ(PinpB

nt )minus χ(PinpZ

nt ))

= maxPinpwt




nt )


CCSI = mintisinθ

maxPinpwt











Pr[

tr(

Wnt (ρ

nXjl

)DtXjl

)


gt 1minus 2minusnβ


sum

tisinθ

Jnsum

j=1

Lnsum

l=1

Dtxjlle I

We define



n)ΠpVtαradica

and Θt =sum



11


Pr

(

Lnsum

l=1

1

LnQt(ρX(t)

jl


ge 1minus 2minusnυ

(62)


mintisinθ

tr(

Wnt (ρ

nxjl

)Dtxjl

)

ge 1minus 2minusnβ

and

maxtisinθ

Lnsum

l=1

1

LnQt(ρxjl

)minusΘt le ǫ

We define Dj =sum

tisinθ

sumLn

l=1Dtxjl then

sumJn

j=1Dj =sum

tisinθ

sumJn

j=1

sumLn



tr(

Wntprime (ρ

nxjlprime

)Dj

)

=sum

tisinθ

Lnsum

l=1

tr(

Wntprime (ρ

nxjlprime

)Dtxjl

)

ge tr(

Wntprime (ρ

otimesnxjlprime

)Dtprimexjlprime

)

ge 1minus 2minusnβ


(radicM1M2

radicM1

)

ge 0

For any γ gt 0 let


1

n

[

mintisinθ

χ(pBnt )minusmax

tisinθχ(pZn

t )

]

minus γ


w(j) =sumLn

l=11Lnρnxjl

j = 1 Jn Dj j =

1 Jn)


maxtisinθ

maxjisin1Jn

tr(


)

le λ


maxtisinθ



CS ge limnrarrinfin

maxUrarrArarr(BZ)t

1

n

(

mintisinθ

χ(U Bnt )minusmax

tisinθχ(U Zn

t )

)

(65)




maxtisinθ

maxjisin1Jn

tr(


t

(

ρ(n)j

))

le λn (66)

maxtisinθ








1

nχ(XuniB

ntprime) + ξ

= mintisinθ

1

nχ(XuniB

nt ) + ξ

le mintisinθ

1

nχ(XuniB

nt )minusmax

tisinθ

1

nχ(XuniZ

nt ) + ξ +

1

nǫ2n

le 1

n

(

mintisinθ


tisinθχ(XuniZ

nt )

)

+ ǫ (68)

Thus

CS le limnrarrinfin

maxUrarrArarr(BZ)t

1

n

(

mintisinθ

χ(U Bnt )minusmax

tisinθχ(U Zn

t )

)





inftisinθ

maxPinpwt

1

n(χ(PinpB

nt )minus χ(PinpZ

nt ))

(70)


Wdiams = supnisinN



12



W (k))K





W (k))K

k=1in the










1

JN

JNsum

j=1

tr(


)





the message




tprime

)

(wtprime(j))]

=

tr(

trHprimeotimesN

[



])

We have

tr∣

∣

(


tprime

)

(wtprime (j))∣

∣

= tr(

trHprimeotimesN

∣



∣

∣

)

= tr∣



∣

∣

=∥



∥

∥

1






tprime

∥

∥

diams=

∥

∥


∥

∥


tiplicative

It follows that

∣

∣

∣

∣

1

JN

JNsum

j=1

tr(


)

minus 1

JN

JNsum

j=1

tr(


)

∣

∣

∣

∣

le 1

JN

JNsum

j=1

∣

∣tr[(


tprime

)


]∣

∣

le 1

JN

JNsum

j=1

tr∣

∣

(


tprime

)


∣

∣

le 1

JN

JNsum

j=1

tr∣

∣

(


tprime

)

(wtprime(j))∣

∣

le 1

JNJNNτ

= Nτ (74)




le

∣

∣

∣

∣

∣

∣

S

JNsum

j=1

1

JNVt(ρ(j))

minus S

JNsum

j=1

1

JNVtprime(ρ(j))

∣

∣

∣

∣

∣

∣

+

∣

∣

∣

∣

∣

∣

JNsum

j=1

1


JNsum

j=1

1


∣

∣

∣

∣

∣

∣




suptisinθ

1

JN

JNsum

j=1

tr(


)




suptisinθ

1

JN

JNsum

j=1

tr(


)

ge 1minus λ

13

suptisinθ

χ(XuniZNt ) le ǫ



inftisinθ

maxPinpwt

1

n(χ(PinpB

nt )minus χ(PinpZ

nt ))








)minus2dprime4









Notimesnt

)




sum




A ge maxp

(

mintisinθ


χ(pEt)

)

(77)



ρPUlowastNt



sum


trE

(

sum

x


Nt

)


trQ

(

sum

x


Nt

)




sum



Nt

)

and VNt HP rarr HQ

sum



Nt

)



and


14


VNt)tisinθ since



xjl




n

t on HEn

such that

tr(

(DQn

txjlotimes IE

n

)UNtρP

n

xjlUlowastNt

)


and

ωEn

jt minus ξEn

t 1 lt ǫ (79)

where ωEn

jt = 1Lnt

sumLnt

l=1 trQn

(

UNtρP

n

xjlUlowastNt

)


xjlmay be pure or

mixed Assume ρPn


sumni=1 p

primejl(i)|κ

(i)xjl〉〈κ(i)

xjl |Pn

then

nsum

i=1

pprimejl(i)tr(

(DQn

txjlotimes IE

n

)UNt|κ(i)

xjl〉〈κ(i)

xjl|Pn

UlowastNt

)

tr

(

(DQn

txjlotimes IE

n

)UNt(

nsum

i=1


〉〈κ(i)xjl

|Pn

)UlowastNt

)

ge 1minus 2minusnβ



tr(

(DQn

txjlotimes IE

n

)UNt|κ(i)

xjl〉〈κ(i)

xjl|Pn

UlowastNt

)

ge 1minus 2minusnβ



there is a ξEn

t on HEn

such that

1

Lnt

Lntsum

l=1

trQn

(

UNt|κ(ilj)


UlowastNt

)

minus ξEn

t 1 lt ǫ

Thus(


xjl |Pn

j l DQn

txjl j l t

)


ρPn

xjl j l DQn

txjl j l t

)



Assume ρPn


n



n isin S(HQn

) we have

V QnMLθ

(


)

(V QnMLθ)lowast

=sum

t

sum

j

sum

l

(

DQn

txjlρQ

n)


We denote

ψQnEnMLθjlt

=(

IEn otimes V QnMLθ

)

(

UN otimes IMLθ)

[

|κjl〉〈κjl|Pn


(


(

IEn otimes V QnMLθ

)lowast


F(

trQnEn

(

ψQnEnMLθjlt

)


ge 1minus ǫ (80)


nEn

on HQnEn

such that

〈0|θ〈0|L〈0|M〈κjl|Pn (

UNtotimes IMLθ

)lowast

(

IEn otimes V QnMLθ

)lowast|ζjlt〉Q


= F

(


nEn


ge 1minus ǫ (81)



n |0〉M|0〉L|0〉θ

and

|bjlt〉PnMLθ =

(

UNtotimes IMLθ

)lowast (IE




We define

|ajk〉PnMLθ =

1radicLn

Lnsum

l=1


nMLθ

15

|bjkt〉PnMLθ =

1radicLn

Lnsum

l=1


nMLθ

and

|bjk〉PnMLθ =

1

|θ|

|θ|sum

t=1

|bjkt〉PnMLθ


1

Ln

Lnsum

k=1

〈ajk|bjk〉PnMLθ

=1

|θ|1

Ln

|θ|sum

t=1

Lnsum

k=1


=1

|θ|1

Ln

|θ|sum

t=1

Lnsum

l=1


ge 1minus ǫ (82)


1minus ǫ


|bjkj〉PnMLθ

=1

|θ|

|θ|sum

t=1

eminusiskj 〈ajkj

|bjkj t〉PnMLθ


∣

∣


|bjkj t〉PnMLθ

∣

∣

∣ le 1 we have

mintisinθ

∣

∣


|bjkj t〉PnMLθ

∣

∣



t isin θ

1minus |θ|ǫ

le∣

∣


|bjkj t〉PnMLθ

∣

∣

∣


|bjkj t〉PnMLθ

= eminusirkj

1

Ln

(

Lnsum

l=1


nMLθ

)

(

Lnsum

l=1


nMLθ

)

(83)


|jt〉QnEnL =

radic

1

Ln

Lnsum

l=1

eminus2πi(l

kjLn


and

|ϑjt〉QnEnMLθ =

radic

1

Ln

Lnsum

l=1

eminus2πilkj

Ln

[

IEn otimes V QnMLθ

]

(UnN |κjl〉P

n

)|0〉M|0〉L|0〉θ


F

(



)

=∣

∣



∣

∣

=1

Ln

(

Lnsum

l=1


nMLθ

)

(

Lnsum

l=1

eminus2πilkjLn e


)



n


such that

∥

∥

∥ξEn

t minus trQn

(


n

UlowastNt

)∥

∥

∥

1le ǫ (85)


∥

∥trQn

(


n

UlowastNt

)

minus trQn

(


∥

∥

1

le 2

[

1minus F

(

trQn

(


n

UlowastNt

)

trQn

(


)]12

le 2

[

1minus F

(


nEn


12

le 2radicǫ (86)


2minus η1 leradic


∥

∥

∥trQnL

(

|jt〉〈jt|QnEnL

)

minus ξEn

t

∥

∥

∥

1

=

∥

∥

∥

∥

∥

1

Ln

Lnsum

l=1

trQn

(


minus ξEn

t

∥

∥

∥

∥

∥

1

le 1

Ln

Lnsum

l=1

∥

∥

∥

∥

trQn

(


n

UlowastNt

)

minus trQn

(


∥

∥

∥

∥

1

+∥

∥

∥ξEn

t minus trQn

(


n

UlowastNt

)∥

∥

∥

1


16



such that if we set

χQnEnMLjjprimet =

(

UQnML(t) otimes IE

n)

(


)(

UQnML(t) otimes IE

n)lowast

then

F(


jjprimet

)


(88)where |ξt〉Q


rification of ξEn

t on HQnEnL



1

Jn

1

Ln

Jnsum

j=1

Lnsum

l=1


n |j〉A

Jnsum

j=1

Lnsum

l=1


n

(89)



1

Jn

1

Ln

(

IA otimes UnNtprime

)

[

Jnsum

j=1

Lnsum

l=1

eminus2πilkj

Ln |j〉A|κjl〉Pn

Jnsum

j=1

Lnsum

l=1

eminus2πilkj

Ln 〈κjl|Pn〈j|A

]

(

IA otimes UnNtprime

)lowast

=1

Jn

1

Ln

Jnsum

j=1

|j〉A(

Lnsum

l=1

eminus2πilkjLn Un

Ntprime|κjl〉P

n

)

Jnsum

j=1

(

Lnsum

l=1


n

(UnNtprime

)lowast)

〈j|A


τQn rarr trQnLθ

[

(

sum

tisinθ


)

V QnMLθ

(


)

V QnMLθlowast(

sum

tisinθ


)lowast]

(90)

to his outcome


We define

ιAQnEnMLθtprime

=

(

sum

tisinθ


)


)

(

1

Jn

1

Ln

Jnsum

j=1

|j〉A(

Lnsum

l=1

eminus2πilkjLn Un

Ntprime|κjl〉P

n

)

Jnsum

j=1

(

Lnsum

l=1


n

(UnNtprime

)lowast)

〈j|A



)lowast

(

sum

tisinθ


)lowast

=

(

sum

tisinθ


)

1

Jn(

Jnsum

j=1


Jnsum

j=1


(

sum

tisinθ


)lowast

(91)


AQnEnMLθtprime )


sumJn

j=1

sumJn



(

sum

tisinθ


)

(

sum

tisinθ

UQnML(t) otimes IE


= IAEn otimessum

tisinθ

UQnML(t) (UQnML


= IAQnEnMLθ

sum




F

ιAQnEnMLθtprime

1

Jn

Jnsum

j=1

Jnsum

jprime=1


= F

(

1

Jn(

Jnsum

j=1


Jnsum

j=1


1

Jn(

Jnsum

j=1


17

(

Jnsum

j=1


)

=1

Jn

∣

∣

∣

∣

( Jnsum

j=1


)

( Jnsum

j=1


)∣

∣

∣

∣



sumJn

j=1

sumJn



F

(

1

Jn

Jnsum

j=1

Jnsum

jprime=1


1

Jn(

Jnsum

j=1


(

Jnsum

j=1


)



1minus F ( η) le 1

2minus η1 le

radic

1minus F ( η)2


F ( η)

ge 1minus 1

2minus η1

ge 1minus 1

2minus υ1 minus

1

2υ minus η1

ge 1minusradic


1minus F (υ η)2


F

(


(

Jnsum

j=1


Jnsum

j=1


)

ge F

(

ιAQnEnMLθtprime

1

Jn(

Jnsum

j=1


(

Jnsum

j=1


)

ge 1minusradic



radicǫminus 8ǫ

(94)

ge 1minusradic

2|θ|radicǫminus



radic


with rate


tχ(X Et)minus 2δ



1

nmintisinθ

maxρisinS(H)Qn



limnrarrinfin

1

nmax

ρisinS(H)QnIC(ρN

otimesnt )




1

nmintisinθ

maxρisinS(H)Qn




18

VII FURTHER NOTES




N(ρ) =

Ksum

i=1

AiρAilowast (98)





we have Ai =sumK

j=1 uijBj (cf [31])







HE Then UN =sumK




)

ρ(


)lowast=

sumKj=1AjρAj

lowast and(


)lowast (sumK


=sumK

j=1 AjlowastAj


UN =sumK



lowast Here A1 AK




1

nmax

ρisinS(H)QnIC(ρN

otimesn) (100)




1

nmax




19


ACKNOWLEDGMENT





























20


















21


middot exp(

minusLntǫ2


radicn(

radicaminus1)

)


middot exp(

minusLntǫ2


radicn(

radicaminus1)

)

le exp(


(34)


S(pt)minus S(Vt|pt)

= S

sum

j

pt(j)sum

l

1

LntVotimesnt (X

(t)jl )

minussum

j

pt(j)S

(

sum

l

1

LntVotimesnt (X

(t)jl )

)

= χ(ptZt)


Pr

Lntsum

l=1

1

LntQt(X



and

Pr

Lntsum

l=1

1

LntQt(X


= 1minus Pr

⋃

t

⋃

j

Lntsum

l=1

1

LntQt(X









⋂

t

maxjisin1Jn

Lntsum

l=1

1

LntW

nt (D

cj(X )|X(t)

jl ) le ǫ

cap

Lntsum

l=1

1

LntQt(X






Lntsum

l=1

1

LntW

nt (D

cj |x

(t)jl ) le ǫ (37)

and

Lntsum

l=1

1

LntQt(x



R = mintisinθ

maxUrarrArarr(BZ)t


1

nχ(U Zn

t ))minusγ


code(


Dj j = 1 Jn)

such that

1

nlog Jn ge R (39)

limnrarrinfin

maxtisinθ

maxjisin1Jn

Lntsum

l=1

1

LntW

nt (D

cj |x

(t)jl ) = 0 (40)


holds Votimesnt (x

(t)jl ) minus Qt(x



Lntsum

l=1

1

LntVotimesnt (x

(t)jprimel)minusΘt

le Lntsum

l=1

1

LntVotimesnt (x

(t)jprimel)minus

Lntsum

l=1

1

LntQt(x

(t)jprimel)

+ Lntsum

l=1

1

LntQt(x

(t)jprimel)minusΘt

leLntsum

l=1

1

LntVotimesn


(t)jprimel)

+ L

(t)ntsum

l=1

1

LntQt(x

(t)jprimel)minusΘt

le 2ǫ (41)

and sumJn

j=11Jn

sumLnt

l=11

LntVotimesnt (x


8


e then




χ(XuniZnt )

= S

Jnsum

j=1

1

Jn

Lntsum

l=1

1

LntVotimesnt (x

(t)jl )

minusJnsum

j=1

1

JnS

Lntsum

l=1

1

LntVotimesnt (x

(t)jl )

le

∣

∣

∣

∣

∣

∣

S

Jnsum

j=1

1

Jn

Lntsum

l=1

1

LntVotimesnt (x

(t)jl )

minus S (Θt)

∣

∣

∣

∣

∣

∣

+

∣

∣

∣

∣

∣

∣

S(Θt)minusJnsum

j=1

1

JnS

Lntsum

l=1

1

LntVotimesnt (x

(t)jl )

∣

∣

∣

∣

∣

∣


+

∣

∣

∣

∣

∣

∣

Jnsum

j=1

1

Jn

S(Θt)minus S

Lntsum

l=1

1

LntVotimesnt (x

(t)jl )

∣

∣

∣

∣

∣

∣



maxtisinθ



Pr(

Et(j) = x(t)jl

)

= 1Lnt



CSCSI ge mintisinθ

maxUrarrArarr(BZ)t


1

nχ(U Zn

t ))

(45)



maxtisinθ

maxjisin1Jn

sum

xnisinAn

E(xn|j)Wnt (D

cj |xn) = ǫ1n (46)

maxtisinθ

χ(J Znt ) = ǫ2n (47)




log Jn = H(Xuni)










I(XuniBtprime) +1

n+

1

nǫ1n log Jn


nχ(XuniZ

ntprime)] +

ǫ1n

n+

1

nlog Jn +

ǫ2n

n


nχ(XuniZ

ntprime)] + ǫ (49)


CSCSI le mintisinθ

maxUrarrArarr(BZ)t


1

nχ(U Zn

t ))






Qt(xn) = ΠpVtα

radicaΠVtα(x


n)ΠpVtαradica

9

and Θt =sum


n)


Pr

(

Lnsum

l=1

1

LnQt(X


)

ge 1minus 2minusnυ

(51)


Lnsum

l=1

1

LnW

nt (D

cj |xjl) le ǫ

Lnsum

l=1

1


For any γ gt 0 let


(

mintisinθ


)

minus γ


1Lnt


1n log Jn ge R and

limnrarrinfin

maxtisinθ

maxjisin1Jn

Lnsum

l=1

1

LnW

nt (D

cj |xjl)) = 0 (52)


maxtisinθ




(mintisinθ


χ(U Zt))


WIRETAP CHANNEL


Section II



mintisinθ

maxPinpwt

1

n(χ(PinpB

nt )minus χ(PinpZ

nt ))



CS = limnrarrinfin

maxUrarrArarr(BZ)t

1

n

(

mintisinθ

χ(U Bnt )

minusmaxtisinθ

χ(U Znt )

)

(55)







mintχ(PinpWt) (56)



log Tmint maxPinp


mint χ(PinpWt) =






log Jn = maxPinpw


n))minus ǫ (57)

10



log Jnt = maxPinpwt


nt ))minus ǫ (58)

Thus


mintisinθ

maxPinpwt

1

n(χ(PinpB

nt )minus χ(PinpZ

nt ))

(59)





C(t)2 = (E(t) D(t)




D(t)j j t isin θ

to decode the





limnrarrinfin

maxPinpwtprime

1

n(χ(PinpB


ntprime))



mintisinθ

maxPinpwt

1

n(χ(PinpB

nt )minus χ(PinpZ

nt ))




nt )


limnrarrinfin

maxPinpwt

1

n(χ(PinpB

nt )minus χ(PinpZ

nt ))

= maxPinpwt




nt )


CCSI = mintisinθ

maxPinpwt











Pr[

tr(

Wnt (ρ

nXjl

)DtXjl

)


gt 1minus 2minusnβ


sum

tisinθ

Jnsum

j=1

Lnsum

l=1

Dtxjlle I

We define



n)ΠpVtαradica

and Θt =sum



11


Pr

(

Lnsum

l=1

1

LnQt(ρX(t)

jl


ge 1minus 2minusnυ

(62)


mintisinθ

tr(

Wnt (ρ

nxjl

)Dtxjl

)

ge 1minus 2minusnβ

and

maxtisinθ

Lnsum

l=1

1

LnQt(ρxjl

)minusΘt le ǫ

We define Dj =sum

tisinθ

sumLn

l=1Dtxjl then

sumJn

j=1Dj =sum

tisinθ

sumJn

j=1

sumLn



tr(

Wntprime (ρ

nxjlprime

)Dj

)

=sum

tisinθ

Lnsum

l=1

tr(

Wntprime (ρ

nxjlprime

)Dtxjl

)

ge tr(

Wntprime (ρ

otimesnxjlprime

)Dtprimexjlprime

)

ge 1minus 2minusnβ


(radicM1M2

radicM1

)

ge 0

For any γ gt 0 let


1

n

[

mintisinθ

χ(pBnt )minusmax

tisinθχ(pZn

t )

]

minus γ


w(j) =sumLn

l=11Lnρnxjl

j = 1 Jn Dj j =

1 Jn)


maxtisinθ

maxjisin1Jn

tr(


)

le λ


maxtisinθ



CS ge limnrarrinfin

maxUrarrArarr(BZ)t

1

n

(

mintisinθ

χ(U Bnt )minusmax

tisinθχ(U Zn

t )

)

(65)




maxtisinθ

maxjisin1Jn

tr(


t

(

ρ(n)j

))

le λn (66)

maxtisinθ








1

nχ(XuniB

ntprime) + ξ

= mintisinθ

1

nχ(XuniB

nt ) + ξ

le mintisinθ

1

nχ(XuniB

nt )minusmax

tisinθ

1

nχ(XuniZ

nt ) + ξ +

1

nǫ2n

le 1

n

(

mintisinθ


tisinθχ(XuniZ

nt )

)

+ ǫ (68)

Thus

CS le limnrarrinfin

maxUrarrArarr(BZ)t

1

n

(

mintisinθ

χ(U Bnt )minusmax

tisinθχ(U Zn

t )

)





inftisinθ

maxPinpwt

1

n(χ(PinpB

nt )minus χ(PinpZ

nt ))

(70)


Wdiams = supnisinN



12



W (k))K





W (k))K

k=1in the










1

JN

JNsum

j=1

tr(


)





the message




tprime

)

(wtprime(j))]

=

tr(

trHprimeotimesN

[



])

We have

tr∣

∣

(


tprime

)

(wtprime (j))∣

∣

= tr(

trHprimeotimesN

∣



∣

∣

)

= tr∣



∣

∣

=∥



∥

∥

1






tprime

∥

∥

diams=

∥

∥


∥

∥


tiplicative

It follows that

∣

∣

∣

∣

1

JN

JNsum

j=1

tr(


)

minus 1

JN

JNsum

j=1

tr(


)

∣

∣

∣

∣

le 1

JN

JNsum

j=1

∣

∣tr[(


tprime

)


]∣

∣

le 1

JN

JNsum

j=1

tr∣

∣

(


tprime

)


∣

∣

le 1

JN

JNsum

j=1

tr∣

∣

(


tprime

)

(wtprime(j))∣

∣

le 1

JNJNNτ

= Nτ (74)




le

∣

∣

∣

∣

∣

∣

S

JNsum

j=1

1

JNVt(ρ(j))

minus S

JNsum

j=1

1

JNVtprime(ρ(j))

∣

∣

∣

∣

∣

∣

+

∣

∣

∣

∣

∣

∣

JNsum

j=1

1


JNsum

j=1

1


∣

∣

∣

∣

∣

∣




suptisinθ

1

JN

JNsum

j=1

tr(


)




suptisinθ

1

JN

JNsum

j=1

tr(


)

ge 1minus λ

13

suptisinθ

χ(XuniZNt ) le ǫ



inftisinθ

maxPinpwt

1

n(χ(PinpB

nt )minus χ(PinpZ

nt ))








)minus2dprime4









Notimesnt

)




sum




A ge maxp

(

mintisinθ


χ(pEt)

)

(77)



ρPUlowastNt



sum


trE

(

sum

x


Nt

)


trQ

(

sum

x


Nt

)




sum



Nt

)

and VNt HP rarr HQ

sum



Nt

)



and


14


VNt)tisinθ since



xjl




n

t on HEn

such that

tr(

(DQn

txjlotimes IE

n

)UNtρP

n

xjlUlowastNt

)


and

ωEn

jt minus ξEn

t 1 lt ǫ (79)

where ωEn

jt = 1Lnt

sumLnt

l=1 trQn

(

UNtρP

n

xjlUlowastNt

)


xjlmay be pure or

mixed Assume ρPn


sumni=1 p

primejl(i)|κ

(i)xjl〉〈κ(i)

xjl |Pn

then

nsum

i=1

pprimejl(i)tr(

(DQn

txjlotimes IE

n

)UNt|κ(i)

xjl〉〈κ(i)

xjl|Pn

UlowastNt

)

tr

(

(DQn

txjlotimes IE

n

)UNt(

nsum

i=1


〉〈κ(i)xjl

|Pn

)UlowastNt

)

ge 1minus 2minusnβ



tr(

(DQn

txjlotimes IE

n

)UNt|κ(i)

xjl〉〈κ(i)

xjl|Pn

UlowastNt

)

ge 1minus 2minusnβ



there is a ξEn

t on HEn

such that

1

Lnt

Lntsum

l=1

trQn

(

UNt|κ(ilj)


UlowastNt

)

minus ξEn

t 1 lt ǫ

Thus(


xjl |Pn

j l DQn

txjl j l t

)


ρPn

xjl j l DQn

txjl j l t

)



Assume ρPn


n



n isin S(HQn

) we have

V QnMLθ

(


)

(V QnMLθ)lowast

=sum

t

sum

j

sum

l

(

DQn

txjlρQ

n)


We denote

ψQnEnMLθjlt

=(

IEn otimes V QnMLθ

)

(

UN otimes IMLθ)

[

|κjl〉〈κjl|Pn


(


(

IEn otimes V QnMLθ

)lowast


F(

trQnEn

(

ψQnEnMLθjlt

)


ge 1minus ǫ (80)


nEn

on HQnEn

such that

〈0|θ〈0|L〈0|M〈κjl|Pn (

UNtotimes IMLθ

)lowast

(

IEn otimes V QnMLθ

)lowast|ζjlt〉Q


= F

(


nEn


ge 1minus ǫ (81)



n |0〉M|0〉L|0〉θ

and

|bjlt〉PnMLθ =

(

UNtotimes IMLθ

)lowast (IE




We define

|ajk〉PnMLθ =

1radicLn

Lnsum

l=1


nMLθ

15

|bjkt〉PnMLθ =

1radicLn

Lnsum

l=1


nMLθ

and

|bjk〉PnMLθ =

1

|θ|

|θ|sum

t=1

|bjkt〉PnMLθ


1

Ln

Lnsum

k=1

〈ajk|bjk〉PnMLθ

=1

|θ|1

Ln

|θ|sum

t=1

Lnsum

k=1


=1

|θ|1

Ln

|θ|sum

t=1

Lnsum

l=1


ge 1minus ǫ (82)


1minus ǫ


|bjkj〉PnMLθ

=1

|θ|

|θ|sum

t=1

eminusiskj 〈ajkj

|bjkj t〉PnMLθ


∣

∣


|bjkj t〉PnMLθ

∣

∣

∣ le 1 we have

mintisinθ

∣

∣


|bjkj t〉PnMLθ

∣

∣



t isin θ

1minus |θ|ǫ

le∣

∣


|bjkj t〉PnMLθ

∣

∣

∣


|bjkj t〉PnMLθ

= eminusirkj

1

Ln

(

Lnsum

l=1


nMLθ

)

(

Lnsum

l=1


nMLθ

)

(83)


|jt〉QnEnL =

radic

1

Ln

Lnsum

l=1

eminus2πi(l

kjLn


and

|ϑjt〉QnEnMLθ =

radic

1

Ln

Lnsum

l=1

eminus2πilkj

Ln

[

IEn otimes V QnMLθ

]

(UnN |κjl〉P

n

)|0〉M|0〉L|0〉θ


F

(



)

=∣

∣



∣

∣

=1

Ln

(

Lnsum

l=1


nMLθ

)

(

Lnsum

l=1

eminus2πilkjLn e


)



n


such that

∥

∥

∥ξEn

t minus trQn

(


n

UlowastNt

)∥

∥

∥

1le ǫ (85)


∥

∥trQn

(


n

UlowastNt

)

minus trQn

(


∥

∥

1

le 2

[

1minus F

(

trQn

(


n

UlowastNt

)

trQn

(


)]12

le 2

[

1minus F

(


nEn


12

le 2radicǫ (86)


2minus η1 leradic


∥

∥

∥trQnL

(

|jt〉〈jt|QnEnL

)

minus ξEn

t

∥

∥

∥

1

=

∥

∥

∥

∥

∥

1

Ln

Lnsum

l=1

trQn

(


minus ξEn

t

∥

∥

∥

∥

∥

1

le 1

Ln

Lnsum

l=1

∥

∥

∥

∥

trQn

(


n

UlowastNt

)

minus trQn

(


∥

∥

∥

∥

1

+∥

∥

∥ξEn

t minus trQn

(


n

UlowastNt

)∥

∥

∥

1


16



such that if we set

χQnEnMLjjprimet =

(

UQnML(t) otimes IE

n)

(


)(

UQnML(t) otimes IE

n)lowast

then

F(


jjprimet

)


(88)where |ξt〉Q


rification of ξEn

t on HQnEnL



1

Jn

1

Ln

Jnsum

j=1

Lnsum

l=1


n |j〉A

Jnsum

j=1

Lnsum

l=1


n

(89)



1

Jn

1

Ln

(

IA otimes UnNtprime

)

[

Jnsum

j=1

Lnsum

l=1

eminus2πilkj

Ln |j〉A|κjl〉Pn

Jnsum

j=1

Lnsum

l=1

eminus2πilkj

Ln 〈κjl|Pn〈j|A

]

(

IA otimes UnNtprime

)lowast

=1

Jn

1

Ln

Jnsum

j=1

|j〉A(

Lnsum

l=1

eminus2πilkjLn Un

Ntprime|κjl〉P

n

)

Jnsum

j=1

(

Lnsum

l=1


n

(UnNtprime

)lowast)

〈j|A


τQn rarr trQnLθ

[

(

sum

tisinθ


)

V QnMLθ

(


)

V QnMLθlowast(

sum

tisinθ


)lowast]

(90)

to his outcome


We define

ιAQnEnMLθtprime

=

(

sum

tisinθ


)


)

(

1

Jn

1

Ln

Jnsum

j=1

|j〉A(

Lnsum

l=1

eminus2πilkjLn Un

Ntprime|κjl〉P

n

)

Jnsum

j=1

(

Lnsum

l=1


n

(UnNtprime

)lowast)

〈j|A



)lowast

(

sum

tisinθ


)lowast

=

(

sum

tisinθ


)

1

Jn(

Jnsum

j=1


Jnsum

j=1


(

sum

tisinθ


)lowast

(91)


AQnEnMLθtprime )


sumJn

j=1

sumJn



(

sum

tisinθ


)

(

sum

tisinθ

UQnML(t) otimes IE


= IAEn otimessum

tisinθ

UQnML(t) (UQnML


= IAQnEnMLθ

sum




F

ιAQnEnMLθtprime

1

Jn

Jnsum

j=1

Jnsum

jprime=1


= F

(

1

Jn(

Jnsum

j=1


Jnsum

j=1


1

Jn(

Jnsum

j=1


17

(

Jnsum

j=1


)

=1

Jn

∣

∣

∣

∣

( Jnsum

j=1


)

( Jnsum

j=1


)∣

∣

∣

∣



sumJn

j=1

sumJn



F

(

1

Jn

Jnsum

j=1

Jnsum

jprime=1


1

Jn(

Jnsum

j=1


(

Jnsum

j=1


)



1minus F ( η) le 1

2minus η1 le

radic

1minus F ( η)2


F ( η)

ge 1minus 1

2minus η1

ge 1minus 1

2minus υ1 minus

1

2υ minus η1

ge 1minusradic


1minus F (υ η)2


F

(


(

Jnsum

j=1


Jnsum

j=1


)

ge F

(

ιAQnEnMLθtprime

1

Jn(

Jnsum

j=1


(

Jnsum

j=1


)

ge 1minusradic



radicǫminus 8ǫ

(94)

ge 1minusradic

2|θ|radicǫminus



radic


with rate


tχ(X Et)minus 2δ



1

nmintisinθ

maxρisinS(H)Qn



limnrarrinfin

1

nmax

ρisinS(H)QnIC(ρN

otimesnt )




1

nmintisinθ

maxρisinS(H)Qn




18

VII FURTHER NOTES




N(ρ) =

Ksum

i=1

AiρAilowast (98)





we have Ai =sumK

j=1 uijBj (cf [31])







HE Then UN =sumK




)

ρ(


)lowast=

sumKj=1AjρAj

lowast and(


)lowast (sumK


=sumK

j=1 AjlowastAj


UN =sumK



lowast Here A1 AK




1

nmax

ρisinS(H)QnIC(ρN

otimesn) (100)




1

nmax




19


ACKNOWLEDGMENT





























20


















21


e then




χ(XuniZnt )

= S

Jnsum

j=1

1

Jn

Lntsum

l=1

1

LntVotimesnt (x

(t)jl )

minusJnsum

j=1

1

JnS

Lntsum

l=1

1

LntVotimesnt (x

(t)jl )

le

∣

∣

∣

∣

∣

∣

S

Jnsum

j=1

1

Jn

Lntsum

l=1

1

LntVotimesnt (x

(t)jl )

minus S (Θt)

∣

∣

∣

∣

∣

∣

+

∣

∣

∣

∣

∣

∣

S(Θt)minusJnsum

j=1

1

JnS

Lntsum

l=1

1

LntVotimesnt (x

(t)jl )

∣

∣

∣

∣

∣

∣


+

∣

∣

∣

∣

∣

∣

Jnsum

j=1

1

Jn

S(Θt)minus S

Lntsum

l=1

1

LntVotimesnt (x

(t)jl )

∣

∣

∣

∣

∣

∣



maxtisinθ



Pr(

Et(j) = x(t)jl

)

= 1Lnt



CSCSI ge mintisinθ

maxUrarrArarr(BZ)t


1

nχ(U Zn

t ))

(45)



maxtisinθ

maxjisin1Jn

sum

xnisinAn

E(xn|j)Wnt (D

cj |xn) = ǫ1n (46)

maxtisinθ

χ(J Znt ) = ǫ2n (47)




log Jn = H(Xuni)










I(XuniBtprime) +1

n+

1

nǫ1n log Jn


nχ(XuniZ

ntprime)] +

ǫ1n

n+

1

nlog Jn +

ǫ2n

n


nχ(XuniZ

ntprime)] + ǫ (49)


CSCSI le mintisinθ

maxUrarrArarr(BZ)t


1

nχ(U Zn

t ))






Qt(xn) = ΠpVtα

radicaΠVtα(x


n)ΠpVtαradica

9

and Θt =sum


n)


Pr

(

Lnsum

l=1

1

LnQt(X


)

ge 1minus 2minusnυ

(51)


Lnsum

l=1

1

LnW

nt (D

cj |xjl) le ǫ

Lnsum

l=1

1


For any γ gt 0 let


(

mintisinθ


)

minus γ


1Lnt


1n log Jn ge R and

limnrarrinfin

maxtisinθ

maxjisin1Jn

Lnsum

l=1

1

LnW

nt (D

cj |xjl)) = 0 (52)


maxtisinθ




(mintisinθ


χ(U Zt))


WIRETAP CHANNEL


Section II



mintisinθ

maxPinpwt

1

n(χ(PinpB

nt )minus χ(PinpZ

nt ))



CS = limnrarrinfin

maxUrarrArarr(BZ)t

1

n

(

mintisinθ

χ(U Bnt )

minusmaxtisinθ

χ(U Znt )

)

(55)







mintχ(PinpWt) (56)



log Tmint maxPinp


mint χ(PinpWt) =






log Jn = maxPinpw


n))minus ǫ (57)

10



log Jnt = maxPinpwt


nt ))minus ǫ (58)

Thus


mintisinθ

maxPinpwt

1

n(χ(PinpB

nt )minus χ(PinpZ

nt ))

(59)





C(t)2 = (E(t) D(t)




D(t)j j t isin θ

to decode the





limnrarrinfin

maxPinpwtprime

1

n(χ(PinpB


ntprime))



mintisinθ

maxPinpwt

1

n(χ(PinpB

nt )minus χ(PinpZ

nt ))




nt )


limnrarrinfin

maxPinpwt

1

n(χ(PinpB

nt )minus χ(PinpZ

nt ))

= maxPinpwt




nt )


CCSI = mintisinθ

maxPinpwt











Pr[

tr(

Wnt (ρ

nXjl

)DtXjl

)


gt 1minus 2minusnβ


sum

tisinθ

Jnsum

j=1

Lnsum

l=1

Dtxjlle I

We define



n)ΠpVtαradica

and Θt =sum



11


Pr

(

Lnsum

l=1

1

LnQt(ρX(t)

jl


ge 1minus 2minusnυ

(62)


mintisinθ

tr(

Wnt (ρ

nxjl

)Dtxjl

)

ge 1minus 2minusnβ

and

maxtisinθ

Lnsum

l=1

1

LnQt(ρxjl

)minusΘt le ǫ

We define Dj =sum

tisinθ

sumLn

l=1Dtxjl then

sumJn

j=1Dj =sum

tisinθ

sumJn

j=1

sumLn



tr(

Wntprime (ρ

nxjlprime

)Dj

)

=sum

tisinθ

Lnsum

l=1

tr(

Wntprime (ρ

nxjlprime

)Dtxjl

)

ge tr(

Wntprime (ρ

otimesnxjlprime

)Dtprimexjlprime

)

ge 1minus 2minusnβ


(radicM1M2

radicM1

)

ge 0

For any γ gt 0 let


1

n

[

mintisinθ

χ(pBnt )minusmax

tisinθχ(pZn

t )

]

minus γ


w(j) =sumLn

l=11Lnρnxjl

j = 1 Jn Dj j =

1 Jn)


maxtisinθ

maxjisin1Jn

tr(


)

le λ


maxtisinθ



CS ge limnrarrinfin

maxUrarrArarr(BZ)t

1

n

(

mintisinθ

χ(U Bnt )minusmax

tisinθχ(U Zn

t )

)

(65)




maxtisinθ

maxjisin1Jn

tr(


t

(

ρ(n)j

))

le λn (66)

maxtisinθ








1

nχ(XuniB

ntprime) + ξ

= mintisinθ

1

nχ(XuniB

nt ) + ξ

le mintisinθ

1

nχ(XuniB

nt )minusmax

tisinθ

1

nχ(XuniZ

nt ) + ξ +

1

nǫ2n

le 1

n

(

mintisinθ


tisinθχ(XuniZ

nt )

)

+ ǫ (68)

Thus

CS le limnrarrinfin

maxUrarrArarr(BZ)t

1

n

(

mintisinθ

χ(U Bnt )minusmax

tisinθχ(U Zn

t )

)





inftisinθ

maxPinpwt

1

n(χ(PinpB

nt )minus χ(PinpZ

nt ))

(70)


Wdiams = supnisinN



12



W (k))K





W (k))K

k=1in the










1

JN

JNsum

j=1

tr(


)





the message




tprime

)

(wtprime(j))]

=

tr(

trHprimeotimesN

[



])

We have

tr∣

∣

(


tprime

)

(wtprime (j))∣

∣

= tr(

trHprimeotimesN

∣



∣

∣

)

= tr∣



∣

∣

=∥



∥

∥

1






tprime

∥

∥

diams=

∥

∥


∥

∥


tiplicative

It follows that

∣

∣

∣

∣

1

JN

JNsum

j=1

tr(


)

minus 1

JN

JNsum

j=1

tr(


)

∣

∣

∣

∣

le 1

JN

JNsum

j=1

∣

∣tr[(


tprime

)


]∣

∣

le 1

JN

JNsum

j=1

tr∣

∣

(


tprime

)


∣

∣

le 1

JN

JNsum

j=1

tr∣

∣

(


tprime

)

(wtprime(j))∣

∣

le 1

JNJNNτ

= Nτ (74)




le

∣

∣

∣

∣

∣

∣

S

JNsum

j=1

1

JNVt(ρ(j))

minus S

JNsum

j=1

1

JNVtprime(ρ(j))

∣

∣

∣

∣

∣

∣

+

∣

∣

∣

∣

∣

∣

JNsum

j=1

1


JNsum

j=1

1


∣

∣

∣

∣

∣

∣




suptisinθ

1

JN

JNsum

j=1

tr(


)




suptisinθ

1

JN

JNsum

j=1

tr(


)

ge 1minus λ

13

suptisinθ

χ(XuniZNt ) le ǫ



inftisinθ

maxPinpwt

1

n(χ(PinpB

nt )minus χ(PinpZ

nt ))








)minus2dprime4









Notimesnt

)




sum




A ge maxp

(

mintisinθ


χ(pEt)

)

(77)



ρPUlowastNt



sum


trE

(

sum

x


Nt

)


trQ

(

sum

x


Nt

)




sum



Nt

)

and VNt HP rarr HQ

sum



Nt

)



and


14


VNt)tisinθ since



xjl




n

t on HEn

such that

tr(

(DQn

txjlotimes IE

n

)UNtρP

n

xjlUlowastNt

)


and

ωEn

jt minus ξEn

t 1 lt ǫ (79)

where ωEn

jt = 1Lnt

sumLnt

l=1 trQn

(

UNtρP

n

xjlUlowastNt

)


xjlmay be pure or

mixed Assume ρPn


sumni=1 p

primejl(i)|κ

(i)xjl〉〈κ(i)

xjl |Pn

then

nsum

i=1

pprimejl(i)tr(

(DQn

txjlotimes IE

n

)UNt|κ(i)

xjl〉〈κ(i)

xjl|Pn

UlowastNt

)

tr

(

(DQn

txjlotimes IE

n

)UNt(

nsum

i=1


〉〈κ(i)xjl

|Pn

)UlowastNt

)

ge 1minus 2minusnβ



tr(

(DQn

txjlotimes IE

n

)UNt|κ(i)

xjl〉〈κ(i)

xjl|Pn

UlowastNt

)

ge 1minus 2minusnβ



there is a ξEn

t on HEn

such that

1

Lnt

Lntsum

l=1

trQn

(

UNt|κ(ilj)


UlowastNt

)

minus ξEn

t 1 lt ǫ

Thus(


xjl |Pn

j l DQn

txjl j l t

)


ρPn

xjl j l DQn

txjl j l t

)



Assume ρPn


n



n isin S(HQn

) we have

V QnMLθ

(


)

(V QnMLθ)lowast

=sum

t

sum

j

sum

l

(

DQn

txjlρQ

n)


We denote

ψQnEnMLθjlt

=(

IEn otimes V QnMLθ

)

(

UN otimes IMLθ)

[

|κjl〉〈κjl|Pn


(


(

IEn otimes V QnMLθ

)lowast


F(

trQnEn

(

ψQnEnMLθjlt

)


ge 1minus ǫ (80)


nEn

on HQnEn

such that

〈0|θ〈0|L〈0|M〈κjl|Pn (

UNtotimes IMLθ

)lowast

(

IEn otimes V QnMLθ

)lowast|ζjlt〉Q


= F

(


nEn


ge 1minus ǫ (81)



n |0〉M|0〉L|0〉θ

and

|bjlt〉PnMLθ =

(

UNtotimes IMLθ

)lowast (IE




We define

|ajk〉PnMLθ =

1radicLn

Lnsum

l=1


nMLθ

15

|bjkt〉PnMLθ =

1radicLn

Lnsum

l=1


nMLθ

and

|bjk〉PnMLθ =

1

|θ|

|θ|sum

t=1

|bjkt〉PnMLθ


1

Ln

Lnsum

k=1

〈ajk|bjk〉PnMLθ

=1

|θ|1

Ln

|θ|sum

t=1

Lnsum

k=1


=1

|θ|1

Ln

|θ|sum

t=1

Lnsum

l=1


ge 1minus ǫ (82)


1minus ǫ


|bjkj〉PnMLθ

=1

|θ|

|θ|sum

t=1

eminusiskj 〈ajkj

|bjkj t〉PnMLθ


∣

∣


|bjkj t〉PnMLθ

∣

∣

∣ le 1 we have

mintisinθ

∣

∣


|bjkj t〉PnMLθ

∣

∣



t isin θ

1minus |θ|ǫ

le∣

∣


|bjkj t〉PnMLθ

∣

∣

∣


|bjkj t〉PnMLθ

= eminusirkj

1

Ln

(

Lnsum

l=1


nMLθ

)

(

Lnsum

l=1


nMLθ

)

(83)


|jt〉QnEnL =

radic

1

Ln

Lnsum

l=1

eminus2πi(l

kjLn


and

|ϑjt〉QnEnMLθ =

radic

1

Ln

Lnsum

l=1

eminus2πilkj

Ln

[

IEn otimes V QnMLθ

]

(UnN |κjl〉P

n

)|0〉M|0〉L|0〉θ


F

(



)

=∣

∣



∣

∣

=1

Ln

(

Lnsum

l=1


nMLθ

)

(

Lnsum

l=1

eminus2πilkjLn e


)



n


such that

∥

∥

∥ξEn

t minus trQn

(


n

UlowastNt

)∥

∥

∥

1le ǫ (85)


∥

∥trQn

(


n

UlowastNt

)

minus trQn

(


∥

∥

1

le 2

[

1minus F

(

trQn

(


n

UlowastNt

)

trQn

(


)]12

le 2

[

1minus F

(


nEn


12

le 2radicǫ (86)


2minus η1 leradic


∥

∥

∥trQnL

(

|jt〉〈jt|QnEnL

)

minus ξEn

t

∥

∥

∥

1

=

∥

∥

∥

∥

∥

1

Ln

Lnsum

l=1

trQn

(


minus ξEn

t

∥

∥

∥

∥

∥

1

le 1

Ln

Lnsum

l=1

∥

∥

∥

∥

trQn

(


n

UlowastNt

)

minus trQn

(


∥

∥

∥

∥

1

+∥

∥

∥ξEn

t minus trQn

(


n

UlowastNt

)∥

∥

∥

1


16



such that if we set

χQnEnMLjjprimet =

(

UQnML(t) otimes IE

n)

(


)(

UQnML(t) otimes IE

n)lowast

then

F(


jjprimet

)


(88)where |ξt〉Q


rification of ξEn

t on HQnEnL



1

Jn

1

Ln

Jnsum

j=1

Lnsum

l=1


n |j〉A

Jnsum

j=1

Lnsum

l=1


n

(89)



1

Jn

1

Ln

(

IA otimes UnNtprime

)

[

Jnsum

j=1

Lnsum

l=1

eminus2πilkj

Ln |j〉A|κjl〉Pn

Jnsum

j=1

Lnsum

l=1

eminus2πilkj

Ln 〈κjl|Pn〈j|A

]

(

IA otimes UnNtprime

)lowast

=1

Jn

1

Ln

Jnsum

j=1

|j〉A(

Lnsum

l=1

eminus2πilkjLn Un

Ntprime|κjl〉P

n

)

Jnsum

j=1

(

Lnsum

l=1


n

(UnNtprime

)lowast)

〈j|A


τQn rarr trQnLθ

[

(

sum

tisinθ


)

V QnMLθ

(


)

V QnMLθlowast(

sum

tisinθ


)lowast]

(90)

to his outcome


We define

ιAQnEnMLθtprime

=

(

sum

tisinθ


)


)

(

1

Jn

1

Ln

Jnsum

j=1

|j〉A(

Lnsum

l=1

eminus2πilkjLn Un

Ntprime|κjl〉P

n

)

Jnsum

j=1

(

Lnsum

l=1


n

(UnNtprime

)lowast)

〈j|A



)lowast

(

sum

tisinθ


)lowast

=

(

sum

tisinθ


)

1

Jn(

Jnsum

j=1


Jnsum

j=1


(

sum

tisinθ


)lowast

(91)


AQnEnMLθtprime )


sumJn

j=1

sumJn



(

sum

tisinθ


)

(

sum

tisinθ

UQnML(t) otimes IE


= IAEn otimessum

tisinθ

UQnML(t) (UQnML


= IAQnEnMLθ

sum




F

ιAQnEnMLθtprime

1

Jn

Jnsum

j=1

Jnsum

jprime=1


= F

(

1

Jn(

Jnsum

j=1


Jnsum

j=1


1

Jn(

Jnsum

j=1


17

(

Jnsum

j=1


)

=1

Jn

∣

∣

∣

∣

( Jnsum

j=1


)

( Jnsum

j=1


)∣

∣

∣

∣



sumJn

j=1

sumJn



F

(

1

Jn

Jnsum

j=1

Jnsum

jprime=1


1

Jn(

Jnsum

j=1


(

Jnsum

j=1


)



1minus F ( η) le 1

2minus η1 le

radic

1minus F ( η)2


F ( η)

ge 1minus 1

2minus η1

ge 1minus 1

2minus υ1 minus

1

2υ minus η1

ge 1minusradic


1minus F (υ η)2


F

(


(

Jnsum

j=1


Jnsum

j=1


)

ge F

(

ιAQnEnMLθtprime

1

Jn(

Jnsum

j=1


(

Jnsum

j=1


)

ge 1minusradic



radicǫminus 8ǫ

(94)

ge 1minusradic

2|θ|radicǫminus



radic


with rate


tχ(X Et)minus 2δ



1

nmintisinθ

maxρisinS(H)Qn



limnrarrinfin

1

nmax

ρisinS(H)QnIC(ρN

otimesnt )




1

nmintisinθ

maxρisinS(H)Qn




18

VII FURTHER NOTES




N(ρ) =

Ksum

i=1

AiρAilowast (98)





we have Ai =sumK

j=1 uijBj (cf [31])







HE Then UN =sumK




)

ρ(


)lowast=

sumKj=1AjρAj

lowast and(


)lowast (sumK


=sumK

j=1 AjlowastAj


UN =sumK



lowast Here A1 AK




1

nmax

ρisinS(H)QnIC(ρN

otimesn) (100)




1

nmax




19


ACKNOWLEDGMENT





























20


















21

and Θt =sum


n)


Pr

(

Lnsum

l=1

1

LnQt(X


)

ge 1minus 2minusnυ

(51)


Lnsum

l=1

1

LnW

nt (D

cj |xjl) le ǫ

Lnsum

l=1

1


For any γ gt 0 let


(

mintisinθ


)

minus γ


1Lnt


1n log Jn ge R and

limnrarrinfin

maxtisinθ

maxjisin1Jn

Lnsum

l=1

1

LnW

nt (D

cj |xjl)) = 0 (52)


maxtisinθ




(mintisinθ


χ(U Zt))


WIRETAP CHANNEL


Section II



mintisinθ

maxPinpwt

1

n(χ(PinpB

nt )minus χ(PinpZ

nt ))



CS = limnrarrinfin

maxUrarrArarr(BZ)t

1

n

(

mintisinθ

χ(U Bnt )

minusmaxtisinθ

χ(U Znt )

)

(55)







mintχ(PinpWt) (56)



log Tmint maxPinp


mint χ(PinpWt) =






log Jn = maxPinpw


n))minus ǫ (57)

10



log Jnt = maxPinpwt


nt ))minus ǫ (58)

Thus


mintisinθ

maxPinpwt

1

n(χ(PinpB

nt )minus χ(PinpZ

nt ))

(59)





C(t)2 = (E(t) D(t)




D(t)j j t isin θ

to decode the





limnrarrinfin

maxPinpwtprime

1

n(χ(PinpB


ntprime))



mintisinθ

maxPinpwt

1

n(χ(PinpB

nt )minus χ(PinpZ

nt ))




nt )


limnrarrinfin

maxPinpwt

1

n(χ(PinpB

nt )minus χ(PinpZ

nt ))

= maxPinpwt




nt )


CCSI = mintisinθ

maxPinpwt











Pr[

tr(

Wnt (ρ

nXjl

)DtXjl

)


gt 1minus 2minusnβ


sum

tisinθ

Jnsum

j=1

Lnsum

l=1

Dtxjlle I

We define



n)ΠpVtαradica

and Θt =sum



11


Pr

(

Lnsum

l=1

1

LnQt(ρX(t)

jl


ge 1minus 2minusnυ

(62)


mintisinθ

tr(

Wnt (ρ

nxjl

)Dtxjl

)

ge 1minus 2minusnβ

and

maxtisinθ

Lnsum

l=1

1

LnQt(ρxjl

)minusΘt le ǫ

We define Dj =sum

tisinθ

sumLn

l=1Dtxjl then

sumJn

j=1Dj =sum

tisinθ

sumJn

j=1

sumLn



tr(

Wntprime (ρ

nxjlprime

)Dj

)

=sum

tisinθ

Lnsum

l=1

tr(

Wntprime (ρ

nxjlprime

)Dtxjl

)

ge tr(

Wntprime (ρ

otimesnxjlprime

)Dtprimexjlprime

)

ge 1minus 2minusnβ


(radicM1M2

radicM1

)

ge 0

For any γ gt 0 let


1

n

[

mintisinθ

χ(pBnt )minusmax

tisinθχ(pZn

t )

]

minus γ


w(j) =sumLn

l=11Lnρnxjl

j = 1 Jn Dj j =

1 Jn)


maxtisinθ

maxjisin1Jn

tr(


)

le λ


maxtisinθ



CS ge limnrarrinfin

maxUrarrArarr(BZ)t

1

n

(

mintisinθ

χ(U Bnt )minusmax

tisinθχ(U Zn

t )

)

(65)




maxtisinθ

maxjisin1Jn

tr(


t

(

ρ(n)j

))

le λn (66)

maxtisinθ








1

nχ(XuniB

ntprime) + ξ

= mintisinθ

1

nχ(XuniB

nt ) + ξ

le mintisinθ

1

nχ(XuniB

nt )minusmax

tisinθ

1

nχ(XuniZ

nt ) + ξ +

1

nǫ2n

le 1

n

(

mintisinθ


tisinθχ(XuniZ

nt )

)

+ ǫ (68)

Thus

CS le limnrarrinfin

maxUrarrArarr(BZ)t

1

n

(

mintisinθ

χ(U Bnt )minusmax

tisinθχ(U Zn

t )

)





inftisinθ

maxPinpwt

1

n(χ(PinpB

nt )minus χ(PinpZ

nt ))

(70)


Wdiams = supnisinN



12



W (k))K





W (k))K

k=1in the










1

JN

JNsum

j=1

tr(


)





the message




tprime

)

(wtprime(j))]

=

tr(

trHprimeotimesN

[



])

We have

tr∣

∣

(


tprime

)

(wtprime (j))∣

∣

= tr(

trHprimeotimesN

∣



∣

∣

)

= tr∣



∣

∣

=∥



∥

∥

1






tprime

∥

∥

diams=

∥

∥


∥

∥


tiplicative

It follows that

∣

∣

∣

∣

1

JN

JNsum

j=1

tr(


)

minus 1

JN

JNsum

j=1

tr(


)

∣

∣

∣

∣

le 1

JN

JNsum

j=1

∣

∣tr[(


tprime

)


]∣

∣

le 1

JN

JNsum

j=1

tr∣

∣

(


tprime

)


∣

∣

le 1

JN

JNsum

j=1

tr∣

∣

(


tprime

)

(wtprime(j))∣

∣

le 1

JNJNNτ

= Nτ (74)




le

∣

∣

∣

∣

∣

∣

S

JNsum

j=1

1

JNVt(ρ(j))

minus S

JNsum

j=1

1

JNVtprime(ρ(j))

∣

∣

∣

∣

∣

∣

+

∣

∣

∣

∣

∣

∣

JNsum

j=1

1


JNsum

j=1

1


∣

∣

∣

∣

∣

∣




suptisinθ

1

JN

JNsum

j=1

tr(


)




suptisinθ

1

JN

JNsum

j=1

tr(


)

ge 1minus λ

13

suptisinθ

χ(XuniZNt ) le ǫ



inftisinθ

maxPinpwt

1

n(χ(PinpB

nt )minus χ(PinpZ

nt ))








)minus2dprime4









Notimesnt

)




sum




A ge maxp

(

mintisinθ


χ(pEt)

)

(77)



ρPUlowastNt



sum


trE

(

sum

x


Nt

)


trQ

(

sum

x


Nt

)




sum



Nt

)

and VNt HP rarr HQ

sum



Nt

)



and


14


VNt)tisinθ since



xjl




n

t on HEn

such that

tr(

(DQn

txjlotimes IE

n

)UNtρP

n

xjlUlowastNt

)


and

ωEn

jt minus ξEn

t 1 lt ǫ (79)

where ωEn

jt = 1Lnt

sumLnt

l=1 trQn

(

UNtρP

n

xjlUlowastNt

)


xjlmay be pure or

mixed Assume ρPn


sumni=1 p

primejl(i)|κ

(i)xjl〉〈κ(i)

xjl |Pn

then

nsum

i=1

pprimejl(i)tr(

(DQn

txjlotimes IE

n

)UNt|κ(i)

xjl〉〈κ(i)

xjl|Pn

UlowastNt

)

tr

(

(DQn

txjlotimes IE

n

)UNt(

nsum

i=1


〉〈κ(i)xjl

|Pn

)UlowastNt

)

ge 1minus 2minusnβ



tr(

(DQn

txjlotimes IE

n

)UNt|κ(i)

xjl〉〈κ(i)

xjl|Pn

UlowastNt

)

ge 1minus 2minusnβ



there is a ξEn

t on HEn

such that

1

Lnt

Lntsum

l=1

trQn

(

UNt|κ(ilj)


UlowastNt

)

minus ξEn

t 1 lt ǫ

Thus(


xjl |Pn

j l DQn

txjl j l t

)


ρPn

xjl j l DQn

txjl j l t

)



Assume ρPn


n



n isin S(HQn

) we have

V QnMLθ

(


)

(V QnMLθ)lowast

=sum

t

sum

j

sum

l

(

DQn

txjlρQ

n)


We denote

ψQnEnMLθjlt

=(

IEn otimes V QnMLθ

)

(

UN otimes IMLθ)

[

|κjl〉〈κjl|Pn


(


(

IEn otimes V QnMLθ

)lowast


F(

trQnEn

(

ψQnEnMLθjlt

)


ge 1minus ǫ (80)


nEn

on HQnEn

such that

〈0|θ〈0|L〈0|M〈κjl|Pn (

UNtotimes IMLθ

)lowast

(

IEn otimes V QnMLθ

)lowast|ζjlt〉Q


= F

(


nEn


ge 1minus ǫ (81)



n |0〉M|0〉L|0〉θ

and

|bjlt〉PnMLθ =

(

UNtotimes IMLθ

)lowast (IE




We define

|ajk〉PnMLθ =

1radicLn

Lnsum

l=1


nMLθ

15

|bjkt〉PnMLθ =

1radicLn

Lnsum

l=1


nMLθ

and

|bjk〉PnMLθ =

1

|θ|

|θ|sum

t=1

|bjkt〉PnMLθ


1

Ln

Lnsum

k=1

〈ajk|bjk〉PnMLθ

=1

|θ|1

Ln

|θ|sum

t=1

Lnsum

k=1


=1

|θ|1

Ln

|θ|sum

t=1

Lnsum

l=1


ge 1minus ǫ (82)


1minus ǫ


|bjkj〉PnMLθ

=1

|θ|

|θ|sum

t=1

eminusiskj 〈ajkj

|bjkj t〉PnMLθ


∣

∣


|bjkj t〉PnMLθ

∣

∣

∣ le 1 we have

mintisinθ

∣

∣


|bjkj t〉PnMLθ

∣

∣



t isin θ

1minus |θ|ǫ

le∣

∣


|bjkj t〉PnMLθ

∣

∣

∣


|bjkj t〉PnMLθ

= eminusirkj

1

Ln

(

Lnsum

l=1


nMLθ

)

(

Lnsum

l=1


nMLθ

)

(83)


|jt〉QnEnL =

radic

1

Ln

Lnsum

l=1

eminus2πi(l

kjLn


and

|ϑjt〉QnEnMLθ =

radic

1

Ln

Lnsum

l=1

eminus2πilkj

Ln

[

IEn otimes V QnMLθ

]

(UnN |κjl〉P

n

)|0〉M|0〉L|0〉θ


F

(



)

=∣

∣



∣

∣

=1

Ln

(

Lnsum

l=1


nMLθ

)

(

Lnsum

l=1

eminus2πilkjLn e


)



n


such that

∥

∥

∥ξEn

t minus trQn

(


n

UlowastNt

)∥

∥

∥

1le ǫ (85)


∥

∥trQn

(


n

UlowastNt

)

minus trQn

(


∥

∥

1

le 2

[

1minus F

(

trQn

(


n

UlowastNt

)

trQn

(


)]12

le 2

[

1minus F

(


nEn


12

le 2radicǫ (86)


2minus η1 leradic


∥

∥

∥trQnL

(

|jt〉〈jt|QnEnL

)

minus ξEn

t

∥

∥

∥

1

=

∥

∥

∥

∥

∥

1

Ln

Lnsum

l=1

trQn

(


minus ξEn

t

∥

∥

∥

∥

∥

1

le 1

Ln

Lnsum

l=1

∥

∥

∥

∥

trQn

(


n

UlowastNt

)

minus trQn

(


∥

∥

∥

∥

1

+∥

∥

∥ξEn

t minus trQn

(


n

UlowastNt

)∥

∥

∥

1


16



such that if we set

χQnEnMLjjprimet =

(

UQnML(t) otimes IE

n)

(


)(

UQnML(t) otimes IE

n)lowast

then

F(


jjprimet

)


(88)where |ξt〉Q


rification of ξEn

t on HQnEnL



1

Jn

1

Ln

Jnsum

j=1

Lnsum

l=1


n |j〉A

Jnsum

j=1

Lnsum

l=1


n

(89)



1

Jn

1

Ln

(

IA otimes UnNtprime

)

[

Jnsum

j=1

Lnsum

l=1

eminus2πilkj

Ln |j〉A|κjl〉Pn

Jnsum

j=1

Lnsum

l=1

eminus2πilkj

Ln 〈κjl|Pn〈j|A

]

(

IA otimes UnNtprime

)lowast

=1

Jn

1

Ln

Jnsum

j=1

|j〉A(

Lnsum

l=1

eminus2πilkjLn Un

Ntprime|κjl〉P

n

)

Jnsum

j=1

(

Lnsum

l=1


n

(UnNtprime

)lowast)

〈j|A


τQn rarr trQnLθ

[

(

sum

tisinθ


)

V QnMLθ

(


)

V QnMLθlowast(

sum

tisinθ


)lowast]

(90)

to his outcome


We define

ιAQnEnMLθtprime

=

(

sum

tisinθ


)


)

(

1

Jn

1

Ln

Jnsum

j=1

|j〉A(

Lnsum

l=1

eminus2πilkjLn Un

Ntprime|κjl〉P

n

)

Jnsum

j=1

(

Lnsum

l=1


n

(UnNtprime

)lowast)

〈j|A



)lowast

(

sum

tisinθ


)lowast

=

(

sum

tisinθ


)

1

Jn(

Jnsum

j=1


Jnsum

j=1


(

sum

tisinθ


)lowast

(91)


AQnEnMLθtprime )


sumJn

j=1

sumJn



(

sum

tisinθ


)

(

sum

tisinθ

UQnML(t) otimes IE


= IAEn otimessum

tisinθ

UQnML(t) (UQnML


= IAQnEnMLθ

sum




F

ιAQnEnMLθtprime

1

Jn

Jnsum

j=1

Jnsum

jprime=1


= F

(

1

Jn(

Jnsum

j=1


Jnsum

j=1


1

Jn(

Jnsum

j=1


17

(

Jnsum

j=1


)

=1

Jn

∣

∣

∣

∣

( Jnsum

j=1


)

( Jnsum

j=1


)∣

∣

∣

∣



sumJn

j=1

sumJn



F

(

1

Jn

Jnsum

j=1

Jnsum

jprime=1


1

Jn(

Jnsum

j=1


(

Jnsum

j=1


)



1minus F ( η) le 1

2minus η1 le

radic

1minus F ( η)2


F ( η)

ge 1minus 1

2minus η1

ge 1minus 1

2minus υ1 minus

1

2υ minus η1

ge 1minusradic


1minus F (υ η)2


F

(


(

Jnsum

j=1


Jnsum

j=1


)

ge F

(

ιAQnEnMLθtprime

1

Jn(

Jnsum

j=1


(

Jnsum

j=1


)

ge 1minusradic



radicǫminus 8ǫ

(94)

ge 1minusradic

2|θ|radicǫminus



radic


with rate


tχ(X Et)minus 2δ



1

nmintisinθ

maxρisinS(H)Qn



limnrarrinfin

1

nmax

ρisinS(H)QnIC(ρN

otimesnt )




1

nmintisinθ

maxρisinS(H)Qn




18

VII FURTHER NOTES




N(ρ) =

Ksum

i=1

AiρAilowast (98)





we have Ai =sumK

j=1 uijBj (cf [31])







HE Then UN =sumK




)

ρ(


)lowast=

sumKj=1AjρAj

lowast and(


)lowast (sumK


=sumK

j=1 AjlowastAj


UN =sumK



lowast Here A1 AK




1

nmax

ρisinS(H)QnIC(ρN

otimesn) (100)




1

nmax




19


ACKNOWLEDGMENT





























20


















21



log Jnt = maxPinpwt


nt ))minus ǫ (58)

Thus


mintisinθ

maxPinpwt

1

n(χ(PinpB

nt )minus χ(PinpZ

nt ))

(59)





C(t)2 = (E(t) D(t)




D(t)j j t isin θ

to decode the





limnrarrinfin

maxPinpwtprime

1

n(χ(PinpB


ntprime))



mintisinθ

maxPinpwt

1

n(χ(PinpB

nt )minus χ(PinpZ

nt ))




nt )


limnrarrinfin

maxPinpwt

1

n(χ(PinpB

nt )minus χ(PinpZ

nt ))

= maxPinpwt




nt )


CCSI = mintisinθ

maxPinpwt











Pr[

tr(

Wnt (ρ

nXjl

)DtXjl

)


gt 1minus 2minusnβ


sum

tisinθ

Jnsum

j=1

Lnsum

l=1

Dtxjlle I

We define



n)ΠpVtαradica

and Θt =sum



11


Pr

(

Lnsum

l=1

1

LnQt(ρX(t)

jl


ge 1minus 2minusnυ

(62)


mintisinθ

tr(

Wnt (ρ

nxjl

)Dtxjl

)

ge 1minus 2minusnβ

and

maxtisinθ

Lnsum

l=1

1

LnQt(ρxjl

)minusΘt le ǫ

We define Dj =sum

tisinθ

sumLn

l=1Dtxjl then

sumJn

j=1Dj =sum

tisinθ

sumJn

j=1

sumLn



tr(

Wntprime (ρ

nxjlprime

)Dj

)

=sum

tisinθ

Lnsum

l=1

tr(

Wntprime (ρ

nxjlprime

)Dtxjl

)

ge tr(

Wntprime (ρ

otimesnxjlprime

)Dtprimexjlprime

)

ge 1minus 2minusnβ


(radicM1M2

radicM1

)

ge 0

For any γ gt 0 let


1

n

[

mintisinθ

χ(pBnt )minusmax

tisinθχ(pZn

t )

]

minus γ


w(j) =sumLn

l=11Lnρnxjl

j = 1 Jn Dj j =

1 Jn)


maxtisinθ

maxjisin1Jn

tr(


)

le λ


maxtisinθ



CS ge limnrarrinfin

maxUrarrArarr(BZ)t

1

n

(

mintisinθ

χ(U Bnt )minusmax

tisinθχ(U Zn

t )

)

(65)




maxtisinθ

maxjisin1Jn

tr(


t

(

ρ(n)j

))

le λn (66)

maxtisinθ








1

nχ(XuniB

ntprime) + ξ

= mintisinθ

1

nχ(XuniB

nt ) + ξ

le mintisinθ

1

nχ(XuniB

nt )minusmax

tisinθ

1

nχ(XuniZ

nt ) + ξ +

1

nǫ2n

le 1

n

(

mintisinθ


tisinθχ(XuniZ

nt )

)

+ ǫ (68)

Thus

CS le limnrarrinfin

maxUrarrArarr(BZ)t

1

n

(

mintisinθ

χ(U Bnt )minusmax

tisinθχ(U Zn

t )

)





inftisinθ

maxPinpwt

1

n(χ(PinpB

nt )minus χ(PinpZ

nt ))

(70)


Wdiams = supnisinN



12



W (k))K





W (k))K

k=1in the










1

JN

JNsum

j=1

tr(


)





the message




tprime

)

(wtprime(j))]

=

tr(

trHprimeotimesN

[



])

We have

tr∣

∣

(


tprime

)

(wtprime (j))∣

∣

= tr(

trHprimeotimesN

∣



∣

∣

)

= tr∣



∣

∣

=∥



∥

∥

1






tprime

∥

∥

diams=

∥

∥


∥

∥


tiplicative

It follows that

∣

∣

∣

∣

1

JN

JNsum

j=1

tr(


)

minus 1

JN

JNsum

j=1

tr(


)

∣

∣

∣

∣

le 1

JN

JNsum

j=1

∣

∣tr[(


tprime

)


]∣

∣

le 1

JN

JNsum

j=1

tr∣

∣

(


tprime

)


∣

∣

le 1

JN

JNsum

j=1

tr∣

∣

(


tprime

)

(wtprime(j))∣

∣

le 1

JNJNNτ

= Nτ (74)




le

∣

∣

∣

∣

∣

∣

S

JNsum

j=1

1

JNVt(ρ(j))

minus S

JNsum

j=1

1

JNVtprime(ρ(j))

∣

∣

∣

∣

∣

∣

+

∣

∣

∣

∣

∣

∣

JNsum

j=1

1


JNsum

j=1

1


∣

∣

∣

∣

∣

∣




suptisinθ

1

JN

JNsum

j=1

tr(


)




suptisinθ

1

JN

JNsum

j=1

tr(


)

ge 1minus λ

13

suptisinθ

χ(XuniZNt ) le ǫ



inftisinθ

maxPinpwt

1

n(χ(PinpB

nt )minus χ(PinpZ

nt ))








)minus2dprime4









Notimesnt

)




sum




A ge maxp

(

mintisinθ


χ(pEt)

)

(77)



ρPUlowastNt



sum


trE

(

sum

x


Nt

)


trQ

(

sum

x


Nt

)




sum



Nt

)

and VNt HP rarr HQ

sum



Nt

)



and


14


VNt)tisinθ since



xjl




n

t on HEn

such that

tr(

(DQn

txjlotimes IE

n

)UNtρP

n

xjlUlowastNt

)


and

ωEn

jt minus ξEn

t 1 lt ǫ (79)

where ωEn

jt = 1Lnt

sumLnt

l=1 trQn

(

UNtρP

n

xjlUlowastNt

)


xjlmay be pure or

mixed Assume ρPn


sumni=1 p

primejl(i)|κ

(i)xjl〉〈κ(i)

xjl |Pn

then

nsum

i=1

pprimejl(i)tr(

(DQn

txjlotimes IE

n

)UNt|κ(i)

xjl〉〈κ(i)

xjl|Pn

UlowastNt

)

tr

(

(DQn

txjlotimes IE

n

)UNt(

nsum

i=1


〉〈κ(i)xjl

|Pn

)UlowastNt

)

ge 1minus 2minusnβ



tr(

(DQn

txjlotimes IE

n

)UNt|κ(i)

xjl〉〈κ(i)

xjl|Pn

UlowastNt

)

ge 1minus 2minusnβ



there is a ξEn

t on HEn

such that

1

Lnt

Lntsum

l=1

trQn

(

UNt|κ(ilj)


UlowastNt

)

minus ξEn

t 1 lt ǫ

Thus(


xjl |Pn

j l DQn

txjl j l t

)


ρPn

xjl j l DQn

txjl j l t

)



Assume ρPn


n



n isin S(HQn

) we have

V QnMLθ

(


)

(V QnMLθ)lowast

=sum

t

sum

j

sum

l

(

DQn

txjlρQ

n)


We denote

ψQnEnMLθjlt

=(

IEn otimes V QnMLθ

)

(

UN otimes IMLθ)

[

|κjl〉〈κjl|Pn


(


(

IEn otimes V QnMLθ

)lowast


F(

trQnEn

(

ψQnEnMLθjlt

)


ge 1minus ǫ (80)


nEn

on HQnEn

such that

〈0|θ〈0|L〈0|M〈κjl|Pn (

UNtotimes IMLθ

)lowast

(

IEn otimes V QnMLθ

)lowast|ζjlt〉Q


= F

(


nEn


ge 1minus ǫ (81)



n |0〉M|0〉L|0〉θ

and

|bjlt〉PnMLθ =

(

UNtotimes IMLθ

)lowast (IE




We define

|ajk〉PnMLθ =

1radicLn

Lnsum

l=1


nMLθ

15

|bjkt〉PnMLθ =

1radicLn

Lnsum

l=1


nMLθ

and

|bjk〉PnMLθ =

1

|θ|

|θ|sum

t=1

|bjkt〉PnMLθ


1

Ln

Lnsum

k=1

〈ajk|bjk〉PnMLθ

=1

|θ|1

Ln

|θ|sum

t=1

Lnsum

k=1


=1

|θ|1

Ln

|θ|sum

t=1

Lnsum

l=1


ge 1minus ǫ (82)


1minus ǫ


|bjkj〉PnMLθ

=1

|θ|

|θ|sum

t=1

eminusiskj 〈ajkj

|bjkj t〉PnMLθ


∣

∣


|bjkj t〉PnMLθ

∣

∣

∣ le 1 we have

mintisinθ

∣

∣


|bjkj t〉PnMLθ

∣

∣



t isin θ

1minus |θ|ǫ

le∣

∣


|bjkj t〉PnMLθ

∣

∣

∣


|bjkj t〉PnMLθ

= eminusirkj

1

Ln

(

Lnsum

l=1


nMLθ

)

(

Lnsum

l=1


nMLθ

)

(83)


|jt〉QnEnL =

radic

1

Ln

Lnsum

l=1

eminus2πi(l

kjLn


and

|ϑjt〉QnEnMLθ =

radic

1

Ln

Lnsum

l=1

eminus2πilkj

Ln

[

IEn otimes V QnMLθ

]

(UnN |κjl〉P

n

)|0〉M|0〉L|0〉θ


F

(



)

=∣

∣



∣

∣

=1

Ln

(

Lnsum

l=1


nMLθ

)

(

Lnsum

l=1

eminus2πilkjLn e


)



n


such that

∥

∥

∥ξEn

t minus trQn

(


n

UlowastNt

)∥

∥

∥

1le ǫ (85)


∥

∥trQn

(


n

UlowastNt

)

minus trQn

(


∥

∥

1

le 2

[

1minus F

(

trQn

(


n

UlowastNt

)

trQn

(


)]12

le 2

[

1minus F

(


nEn


12

le 2radicǫ (86)


2minus η1 leradic


∥

∥

∥trQnL

(

|jt〉〈jt|QnEnL

)

minus ξEn

t

∥

∥

∥

1

=

∥

∥

∥

∥

∥

1

Ln

Lnsum

l=1

trQn

(


minus ξEn

t

∥

∥

∥

∥

∥

1

le 1

Ln

Lnsum

l=1

∥

∥

∥

∥

trQn

(


n

UlowastNt

)

minus trQn

(


∥

∥

∥

∥

1

+∥

∥

∥ξEn

t minus trQn

(


n

UlowastNt

)∥

∥

∥

1


16



such that if we set

χQnEnMLjjprimet =

(

UQnML(t) otimes IE

n)

(


)(

UQnML(t) otimes IE

n)lowast

then

F(


jjprimet

)


(88)where |ξt〉Q


rification of ξEn

t on HQnEnL



1

Jn

1

Ln

Jnsum

j=1

Lnsum

l=1


n |j〉A

Jnsum

j=1

Lnsum

l=1


n

(89)



1

Jn

1

Ln

(

IA otimes UnNtprime

)

[

Jnsum

j=1

Lnsum

l=1

eminus2πilkj

Ln |j〉A|κjl〉Pn

Jnsum

j=1

Lnsum

l=1

eminus2πilkj

Ln 〈κjl|Pn〈j|A

]

(

IA otimes UnNtprime

)lowast

=1

Jn

1

Ln

Jnsum

j=1

|j〉A(

Lnsum

l=1

eminus2πilkjLn Un

Ntprime|κjl〉P

n

)

Jnsum

j=1

(

Lnsum

l=1


n

(UnNtprime

)lowast)

〈j|A


τQn rarr trQnLθ

[

(

sum

tisinθ


)

V QnMLθ

(


)

V QnMLθlowast(

sum

tisinθ


)lowast]

(90)

to his outcome


We define

ιAQnEnMLθtprime

=

(

sum

tisinθ


)


)

(

1

Jn

1

Ln

Jnsum

j=1

|j〉A(

Lnsum

l=1

eminus2πilkjLn Un

Ntprime|κjl〉P

n

)

Jnsum

j=1

(

Lnsum

l=1


n

(UnNtprime

)lowast)

〈j|A



)lowast

(

sum

tisinθ


)lowast

=

(

sum

tisinθ


)

1

Jn(

Jnsum

j=1


Jnsum

j=1


(

sum

tisinθ


)lowast

(91)


AQnEnMLθtprime )


sumJn

j=1

sumJn



(

sum

tisinθ


)

(

sum

tisinθ

UQnML(t) otimes IE


= IAEn otimessum

tisinθ

UQnML(t) (UQnML


= IAQnEnMLθ

sum




F

ιAQnEnMLθtprime

1

Jn

Jnsum

j=1

Jnsum

jprime=1


= F

(

1

Jn(

Jnsum

j=1


Jnsum

j=1


1

Jn(

Jnsum

j=1


17

(

Jnsum

j=1


)

=1

Jn

∣

∣

∣

∣

( Jnsum

j=1


)

( Jnsum

j=1


)∣

∣

∣

∣



sumJn

j=1

sumJn



F

(

1

Jn

Jnsum

j=1

Jnsum

jprime=1


1

Jn(

Jnsum

j=1


(

Jnsum

j=1


)



1minus F ( η) le 1

2minus η1 le

radic

1minus F ( η)2


F ( η)

ge 1minus 1

2minus η1

ge 1minus 1

2minus υ1 minus

1

2υ minus η1

ge 1minusradic


1minus F (υ η)2


F

(


(

Jnsum

j=1


Jnsum

j=1


)

ge F

(

ιAQnEnMLθtprime

1

Jn(

Jnsum

j=1


(

Jnsum

j=1


)

ge 1minusradic



radicǫminus 8ǫ

(94)

ge 1minusradic

2|θ|radicǫminus



radic


with rate


tχ(X Et)minus 2δ



1

nmintisinθ

maxρisinS(H)Qn



limnrarrinfin

1

nmax

ρisinS(H)QnIC(ρN

otimesnt )




1

nmintisinθ

maxρisinS(H)Qn




18

VII FURTHER NOTES




N(ρ) =

Ksum

i=1

AiρAilowast (98)





we have Ai =sumK

j=1 uijBj (cf [31])







HE Then UN =sumK




)

ρ(


)lowast=

sumKj=1AjρAj

lowast and(


)lowast (sumK


=sumK

j=1 AjlowastAj


UN =sumK



lowast Here A1 AK




1

nmax

ρisinS(H)QnIC(ρN

otimesn) (100)




1

nmax




19


ACKNOWLEDGMENT





























20


















21


Pr

(

Lnsum

l=1

1

LnQt(ρX(t)

jl


ge 1minus 2minusnυ

(62)


mintisinθ

tr(

Wnt (ρ

nxjl

)Dtxjl

)

ge 1minus 2minusnβ

and

maxtisinθ

Lnsum

l=1

1

LnQt(ρxjl

)minusΘt le ǫ

We define Dj =sum

tisinθ

sumLn

l=1Dtxjl then

sumJn

j=1Dj =sum

tisinθ

sumJn

j=1

sumLn



tr(

Wntprime (ρ

nxjlprime

)Dj

)

=sum

tisinθ

Lnsum

l=1

tr(

Wntprime (ρ

nxjlprime

)Dtxjl

)

ge tr(

Wntprime (ρ

otimesnxjlprime

)Dtprimexjlprime

)

ge 1minus 2minusnβ


(radicM1M2

radicM1

)

ge 0

For any γ gt 0 let


1

n

[

mintisinθ

χ(pBnt )minusmax

tisinθχ(pZn

t )

]

minus γ


w(j) =sumLn

l=11Lnρnxjl

j = 1 Jn Dj j =

1 Jn)


maxtisinθ

maxjisin1Jn

tr(


)

le λ


maxtisinθ



CS ge limnrarrinfin

maxUrarrArarr(BZ)t

1

n

(

mintisinθ

χ(U Bnt )minusmax

tisinθχ(U Zn

t )

)

(65)




maxtisinθ

maxjisin1Jn

tr(


t

(

ρ(n)j

))

le λn (66)

maxtisinθ








1

nχ(XuniB

ntprime) + ξ

= mintisinθ

1

nχ(XuniB

nt ) + ξ

le mintisinθ

1

nχ(XuniB

nt )minusmax

tisinθ

1

nχ(XuniZ

nt ) + ξ +

1

nǫ2n

le 1

n

(

mintisinθ


tisinθχ(XuniZ

nt )

)

+ ǫ (68)

Thus

CS le limnrarrinfin

maxUrarrArarr(BZ)t

1

n

(

mintisinθ

χ(U Bnt )minusmax

tisinθχ(U Zn

t )

)





inftisinθ

maxPinpwt

1

n(χ(PinpB

nt )minus χ(PinpZ

nt ))

(70)


Wdiams = supnisinN



12



W (k))K





W (k))K

k=1in the










1

JN

JNsum

j=1

tr(


)





the message




tprime

)

(wtprime(j))]

=

tr(

trHprimeotimesN

[



])

We have

tr∣

∣

(


tprime

)

(wtprime (j))∣

∣

= tr(

trHprimeotimesN

∣



∣

∣

)

= tr∣



∣

∣

=∥



∥

∥

1






tprime

∥

∥

diams=

∥

∥


∥

∥


tiplicative

It follows that

∣

∣

∣

∣

1

JN

JNsum

j=1

tr(


)

minus 1

JN

JNsum

j=1

tr(


)

∣

∣

∣

∣

le 1

JN

JNsum

j=1

∣

∣tr[(


tprime

)


]∣

∣

le 1

JN

JNsum

j=1

tr∣

∣

(


tprime

)


∣

∣

le 1

JN

JNsum

j=1

tr∣

∣

(


tprime

)

(wtprime(j))∣

∣

le 1

JNJNNτ

= Nτ (74)




le

∣

∣

∣

∣

∣

∣

S

JNsum

j=1

1

JNVt(ρ(j))

minus S

JNsum

j=1

1

JNVtprime(ρ(j))

∣

∣

∣

∣

∣

∣

+

∣

∣

∣

∣

∣

∣

JNsum

j=1

1


JNsum

j=1

1


∣

∣

∣

∣

∣

∣




suptisinθ

1

JN

JNsum

j=1

tr(


)




suptisinθ

1

JN

JNsum

j=1

tr(


)

ge 1minus λ

13

suptisinθ

χ(XuniZNt ) le ǫ



inftisinθ

maxPinpwt

1

n(χ(PinpB

nt )minus χ(PinpZ

nt ))








)minus2dprime4









Notimesnt

)




sum




A ge maxp

(

mintisinθ


χ(pEt)

)

(77)



ρPUlowastNt



sum


trE

(

sum

x


Nt

)


trQ

(

sum

x


Nt

)




sum



Nt

)

and VNt HP rarr HQ

sum



Nt

)



and


14


VNt)tisinθ since



xjl




n

t on HEn

such that

tr(

(DQn

txjlotimes IE

n

)UNtρP

n

xjlUlowastNt

)


and

ωEn

jt minus ξEn

t 1 lt ǫ (79)

where ωEn

jt = 1Lnt

sumLnt

l=1 trQn

(

UNtρP

n

xjlUlowastNt

)


xjlmay be pure or

mixed Assume ρPn


sumni=1 p

primejl(i)|κ

(i)xjl〉〈κ(i)

xjl |Pn

then

nsum

i=1

pprimejl(i)tr(

(DQn

txjlotimes IE

n

)UNt|κ(i)

xjl〉〈κ(i)

xjl|Pn

UlowastNt

)

tr

(

(DQn

txjlotimes IE

n

)UNt(

nsum

i=1


〉〈κ(i)xjl

|Pn

)UlowastNt

)

ge 1minus 2minusnβ



tr(

(DQn

txjlotimes IE

n

)UNt|κ(i)

xjl〉〈κ(i)

xjl|Pn

UlowastNt

)

ge 1minus 2minusnβ



there is a ξEn

t on HEn

such that

1

Lnt

Lntsum

l=1

trQn

(

UNt|κ(ilj)


UlowastNt

)

minus ξEn

t 1 lt ǫ

Thus(


xjl |Pn

j l DQn

txjl j l t

)


ρPn

xjl j l DQn

txjl j l t

)



Assume ρPn


n



n isin S(HQn

) we have

V QnMLθ

(


)

(V QnMLθ)lowast

=sum

t

sum

j

sum

l

(

DQn

txjlρQ

n)


We denote

ψQnEnMLθjlt

=(

IEn otimes V QnMLθ

)

(

UN otimes IMLθ)

[

|κjl〉〈κjl|Pn


(


(

IEn otimes V QnMLθ

)lowast


F(

trQnEn

(

ψQnEnMLθjlt

)


ge 1minus ǫ (80)


nEn

on HQnEn

such that

〈0|θ〈0|L〈0|M〈κjl|Pn (

UNtotimes IMLθ

)lowast

(

IEn otimes V QnMLθ

)lowast|ζjlt〉Q


= F

(


nEn


ge 1minus ǫ (81)



n |0〉M|0〉L|0〉θ

and

|bjlt〉PnMLθ =

(

UNtotimes IMLθ

)lowast (IE




We define

|ajk〉PnMLθ =

1radicLn

Lnsum

l=1


nMLθ

15

|bjkt〉PnMLθ =

1radicLn

Lnsum

l=1


nMLθ

and

|bjk〉PnMLθ =

1

|θ|

|θ|sum

t=1

|bjkt〉PnMLθ


1

Ln

Lnsum

k=1

〈ajk|bjk〉PnMLθ

=1

|θ|1

Ln

|θ|sum

t=1

Lnsum

k=1


=1

|θ|1

Ln

|θ|sum

t=1

Lnsum

l=1


ge 1minus ǫ (82)


1minus ǫ


|bjkj〉PnMLθ

=1

|θ|

|θ|sum

t=1

eminusiskj 〈ajkj

|bjkj t〉PnMLθ


∣

∣


|bjkj t〉PnMLθ

∣

∣

∣ le 1 we have

mintisinθ

∣

∣


|bjkj t〉PnMLθ

∣

∣



t isin θ

1minus |θ|ǫ

le∣

∣


|bjkj t〉PnMLθ

∣

∣

∣


|bjkj t〉PnMLθ

= eminusirkj

1

Ln

(

Lnsum

l=1


nMLθ

)

(

Lnsum

l=1


nMLθ

)

(83)


|jt〉QnEnL =

radic

1

Ln

Lnsum

l=1

eminus2πi(l

kjLn


and

|ϑjt〉QnEnMLθ =

radic

1

Ln

Lnsum

l=1

eminus2πilkj

Ln

[

IEn otimes V QnMLθ

]

(UnN |κjl〉P

n

)|0〉M|0〉L|0〉θ


F

(



)

=∣

∣



∣

∣

=1

Ln

(

Lnsum

l=1


nMLθ

)

(

Lnsum

l=1

eminus2πilkjLn e


)



n


such that

∥

∥

∥ξEn

t minus trQn

(


n

UlowastNt

)∥

∥

∥

1le ǫ (85)


∥

∥trQn

(


n

UlowastNt

)

minus trQn

(


∥

∥

1

le 2

[

1minus F

(

trQn

(


n

UlowastNt

)

trQn

(


)]12

le 2

[

1minus F

(


nEn


12

le 2radicǫ (86)


2minus η1 leradic


∥

∥

∥trQnL

(

|jt〉〈jt|QnEnL

)

minus ξEn

t

∥

∥

∥

1

=

∥

∥

∥

∥

∥

1

Ln

Lnsum

l=1

trQn

(


minus ξEn

t

∥

∥

∥

∥

∥

1

le 1

Ln

Lnsum

l=1

∥

∥

∥

∥

trQn

(


n

UlowastNt

)

minus trQn

(


∥

∥

∥

∥

1

+∥

∥

∥ξEn

t minus trQn

(


n

UlowastNt

)∥

∥

∥

1


16



such that if we set

χQnEnMLjjprimet =

(

UQnML(t) otimes IE

n)

(


)(

UQnML(t) otimes IE

n)lowast

then

F(


jjprimet

)


(88)where |ξt〉Q


rification of ξEn

t on HQnEnL



1

Jn

1

Ln

Jnsum

j=1

Lnsum

l=1


n |j〉A

Jnsum

j=1

Lnsum

l=1


n

(89)



1

Jn

1

Ln

(

IA otimes UnNtprime

)

[

Jnsum

j=1

Lnsum

l=1

eminus2πilkj

Ln |j〉A|κjl〉Pn

Jnsum

j=1

Lnsum

l=1

eminus2πilkj

Ln 〈κjl|Pn〈j|A

]

(

IA otimes UnNtprime

)lowast

=1

Jn

1

Ln

Jnsum

j=1

|j〉A(

Lnsum

l=1

eminus2πilkjLn Un

Ntprime|κjl〉P

n

)

Jnsum

j=1

(

Lnsum

l=1


n

(UnNtprime

)lowast)

〈j|A


τQn rarr trQnLθ

[

(

sum

tisinθ


)

V QnMLθ

(


)

V QnMLθlowast(

sum

tisinθ


)lowast]

(90)

to his outcome


We define

ιAQnEnMLθtprime

=

(

sum

tisinθ


)


)

(

1

Jn

1

Ln

Jnsum

j=1

|j〉A(

Lnsum

l=1

eminus2πilkjLn Un

Ntprime|κjl〉P

n

)

Jnsum

j=1

(

Lnsum

l=1


n

(UnNtprime

)lowast)

〈j|A



)lowast

(

sum

tisinθ


)lowast

=

(

sum

tisinθ


)

1

Jn(

Jnsum

j=1


Jnsum

j=1


(

sum

tisinθ


)lowast

(91)


AQnEnMLθtprime )


sumJn

j=1

sumJn



(

sum

tisinθ


)

(

sum

tisinθ

UQnML(t) otimes IE


= IAEn otimessum

tisinθ

UQnML(t) (UQnML


= IAQnEnMLθ

sum




F

ιAQnEnMLθtprime

1

Jn

Jnsum

j=1

Jnsum

jprime=1


= F

(

1

Jn(

Jnsum

j=1


Jnsum

j=1


1

Jn(

Jnsum

j=1


17

(

Jnsum

j=1


)

=1

Jn

∣

∣

∣

∣

( Jnsum

j=1


)

( Jnsum

j=1


)∣

∣

∣

∣



sumJn

j=1

sumJn



F

(

1

Jn

Jnsum

j=1

Jnsum

jprime=1


1

Jn(

Jnsum

j=1


(

Jnsum

j=1


)



1minus F ( η) le 1

2minus η1 le

radic

1minus F ( η)2


F ( η)

ge 1minus 1

2minus η1

ge 1minus 1

2minus υ1 minus

1

2υ minus η1

ge 1minusradic


1minus F (υ η)2


F

(


(

Jnsum

j=1


Jnsum

j=1


)

ge F

(

ιAQnEnMLθtprime

1

Jn(

Jnsum

j=1


(

Jnsum

j=1


)

ge 1minusradic



radicǫminus 8ǫ

(94)

ge 1minusradic

2|θ|radicǫminus



radic


with rate


tχ(X Et)minus 2δ



1

nmintisinθ

maxρisinS(H)Qn



limnrarrinfin

1

nmax

ρisinS(H)QnIC(ρN

otimesnt )




1

nmintisinθ

maxρisinS(H)Qn




18

VII FURTHER NOTES




N(ρ) =

Ksum

i=1

AiρAilowast (98)





we have Ai =sumK

j=1 uijBj (cf [31])







HE Then UN =sumK




)

ρ(


)lowast=

sumKj=1AjρAj

lowast and(


)lowast (sumK


=sumK

j=1 AjlowastAj


UN =sumK



lowast Here A1 AK




1

nmax

ρisinS(H)QnIC(ρN

otimesn) (100)




1

nmax




19


ACKNOWLEDGMENT





























20


















21



W (k))K





W (k))K

k=1in the










1

JN

JNsum

j=1

tr(


)





the message




tprime

)

(wtprime(j))]

=

tr(

trHprimeotimesN

[



])

We have

tr∣

∣

(


tprime

)

(wtprime (j))∣

∣

= tr(

trHprimeotimesN

∣



∣

∣

)

= tr∣



∣

∣

=∥



∥

∥

1






tprime

∥

∥

diams=

∥

∥


∥

∥


tiplicative

It follows that

∣

∣

∣

∣

1

JN

JNsum

j=1

tr(


)

minus 1

JN

JNsum

j=1

tr(


)

∣

∣

∣

∣

le 1

JN

JNsum

j=1

∣

∣tr[(


tprime

)


]∣

∣

le 1

JN

JNsum

j=1

tr∣

∣

(


tprime

)


∣

∣

le 1

JN

JNsum

j=1

tr∣

∣

(


tprime

)

(wtprime(j))∣

∣

le 1

JNJNNτ

= Nτ (74)




le

∣

∣

∣

∣

∣

∣

S

JNsum

j=1

1

JNVt(ρ(j))

minus S

JNsum

j=1

1

JNVtprime(ρ(j))

∣

∣

∣

∣

∣

∣

+

∣

∣

∣

∣

∣

∣

JNsum

j=1

1


JNsum

j=1

1


∣

∣

∣

∣

∣

∣




suptisinθ

1

JN

JNsum

j=1

tr(


)




suptisinθ

1

JN

JNsum

j=1

tr(


)

ge 1minus λ

13

suptisinθ

χ(XuniZNt ) le ǫ



inftisinθ

maxPinpwt

1

n(χ(PinpB

nt )minus χ(PinpZ

nt ))








)minus2dprime4









Notimesnt

)




sum




A ge maxp

(

mintisinθ


χ(pEt)

)

(77)



ρPUlowastNt



sum


trE

(

sum

x


Nt

)


trQ

(

sum

x


Nt

)




sum



Nt

)

and VNt HP rarr HQ

sum



Nt

)



and


14


VNt)tisinθ since



xjl




n

t on HEn

such that

tr(

(DQn

txjlotimes IE

n

)UNtρP

n

xjlUlowastNt

)


and

ωEn

jt minus ξEn

t 1 lt ǫ (79)

where ωEn

jt = 1Lnt

sumLnt

l=1 trQn

(

UNtρP

n

xjlUlowastNt

)


xjlmay be pure or

mixed Assume ρPn


sumni=1 p

primejl(i)|κ

(i)xjl〉〈κ(i)

xjl |Pn

then

nsum

i=1

pprimejl(i)tr(

(DQn

txjlotimes IE

n

)UNt|κ(i)

xjl〉〈κ(i)

xjl|Pn

UlowastNt

)

tr

(

(DQn

txjlotimes IE

n

)UNt(

nsum

i=1


〉〈κ(i)xjl

|Pn

)UlowastNt

)

ge 1minus 2minusnβ



tr(

(DQn

txjlotimes IE

n

)UNt|κ(i)

xjl〉〈κ(i)

xjl|Pn

UlowastNt

)

ge 1minus 2minusnβ



there is a ξEn

t on HEn

such that

1

Lnt

Lntsum

l=1

trQn

(

UNt|κ(ilj)


UlowastNt

)

minus ξEn

t 1 lt ǫ

Thus(


xjl |Pn

j l DQn

txjl j l t

)


ρPn

xjl j l DQn

txjl j l t

)



Assume ρPn


n



n isin S(HQn

) we have

V QnMLθ

(


)

(V QnMLθ)lowast

=sum

t

sum

j

sum

l

(

DQn

txjlρQ

n)


We denote

ψQnEnMLθjlt

=(

IEn otimes V QnMLθ

)

(

UN otimes IMLθ)

[

|κjl〉〈κjl|Pn


(


(

IEn otimes V QnMLθ

)lowast


F(

trQnEn

(

ψQnEnMLθjlt

)


ge 1minus ǫ (80)


nEn

on HQnEn

such that

〈0|θ〈0|L〈0|M〈κjl|Pn (

UNtotimes IMLθ

)lowast

(

IEn otimes V QnMLθ

)lowast|ζjlt〉Q


= F

(


nEn


ge 1minus ǫ (81)



n |0〉M|0〉L|0〉θ

and

|bjlt〉PnMLθ =

(

UNtotimes IMLθ

)lowast (IE




We define

|ajk〉PnMLθ =

1radicLn

Lnsum

l=1


nMLθ

15

|bjkt〉PnMLθ =

1radicLn

Lnsum

l=1


nMLθ

and

|bjk〉PnMLθ =

1

|θ|

|θ|sum

t=1

|bjkt〉PnMLθ


1

Ln

Lnsum

k=1

〈ajk|bjk〉PnMLθ

=1

|θ|1

Ln

|θ|sum

t=1

Lnsum

k=1


=1

|θ|1

Ln

|θ|sum

t=1

Lnsum

l=1


ge 1minus ǫ (82)


1minus ǫ


|bjkj〉PnMLθ

=1

|θ|

|θ|sum

t=1

eminusiskj 〈ajkj

|bjkj t〉PnMLθ


∣

∣


|bjkj t〉PnMLθ

∣

∣

∣ le 1 we have

mintisinθ

∣

∣


|bjkj t〉PnMLθ

∣

∣



t isin θ

1minus |θ|ǫ

le∣

∣


|bjkj t〉PnMLθ

∣

∣

∣


|bjkj t〉PnMLθ

= eminusirkj

1

Ln

(

Lnsum

l=1


nMLθ

)

(

Lnsum

l=1


nMLθ

)

(83)


|jt〉QnEnL =

radic

1

Ln

Lnsum

l=1

eminus2πi(l

kjLn


and

|ϑjt〉QnEnMLθ =

radic

1

Ln

Lnsum

l=1

eminus2πilkj

Ln

[

IEn otimes V QnMLθ

]

(UnN |κjl〉P

n

)|0〉M|0〉L|0〉θ


F

(



)

=∣

∣



∣

∣

=1

Ln

(

Lnsum

l=1


nMLθ

)

(

Lnsum

l=1

eminus2πilkjLn e


)



n


such that

∥

∥

∥ξEn

t minus trQn

(


n

UlowastNt

)∥

∥

∥

1le ǫ (85)


∥

∥trQn

(


n

UlowastNt

)

minus trQn

(


∥

∥

1

le 2

[

1minus F

(

trQn

(


n

UlowastNt

)

trQn

(


)]12

le 2

[

1minus F

(


nEn


12

le 2radicǫ (86)


2minus η1 leradic


∥

∥

∥trQnL

(

|jt〉〈jt|QnEnL

)

minus ξEn

t

∥

∥

∥

1

=

∥

∥

∥

∥

∥

1

Ln

Lnsum

l=1

trQn

(


minus ξEn

t

∥

∥

∥

∥

∥

1

le 1

Ln

Lnsum

l=1

∥

∥

∥

∥

trQn

(


n

UlowastNt

)

minus trQn

(


∥

∥

∥

∥

1

+∥

∥

∥ξEn

t minus trQn

(


n

UlowastNt

)∥

∥

∥

1


16



such that if we set

χQnEnMLjjprimet =

(

UQnML(t) otimes IE

n)

(


)(

UQnML(t) otimes IE

n)lowast

then

F(


jjprimet

)


(88)where |ξt〉Q


rification of ξEn

t on HQnEnL



1

Jn

1

Ln

Jnsum

j=1

Lnsum

l=1


n |j〉A

Jnsum

j=1

Lnsum

l=1


n

(89)



1

Jn

1

Ln

(

IA otimes UnNtprime

)

[

Jnsum

j=1

Lnsum

l=1

eminus2πilkj

Ln |j〉A|κjl〉Pn

Jnsum

j=1

Lnsum

l=1

eminus2πilkj

Ln 〈κjl|Pn〈j|A

]

(

IA otimes UnNtprime

)lowast

=1

Jn

1

Ln

Jnsum

j=1

|j〉A(

Lnsum

l=1

eminus2πilkjLn Un

Ntprime|κjl〉P

n

)

Jnsum

j=1

(

Lnsum

l=1


n

(UnNtprime

)lowast)

〈j|A


τQn rarr trQnLθ

[

(

sum

tisinθ


)

V QnMLθ

(


)

V QnMLθlowast(

sum

tisinθ


)lowast]

(90)

to his outcome


We define

ιAQnEnMLθtprime

=

(

sum

tisinθ


)


)

(

1

Jn

1

Ln

Jnsum

j=1

|j〉A(

Lnsum

l=1

eminus2πilkjLn Un

Ntprime|κjl〉P

n

)

Jnsum

j=1

(

Lnsum

l=1


n

(UnNtprime

)lowast)

〈j|A



)lowast

(

sum

tisinθ


)lowast

=

(

sum

tisinθ


)

1

Jn(

Jnsum

j=1


Jnsum

j=1


(

sum

tisinθ


)lowast

(91)


AQnEnMLθtprime )


sumJn

j=1

sumJn



(

sum

tisinθ


)

(

sum

tisinθ

UQnML(t) otimes IE


= IAEn otimessum

tisinθ

UQnML(t) (UQnML


= IAQnEnMLθ

sum




F

ιAQnEnMLθtprime

1

Jn

Jnsum

j=1

Jnsum

jprime=1


= F

(

1

Jn(

Jnsum

j=1


Jnsum

j=1


1

Jn(

Jnsum

j=1


17

(

Jnsum

j=1


)

=1

Jn

∣

∣

∣

∣

( Jnsum

j=1


)

( Jnsum

j=1


)∣

∣

∣

∣



sumJn

j=1

sumJn



F

(

1

Jn

Jnsum

j=1

Jnsum

jprime=1


1

Jn(

Jnsum

j=1


(

Jnsum

j=1


)



1minus F ( η) le 1

2minus η1 le

radic

1minus F ( η)2


F ( η)

ge 1minus 1

2minus η1

ge 1minus 1

2minus υ1 minus

1

2υ minus η1

ge 1minusradic


1minus F (υ η)2


F

(


(

Jnsum

j=1


Jnsum

j=1


)

ge F

(

ιAQnEnMLθtprime

1

Jn(

Jnsum

j=1


(

Jnsum

j=1


)

ge 1minusradic



radicǫminus 8ǫ

(94)

ge 1minusradic

2|θ|radicǫminus



radic


with rate


tχ(X Et)minus 2δ



1

nmintisinθ

maxρisinS(H)Qn



limnrarrinfin

1

nmax

ρisinS(H)QnIC(ρN

otimesnt )




1

nmintisinθ

maxρisinS(H)Qn




18

VII FURTHER NOTES




N(ρ) =

Ksum

i=1

AiρAilowast (98)





we have Ai =sumK

j=1 uijBj (cf [31])







HE Then UN =sumK




)

ρ(


)lowast=

sumKj=1AjρAj

lowast and(


)lowast (sumK


=sumK

j=1 AjlowastAj


UN =sumK



lowast Here A1 AK




1

nmax

ρisinS(H)QnIC(ρN

otimesn) (100)




1

nmax




19


ACKNOWLEDGMENT





























20


















21

suptisinθ

χ(XuniZNt ) le ǫ



inftisinθ

maxPinpwt

1

n(χ(PinpB

nt )minus χ(PinpZ

nt ))








)minus2dprime4









Notimesnt

)




sum




A ge maxp

(

mintisinθ


χ(pEt)

)

(77)



ρPUlowastNt



sum


trE

(

sum

x


Nt

)


trQ

(

sum

x


Nt

)




sum



Nt

)

and VNt HP rarr HQ

sum



Nt

)



and


14


VNt)tisinθ since



xjl




n

t on HEn

such that

tr(

(DQn

txjlotimes IE

n

)UNtρP

n

xjlUlowastNt

)


and

ωEn

jt minus ξEn

t 1 lt ǫ (79)

where ωEn

jt = 1Lnt

sumLnt

l=1 trQn

(

UNtρP

n

xjlUlowastNt

)


xjlmay be pure or

mixed Assume ρPn


sumni=1 p

primejl(i)|κ

(i)xjl〉〈κ(i)

xjl |Pn

then

nsum

i=1

pprimejl(i)tr(

(DQn

txjlotimes IE

n

)UNt|κ(i)

xjl〉〈κ(i)

xjl|Pn

UlowastNt

)

tr

(

(DQn

txjlotimes IE

n

)UNt(

nsum

i=1


〉〈κ(i)xjl

|Pn

)UlowastNt

)

ge 1minus 2minusnβ



tr(

(DQn

txjlotimes IE

n

)UNt|κ(i)

xjl〉〈κ(i)

xjl|Pn

UlowastNt

)

ge 1minus 2minusnβ



there is a ξEn

t on HEn

such that

1

Lnt

Lntsum

l=1

trQn

(

UNt|κ(ilj)


UlowastNt

)

minus ξEn

t 1 lt ǫ

Thus(


xjl |Pn

j l DQn

txjl j l t

)


ρPn

xjl j l DQn

txjl j l t

)



Assume ρPn


n



n isin S(HQn

) we have

V QnMLθ

(


)

(V QnMLθ)lowast

=sum

t

sum

j

sum

l

(

DQn

txjlρQ

n)


We denote

ψQnEnMLθjlt

=(

IEn otimes V QnMLθ

)

(

UN otimes IMLθ)

[

|κjl〉〈κjl|Pn


(


(

IEn otimes V QnMLθ

)lowast


F(

trQnEn

(

ψQnEnMLθjlt

)


ge 1minus ǫ (80)


nEn

on HQnEn

such that

〈0|θ〈0|L〈0|M〈κjl|Pn (

UNtotimes IMLθ

)lowast

(

IEn otimes V QnMLθ

)lowast|ζjlt〉Q


= F

(


nEn


ge 1minus ǫ (81)



n |0〉M|0〉L|0〉θ

and

|bjlt〉PnMLθ =

(

UNtotimes IMLθ

)lowast (IE




We define

|ajk〉PnMLθ =

1radicLn

Lnsum

l=1


nMLθ

15

|bjkt〉PnMLθ =

1radicLn

Lnsum

l=1


nMLθ

and

|bjk〉PnMLθ =

1

|θ|

|θ|sum

t=1

|bjkt〉PnMLθ


1

Ln

Lnsum

k=1

〈ajk|bjk〉PnMLθ

=1

|θ|1

Ln

|θ|sum

t=1

Lnsum

k=1


=1

|θ|1

Ln

|θ|sum

t=1

Lnsum

l=1


ge 1minus ǫ (82)


1minus ǫ


|bjkj〉PnMLθ

=1

|θ|

|θ|sum

t=1

eminusiskj 〈ajkj

|bjkj t〉PnMLθ


∣

∣


|bjkj t〉PnMLθ

∣

∣

∣ le 1 we have

mintisinθ

∣

∣


|bjkj t〉PnMLθ

∣

∣



t isin θ

1minus |θ|ǫ

le∣

∣


|bjkj t〉PnMLθ

∣

∣

∣


|bjkj t〉PnMLθ

= eminusirkj

1

Ln

(

Lnsum

l=1


nMLθ

)

(

Lnsum

l=1


nMLθ

)

(83)


|jt〉QnEnL =

radic

1

Ln

Lnsum

l=1

eminus2πi(l

kjLn


and

|ϑjt〉QnEnMLθ =

radic

1

Ln

Lnsum

l=1

eminus2πilkj

Ln

[

IEn otimes V QnMLθ

]

(UnN |κjl〉P

n

)|0〉M|0〉L|0〉θ


F

(



)

=∣

∣



∣

∣

=1

Ln

(

Lnsum

l=1


nMLθ

)

(

Lnsum

l=1

eminus2πilkjLn e


)



n


such that

∥

∥

∥ξEn

t minus trQn

(


n

UlowastNt

)∥

∥

∥

1le ǫ (85)


∥

∥trQn

(


n

UlowastNt

)

minus trQn

(


∥

∥

1

le 2

[

1minus F

(

trQn

(


n

UlowastNt

)

trQn

(


)]12

le 2

[

1minus F

(


nEn


12

le 2radicǫ (86)


2minus η1 leradic


∥

∥

∥trQnL

(

|jt〉〈jt|QnEnL

)

minus ξEn

t

∥

∥

∥

1

=

∥

∥

∥

∥

∥

1

Ln

Lnsum

l=1

trQn

(


minus ξEn

t

∥

∥

∥

∥

∥

1

le 1

Ln

Lnsum

l=1

∥

∥

∥

∥

trQn

(


n

UlowastNt

)

minus trQn

(


∥

∥

∥

∥

1

+∥

∥

∥ξEn

t minus trQn

(


n

UlowastNt

)∥

∥

∥

1


16



such that if we set

χQnEnMLjjprimet =

(

UQnML(t) otimes IE

n)

(


)(

UQnML(t) otimes IE

n)lowast

then

F(


jjprimet

)


(88)where |ξt〉Q


rification of ξEn

t on HQnEnL



1

Jn

1

Ln

Jnsum

j=1

Lnsum

l=1


n |j〉A

Jnsum

j=1

Lnsum

l=1


n

(89)



1

Jn

1

Ln

(

IA otimes UnNtprime

)

[

Jnsum

j=1

Lnsum

l=1

eminus2πilkj

Ln |j〉A|κjl〉Pn

Jnsum

j=1

Lnsum

l=1

eminus2πilkj

Ln 〈κjl|Pn〈j|A

]

(

IA otimes UnNtprime

)lowast

=1

Jn

1

Ln

Jnsum

j=1

|j〉A(

Lnsum

l=1

eminus2πilkjLn Un

Ntprime|κjl〉P

n

)

Jnsum

j=1

(

Lnsum

l=1


n

(UnNtprime

)lowast)

〈j|A


τQn rarr trQnLθ

[

(

sum

tisinθ


)

V QnMLθ

(


)

V QnMLθlowast(

sum

tisinθ


)lowast]

(90)

to his outcome


We define

ιAQnEnMLθtprime

=

(

sum

tisinθ


)


)

(

1

Jn

1

Ln

Jnsum

j=1

|j〉A(

Lnsum

l=1

eminus2πilkjLn Un

Ntprime|κjl〉P

n

)

Jnsum

j=1

(

Lnsum

l=1


n

(UnNtprime

)lowast)

〈j|A



)lowast

(

sum

tisinθ


)lowast

=

(

sum

tisinθ


)

1

Jn(

Jnsum

j=1


Jnsum

j=1


(

sum

tisinθ


)lowast

(91)


AQnEnMLθtprime )


sumJn

j=1

sumJn



(

sum

tisinθ


)

(

sum

tisinθ

UQnML(t) otimes IE


= IAEn otimessum

tisinθ

UQnML(t) (UQnML


= IAQnEnMLθ

sum




F

ιAQnEnMLθtprime

1

Jn

Jnsum

j=1

Jnsum

jprime=1


= F

(

1

Jn(

Jnsum

j=1


Jnsum

j=1


1

Jn(

Jnsum

j=1


17

(

Jnsum

j=1


)

=1

Jn

∣

∣

∣

∣

( Jnsum

j=1


)

( Jnsum

j=1


)∣

∣

∣

∣



sumJn

j=1

sumJn



F

(

1

Jn

Jnsum

j=1

Jnsum

jprime=1


1

Jn(

Jnsum

j=1


(

Jnsum

j=1


)



1minus F ( η) le 1

2minus η1 le

radic

1minus F ( η)2


F ( η)

ge 1minus 1

2minus η1

ge 1minus 1

2minus υ1 minus

1

2υ minus η1

ge 1minusradic


1minus F (υ η)2


F

(


(

Jnsum

j=1


Jnsum

j=1


)

ge F

(

ιAQnEnMLθtprime

1

Jn(

Jnsum

j=1


(

Jnsum

j=1


)

ge 1minusradic



radicǫminus 8ǫ

(94)

ge 1minusradic

2|θ|radicǫminus



radic


with rate


tχ(X Et)minus 2δ



1

nmintisinθ

maxρisinS(H)Qn



limnrarrinfin

1

nmax

ρisinS(H)QnIC(ρN

otimesnt )




1

nmintisinθ

maxρisinS(H)Qn




18

VII FURTHER NOTES




N(ρ) =

Ksum

i=1

AiρAilowast (98)





we have Ai =sumK

j=1 uijBj (cf [31])







HE Then UN =sumK




)

ρ(


)lowast=

sumKj=1AjρAj

lowast and(


)lowast (sumK


=sumK

j=1 AjlowastAj


UN =sumK



lowast Here A1 AK




1

nmax

ρisinS(H)QnIC(ρN

otimesn) (100)




1

nmax




19


ACKNOWLEDGMENT





























20


















21


VNt)tisinθ since



xjl




n

t on HEn

such that

tr(

(DQn

txjlotimes IE

n

)UNtρP

n

xjlUlowastNt

)


and

ωEn

jt minus ξEn

t 1 lt ǫ (79)

where ωEn

jt = 1Lnt

sumLnt

l=1 trQn

(

UNtρP

n

xjlUlowastNt

)


xjlmay be pure or

mixed Assume ρPn


sumni=1 p

primejl(i)|κ

(i)xjl〉〈κ(i)

xjl |Pn

then

nsum

i=1

pprimejl(i)tr(

(DQn

txjlotimes IE

n

)UNt|κ(i)

xjl〉〈κ(i)

xjl|Pn

UlowastNt

)

tr

(

(DQn

txjlotimes IE

n

)UNt(

nsum

i=1


〉〈κ(i)xjl

|Pn

)UlowastNt

)

ge 1minus 2minusnβ



tr(

(DQn

txjlotimes IE

n

)UNt|κ(i)

xjl〉〈κ(i)

xjl|Pn

UlowastNt

)

ge 1minus 2minusnβ



there is a ξEn

t on HEn

such that

1

Lnt

Lntsum

l=1

trQn

(

UNt|κ(ilj)


UlowastNt

)

minus ξEn

t 1 lt ǫ

Thus(


xjl |Pn

j l DQn

txjl j l t

)


ρPn

xjl j l DQn

txjl j l t

)



Assume ρPn


n



n isin S(HQn

) we have

V QnMLθ

(


)

(V QnMLθ)lowast

=sum

t

sum

j

sum

l

(

DQn

txjlρQ

n)


We denote

ψQnEnMLθjlt

=(

IEn otimes V QnMLθ

)

(

UN otimes IMLθ)

[

|κjl〉〈κjl|Pn


(


(

IEn otimes V QnMLθ

)lowast


F(

trQnEn

(

ψQnEnMLθjlt

)


ge 1minus ǫ (80)


nEn

on HQnEn

such that

〈0|θ〈0|L〈0|M〈κjl|Pn (

UNtotimes IMLθ

)lowast

(

IEn otimes V QnMLθ

)lowast|ζjlt〉Q


= F

(


nEn


ge 1minus ǫ (81)



n |0〉M|0〉L|0〉θ

and

|bjlt〉PnMLθ =

(

UNtotimes IMLθ

)lowast (IE




We define

|ajk〉PnMLθ =

1radicLn

Lnsum

l=1


nMLθ

15

|bjkt〉PnMLθ =

1radicLn

Lnsum

l=1


nMLθ

and

|bjk〉PnMLθ =

1

|θ|

|θ|sum

t=1

|bjkt〉PnMLθ


1

Ln

Lnsum

k=1

〈ajk|bjk〉PnMLθ

=1

|θ|1

Ln

|θ|sum

t=1

Lnsum

k=1


=1

|θ|1

Ln

|θ|sum

t=1

Lnsum

l=1


ge 1minus ǫ (82)


1minus ǫ


|bjkj〉PnMLθ

=1

|θ|

|θ|sum

t=1

eminusiskj 〈ajkj

|bjkj t〉PnMLθ


∣

∣


|bjkj t〉PnMLθ

∣

∣

∣ le 1 we have

mintisinθ

∣

∣


|bjkj t〉PnMLθ

∣

∣



t isin θ

1minus |θ|ǫ

le∣

∣


|bjkj t〉PnMLθ

∣

∣

∣


|bjkj t〉PnMLθ

= eminusirkj

1

Ln

(

Lnsum

l=1


nMLθ

)

(

Lnsum

l=1


nMLθ

)

(83)


|jt〉QnEnL =

radic

1

Ln

Lnsum

l=1

eminus2πi(l

kjLn


and

|ϑjt〉QnEnMLθ =

radic

1

Ln

Lnsum

l=1

eminus2πilkj

Ln

[

IEn otimes V QnMLθ

]

(UnN |κjl〉P

n

)|0〉M|0〉L|0〉θ


F

(



)

=∣

∣



∣

∣

=1

Ln

(

Lnsum

l=1


nMLθ

)

(

Lnsum

l=1

eminus2πilkjLn e


)



n


such that

∥

∥

∥ξEn

t minus trQn

(


n

UlowastNt

)∥

∥

∥

1le ǫ (85)


∥

∥trQn

(


n

UlowastNt

)

minus trQn

(


∥

∥

1

le 2

[

1minus F

(

trQn

(


n

UlowastNt

)

trQn

(


)]12

le 2

[

1minus F

(


nEn


12

le 2radicǫ (86)


2minus η1 leradic


∥

∥

∥trQnL

(

|jt〉〈jt|QnEnL

)

minus ξEn

t

∥

∥

∥

1

=

∥

∥

∥

∥

∥

1

Ln

Lnsum

l=1

trQn

(


minus ξEn

t

∥

∥

∥

∥

∥

1

le 1

Ln

Lnsum

l=1

∥

∥

∥

∥

trQn

(


n

UlowastNt

)

minus trQn

(


∥

∥

∥

∥

1

+∥

∥

∥ξEn

t minus trQn

(


n

UlowastNt

)∥

∥

∥

1


16



such that if we set

χQnEnMLjjprimet =

(

UQnML(t) otimes IE

n)

(


)(

UQnML(t) otimes IE

n)lowast

then

F(


jjprimet

)


(88)where |ξt〉Q


rification of ξEn

t on HQnEnL



1

Jn

1

Ln

Jnsum

j=1

Lnsum

l=1


n |j〉A

Jnsum

j=1

Lnsum

l=1


n

(89)



1

Jn

1

Ln

(

IA otimes UnNtprime

)

[

Jnsum

j=1

Lnsum

l=1

eminus2πilkj

Ln |j〉A|κjl〉Pn

Jnsum

j=1

Lnsum

l=1

eminus2πilkj

Ln 〈κjl|Pn〈j|A

]

(

IA otimes UnNtprime

)lowast

=1

Jn

1

Ln

Jnsum

j=1

|j〉A(

Lnsum

l=1

eminus2πilkjLn Un

Ntprime|κjl〉P

n

)

Jnsum

j=1

(

Lnsum

l=1


n

(UnNtprime

)lowast)

〈j|A


τQn rarr trQnLθ

[

(

sum

tisinθ


)

V QnMLθ

(


)

V QnMLθlowast(

sum

tisinθ


)lowast]

(90)

to his outcome


We define

ιAQnEnMLθtprime

=

(

sum

tisinθ


)


)

(

1

Jn

1

Ln

Jnsum

j=1

|j〉A(

Lnsum

l=1

eminus2πilkjLn Un

Ntprime|κjl〉P

n

)

Jnsum

j=1

(

Lnsum

l=1


n

(UnNtprime

)lowast)

〈j|A



)lowast

(

sum

tisinθ


)lowast

=

(

sum

tisinθ


)

1

Jn(

Jnsum

j=1


Jnsum

j=1


(

sum

tisinθ


)lowast

(91)


AQnEnMLθtprime )


sumJn

j=1

sumJn



(

sum

tisinθ


)

(

sum

tisinθ

UQnML(t) otimes IE


= IAEn otimessum

tisinθ

UQnML(t) (UQnML


= IAQnEnMLθ

sum




F

ιAQnEnMLθtprime

1

Jn

Jnsum

j=1

Jnsum

jprime=1


= F

(

1

Jn(

Jnsum

j=1


Jnsum

j=1


1

Jn(

Jnsum

j=1


17

(

Jnsum

j=1


)

=1

Jn

∣

∣

∣

∣

( Jnsum

j=1


)

( Jnsum

j=1


)∣

∣

∣

∣



sumJn

j=1

sumJn



F

(

1

Jn

Jnsum

j=1

Jnsum

jprime=1


1

Jn(

Jnsum

j=1


(

Jnsum

j=1


)



1minus F ( η) le 1

2minus η1 le

radic

1minus F ( η)2


F ( η)

ge 1minus 1

2minus η1

ge 1minus 1

2minus υ1 minus

1

2υ minus η1

ge 1minusradic


1minus F (υ η)2


F

(


(

Jnsum

j=1


Jnsum

j=1


)

ge F

(

ιAQnEnMLθtprime

1

Jn(

Jnsum

j=1


(

Jnsum

j=1


)

ge 1minusradic



radicǫminus 8ǫ

(94)

ge 1minusradic

2|θ|radicǫminus



radic


with rate


tχ(X Et)minus 2δ



1

nmintisinθ

maxρisinS(H)Qn



limnrarrinfin

1

nmax

ρisinS(H)QnIC(ρN

otimesnt )




1

nmintisinθ

maxρisinS(H)Qn




18

VII FURTHER NOTES




N(ρ) =

Ksum

i=1

AiρAilowast (98)





we have Ai =sumK

j=1 uijBj (cf [31])







HE Then UN =sumK




)

ρ(


)lowast=

sumKj=1AjρAj

lowast and(


)lowast (sumK


=sumK

j=1 AjlowastAj


UN =sumK



lowast Here A1 AK




1

nmax

ρisinS(H)QnIC(ρN

otimesn) (100)




1

nmax




19


ACKNOWLEDGMENT





























20


















21

|bjkt〉PnMLθ =

1radicLn

Lnsum

l=1


nMLθ

and

|bjk〉PnMLθ =

1

|θ|

|θ|sum

t=1

|bjkt〉PnMLθ


1

Ln

Lnsum

k=1

〈ajk|bjk〉PnMLθ

=1

|θ|1

Ln

|θ|sum

t=1

Lnsum

k=1


=1

|θ|1

Ln

|θ|sum

t=1

Lnsum

l=1


ge 1minus ǫ (82)


1minus ǫ


|bjkj〉PnMLθ

=1

|θ|

|θ|sum

t=1

eminusiskj 〈ajkj

|bjkj t〉PnMLθ


∣

∣


|bjkj t〉PnMLθ

∣

∣

∣ le 1 we have

mintisinθ

∣

∣


|bjkj t〉PnMLθ

∣

∣



t isin θ

1minus |θ|ǫ

le∣

∣


|bjkj t〉PnMLθ

∣

∣

∣


|bjkj t〉PnMLθ

= eminusirkj

1

Ln

(

Lnsum

l=1


nMLθ

)

(

Lnsum

l=1


nMLθ

)

(83)


|jt〉QnEnL =

radic

1

Ln

Lnsum

l=1

eminus2πi(l

kjLn


and

|ϑjt〉QnEnMLθ =

radic

1

Ln

Lnsum

l=1

eminus2πilkj

Ln

[

IEn otimes V QnMLθ

]

(UnN |κjl〉P

n

)|0〉M|0〉L|0〉θ


F

(



)

=∣

∣



∣

∣

=1

Ln

(

Lnsum

l=1


nMLθ

)

(

Lnsum

l=1

eminus2πilkjLn e


)



n


such that

∥

∥

∥ξEn

t minus trQn

(


n

UlowastNt

)∥

∥

∥

1le ǫ (85)


∥

∥trQn

(


n

UlowastNt

)

minus trQn

(


∥

∥

1

le 2

[

1minus F

(

trQn

(


n

UlowastNt

)

trQn

(


)]12

le 2

[

1minus F

(


nEn


12

le 2radicǫ (86)


2minus η1 leradic


∥

∥

∥trQnL

(

|jt〉〈jt|QnEnL

)

minus ξEn

t

∥

∥

∥

1

=

∥

∥

∥

∥

∥

1

Ln

Lnsum

l=1

trQn

(


minus ξEn

t

∥

∥

∥

∥

∥

1

le 1

Ln

Lnsum

l=1

∥

∥

∥

∥

trQn

(


n

UlowastNt

)

minus trQn

(


∥

∥

∥

∥

1

+∥

∥

∥ξEn

t minus trQn

(


n

UlowastNt

)∥

∥

∥

1


16



such that if we set

χQnEnMLjjprimet =

(

UQnML(t) otimes IE

n)

(


)(

UQnML(t) otimes IE

n)lowast

then

F(


jjprimet

)


(88)where |ξt〉Q


rification of ξEn

t on HQnEnL



1

Jn

1

Ln

Jnsum

j=1

Lnsum

l=1


n |j〉A

Jnsum

j=1

Lnsum

l=1


n

(89)



1

Jn

1

Ln

(

IA otimes UnNtprime

)

[

Jnsum

j=1

Lnsum

l=1

eminus2πilkj

Ln |j〉A|κjl〉Pn

Jnsum

j=1

Lnsum

l=1

eminus2πilkj

Ln 〈κjl|Pn〈j|A

]

(

IA otimes UnNtprime

)lowast

=1

Jn

1

Ln

Jnsum

j=1

|j〉A(

Lnsum

l=1

eminus2πilkjLn Un

Ntprime|κjl〉P

n

)

Jnsum

j=1

(

Lnsum

l=1


n

(UnNtprime

)lowast)

〈j|A


τQn rarr trQnLθ

[

(

sum

tisinθ


)

V QnMLθ

(


)

V QnMLθlowast(

sum

tisinθ


)lowast]

(90)

to his outcome


We define

ιAQnEnMLθtprime

=

(

sum

tisinθ


)


)

(

1

Jn

1

Ln

Jnsum

j=1

|j〉A(

Lnsum

l=1

eminus2πilkjLn Un

Ntprime|κjl〉P

n

)

Jnsum

j=1

(

Lnsum

l=1


n

(UnNtprime

)lowast)

〈j|A



)lowast

(

sum

tisinθ


)lowast

=

(

sum

tisinθ


)

1

Jn(

Jnsum

j=1


Jnsum

j=1


(

sum

tisinθ


)lowast

(91)


AQnEnMLθtprime )


sumJn

j=1

sumJn



(

sum

tisinθ


)

(

sum

tisinθ

UQnML(t) otimes IE


= IAEn otimessum

tisinθ

UQnML(t) (UQnML


= IAQnEnMLθ

sum




F

ιAQnEnMLθtprime

1

Jn

Jnsum

j=1

Jnsum

jprime=1


= F

(

1

Jn(

Jnsum

j=1


Jnsum

j=1


1

Jn(

Jnsum

j=1


17

(

Jnsum

j=1


)

=1

Jn

∣

∣

∣

∣

( Jnsum

j=1


)

( Jnsum

j=1


)∣

∣

∣

∣



sumJn

j=1

sumJn



F

(

1

Jn

Jnsum

j=1

Jnsum

jprime=1


1

Jn(

Jnsum

j=1


(

Jnsum

j=1


)



1minus F ( η) le 1

2minus η1 le

radic

1minus F ( η)2


F ( η)

ge 1minus 1

2minus η1

ge 1minus 1

2minus υ1 minus

1

2υ minus η1

ge 1minusradic


1minus F (υ η)2


F

(


(

Jnsum

j=1


Jnsum

j=1


)

ge F

(

ιAQnEnMLθtprime

1

Jn(

Jnsum

j=1


(

Jnsum

j=1


)

ge 1minusradic



radicǫminus 8ǫ

(94)

ge 1minusradic

2|θ|radicǫminus



radic


with rate


tχ(X Et)minus 2δ



1

nmintisinθ

maxρisinS(H)Qn



limnrarrinfin

1

nmax

ρisinS(H)QnIC(ρN

otimesnt )




1

nmintisinθ

maxρisinS(H)Qn




18

VII FURTHER NOTES




N(ρ) =

Ksum

i=1

AiρAilowast (98)





we have Ai =sumK

j=1 uijBj (cf [31])







HE Then UN =sumK




)

ρ(


)lowast=

sumKj=1AjρAj

lowast and(


)lowast (sumK


=sumK

j=1 AjlowastAj


UN =sumK



lowast Here A1 AK




1

nmax

ρisinS(H)QnIC(ρN

otimesn) (100)




1

nmax




19


ACKNOWLEDGMENT





























20


















21



such that if we set

χQnEnMLjjprimet =

(

UQnML(t) otimes IE

n)

(


)(

UQnML(t) otimes IE

n)lowast

then

F(


jjprimet

)


(88)where |ξt〉Q


rification of ξEn

t on HQnEnL



1

Jn

1

Ln

Jnsum

j=1

Lnsum

l=1


n |j〉A

Jnsum

j=1

Lnsum

l=1


n

(89)



1

Jn

1

Ln

(

IA otimes UnNtprime

)

[

Jnsum

j=1

Lnsum

l=1

eminus2πilkj

Ln |j〉A|κjl〉Pn

Jnsum

j=1

Lnsum

l=1

eminus2πilkj

Ln 〈κjl|Pn〈j|A

]

(

IA otimes UnNtprime

)lowast

=1

Jn

1

Ln

Jnsum

j=1

|j〉A(

Lnsum

l=1

eminus2πilkjLn Un

Ntprime|κjl〉P

n

)

Jnsum

j=1

(

Lnsum

l=1


n

(UnNtprime

)lowast)

〈j|A


τQn rarr trQnLθ

[

(

sum

tisinθ


)

V QnMLθ

(


)

V QnMLθlowast(

sum

tisinθ


)lowast]

(90)

to his outcome


We define

ιAQnEnMLθtprime

=

(

sum

tisinθ


)


)

(

1

Jn

1

Ln

Jnsum

j=1

|j〉A(

Lnsum

l=1

eminus2πilkjLn Un

Ntprime|κjl〉P

n

)

Jnsum

j=1

(

Lnsum

l=1


n

(UnNtprime

)lowast)

〈j|A



)lowast

(

sum

tisinθ


)lowast

=

(

sum

tisinθ


)

1

Jn(

Jnsum

j=1


Jnsum

j=1


(

sum

tisinθ


)lowast

(91)


AQnEnMLθtprime )


sumJn

j=1

sumJn



(

sum

tisinθ


)

(

sum

tisinθ

UQnML(t) otimes IE


= IAEn otimessum

tisinθ

UQnML(t) (UQnML


= IAQnEnMLθ

sum




F

ιAQnEnMLθtprime

1

Jn

Jnsum

j=1

Jnsum

jprime=1


= F

(

1

Jn(

Jnsum

j=1


Jnsum

j=1


1

Jn(

Jnsum

j=1


17

(

Jnsum

j=1


)

=1

Jn

∣

∣

∣

∣

( Jnsum

j=1


)

( Jnsum

j=1


)∣

∣

∣

∣



sumJn

j=1

sumJn



F

(

1

Jn

Jnsum

j=1

Jnsum

jprime=1


1

Jn(

Jnsum

j=1


(

Jnsum

j=1


)



1minus F ( η) le 1

2minus η1 le

radic

1minus F ( η)2


F ( η)

ge 1minus 1

2minus η1

ge 1minus 1

2minus υ1 minus

1

2υ minus η1

ge 1minusradic


1minus F (υ η)2


F

(


(

Jnsum

j=1


Jnsum

j=1


)

ge F

(

ιAQnEnMLθtprime

1

Jn(

Jnsum

j=1


(

Jnsum

j=1


)

ge 1minusradic



radicǫminus 8ǫ

(94)

ge 1minusradic

2|θ|radicǫminus



radic


with rate


tχ(X Et)minus 2δ



1

nmintisinθ

maxρisinS(H)Qn



limnrarrinfin

1

nmax

ρisinS(H)QnIC(ρN

otimesnt )




1

nmintisinθ

maxρisinS(H)Qn




18

VII FURTHER NOTES




N(ρ) =

Ksum

i=1

AiρAilowast (98)





we have Ai =sumK

j=1 uijBj (cf [31])







HE Then UN =sumK




)

ρ(


)lowast=

sumKj=1AjρAj

lowast and(


)lowast (sumK


=sumK

j=1 AjlowastAj


UN =sumK



lowast Here A1 AK




1

nmax

ρisinS(H)QnIC(ρN

otimesn) (100)




1

nmax




19


ACKNOWLEDGMENT





























20


















21

(

Jnsum

j=1


)

=1

Jn

∣

∣

∣

∣

( Jnsum

j=1


)

( Jnsum

j=1


)∣

∣

∣

∣



sumJn

j=1

sumJn



F

(

1

Jn

Jnsum

j=1

Jnsum

jprime=1


1

Jn(

Jnsum

j=1


(

Jnsum

j=1


)



1minus F ( η) le 1

2minus η1 le

radic

1minus F ( η)2


F ( η)

ge 1minus 1

2minus η1

ge 1minus 1

2minus υ1 minus

1

2υ minus η1

ge 1minusradic


1minus F (υ η)2


F

(


(

Jnsum

j=1


Jnsum

j=1


)

ge F

(

ιAQnEnMLθtprime

1

Jn(

Jnsum

j=1


(

Jnsum

j=1


)

ge 1minusradic



radicǫminus 8ǫ

(94)

ge 1minusradic

2|θ|radicǫminus



radic


with rate


tχ(X Et)minus 2δ



1

nmintisinθ

maxρisinS(H)Qn



limnrarrinfin

1

nmax

ρisinS(H)QnIC(ρN

otimesnt )




1

nmintisinθ

maxρisinS(H)Qn




18

VII FURTHER NOTES




N(ρ) =

Ksum

i=1

AiρAilowast (98)





we have Ai =sumK

j=1 uijBj (cf [31])







HE Then UN =sumK




)

ρ(


)lowast=

sumKj=1AjρAj

lowast and(


)lowast (sumK


=sumK

j=1 AjlowastAj


UN =sumK



lowast Here A1 AK




1

nmax

ρisinS(H)QnIC(ρN

otimesn) (100)




1

nmax




19


ACKNOWLEDGMENT





























20


















21

VII FURTHER NOTES




N(ρ) =

Ksum

i=1

AiρAilowast (98)





we have Ai =sumK

j=1 uijBj (cf [31])







HE Then UN =sumK




)

ρ(


)lowast=

sumKj=1AjρAj

lowast and(


)lowast (sumK


=sumK

j=1 AjlowastAj


UN =sumK



lowast Here A1 AK




1

nmax

ρisinS(H)QnIC(ρN

otimesn) (100)




1

nmax




19


ACKNOWLEDGMENT





























20


















21


ACKNOWLEDGMENT





























20


















21


















21

secrecy capacities of compound quantum wiretap channels and applications

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