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Quantum Communication and Cryptography
Quantum Communication and Cryptography
Joseph M. Renes
Theoretical Quantum Physics, Institut für Angewandte PhysikTechnische Universität Darmstadt
IQING 5 Innsbruck 2007 April 12
J. M. Renes IQING 5
Quantum Communication and Cryptography
Introduction
Overview
Alice Bobquantum channel
classical channel
shared quantum states
shared classical randomness
Communication between distant partiesSend messagesDistributed computationCrytographic useCreate entanglement
Major point: Interchangeability of resourcesnoisy channel −→ noiseless channel; noiseless + randomness −→ noisylow-fielity EPR pairs + classical communication −→ maximal entanglement. . .
J. M. Renes IQING 5
Quantum Communication and Cryptography
Introduction
Outline
Overview recent work in quantum information theory. (in the strict sense)
1 Resource framework: how the myriad of protocols fit together.Expressed as inequalities, for instance teleportation:[qq] + 2[c → c] ≤ [q → q]Quantum family tree: Fully-Quantum Slepian-Wolf
2 Connection between Entanglement and Secret Keysstatic/dynamic & quantum/classical:
entanglement generation & distillationsecret key generation & distillation
protocols have essentially the same proof3 Erasure is Fundamental
Closer look at quantum capacityDecoupling from the environment is enoughFor coding, just pick a random subspace
J. M. Renes IQING 5
Quantum Communication and Cryptography
Resource Framework
Unit Resources and Simple Protocols
Unit Resources
Ideal cbit channel [c → c][q → q] Ideal qbit channel
Private cbit channel (c → c)
Shared randomness [cc] [qq] Shared entanglement
Simple Protocols
Teleportation [qq] + 2[c → c] ≥ [q → q]
Dense Coding [q → q] + [qq] ≥ 2[c → c]
Entanglement Distribution [q → q] ≥ [qq]
Private Communication [qq] + [c → c] ≥ (c → c)[q → q] ≥ (c → c)
I. Devetak, A. W. Harrow, and A. Winter, quant-ph/0512015
J. M. Renes IQING 5
Quantum Communication and Cryptography
Resource Framework
General Resources and Standard Protocols
General Resources
noisy channel 〈N A→B〉 shared state 〈ϕAB〉
Standard Protocols
Schumacher compression S(ρA)[q → q] ≥ 〈idA→B〉HSW theorem 〈N T→B〉 ≥ I(X: B)[c → c]
Quantum capacity 〈N T→B〉 ≥ I(A〉B)[q → q]
Entanglement distillation 〈ρAB〉+ I(A:E)[c → c] ≥ I(A〉B)[qq]
J. M. Renes IQING 5
Quantum Communication and Cryptography
Resource Framework
General Resources and Standard Protocols
General Resources
noisy channel 〈N A→B〉 shared state 〈ϕAB〉
Standard Protocols
Schumacher compression S(ρA)[q → q] ≥ 〈idA→B〉HSW theorem 〈N T→B〉 ≥ I(X: B)[c → c]
Quantum capacity 〈N T→B〉 ≥ I(A〉B)[q → q]
Entanglement distillation 〈ρAB〉+ I(A:E)[c → c] ≥ I(A〉B)[qq]
Transmit ρA by first compressing it, then using a noiseless channel
J. M. Renes IQING 5
Quantum Communication and Cryptography
Resource Framework
General Resources and Standard Protocols
General Resources
noisy channel 〈N A→B〉 shared state 〈ϕAB〉
Standard Protocols
Schumacher compression S(ρA)[q → q] ≥ 〈idA→B〉HSW theorem 〈N T→B〉 ≥ I(X: B)[c → c]
Quantum capacity 〈N T→B〉 ≥ I(A〉B)[q → q]
Entanglement distillation 〈ρAB〉+ I(A:E)[c → c] ≥ I(A〉B)[qq]
Transmit classical message over a quantum channel
Start with the state ρXT =∑
x px |x〉〈x|X ⊗ ρT
x
Send T to obtain σXB = N T→B(ρXT)
I(X: B) is the Holevo quantity χ = S(∑
x pxσBx ) −
∑x pxS(σB
x ).
J. M. Renes IQING 5
Quantum Communication and Cryptography
Resource Framework
General Resources and Standard Protocols
General Resources
noisy channel 〈N A→B〉 shared state 〈ϕAB〉
Standard Protocols
Schumacher compression S(ρA)[q → q] ≥ 〈idA→B〉HSW theorem 〈N T→B〉 ≥ I(X: B)[c → c]
Quantum capacity 〈N T→B〉 ≥ I(A〉B)[q → q]
Entanglement distillation 〈ρAB〉+ I(A:E)[c → c] ≥ I(A〉B)[qq]
Encode quantum messages for transmission over a noisy channel
Input ensemble state ρT has purification |ψ〉〈ψ|TA
Channel outputs σAB = N T→B(ρTA)
I(A〉B) = −H(A|B) is the coherent information.
J. M. Renes IQING 5
Quantum Communication and Cryptography
Resource Framework
General Resources and Standard Protocols
General Resources
noisy channel 〈N A→B〉 shared state 〈ϕAB〉
Standard Protocols
Schumacher compression S(ρA)[q → q] ≥ 〈idA→B〉HSW theorem 〈N T→B〉 ≥ I(X: B)[c → c]
Quantum capacity 〈N T→B〉 ≥ I(A〉B)[q → q]
Entanglement distillation 〈ρAB〉+ I(A:E)[c → c] ≥ I(A〉B)[qq]
Distill maximal entanglement from ϕAB
use only local operations and classical communication
E comes from purifying ϕAB to |ψ〉〈ψ|ABE
J. M. Renes IQING 5
Quantum Communication and Cryptography
Resource Framework
Quantum Family Tree
FQSWFQRS
♀
EntanglementDistillation
Noisy Dense Coding
Noisy Teleportation
♂Entanglement-
assisted Capacity
Quantum Capacity
State Merging
Broadcast
QMAC
Environment-assistedCapacity Fully-Quantum Slepian-Wolf
Given: state |ϕ〉ABR and noiselessquantum communication
Goal: distill EPR pairs andtransfer ϕA completely to Bob
R A
B
|ϕ〉
R A
BB
|Φ〉|ϕ〉
〈WS→AB : ϕS〉+ 12 I(A:R)[q → q] ≥
12 I(A:B)[qq] + 〈idS→B
: ϕS〉.A. Abeyesinghe, I. Devetak, P. Hayden, and A. Winter,quant-ph/0606225
J. M. Renes IQING 5
Quantum Communication and Cryptography
Entanglement and Secret Keys
Four Protocols
r = I(A〉B) = H(B) − H(AB)I. Devetak and A. Winter, PRL 93 080501 (2004).
Entanglement Distillation
Convert (ρAB)⊗n into nr ebitsΦ⊗nr using 1-LOCC
Secret Key Distillation
Convert (ρAB)⊗n into nr secretkey bits κ⊗nr using 1-LOPC
Entanglement Generation
Use n copies of the channel Nto generate nr bits of sharedentanglement, (ΦAB)⊗nr
Secret Key Generation
Use n copies of the channel Nto generate nr bits of sharedsecret key, (κABE)⊗nr
Rows: Static vs. DynamicColumns: Classical vs. Quantum
J. M. Renes IQING 5
Quantum Communication and Cryptography
Entanglement and Secret Keys
Four Protocols
r = I(A〉B) = H(B) − H(AB)I. Devetak and A. Winter, PRL 93 080501 (2004).
Entanglement Distillation
Convert (ρAB)⊗n into nr ebitsΦ⊗nr using 1-LOCC
Secret Key Distillation
Convert (ρAB)⊗n into nr secretkey bits κ⊗nr using 1-LOPC
Entanglement Generation
Use n copies of the channel Nto generate nr bits of sharedentanglement, (ΦAB)⊗nr
Secret Key Generation
Use n copies of the channel Nto generate nr bits of sharedsecret key, (κABE)⊗nr
A T E N D B
Specify Encoding for Alice and Decoding for Bob.Same as quantum channel capacity
J. M. Renes IQING 5
Quantum Communication and Cryptography
Entanglement and Secret Keys
Four Protocols
r = I(A〉B) = H(B) − H(AB)I. Devetak and A. Winter, PRL 93 080501 (2004).
Entanglement Distillation
Convert (ρAB)⊗n into nr ebitsΦ⊗nr using 1-LOCC
Secret Key Distillation
Convert (ρAB)⊗n into nr secretkey bits κ⊗nr using 1-LOPC
Entanglement Generation
Use n copies of the channel Nto generate nr bits of sharedentanglement, (ΦAB)⊗nr
Secret Key Generation
Use n copies of the channel Nto generate nr bits of sharedsecret key, (κABE)⊗nr
T E N M
Specify Encoding for Alice and Measurement for Bob.κABE = 1
2∑
k |kk〉〈kk|AB ⊗ ρE
J. M. Renes IQING 5
Quantum Communication and Cryptography
Entanglement and Secret Keys
Four Protocols
r = I(A〉B) = H(B) − H(AB)I. Devetak and A. Winter, PRL 93 080501 (2004).
Entanglement Distillation
Convert (ρAB)⊗n into nr ebitsΦ⊗nr using 1-LOCC
Secret Key Distillation
Convert (ρAB)⊗n into nr secretkey bits κ⊗nr using 1-LOPC
Entanglement Generation
Use n copies of the channel Nto generate nr bits of sharedentanglement, (ΦAB)⊗nr
Secret Key Generation
Use n copies of the channel Nto generate nr bits of sharedsecret key, (κABE)⊗nr
ρAB is purified to |ψ〉ABE =∑
x√
px |x〉A ⊗ |ψk〉BE
regard |ψk〉BE as N ◦ E(|ϕk〉T)
rate bound is known as the Hashing inequality
J. M. Renes IQING 5
Quantum Communication and Cryptography
Entanglement and Secret Keys
Four Protocols
r = I(A〉B) = H(B) − H(AB)I. Devetak and A. Winter, PRL 93 080501 (2004).
Entanglement Distillation
Convert (ρAB)⊗n into nr ebitsΦ⊗nr using 1-LOCC
Secret Key Distillation
Convert (ρAB)⊗n into nr secretkey bits κ⊗nr using 1-LOPC
Entanglement Generation
Use n copies of the channel Nto generate nr bits of sharedentanglement, (ΦAB)⊗nr
Secret Key Generation
Use n copies of the channel Nto generate nr bits of sharedsecret key, (κABE)⊗nr
Starting point is ρAB =∑
k pk |k〉〈k|A ⊗ψBk , the decohered version.
Use only one-way local operations and public communication
J. M. Renes IQING 5
Quantum Communication and Cryptography
Entanglement and Secret Keys
Four Protocols
r = I(A〉B) = H(B) − H(AB)I. Devetak and A. Winter, PRL 93 080501 (2004).
Entanglement Distillation
Convert (ρAB)⊗n into nr ebitsΦ⊗nr using 1-LOCC
Secret Key Distillation
Convert (ρAB)⊗n into nr secretkey bits κ⊗nr using 1-LOPC
Entanglement Generation
Use n copies of the channel Nto generate nr bits of sharedentanglement, (ΦAB)⊗nr
Secret Key Generation
Use n copies of the channel Nto generate nr bits of sharedsecret key, (κABE)⊗nr
J. M. Renes IQING 5
Quantum Communication and Cryptography
Entanglement and Secret Keys
Typical Sequences and Subspaces
In the asymptotic case we use the power of typical sequences and sets.
Consider coin with Prheads = p:
After n → ∞ tosses, only the set of sequences for which#heads ≈ np has nonnegligible probability
Moreover, sequences occur with essentially equal probability
Roughly 2nH2(p) such sequences, for binary entropy H2
Apply the same to density matrices by working in the eigenbasis:
ρ has eigenvalues {p,1−p}
à ρ⊗n has support on subspace of dimension 2nH2(p) = 2nS(ρ)
à ρ⊗n is essentially a projector onto this subspace
J. M. Renes IQING 5
Quantum Communication and Cryptography
Entanglement and Secret Keys
Secret Key Distillation: Information Reconciliation
Start with n copies of ψABE =∑
k pk |k〉〈k|A ⊗ψBEk
First correct errors: Bob doesn’t know k exactly. ⇒ HSW Theorem!
Alice k strings Bob state support
Bob cannot reliably distinguish the ψBk . Can distinguish elements of a random subset!
The ψBk are essentially projectors of dimension ≈ 2nH(B| K)
Average state ψB lives on a space of size ≈ 2nH(B)
à Intuitively we can pack in 2nI(K:B) disjoint ψBk
Hence, Alice projects onto a random subset of this size, and tells Bob which one.
J. M. Renes IQING 5
Quantum Communication and Cryptography
Entanglement and Secret Keys
Secret Key Distillation: Information Reconciliation
Start with n copies of ψABE =∑
k pk |k〉〈k|A ⊗ψBEk
First correct errors: Bob doesn’t know k exactly. ⇒ HSW Theorem!
Alice k strings Bob state support
Bob cannot reliably distinguish the ψBk . Can distinguish elements of a random subset!
The ψBk are essentially projectors of dimension ≈ 2nH(B| K)
Average state ψB lives on a space of size ≈ 2nH(B)
à Intuitively we can pack in 2nI(K:B) disjoint ψBk
Hence, Alice projects onto a random subset of this size, and tells Bob which one.
J. M. Renes IQING 5
Quantum Communication and Cryptography
Entanglement and Secret Keys
Secret Key Distillation: Privacy Amplification
Now perform privacy amplification to remove Eve’s information.
Alice k strings Eve state support
ψEk partially distinguishable; Eve can learn some information about the string k.
Average state ψE is supported on a subspace of size ≈ 2nH(E)
Each ψEk is a projector of dimension ≈ 2nH(E| K)
à Intuitively, only ≈ 2nI(K:E) states are needed to cover the support of the average
Hence, if we label every state in the “cover” with the same secret key letter, Eve willhave no information about it.
J. M. Renes IQING 5
Quantum Communication and Cryptography
Entanglement and Secret Keys
Secret Key Distillation: Privacy Amplification
Now perform privacy amplification to remove Eve’s information.
Alice k strings Eve state support
ψEk partially distinguishable; Eve can learn some information about the string k.
Average state ψE is supported on a subspace of size ≈ 2nH(E)
Each ψEk is a projector of dimension ≈ 2nH(E| K)
à Intuitively, only ≈ 2nI(K:E) states are needed to cover the support of the average
Hence, if we label every state in the “cover” with the same secret key letter, Eve willhave no information about it.
J. M. Renes IQING 5
Quantum Communication and Cryptography
Entanglement and Secret Keys
Putting it all together
+ How do we know these sorts of HSW and PA codes really exist?
Invoke the usual trick due to Shannon: Show that random codingworks with high probability! The codewords km,s are such that:
Bob can distinguish the ψBm,s
Coarse-graining over s leaves Eve with indistinguishable ψEm
Rate is r = I(A〉B) = I(K : B) − I(K : E)Moreover, Alice only has to communicate nH(K|B) bits to Bob.
+ For secret key generation, just feed a code into to the channel.
+ Entanglement case: make states & operations coherentStart with |Ψ〉ABE
=∑
k√
pk |k〉A ⊗ |ψk〉BE in the static casecoherently project onto one of the random HSW subsets. . .convert static to dynamic as before
J. M. Renes IQING 5
Quantum Communication and Cryptography
Entanglement and Secret Keys
Private State Distillation
Can instead construct PA codes based on the uncertainty principle!
1 From ρAA ′BB ′, measure ZA and ZB to create a key
Key is private iff Bob+A ′ can predict XA mmt (think entanglement)Defines the set of private states Horodecki3 and Oppenheim, PRL 94, 160502 (2005)
UP: Bob’s knowledge of XA constrains Eve’s knowledge of ZA.
2 If ρBB ′x gives only partial information. . .
Use HSW theorem again!Alice picks a random subset, erasing info Bob doesn’t have
3 Watch out!XA measurement is hypothetical, isn’t actually performedFigure out what key would have been. . .If actual key distillation and XA measurement commute, okNoisy preprocessing examined in: J. M. Renes and G. Smith, PRL 98, 020502 (2007).General shield (but using only linear functions) considered in:J. M. Renes and J.-C. Boileau, quant-ph/0702187.
J. M. Renes IQING 5
Quantum Communication and Cryptography
Erasure is Fundamental
Destruction is a relatively indiscriminate goal
From FQSW:“In contrast to most proofs in information theory, instead of showing how to establish perfectcorrelation of some kind between the sender and the receiver, our proof proceeds by showing thatthe protocol destroys all correlation between the sender and a reference system. Since destructionis a relatively indiscriminate goal, the resulting proof is correspondingly simple.”
Quantum capacity strategy: P. Hayden, M. Horodecki, J. Yard, and A. Winter, quant-ph/0702005
View problem as sending entanglement with high fidelityPick a random subspace at the encoderDecoupling from the environment ensures a decoder
Decoupling implies (good) decoding
Given a channel N T→R, purify it to a unitary VT→REN by using an additional
system E. Now put half a maximally entangled state |Φ〉AT through thechannel, producing |ψ〉ARE
= VT→REN |Φ〉AT.
Then there exists a decoding map DR→B such that
F(|Φ〉AB ,1A ⊗D ◦ N (ΦAT)
)≥ 1 −
∥∥ψAE − 1A/dA ⊗ψE∥∥
1
J. M. Renes IQING 5
Quantum Communication and Cryptography
Erasure is Fundamental
One-shot Version to Memoryless Channels
Use a random subspace of the appropriate size.
Suppose |ψ〉ARE comes out of the channel such that ψA ismaximally-mixed. (Put in an entangled state. . . )
Now project onto a subspace A using ΠA→A, obtaining ψAEΠ .
By first applying a unitary we can use Π to project onto any subspacewe like: ψAE
U ∝ (ΠU ⊗ 1E)ψAE(U†Π⊗ 1
E)
Averaging over all choices of Π yields∫U(A)
dU∥∥∥ψAE
U − 1A/dA ⊗ψ
EU
∥∥∥1≤
√|A| |E| Tr[(ψAE)2]
à RHS small ⇒ channel output decoupled ⇒ decoding operation
à Encoding operation is UᵀΠ, since original input maximally-entangled.
à For the case of block inputs to a memoryless channel, we can evaluatethe rhs by using typical sequences and subspaces. . .
J. M. Renes IQING 5