Postselection technique for quantum channels with application to QKDchannels with application to QKD

Matthias Christandl, University of Munichjoint with Robert König and Renato Renner

2

Outline

• MotivationMotivation

• FormalismFormalism

• Main Result• Main Result

• Proof• Proof

• Application to Quantum Cryptography• Application to Quantum Cryptography

Summary and Extension• Summary and Extension

3

Motivation: real versus ideal

• Car rideCar ride– In the ideal car ride we have no accident– In the real car ride we might have an accidentg– We still take the car, if real ≈ ideal

• Quantum Key Distributiony– In the ideal QKD scheme, Alice and Bob obtain

identical and perfectly secure strings (a key)– In the real QKD scheme, Alice and Bob may obtain

non-identical and compromised stringsWe still like to use QKD as long as real ≈ ideal– We still like to use QKD as long as real ≈ ideal

– Proving real ≈ ideal is a security proof• Provide tool for comparing real and ideal processes• Provide tool for comparing real and ideal processes

4

Formalism: quantum evolutionq

Eρ σ

E positive and trace preserving

E

ρ σ

id

E ⊗ id positive and trace preserving

id

E ⊗ id positive and trace preservingE completely positive and trace preserving (CPTP)

5

Formalism: quantum evolutionq

• Quantum Evolution– completely positive and trace preserving (CPTP) map E

• Examples• Examples– Quantum protocols (e.g. for QKD) (ideal or real)– Quantum circuits– Time evolution of a system with Hamiltonian H– Car ride (ideal or real)–– …

• Proving real ≈ ideal is done via proving that E ≈ F

– E and F are CPTP maps

• Need for distance measure on CPTP mapsNeed for distance measure on CPTP maps– Diamond norm (Kitaev)

6

Formalism: diamond norm

• Maximal probability to decide between E and F Maximal probability to decide between E and F

p = ½ + ¼ ||E-F||

|| ||E F||E-F||:=maxρ || - ||1

E

id

F

id

ρ ρ

id id

=maxρ || ||1

E-Fρρ || ||1

idρ

7

Formalism: diamond norm

• "If we cannot see a difference they are identical"If we cannot see a difference, they are identical• Operational definition• Strongest notion of distanceStrongest notion of distance• Maximum is difficult to evaluate• Diamond norm is related to completely bounded• Diamond norm is related to completely bounded

norm by duality

8

Our situation: map on n particlesp p

• is CPTPE : B(H⊗n)→ B(H⊗n) is CPTPE : B(H )→ B(H )

Eρn σn

• State of n particles as input• State of n particles as output• State of n particles as output• State space of one particle H ∼= Cd

9

Our situation: permutation-covariancep

• E is permutation –covariant ifE is permutation covariant if

E = π† E π

†πρn π ρn π†

10

Main result

• For E, F permutation-covariant on n particlesFor E, F permutation covariant on n particles

|| ||E-F

• ||E-F|| ≤ poly(n) || ||1

Φid

id

• Φ is maximally entangled state between symmetric subspace of andsubspace of and |Φi = 1

poly(n)

Xi

|ii|ii where |ii o.n. basis of Symn(H⊗H)

11

Proof

• Lemma 1: The maximisation in the diamond norm isLemma 1: The maximisation in the diamond norm is achieved on (purifications of) permutation-invariant states

• Lemma 2: permutation-invariant states have bosonic purifications

• Lemma 3: every bosonic state can be obtained by post-selecting from a fixed state (with probability 1/ l ( ))1/poly(n))

f f f• Lemma 4: this fixed state is the purification of a de Finetti state

12Lemma 1: Maximum is taken on

i t tperm.-inv. states

∆:= !|| || || ||

π∆E-F

idρn!|| ||1

idρ|| ||1= π

ρ|| ||1

π ∆

= π || π id

π ∆

id ||1=ρ

id|| ||1 || id id

π

||1

13Lemma 2: Permutation-invariant t t h B i ifi tistates have Bosonic purifications

• ρ= π ρ π† for all πρ π ρ π for all π• Define |Ψi = (ρ1/2⊗1) |Φi |Φi = |ii|ii• ThenThen

π ⊗ π |Ψi = (π ⊗ π) (ρ1/2⊗1)|Φi

= (ρ1/2⊗1) (π ⊗ π) |Φi= (ρ ⊗1) (π ⊗ π) |Φi

= (ρ1/2⊗1) |Φi

= |Ψi= |Ψi

• Hence |Ψi is bosonic• Hence, |Ψi is bosonic• |Ψi is also a purification of ρ

14

Purifications are equivalentq

|| π id

π ∆

id ||1

ρ || π id

π ∆

id ||1

ρ|| id idπ

||1 || id idπ

||1

||∆

||||π

||ρ∆

||id

||1= Ψ= || π id idπ

||1

ρ

π

15Lemma 3: Post-selection in t l t titeleportation

Tr Ψ ·Φ

Ψ

• Probability of success =1/dim Symn(Cd⊗Cd)=1/poly(n)1/poly(n)

16

Post-selection

∆

||∆

||Ψ

|| ||id

||id

||1Ψ

Tr Ψ ·||

id||1

=poly(n) Φ

||

∆

id||≤ poly(n)

id||1

Φid

id

17

Lemma 4

• Φ is the maximally entangled state fromΦ is the maximally entangled state from Symn(Cd⊗Cd) to a purifying system R

∆

Φid

id

18

Altogetherg

∆ρ || ||

π ∆

||ρ||∆||=max || =max

idρ ||1 || π id id

π

||1||∆||=max || =max

∆

∆

||Ψ ||≤ poly(n)

∆

id≤ ||id

||1Ψ ||≤ poly(n)

id||Φ

id≤ max ||

19

QKD: real protocolQ pEveDistribution

BobAliceρn

Permutation• chosen at random• communicated to Bob

Measurement

Classical Communication• Parameter Est.• Error Correction• Privacy Amplif.

(SA, SB)

20

QKD: real protocolQ p

EveDistribution

BobAlice

Distribution

ρn Input

Permutation

Measurement

Cl i lProtocolE

Permutation

Classical Communication• Parameter Est.

E C ti• Error Correction• Privacy Amplif.

(SA, SB) Output

21

QKD: ideal protocolQ p

EveDistribution

BobAlice

Distribution

ρn Input

Permutation

Cl i lProtocolE

Measurement

Permutation

Classical Communication• Parameter Est.

E C ti• Error Correction• Privacy Amplif.

(SA, SB) OutputS

(S, S) Perfect key

22

QKD: application of main resultQ pp

• For E, F permutation-covariant on n particlesFor E, F permutation covariant on n particles

|| ||E-F

• ||E-F|| ≤ poly(n) || ||1

Φid

id

• We want a bound in terms of tensor product states, not purifications of convex combinations of tensornot purifications of convex combinations of tensor product states → remove second purification

23

QKD: removal of second purificationQ p

• The dimension of the second purification is poly(n)y( )• Shortening the key by 2 log poly(n) bits with privacy

amplification gives

E-FE ' -F '

|| ||Φ

idΦ

id|| ||||||≤trid

E-F

|||| id ||||≤max

24

QKD: collective vs general attacksQ g

• ||E'-F'|| ≤ poly(n) max || ||1

E-F

idid

• This shows that Eve’s optimal strategy is a collective attack (attack each system in the same way)

• The same security parameter by only reducing the k l th b O(l ) bitkey length by O(log n) bits

• Improves over previous analyses using Renner’s exponential de Finetti theoremexponential de Finetti theorem

• Practical relevance (finite key analysis)

25

Summaryy

• Real versus idealReal versus ideal

• perm covariant

E-F

E F perm. covariant

• ||E-F|| ≤ poly(n) || ||1 idΦ

idE, F

id

E-F

||E' F'|| ≤ poly(n) max || ||id• ||E'-F'|| ≤ poly(n) max || ||1

• Security against collective attack implies security against general attacks

26

Generalisation: arbitrary group actiony g p

• For ∆ group-covariant (with Haar measure)For ∆ group covariant (with Haar measure)

|| ||

∆

id• ||∆|| ≤ poly(n) || ||1

idΦ

id

id

27

Generalisation: arbitrary group actiony g p

• For ∆ group-covariant (with Haar measure)For ∆ group covariant (with Haar measure)

|| ||

∆

id• ||∆|| ≤ dim || ||1

idΦ

id

id

Phys. Rev. Lett. 102, 020504 (2009)arXiv:0809.3019