Postselection technique for quantum channels and applications for qkd

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Quantum Key Distribution

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Postselection technique for quantum channels and applications for qkd

  1. 1. Postselection technique for quantum channels with application to QKDchannels with application to QKD Matthias Christandl, University of Munich joint with Robert König and Renato Renner
  2. 2. 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. 3. 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 strings We 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. 4. 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 preserving E completely positive and trace preserving (CPTP)
  5. 5. 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. 6. 6 Formalism: diamond norm • Maximal probability to decide between E and FMaximal 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-F ρρ || ||1 id ρ
  7. 7. 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. 8. 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. 9. 9 Our situation: permutation-covariancep • E is permutation –covariant ifE is permutation covariant if E = π† E π † πρn π ρn π†
  10. 10. 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 Φ id id • Φ is maximally entangled state between symmetric subspace of andsubspace of and |Φi = 1 poly(n) X i |ii|ii where |ii o.n. basis of Symn(H⊗H)
  11. 11. 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
  12. 12. 12 Lemma 1: Maximum is taken on i t tperm.-inv. states ∆:= !|| || || || π∆E-F id ρn!|| ||1 id ρ|| ||1= π ρ|| ||1 π ∆ = π || π id π ∆ id ||1= ρ id || ||1 || id id π ||1
  13. 13. 13 Lemma 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. 14. 14 Purifications are equivalentq || π id π ∆ id ||1 ρ || π id π ∆ id ||1 ρ || id id π ||1 || id id π ||1 || ∆ |||| π || ρ ∆ || id ||1= Ψ= || π id id π ||1 ρ π
  15. 15. 15 Lemma 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. 16. 16 Post-selection ∆ || ∆ || Ψ || || id || id ||1 Ψ Tr Ψ · || id ||1 =poly(n) Φ || ∆ id||≤ poly(n) id ||1 Φ id id
  17. 17. 17 Lemma 4 • Φ is the maximally entangled state fromΦ is the maximally entangled state from Symn(Cd⊗Cd) to a purifying system R ∆ Φ id id
  18. 18. 18 Altogetherg ∆ ρ || ||  π ∆ || ρ ||∆||=max || =max id ρ ||1 || π id id π ||1 ||∆||=max || =max ∆ ∆ ||Ψ ||≤ poly(n) ∆ id≤ || id ||1Ψ ||≤ poly(n) id ||Φ id≤ max ||
  19. 19. 19 QKD: real protocolQ p EveDistribution BobAlice ρn Permutation • chosen at random • communicated to Bob Measurement Classical Communication • Parameter Est. • Error Correction • Privacy Amplif. (SA, SB)
  20. 20. 20 QKD: real protocolQ p EveDistribution BobAlice Distribution ρn Input Permutation Measurement Cl i l ProtocolE Permutation Classical Communication • Parameter Est. E C ti• Error Correction • Privacy Amplif. (SA, SB) Output
  21. 21. 21 QKD: ideal protocolQ p EveDistribution BobAlice Distribution ρn Input Permutation Cl i l ProtocolE Measurement Permutation Classical Communication • Parameter Est. E C ti• Error Correction • Privacy Amplif. (SA, SB) Output S (S, S) Perfect key
  22. 22. 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 Φ id 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. 23. 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 id Φ id|| ||||||≤ trid E-F |||| id ||||≤max
  24. 24. 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. 25. 25 Summaryy • Real versus idealReal versus ideal • perm covariant E-F E F perm. covariant • ||E-F|| ≤ poly(n) || ||1 id Φ id E, 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. 26. 26 Generalisation: arbitrary group actiony g p • For ∆ group-covariant (with Haar measure)For ∆ group covariant (with Haar measure) || || ∆ id• ||∆|| ≤ poly(n) || ||1 id Φ id id
  27. 27. 27 Generalisation: arbitrary group actiony g p • For ∆ group-covariant (with Haar measure)For ∆ group covariant (with Haar measure) || || ∆ id• ||∆|| ≤ dim || ||1 id Φ id id Phys. Rev. Lett. 102, 020504 (2009) arXiv:0809.3019

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