Postselection technique for quantum channels and applications for qkd
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Postselection technique for quantum channels and applications for qkd

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Quantum Key Distribution

Quantum Key Distribution

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Postselection technique for quantum channels and applications for qkd Postselection technique for quantum channels and applications for qkd Presentation Transcript

  • 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 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 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 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 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 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 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 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 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 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 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 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 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 Post-selection ∆ || ∆ || Ψ || || id || id ||1 Ψ Tr Ψ · || id ||1 =poly(n) Φ || ∆ id||≤ poly(n) id ||1 Φ id 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 ∆ Φ id id
  • 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 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 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 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 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 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 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 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 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 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