Qkd and de finetti theorem

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Qkd and de finetti theorem

  1. 1. Quantum Key Distribution and de Finetti’s Theorem Matthias Christandl Institute for Theoretical Physics, ETH Zurich June 2010 Matthias Christandl Quantum Key Distribution and de Finetti’s Theorem
  2. 2. Overview Introduction to Quantum Key Distribution Two tools for proving security: De Finetti’s Theorem Post-Selection Technique Summary Matthias Christandl Quantum Key Distribution and de Finetti’s Theorem
  3. 3. Quantum Key Distribution Alice und Bob want to communicate in secrecy, but their phone is tapped. Alice Eve Bob phone If they share key (string of secret random numbers), Alice Eve Bob message +key -------------- cipher cipher - key -------------- message cipher is random and message secure (Vernam, 1926) Matthias Christandl Quantum Key Distribution and de Finetti’s Theorem
  4. 4. Quantum Key Distribution key is as long as the message Shannon (1949): this is optimal secret communication ˆ= key distribution possible key distribution schemes: Alice and Bob meet ⇒ impractical Weaker level of security assumptions on speed of Eve’s computer (public key cryptography) assumptions on size of Eve’s harddrive (bounded storage model) Use quantum mechanical effects (Bennett & Brassard 1984, Ekert 1991) Matthias Christandl Quantum Key Distribution and de Finetti’s Theorem
  5. 5. Quantum Key Distribution Quantum mechanics governs atoms and photons Spin-1 2 system: points on the sphere cos θ eiϕ sin θ = cos θ|0 + eiϕ sin θ|1 ∈ C2 unit of information, the quantum bit or ”qubit” we measure a qubit along a basis if basis is {|0 , |1 }, we obtain ’0’ with probability cos2 θ. in general: express qubit in basis and consider |amplitude|2 . Matthias Christandl Quantum Key Distribution and de Finetti’s Theorem
  6. 6. Quantum Key Distribution The state of two qubits: |ψ ∈ C2 ⊗ C2 , ψ||ψ = 1 |ψ = ψ00|0 ⊗ |0 + ψ01|0 ⊗ |1 + ψ10|1 ⊗ |0 + ψ11|1 ⊗ |1 = ψ00|0 |0 + ψ01|0 |1 + ψ10|1 |0 + ψ11|1 |1 each qubit is measured in basis {|0 , |1 } measurement basis {|0 |0 , |0 |1 , |1 |0 , |1 |1 } obtain ’ij’ with probability |ψij |2 . Matthias Christandl Quantum Key Distribution and de Finetti’s Theorem
  7. 7. Quantum Key Distribution Entangled state of two qubits 1√ 2 (|0 A|0 B + |1 A|1 B) New basis |+ = 1 √ 2 (|0 + |1 ) |− = 1 √ 2 (|0 − |1 ) easy calculation 1 √ 2 (|0 A|0 B + |1 A|1 B) = 1 √ 2 (|+ A|+ B + |− A|− B) Alice and Bob measure in basis {|0 , |1 } ⇒ same result Alice and Bob measure in basis {|+ , |− } ⇒ same result Converse is true, too: same measurement result ⇒ they have state 1√ 2 (|0 A|0 B + |1 A|1 B) Alice and Bob can test whether or not they have the state 1√ 2 (|0 A|0 B + |1 A|1 B)! Matthias Christandl Quantum Key Distribution and de Finetti’s Theorem
  8. 8. Quantum Key Distribution Assume that Alice and Bob have the state |φ AB = 1√ 2 (|0 A|0 B + |1 A|1 B) and measure in the same basis. Can someone else guess the result? No! The measurement result is secure! Total state of Alice, Bob and Eve |ψ ABE = |φ AB ⊗ |φ E , because Alice and Bob have a pure state Eve is not at all correlated with Alice and Bob! Monogamy of entanglement Matthias Christandl Quantum Key Distribution and de Finetti’s Theorem
  9. 9. Quantum Key Distribution The Quantum Key Distribution Protocol Distribution Alice Eve Bob glass fibre 1 1 1 Matthias Christandl Quantum Key Distribution and de Finetti’s Theorem
  10. 10. Quantum Key Distribution The Quantum Key Distribution Protocol Distribution Alice Eve Bob glass fibre 1 2 1 2 1 2 Matthias Christandl Quantum Key Distribution and de Finetti’s Theorem
  11. 11. Quantum Key Distribution The Quantum Key Distribution Protocol Distribution Alice Eve Bob glass fibre 1 2 n 1 2 n 1 2 n Matthias Christandl Quantum Key Distribution and de Finetti’s Theorem
  12. 12. Quantum Key Distribution The Quantum Key Distribution Protocol Distribution Alice Eve Bob glass fibre 1 2 n 1 2 n 1 2 n Measurement with {|0 , |1 } or {|+ , |− } 0 1 1 0 0 1 FE FE FE FE FE FE Matthias Christandl Quantum Key Distribution and de Finetti’s Theorem
  13. 13. Quantum Key Distribution The Quantum Key Distribution Protocol Distribution Alice Eve Bob glass fibre 1 2 n 1 2 n 1 2 n Measurement with {|0 , |1 } or {|+ , |− } 0 1 1 0 0 1 FE FE FE FE FE FE Error-free? |φ AB ? = 1√ 2 (|0 A|0 B + |1 A|1 B ) phone Matthias Christandl Quantum Key Distribution and de Finetti’s Theorem
  14. 14. Quantum Key Distribution The Quantum Key Distribution Protocol Distribution Alice Eve Bob glass fibre 1 2 n 1 2 n 1 2 n Measurement with {|0 , |1 } or {|+ , |− } 0 1 1 0 0 1 FE FE FE FE FE FE Error-free? |φ AB ? = 1√ 2 (|0 A|0 B + |1 A|1 B ) phone If YES: key. If NO: no key Matthias Christandl Quantum Key Distribution and de Finetti’s Theorem
  15. 15. Quantum Key Distribution Proof works as long as |Ψ n ABC = |ψ ⊗n ABE . Alice Bob But why should Eve prepare such a state? Why not the following? Alice Bob We can assume: π|Ψ n ABC = |Ψ n ABC for all π ∈ Sn. Goal: two methods that reduce second to first case! Matthias Christandl Quantum Key Distribution and de Finetti’s Theorem
  16. 16. De Finetti’s Theorem De Finetti’s Theorem (Diaconis and Freedman, 1980) Drawing balls from an urn with or without replacement results in almost the same probability distribution. If k are drawn out of n, then ||Pk − i pi Q×k i ||1 ≤ const k n . Matthias Christandl Quantum Key Distribution and de Finetti’s Theorem
  17. 17. De Finetti’s Theorem Quantum generalisations by Størmer, Hudson & Moody, and Werner et al.. (n = ∞) Quantum De Finetti Theorem Christandl, K¨onig, Mitchison, Renner, Comm. Math. Phys. 273, 473498 (2007) Let |Ψ n be a permutation-invariant state: π|Ψ n = |Ψ n for all π ∈ Sn, then ||ρk − i pi |ψ ψ|⊗k i ||1 ≤ const k n Matthias Christandl Quantum Key Distribution and de Finetti’s Theorem
  18. 18. De Finetti’s Theorem Alice and Bob select a random sample of pairs (after pairs have been distributed!) Alice BobAlice Bob Quantum de Finetti ⇒ can use proof from before (tensor product) ⇒ proof of the security of Quantum Key Distribution! Matthias Christandl Quantum Key Distribution and de Finetti’s Theorem
  19. 19. De Finetti’s Theorem Closer look: deviation from perfect key (due to quantum de Finetti theorem) ≈ k/n n: number of pairs that Eve distributed k: number of bits of key key rate r ≈ k/n ≈ ≈ 0 ⇒ not good enough need replacement for de Finetti theorem Renner’s exp. de Finetti theorem, involved, non-optimal Matthias Christandl Quantum Key Distribution and de Finetti’s Theorem
  20. 20. Post-Selection Technique Statistical process Λ Input: n-bit string Output: success or failure Lemma If for any i.i.d. distribution Prob[failure] ≤ , then Prob[failure] ≤ (n + 1) for any permutation-invariant distribution. Typically, ≈ 2−αn , in information-theoretic tasks. Matthias Christandl Quantum Key Distribution and de Finetti’s Theorem
  21. 21. Post-Selection Technique Proof: P permutation-invariant distribution on n bits: P = n k=0 pkQk. Qk equal probability for all strings with k zeros. Take P as n-fold i.i.d distribution, where ’0’ has probability r. r*n k p_k Certainly, prn ≥ 1/(n + 1) ⇒ Qrn ≤ (n + 1)Pr . Matthias Christandl Quantum Key Distribution and de Finetti’s Theorem
  22. 22. Post-Selection Technique Qrn ≤ (n + 1)Pr . Prob[failure]P = k pkProb[failure]Qk ≤ k pk(n + 1)Prob[failure]Pk/n ≤ (n + 1) Matthias Christandl Quantum Key Distribution and de Finetti’s Theorem
  23. 23. Post-Selection Technique Quantum operation Λ Input: n-qubit state Output: success or failure (classical bit) Post-selection Technique Christandl, K¨onig, Renner, Phys. Rev. Lett. 102, 020504 (2009) For any input |Ψ n = |ψ ⊗n Prob[failure] ≤ , then Prob[failure] ≤ n3 for any state permutation-invariant state |Ψ . Typically, ≈ 2−αn , in information-theoretic tasks. Matthias Christandl Quantum Key Distribution and de Finetti’s Theorem
  24. 24. Post-Selection Technique The Quantum Key Distribution Protocol Distribution |Ψ n ABC = |ψ ⊗n ABE Alice Eve Bob glass fibre 1 2 n 1 2 n 1 2 n Measurement with {|0 , |1 } or {|+ , |− } 0 1 1 0 0 1 FE FE FE FE FE FE Error-free? phone If YES: key. If NO: no key By assumption: Prob[failure] ≤ Matthias Christandl Quantum Key Distribution and de Finetti’s Theorem
  25. 25. Post-Selection Technique The Quantum Key Distribution Protocol Distribution |Ψ n ABC permutation-invariant (or general) Alice Eve Bob glass fibre 1 2 n 1 2 n 1 2 n Measurement with {|0 , |1 } or {|+ , |− } 0 1 1 0 0 1 FE FE FE FE FE FE Error-free? phone If YES: key. If NO: no key Post-selection tech.: Prob[failure] ≤ poly(n) ≈ poly(n)2−δ2n Matthias Christandl Quantum Key Distribution and de Finetti’s Theorem
  26. 26. Post-Selection Technique security proof against the most general attacks optimal security parameters relevant in current experiments (since n ≈ 105 ) Eve’s best attack |Ψn ABE = |ψABE ⊗n conceptual and technical simplification of security proofs Other applications: Quantum Reverse Shannon Theorem Berta, Christandl and Renner, arXiv:0912.3805 Matthias Christandl Quantum Key Distribution and de Finetti’s Theorem
  27. 27. Summary Quantum Key Distribution Alice BobAlice Bob Quantum de Finetti De Finetti’s Theorem Post-Selection Technique optimal security parameters applications outside quantum cryptography: quantum Shannon theory and quantum tomography Matthias Christandl Quantum Key Distribution and de Finetti’s Theorem

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