Unconditional Security of the Bennett 1992 quantum key-distribution protocol over a lossy and noisy channel

Transcription

1 Unconditional Security of the Bennett 1992 quantum key-distribution protocol over a lossy and noisy channel Kiyoshi Tamaki *Perimeter Institute for Theoretical Physics Collaboration with Masato Koashi (Osaka Univ, Creat, Sorst), Norbert Lütkenhaus (Univ. of Erlangen-Nürnberg, Max Plank Research Group), and Nobuyuki Imoto (Sokendai, Creat, Sorst, NTT) * 1

2 Summary of my talk B 92 QKD Protocol Outline of the proof Examples of the security Summary and Conclusion. 2

3 No Eve, noises and losses case (B92) 0 or 1 Alice encoder Quantum Ch Bob? 1 or 0? Bob tells Alice whether the outcome is conclusive or not over the public ch. 0,1: conclusive Alice and Bob share identical bit values! 3

4 The effects of noises or Eve Alice noise noise Bob 0 encoder noise 0? Eve or Noises, Eavesdropping error, information leakage For security All noises are induced by Eve 4

5 Security proof of the B92 protocol Is the B92 really unconditionally secure? Is the B92 secure against Eve who has unlimited computational power and unlimited technology for state preparations, measurements and manipulations? Assumptions on Alice and Bob Alice: Bob: A single photon source. An ideal photon counter that discriminates single photon one hand and multi-photon or single photon on the other hand. 5

6 Outline of the security proof of the B92 Protocol 1 (Secure) Key words: Error correction, Bell state, Entanglement distillation protocol (EDP) (Equivalent with respect to key distribution) The B92 6

7 Entanglement Distillation Protocol (By CSS Code) (by Shor and Preskill 2000) Alice Bob Syndrome measurement Relative bit error position Public ch Relative phase error position Error correction Syndrome measurement Sharing pairs of a Bell state 7

8 Protocol 1 (Secure) Alice Bob Eve Single photon state Broadcasting the filtering succeeded or not Bit and phase error estimation Quantum error correction 8

9 Error estimations on the Protocol 1 Alice Test bits Eve Test bits Bob Phase error rate and bit error rate is not independent Phase error rate is estimated by bit error rate (the Protocol 1 is secure) 9

10 Outline of the security proof of the B92 Protocol 1 (Secure) Key words: Error correction, Bell state, Entanglement distillation protocol (EDP) (Equivalent with respect to key distribution) The B92 10

11 A brief explanation of the equivalence Main Observation (by shor and Preskill) Only the bit values are important No need for phase error correction Commute! Commute! Alice and Bob are allowed to measure before. 11

12 Protocol 1 (Secure) No need for phase error correction (Shor and Preskill) Alice Eve Bob Randomly chosen Equivalent! Classical data processing (error correction, privacy amplification) Eve Classical data processing (error correction, privacy amplification) 12

13 Example of the security and estimation : L = 0 : L = 0.2 : L = 0.5 : Optimal net growth rate of secret key per pulse : depolarizing rate : the prob that Bob detects vacuum (Loss rate) The vacuum state 13

14 Summary and conclusion We have estimated the unconditionally security of the B92 protocol with single photon source and ideal photon counter. We have shown the B92 protocol can be regarded as an EPP initiated by a filtering process. Thanks to the filtering, we can estimate the phase error rate. Future study Relaxation of the assumptions. Security estimation of B92 with coherent state. 14

15 Derivation of the B92 measurement from that in the Protocol 1 15

16 The phase error rate estimation from the bit error rate Alice Test bits gedanken Test bits Bob gedanken Note: It is dangerous to put some assumptions on the state. 16

17 Nonorthogonal The bit error and the phase error have a correlation!! : subspace spanned by : subspace spanned by Qubit space Upper bound of for given? 17

18 Consider any -qubit state that is symmetric under any permutation Question ANS, Test bit Untested bit For given, how much is the upper bound of? For the estimation, we are allowed to regard the state as having stemmed from Independently and Identically Distributed quantum source! 18

19 : unitary operator corresponds to permutation of M qubit M qubit state that is symmetric under any permutation 19

20 M qubit space can be decomposed as : unitary operator corresponds to permutation of M qubit M qubit state that is symmetric under any permutation j=0 j=1 b= b= : number of qubits measured in b basis 20

21 The class of the eavesdropping Individual Attack Coherent Attack (General Attack),, and Eve s measurement is arbitrary. 21

22 Quantum Key Distribution (QKD) A way to share a random bit string between sender (Alice) and receiver (Bob) whose info leaks arbitrary small to Eve. Quantum Ch Public Ch?? Alice Eve Bob 22