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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) ,

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slide1

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)

summary of my talk
Summary of my talk
  • B 92 QKD Protocol
  • Outline of the proof
  • Examples of the security
  • Summary and Conclusion.
slide3

No Eve, noises and losses case (B92)

Alice

Bob

?

0

1

or

or

encoder

0

1

Quantum Ch

?

Bob tells Alice whether the outcome is conclusive

or not over the public ch.

0,1: conclusive

Alice and Bob share identical bit values !

slide4

The effects of noises or Eve

noise

noise

Bob

Alice

0

encoder

or

0?

noise

Eve

Noises, Eavesdropping → error, information leakage

For security

All noises are induced by Eve

slide5

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: A single photon source.

Bob:

An ideal photon counter that discriminates single photon one hand and multi-photon or single photon on the other hand.

slide6

Outline of the security proof of the B92

Key words: Error correction, Bell state,

Entanglement distillation protocol (EDP)

Protocol 1 (Secure)

(Equivalent with respect to key distribution)

The B92

slide7

Entanglement Distillation Protocol (By CSS Code)

(by Shor and Preskill 2000)

Alice

Bob

Relative bit error position

Syndrome

measurement

Public ch

Syndrome

measurement

Relative phase error position

Error correction

Sharing

pairs of a Bell state

slide8

Protocol 1 (Secure)

Alice

Bob

Eve

Single photon state

Broadcasting the filtering succeeded or not

Bit and phase error estimation

Quantum error correction

slide9

Error estimations on the Protocol 1

Alice

Bob

Test bits

Test bits

Eve

Phase error rate and bit error rate is not independent

Phase error rate is estimated by bit error rate (the Protocol 1 is secure)

slide10

Outline of the security proof of the B92

Key words: Error correction, Bell state,

Entanglement distillation protocol (EDP)

Protocol 1 (Secure)

(Equivalent with respect to key distribution)

The B92

slide11

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 .

slide12

Protocol 1 (Secure)

No need for phase error correction (Shor and Preskill)

Alice

Eve

Bob

Equivalent !

Randomly chosen

Classical data processing

(error correction, privacy

amplification)

Classical data processing

(error correction, privacy

amplification)

Eve

slide13

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

slide14

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.

slide16

The phase error rate estimation from the bit error rate

0

1

Test bits

0

0

Test bits

1

0

Alice

Bob

1

1

0

1

gedanken

gedanken

0

0

Note: It is dangerous to put some assumptions on the state.

slide17

The bit error and the phase error

have a correlation !!

Nonorthogonal

: subspace spanned by

Qubit space

: subspace spanned by

for given

Upper bound of

?

slide18

Consider any -qubit state that is symmetric under any permutation

Question

σ

σ

σ

σ

σ

σ

σ

σ

α

α

α

α

β

β

β

β

Untested bit

Test bit

σ

, how much is

For given

α

?

σ

the upper bound of

ANS,

β

α

β

For the estimation, we are allowed to regard the state as having stemmed from Independently and Identically Distributed quantum source !

slide19

σ

σ

σ

σ

σ

σ

σ

σ

α

α

α

α

β

β

β

β

: unitary operator corresponds to permutation of M qubit

M qubit state that is symmetric under any permutation

slide20

M qubit space can be decomposed as

: unitary operator corresponds to permutation of M qubit

M qubit state that is symmetric under any permutation

σ

σ

σ

σ

σ

σ

σ

σ

j=1

j=0

α

α

α

α

β

β

β

β

b=α

: number of qubits measured

in b basis

b=Β

slide21

The class of the eavesdropping

Individual

Attack

Coherent

Attack

(General

Attack)

・ ・

・ ・

, and Eve’s measurement is arbitrary.

,

slide22

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

??

0100110101101

0100110101101

Alice

Bob

Eve