Unconditional%20Security%20of%20the%20Bennett%201992%20quantum%20key-distribution%20protocol%20over%20a%20lossy%20and%20noisy%20channel

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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),
Norbert Lütkenhaus (Univ. of Erlangen-Nürnberg,
Max Plank Research Group), and Nobuyuki Imoto
(Sokendai, Creat, Sorst, NTT)
2
Summary of my talk

• B 92 QKD Protocol
• Outline of the proof
• Examples of the security
• Summary and Conclusion.

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No Eve, noises and losses case (B92)
Alice
Bob
?
0
1
or
encoder
or
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 !
4
The effects of noises or Eve
noise
Bob
noise
Alice
0
encoder
or
0?
noise
Eve
Noises, Eavesdropping ? error, information
leakage
For security
All noises are induced by Eve
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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.
An ideal photon counter that discriminates single
photon one hand and multi-photon or single photon
on the other hand.
Bob
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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
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Entanglement Distillation Protocol (By CSS Code)
(by Shor and Preskill 2000)
Alice
Bob
Relative bit error position
Public ch
Syndrome measurement
Syndrome measurement
Relative phase error position
Error correction
pairs of a Bell state
Sharing
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Protocol 1 (Secure)
Alice
Bob
Eve
Single photon state
Broadcasting the filtering succeeded or not
Bit and phase error estimation
Quantum error correction
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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)
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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
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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
.
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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
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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
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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.
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Derivation of the B92 measurement from that in
the Protocol 1
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The phase error rate estimation from the bit
error rate
1
0
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.
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The bit error and the phase error have a
correlation !!
Nonorthogonal
subspace spanned by
Qubit space
subspace spanned by
for given
Upper bound of
?
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Consider any -qubit state that is symmetric
under any permutation
Question
s
s
s
s
s
s
s
s
a
a
a
a
ß
ß
ß
ß
Untested bit
Test bit
s
, how much is
For given
a
s
?
the upper bound of
ANS,
ß
a
ß
For the estimation, we are allowed to regard the
state as having stemmed from Independently and
Identically Distributed quantum source !
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s
s
s
s
s
s
s
s
a
a
a
a
ß
ß
ß
ß
unitary operator corresponds to permutation of
M qubit
M qubit state that is symmetric under any
permutation
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M qubit space can be decomposed as
unitary operator corresponds to permutation of
M qubit
M qubit state that is symmetric under any
permutation
s
s
s
s
s
s
s
s
j1
j0
a
a
a
a
ß
ß
ß
ß
ba
number of qubits measured in b basis
b?
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The class of the eavesdropping
Individual Attack
Coherent Attack (General Attack)


, and Eves measurement is arbitrary.
,
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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