Security of practical quantum cryptography with heralded single

photon sources

Mikołaj Lasota1, Rafał Demkowicz-Dobrzański2, Konrad Banaszek2

1Nicolaus Copernicus University, Torun, Poland2University of Warsaw, Warsaw, Poland

Problems with practical realisation of quantum cryptography protocols

Setup imperfection:- fibers: photon losses

- detectors: dark counts, limited detection efficiency

- Single photon sources: multiphoton pulses

• G. Brassard, N. Lutkenhaus, T. Mor, B. Sanders; Phys. Rev. Lett. 85, 1330 (2000)

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Problems with practical realisation of quantum cryptography protocols

Setup imperfection:

- fibers: photon losses

- detectors: dark counts,

limited detection efficiency

- Single photon sources:

multiphoton pulses

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n

np

n

3.0n

Using heralded single photon source in quantum cryptography

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In the case of multimode SPDC process:

Using heralded single photon source in quantum cryptography

• Definition: - probability of exactly one click in the heralding detection system, while there were „i” pairs of photons generated by Alice’s source

• Ideally: - -

• In reality we have due to: - dark counts- limited detection efficiency- losses- partial photon number resolution

iq

020 qq11 q

0,1,0 210 qqq

Minimal transmission of the channel, required for QKD security

• Explicit formula for depends on:- the protocol used by Alice and Bob- the list of assumptions about Eve’s possibilities of attack

• Ideal single photon source:~ probability of a dark count in Bob’s detector

• Attenuated laser as a source of single photons:~ (probability of a dark count in Bob’s detector)1/2

• Heralded single photon source:

21

20minminmin q

qqTTT WCPSGLHSPS

minT

SGLTmin

WCPTmin

Key generation rate• Definition: the amount of bits of secure key produced by a given

setup per unit of time

• Motivation: not only the maximal distance, but also the speed of QKD is important

• General formula for key generation rate:

- - repetition rate of Alice’s source

- - probability of a click in Bob’s detector when Alice’s source emits a pulse

- - probability of accepting the bit by Alice and Bob during the stage of sifting (basis reconciliation)

- - mutual information between X and Y

AEABsift IIppRk exp

expp

siftp

R

XYI

Key generation rate – dependence on complete transmission of the channel

(Alice’s detector: efficiency - 60%, dark counts probability – 10-6, Bob’s detector: dark counts probability – 10-5)

Multiplexing detector with n stages as additional detection system

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1. stage 2. stage

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Effective detection efficiency:

Key generation rate for a multiplexing detection system with n stages

Key generation rate for a multiplexing detection system with n stages

Key generation rate – comparison between WCP and HSPS

• Approximately, in the absence of dark counts:

• For the multiplexing detection system considered here:

• Conclusion: for we can increase key generation rate for large values of using HSPS source with multiplexing detection system only if we have

WCPHSPS Kq

qK

2

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Key generation rate for WCP and HSPS cryptography

Key generation rate for WCP and HSPS cryptography

Conclusions

• For short distances HSPS cryptography with multiplexing can beat WCP only if we have binary detectors with very good detection efficiency

• For intermediate distances HSPS cryptography with multiplexing is better than HSPS with single binary detector

• For long distances (close to the maximal distance of security) HSPS cryptography with single binary detector is the best

WCPHSPS Kq

qK

2

21

21

20minminmin q

qqTTT WCPSGLHSPS

Large transmissions

Short distances

Low transmissions

Long distances