Key-rate enhancement using qutrit states for quantum key distribution with askew aligned sources

Yonggi Jo and Wonmin Son

Phys. Rev. A 94, 052316 – Published 14 November 2016

Abstract

It is known that measurement-device-independent quantum key distribution (MDI-QKD) provides ultimate security from all types of side-channel attack on detectors at the expense of low key rate. In the present study, we propose MDI-QKD using three-dimensional quantum states and show that the protocol improves the secret key rate under the analysis of mismatched-basis statistics. Specifically, we analyze security of the $3 d$ -MDI-QKD protocol with askew aligned sources, meaning that the original sources contain unwanted states instead of the expected one. We evaluate the secret key rate of the protocol and identify the regime in which the key rate is higher than the protocol with the qubit MDI-QKD.

Received 11 June 2016

DOI:https://doi.org/10.1103/PhysRevA.94.052316

Physics Subject Headings (PhySH)

Quantum cryptography

Quantum Information, Science & Technology

Authors & Affiliations

Yonggi Jo¹ and Wonmin Son^1,2

¹Department of Physics, Sogang University, 35, Baekbeom-ro, Mapo-gu, Seoul 04107, Republic of Korea
²Department of Physics, University of Oxford, Parks Road, Oxford OX13PU, United Kingdom

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Issue

Vol. 94, Iss. 5 — November 2016

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Images

Figure 1
A schematic setup to generate three-dimensional quantum states of two bases [33]. In time basis (a), Alice and Bob encode information in time states of the photon. In this setup, the time states are generated with three different delay lines and switch. The time that the photon injected into the output port can be controlled with choice of delay line. $| 0 〉, | 1 〉$ , and $| 2 〉$ denote time states when photon passes through the shortest, the middle, and the longest delay line, respectively. In energy basis (b), Alice and Bob prepare similar settings with phase modulators on delay lines and $1 \times 3$ coupler instead of switch. From Eq. (1), in order to generate the states in the energy basis, ${e^{i α}, e^{i β}}$ should be ${1, 1}, {ω^{2}, ω}$ , and ${ω, ω^{2}}$ to generate $| \bar{0} 〉, | \bar{1} 〉$ , and $| \bar{2} 〉$ , respectively.
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Figure 2
A schematic diagram of MDI-QKD using three-dimensional quantum states ( $3 d$ -MDI-QKD). BSM means Bell-state measurement. In 3-dim MDI-QKD, BSM should be able to discriminate three-dimensional maximally entangled states.
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Figure 3
(a) The secret key rate per sifted key $r$ of original MDI-QKD (black, dashed line) and $3 d$ -MDI-QKD (red line). (b) Secret key rate $R$ per total signal of original qubit MDI-QKD (black, dashed line) and $3 d$ -MDI-QKD (red line). We assumed $3 d$ -BSM can discriminate three maximally entangled states among nine, and qubit BSM can discriminate two Bell states among four. $R$ can be obtained from (sifted key rate) $\times r$ , so from the consideration of success probability of BSM and the probability that Alice and Bob choose same bases, the key rate $R$ of qubit MDI-QKD is $\frac{1}{4} r_{2}$ and that of $3 d$ -MDI-QKD is $\frac{1}{6} r_{3}$ . Details are described in the text. (c) The difference between the secret key rate $R$ of original MDI-QKD and $3 d$ -MDI-QKD. Both secret key rates are obtained under the assumption of askew aligned sources. $ɛ$ is the factor which is defined in Eq. (16). Blue, dashed line shows difference between the secret key rates when the state error rate is fixed as $Q_{s} = 0.01$ ; red line shows the difference when $Q_{s} = 0.05$ .
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Figure 4
The secret key rate of MDI-QKD vs total transmission loss of optical channel. The dark count rate of the single photon detector is assumed as $10^{- 5}$ per pulse. We assume there is no Eve. (a) The secret key rate $r$ per sifted key of $3 d$ -MDI-QKD (blue, dashed line) and $3 d$ -MDI-QKD with askew aligned sources (red line). (b) The secret key rate $r$ per sifted key of original MDI-QKD (black, dashed line) and $3 d$ -MDI-QKD (red line). Both secret key rates are obtained under the assumption of askew aligned sources. The intersection point of the secret key rates appears at the transmission loss 20 dB approximately. (c) The secret key rate $R$ per total signal of original MDI-QKD (black, dashed line) and $3 d$ -MDI-QKD (red line) under the assumption of askew aligned sources. We assume that qubit BSM discriminates two Bell states among four and $3 d$ -BSM does three maximally entangled states among nine. The intersection point appears at transmission loss 10.5 dB approximately.
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