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Asymptotic security of the four-state continuous-variable quantum key distribution protocol

Shouvik Ghorai, Philippe Grangier, Eleni Diamanti, Anthony Leverrier

Published Feb 4, 2019• 0 citations
OpenAlex

Abstract

We establish a lower bound on the asymptotic secret key rate of continuous-variable quantum key distribution with a discrete modulation of coherent states. The bound is valid against collective attacks and is obtained by formulating the problem as a semidefinite program. We illustrate our general approach with the quadrature phase-shift keying (QPSK) modulation scheme and show that distances over 100 km are achievable for realistic values of noise. We also discuss the application to more complex quadrature amplitude modulations (QAM) schemes. This work is a major step towards establishing the full security of continuous-variable protocols with a discrete modulation in the finite-size regime and opens the way to large-scale deployment of these protocols for quantum key distribution.

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arXiv:2112.00214•New Journal of Physics•Feb 1, 2023•34 citations
We establish a lower bound on the asymptotic secret key rate of continuous-variable quantum key distribution with a discrete modulation of coherent states. The bound is valid against collective attacks and is obtained by formulating the problem as a semidefinite program. We illustrate our general approach with the quadrature phase-shift keying (QPSK) modulation scheme and show that distances over 100 km are achievable for realistic values of noise. We also discuss the application to more complex quadrature amplitude modulations (QAM) schemes. This work is a major step towards establishing the full security of continuous-variable protocols with a discrete modulation in the finite-size regime and opens the way to large-scale deployment of these protocols for quantum key distribution.
We consider discrete-modulation protocols for continuous-variable quantum key distribution (CV-QKD) that employ a modulation constellation consisting of a finite number of coherent states and that use a homodyne- or a heterodyne-detection receiver. We establish a security proof for collective attacks in the asymptotic regime, and we provide a formula for an achievable secret-key rate. Previous works established security proofs for discrete-modulation CV-QKD protocols that use two or three coherent states. The main constituents of our approach include approximating a complex, isotropic Gaussian probability distribution by a finite-size Gauss-Hermite constellation, applying entropic continuity bounds, and leveraging previous security proofs for Gaussian-modulation protocols. As an application of our method, we calculate secret-key rates achievable over a lossy thermal bosonic channel. We show that the rates for discrete-modulation protocols approach the rates achieved by a Gaussian-modulation protocol as the constellation size is increased. For pure-loss channels, our results indicate that in the high-loss regime and for sufficiently large constellation size, the achievable key rates scale optimally, i.e., proportional to the channel's transmissivity.
We establish an analytical lower bound on the asymptotic secret key rate of continuous-variable quantum key distribution with an arbitrary modulation of coherent states. Previously, such bounds were only available for protocols with a Gaussian modulation, and numerical bounds existed in the case of simple phase-shift-keying modulations. The latter bounds were obtained as a solution of convex optimization problems and our new analytical bound matches the results of Ghorai et al. (2019), up to numerical precision. The more relevant case of quadrature amplitude modulation (QAM) could not be analyzed with the previous techniques, due to their large number of coherent states. Our bound shows that relatively small constellation sizes, with say 64 states, are essentially sufficient to obtain a performance close to a true Gaussian modulation and are therefore an attractive solution for large-scale deployment of continuous-variable quantum key distribution. We also derive similar bounds when the modulation consists of arbitrary states, not necessarily pure.
Continuous variable quantum key distribution with discrete modulation has the potential to provide information-theoretic security using widely available optical elements and existing telecom infrastructure. While their implementation is significantly simpler than that for protocols based on Gaussian modulation, proving their finite-size security against coherent attacks poses a challenge. In this work we prove finite-size security against coherent attacks for a discrete-modulated quantum key distribution protocol involving four coherent states and heterodyne detection. To do so, and contrary to most of the existing schemes, we first discretize all the continuous variables generated during the protocol. This allows us to use the entropy accumulation theorem, a tool that has previously been used in the setting of discrete variables, to construct the finite-size security proof. We then compute the corresponding finite-key rates through semi-definite programming and under a photon-number cutoff. Our analysis provides asymptotic rates in the range of 0.1 − 10 − 4 bits per round for distances up to hundred kilometres, while in the finite case and for realistic parameters, we get of the order of 10 Gbits of secret key after n ∼ 10 11 rounds and distances of few tens of kilometres.
This paper introduces an approach to solving the convex optimization problem for subcarrier wave continuous-variable quantum key distribution (QKD) protocol using discrete modulation. For the first time, we provide a solution to this problem. By employing semidefinite programming, we have estimated the lower bound of the secure key generation rate in the asymptotic limit, taking into account an optimal attack from potential eavesdroppers. We considered an optical scheme that involves the use of quadrature phase modulation of coherent states and coherent detection based on the interference between the carrier and side frequencies. Our findings demonstrate the feasibility of distributing secure key generation rates over distances exceeding 100 km, even in the presence of realistic excess noise.
Discrete-modulated (DM) continuous-variable quantum key distribution (CV-QKD) protocols are promising candidates for commercial implementations of quantum communication networks due to their experimental simplicity.While tight security analyses in the asymptotic limit exist, proofs in the finitesize regime are still subject to active research.We present a composable finite-size security proof against independently and identically distributed collective attacks for a general DM CV-QKD protocol.We introduce a new energy testing theorem to bound the effective dimension of Bob's system and rigorously prove security within Renner's -security framework and address the issue of acceptance sets in protocols and their security proof.We want to highlight that our method also allows for nonunique acceptance statistics, which is necessary in practise.Finally, we extend and apply a numerical security proof technique to calculate tight lower bounds on the secure key rate.To demonstrate our method, we apply it to a quadrature phase-shift keying protocol for both untrusted, ideal and trusted, nonideal detectors.The results show that our security proof method yields secure finite-size key rates under experimentally viable conditions up to at least 72 km transmission distance.
Abstract Discrete-modulation continuous-variable (CV) quantum key distribution has the potential for large-scale deployment in secure quantum communication networks owing to its low implementation complexity and compatibility with the current coherent optical telecommunication. However, current discrete-modulation protocols require relatively large constellation sizes to achieve a key rate comparable to that of the Gaussian modulation. Here, we show that a high key rate comparable to the Gaussian modulation can be achieved using only ten or so coherent states by implementing suitable key map and numerical convex optimization techniques. Specifically, the key rate of the two-ring constellation with 12 coherent states (four states in the inner ring and eight states in the outer ring) can reach 2.4 times of that of original quadrature phase shift keying and 70% of the Gaussian modulation protocol at 50 km. Such an approach can easily be applied to existing systems, making the discrete-modulation protocol an attractive alternative for high-rate and low-cost applications in secure quantum communication networks.