Security of discrete-modulated continuous-variable quantum key distribution
1ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, Av. Carl Friedrich Gauss 3, 08860 Castelldefels (Barcelona), Spain.
2Université de Lyon, Inria, ENS de Lyon, UCBL, LIP, F-69342, Lyon Cedex 07, France.
3ICREA - Institució Catalana de Recerca i Estudis Avançats, 08010 Barcelona, Spain.
| Published: | 2024-07-18, volume 8, page 1418 |
| Eprint: | arXiv:2303.09255v4 |
| Doi: | https://doi.org/10.22331/q-2024-07-18-1418 |
| Citation: | Quantum 8, 1418 (2024). |
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Abstract
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 $n\sim10^{11}$ rounds and distances of few tens of kilometres.
Featured image: Left: Our novel scheme of discretisation of the phase space, which is performed by Bob in parameter estimation rounds. In this figure, the modulation for parameter estimation in phases and amplitudes is given by ∆/δ = 2, with the outmost modules extending to infinity. Right: Bob's modulation in key rounds.
Popular summary
Efficiently distributing such a key between Alice and Bob in a way that they can test whether Eve has any correlations with it, is where quantum physics comes in. Namely, its has been shown by Bennett and Brassard in 1984 [Bennett, Charles H., and Gilles Brassard. "Quantum cryptography: Public key distribution and coin tossing." Theoretical computer science 560 (2014): 7-11.] that a QKD protocol where Alice sends quantum states randomly chosen from two incompatible bases to Bob, who measures the states in one of the bases, which is followed by classical post-processing, can provide a secure key for the one-time-pad protocol. Its security relies on the incompatibility of the bases, which is a basic quantum mechanical property, that prevents Eve from obtaining a significant amount of key without being detected. A number of rigorous security proofs of this so-called BB84 protocol, and many variants, have been provided, even in the case of the most general attacks by Eve, also known as coherent attacks.
While being surprisingly simple in theory, the implementation of QKD has faced, and is still facing, a number of technical challenges. One challenge has been the fact that in its simplest form, QKD relies on single photon states, which are difficult to obtain experimentally. This challenge has been overcome by the use of weak laser pulses, which can be described by coherent states. One way to make use of coherent states is to simulate single photon protocols, such as BB84. A different approach has been taken by Grosshans and Grangier [Grosshans, Frédéric, and Philippe Grangier. "Continuous variable quantum cryptography using coherent states." Physical review letters 88.5 (2002): 057902.], who have devised a protocol completely based on coherent states. In such a protocol, also known as continuous variable QKD (CVQKD), Alice chooses a coherent state, either by a Gaussian distribution, or uniformly from a discrete set, sends it to Bob who the performs either homodyne or heterodyne measurements. The advantage of using such an approach is that it can be implemented using readily available optical elements and existing fibre optical cables.
Proving the security of CVQKD in the case where Alice chooses from a discrete set of coherent states, also known as discrete modulated CVQKD (DMCVQKD), against the most general coherent attacks is a challenging task. In the present work we derive such a security proof. Our main ingredients are a numerical method that has previously only been used to prove security of DMCVQKD against a more restricted class of attacks, known as collective attacks, as well as the entropy accumulation theorem (EAT), which has been derived to proof coherent attack security for a different class of QKD protocols, which relies on entangled states rather than a prepare-and-measure setting. Adapting the EAT to the wide class of prepare-and-measure protocols, which includes our DMCVQKD protocol, has been a challenge on its own. We here overcome this challenge by introducing a method, involving a virtual state tomography, that could be more generally useful to apply the EAT to other prepare-and-measure protocols.
► BibTeX data
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