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arXiv:2204.02691 · Physical Review Research

Scalable implementation of (d+1) mutually unbiased bases for d -dimensional quantum key distribution

Takuya Ikuta, Seiseki Akibue, Yuya Yonezu, Toshimori Honjo, Hiroki Takesue, Kyo Inoue

Published Oct 12, 2022• 15 citations
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Abstract

A scalable implementation of ($d$ + 1) mutually unbiased bases is proposed for prime power dimensions, where these bases can ensure a security of a robust $d$-dimensional quantum key distribution protocol.

DOI
10.1103/physrevresearch.4.l042007

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A scalable implementation of (ddd + 1) mutually unbiased bases is proposed for prime power dimensions, where these bases can ensure a security of a robust ddd-dimensional quantum key distribution protocol.
We investigate the advantages of high-dimensional encoding for a quantum key distribution protocol. In particular, we address a BBM92-like protocol where the dimension of the systems can be larger than two and more than two mutually unbiased bases (MUBs) can be employed. Indeed, it is known that, for a system whose dimension ddd is a prime or the power of a prime, up to d+1d+1d+1 MUBs can be found. We derive an analytic expression for the asymptotic key rate when d+1d+1d+1 MUBs are exploited and show the effects of using different numbers of MUBs on the performance of the protocol. Then, we move to the non-asymptotic case and optimize the finite key rate against collective and coherent attacks for generic dimension of the systems and all possible numbers of MUBs. In the finite-key scenario, we find that, if the number of rounds is small enough, the highest key rate is obtained by exploiting three MUBs, instead of d+1d+1d+1 as one may expect.
We study a general quantum key distribution protocol in higher dimension. In this protocol, quantum states in arbitrary g+1g+1g+1 ($1\ensuremath{\le}g\ensuremath{\le}d$) out of all d+1d+1d+1 mutually unbiased bases in a ddd-dimensional system can be used for the key encoding. This provides a natural generalization of the quantum key distribution in higher dimension and recovers the previously known results for g=1g=1g=1 and ddd. In our investigation, we study Eve's attack by two slightly different approaches. One is considering the optimal cloner of Eve, and the other, defined as the optimal attack, is maximizing Eve's information. We derive results for both approaches and show the deviation of the optimal cloner from the optimal attack. With our systematic investigation of the quantum key distribution protocols in higher dimension, one may balance the security gain and the implementation cost by changing the number of bases in the key encoding. As a side product, we also prove the equivalency between the optimal phase covariant quantum cloning machine and the optimal cloner for the $g=d\ensuremath{-}1$ quantum key distribution.
We present an experimental study of higher-dimensional quantum key distribution protocols based on mutually unbiased bases, implemented by means of photons carrying orbital angular momentum. We perform (d+1d+1d+1) mutually unbiased measurements in a classically simulated prepare-and-measure scheme and on a pair of entangled photons for dimensions ranging from d=2d=2d=2 to 5. In our analysis, we pay attention to the detection efficiency and photon pair creation probability. As security measures, we determine from experimental data the average error rate, the mutual information shared between the sender and receiver, and the secret key generation rate per photon. We demonstrate that increasing the dimension leads to an increased information capacity as well as higher key generation rates per photon. However, we find that the benefit of increasing the dimension is limited by practical implementation considerations, which in our case results in deleterious effects observed beyond a dimension of d=4d=4d=4.