Security of quantum key distribution with detection-efficiency mismatch in the multiphoton case

Anton Trushechkin

Steklov Mathematical Institute of RAS, Steklov International Mathematical Center, Moscow 119991, Russia
Department of Mathematics and NTI Center for Quantum Communications, National University of Science and Technology MISIS, Moscow 119049, Russia
QRate, Skolkovo, Moscow 143025, Russia

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Detection-efficiency mismatch is a common problem in practical quantum key distribution (QKD) systems. Current security proofs of QKD with detection-efficiency mismatch rely either on the assumption of the single-photon light source on the sender side or on the assumption of the single-photon input of the receiver side. These assumptions impose restrictions on the class of possible eavesdropping strategies. Here we present a rigorous security proof without these assumptions and, thus, solve this important problem and prove the security of QKD with detection-efficiency mismatch against general attacks (in the asymptotic regime). In particular, we adapt the decoy state method to the case of detection-efficiency mismatch.

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Quantum key distribution (QKD) allows two distant parties to establish a common secret key for confidential messaging. In view of possible threat to the traditional public key from fault-tolerant quantum computers, QKD is believed to be an important part of future secure communication infrastructure. The first QKD protocol (named BB84) was discovered by Bennett and Brassard in 1984. Later first experiments and security proofs were proposed. Now QKD is a commercial technology with developed security proofs.

However, security proofs which take into account certain imperfection of hardware devices are still challenging. One of such imperfections is so called detection-efficiency mismatch, where two single-photon detectors have different quantum efficiencies, i.e., different probabilities of photon detection. Such a problem should be taken into account because it is practically impossible to make two absolutely identical detectors.

Mathematically, security proof for QKD with detection-efficiency mismatch for the general case is challenging because the Hilbert space we deal with is infinite-dimensional (a reduction to a finite-dimensional space that is possible for the case of identical detectors does not work here). So, fundamentally new approaches to prove the security was required. The main new method proposed in this work is an analytical bound of the number of multiphoton detection events using the entropic uncertainty relations. This allows us to reduce the problem to a finite-dimensional one. For the analytical solution of the finite-dimensional problem (which is still non-trivial), we propose to use symmetries of the problem.

Thus, in this paper, we proof the security of the BB84 protocol with detection-efficiency mismatch and analytically derive bounds for the secret key rate in this case. Also we adapt the decoy state method to the case of detection-efficiency mismatch.

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► References

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[3] Aleksandr Khmelev, Alexey Duplinsky, Ruslan Bakhshaliev, Egor Ivchenko, Liubov Pismeniuk, Vladimir Mayboroda, Ivan Nesterov, Arkadiy Chernov, Anton Trushechkin, Evgeniy Kiktenko, Vladimir Kurochkin, and Aleksey Fedorov, "Eurasian-scale experimental satellite-based quantum key distribution with detector efficiency mismatch analysis", Optics Express 32 7, 11964 (2024).

[4] Sukhpal Singh Gill, Adarsh Kumar, Harvinder Singh, Manmeet Singh, Kamalpreet Kaur, Muhammad Usman, and Rajkumar Buyya, "Quantum Computing: A Taxonomy, Systematic Review and Future Directions", arXiv:2010.15559, (2020).

[5] Yanbao Zhang, Patrick J. Coles, Adam Winick, Jie Lin, and Norbert Lütkenhaus, "Security proof of practical quantum key distribution with detection-efficiency mismatch", Physical Review Research 3 1, 013076 (2021).

[6] Timofei Rusalev, Daniil Stepanenko, and Igor Volovich, "No Violation of Bell-CHSH Inequalities at Large Distances", arXiv:2312.07499, (2023).

[7] Mathieu Bozzio, Adrien Cavaillès, Eleni Diamanti, Adrian Kent, and Damián Pitalúa-García, "Multiphoton and Side-Channel Attacks in Mistrustful Quantum Cryptography", PRX Quantum 2 3, 030338 (2021).

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