# Security of quantum key distribution with intensity correlations

Víctor Zapatero1, Álvaro Navarrete1, Kiyoshi Tamaki2, and Marcos Curty1

1Escuela de Ingeniería de Telecomunicación, Department of Signal Theory and Communications, ­University of Vigo, Vigo E-36310, Spain
2Faculty of Engineering, University of Toyama, Gofuku 3190, Toyama 930-8555, Japan

### Abstract

The decoy-state method in quantum key distribution (QKD) is a popular technique to approximately achieve the performance of ideal single-photon sources by means of simpler and practical laser sources. In high-speed decoy-state QKD systems, however, intensity correlations between succeeding pulses leak information about the users' intensity settings, thus invalidating a key assumption of this approach. Here, we solve this pressing problem by developing a general technique to incorporate arbitrary intensity correlations to the security analysis of decoy-state QKD. This technique only requires to experimentally quantify two main parameters: the correlation range and the maximum relative deviation between the selected and the actually emitted intensities. As a side contribution, we provide a non-standard derivation of the asymptotic secret key rate formula from the non-asymptotic one, in so revealing a necessary condition for the significance of the former.

Quantum key distribution enables secure and remote delivery of cryptographic keys based on the laws of quantum mechanics. The main practical interest of QKD is that, when combined with the one-time-pad encryption scheme, it provides information-theoretic security, unconcerned about future (classical or quantum) computational advances of the adversary. For this reason, QKD has experienced a tremendous development both in theory and in practice since its conception, in so becoming the most mature application of quantum information science.

Nevertheless, despite this progress, various challenges must be addressed in order to achieve the widespread adoption of QKD. In particular, fiber-based QKD schemes are the most widely deployed, and due to the low transmissivity of the signals in the optical fibers (which decreases exponentially with the fiber length), one major challenge is to achieve high secret key rates at long distances. For this purpose, one possibility is to increase the repetition rate of the laser source in the transmitter station. Nevertheless, for clock rates of the order of GHz, intensity correlations between succeeding pulses appear, possibly opening a security loophole. To be precise, in the absence of ideal single-photon sources, most QKD protocols use lasers that operate emitting phase-randomised weak coherent pulses (PRWCPs). These sources allow the QKD users to implement the so-called decoy-state method, a standard procedure to tightly lower bound the extractable secret key length of a QKD session. Importantly, the central assumption of decoy-state-based QKD is that, for any given signal, its detection probability conditioned on the emission of a certain number of photons does not depend on the intensity of the pulse, i.e., on its mean photon number. However, this assumption fails in the presence of a side channel leaking information about the intensity setting selected in each protocol round, and for GHz (or higher frequency) repetition rates, one such side channel is represented by intensity correlations. Intuitively, an adversary could exploit the correlations to gain information about previous intensity settings, which would allow her to make the $n$-photon click and error probabilities dependent on the setting. This being the case, and aiming to develop ultrafast decoy-state-based QKD systems, the question arises of how to account for arbitrary intensity correlations in the security analysis. In this regard, the existing results are notably restricted, as they mostly consider setting-choice-independent correlations which do not capture the major threat.

In this work, we solve this pressing problem by developing the missing security analysis for decoy-state QKD with arbitrary intensity correlations. The central idea —originally used in the context of Trojan horse attacks in QKD— is to pose a restriction on the maximum bias that Eve can induce between the $n$-photon click and error probabilities associated to different intensity settings. Particularly, we accomplish this task using a fundamental result that essentially follows from the Cauchy-Schwarz (CS) inequality in complex Hilbert spaces. This result, which we refer to as the CS constraint, provides tighter bounds on the indistinguishability of non-orthogonal quantum states than the well-known trace distance argument. Also, despite being asymptotic, our approach is experimental friendly because of two reasons. On the one hand, it fully characterizes the intensity correlations via two well-understood physical parameters: the correlation range, $\xi$, and the maximum relative deviation, $\delta_{\rm max}$, between the selected intensity settings and the actually emitted intensities. On the other hand, it allows for improved secret key rates in case a specific correlations model is known to describe the intensity modulator.

As a related contribution of this work, we provide a non-standard derivation of the asymptotic secret key rate formula, relying on a statistical condition that becomes non-trivial in the most general context of coherent attacks and arbitrary intensity correlations. For the problem at hand, the asymptotic formula tolerates a restricted type of coherent attacks that we characterize in detail.

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### Cited by

[1] Akihiro Mizutani and Go Kato, "Security of round-robin differential-phase-shift quantum-key-distribution protocol with correlated light sources", Physical Review A 104 6, 062611 (2021).

[2] Hua-Jian Ding, Xing-Yu Zhou, Chun-Hui Zhang, Jian Li, and Qin Wang, "Measurement-device-independent quantum key distribution with insecure sources", arXiv:2107.07803.

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