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Lobachevskii Journal of Mathematics

The Derivation of Operator Inequalities for the BB84 Coherent Protocol in the Three-photon and Two-photon Cases

V. V. Yatsulevich

Published Jun 1, 2025• 0 citations
Publisher

Abstract

Abstract The BB84 quantum key distribution protocol using coherent states with detection efficiency mismatch in active and passive bases is important, as it is used in real-world experiments. It is therefore essential to demonstrate the resilience of various protocol variations against different attacks. However, this is not always straightforward. To demonstrate stability under certain conditions, we need operator inequalities. Therefore, this study aims to derive some operator inequalities for quantum observables in three- and two-photon scenarios.

DOI
10.1134/s199508022560788x

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Abstract The BB84 quantum key distribution protocol using coherent states with detection efficiency mismatch in active and passive bases is important, as it is used in real-world experiments. It is therefore essential to demonstrate the resilience of various protocol variations against different attacks. However, this is not always straightforward. To demonstrate stability under certain conditions, we need operator inequalities. Therefore, this study aims to derive some operator inequalities for quantum observables in three- and two-photon scenarios.
Quantum coherence was recently formalized as a physical resource to measure the strength of superposition. Based on the resource theory, we present a systematic framework that connects a coherence measure to the security of quantum key distribution. By considering a generic entanglement-based key distribution protocol under the collective attack scenario, we show that the key rate can be quantified by the coherence of the shared bipartite states. This framework allows us to derive the key rate of the BB84 and six-state protocols. By utilizing fine-grained parameters, we also derive the improved key rates of both protocols within the coherence-based framework. Furthermore, we apply it to a practical issue, detection efficiency mismatch, and obtain an improved result. In conclusion, this framework demonstrates the interplay among coherence, entanglement, and quantum key distribution at a fundamental level.
One of the challenges in practical quantum key distribution is dealing with efficiency mismatch between different threshold single-photon detectors. There are known bounds for the secret key rate for the BB84 protocol with detection-efficiency mismatch provided that the eavesdropper sends no more than one photon to the legitimate receiver. Here we improve these bounds and give tight bounds for the secret key rate with a constant detection-efficiency mismatch under the same single-photon assumption. We propose a method based on the analytical minimization of the relative entropy of coherence, which can be used in other problems in quantum key distribution. Also we propose an adaptation of the decoy state method to prove the security in the case of weak coherent pulses on the source side.
Most experimental realizations of quantum key distribution are based on the Bennett-Brassard 1984 (the so-called BB84) protocol. In a typical optical implementation of this scheme, the sender uses an active source to produce the required BB84 signal states. While active state preparation of BB84 signals is a simple and elegant solution in principle, in practice passive state preparation might be desirable in some scenarios, for instance, in those experimental setups operating at high transmission rates. Passive schemes might also be more robust against side-channel attacks than active sources. Typical passive devices involve parametric down-conversion. In this paper, we show that both coherent light and practical single-photon sources are also suitable for passive generation of BB84 signal states. Our method does not require any externally driven element, but only linear optical components and photodetectors. In the case of coherent light, the resulting key rate is similar to the one delivered by an active source. When the sender uses practical single-photon sources, however, the distance covered by a passive transmitter might be longer than that of an active configuration.
Decoy-state quantum key distribution (QKD) has convincingly been shown the core solution to secure key exchange. While standard BB84 protocol needs to prepare and measure all states of two complementary bases, which seriously restricts its real life applications in photonic integration and high speed system. Fortunately, a simplified BB84, only sending and receiving three states, achieves the same asymptotic secret key rate compared with standard BB84 under coherent attacks. In this work, by skillfully exploiting uncertainty relation for smooth entropies, we provide the finite-key security bounds for decoy-state simplified BB84 QKD against coherent attacks in the universally composable framework. Simulation results show that it can also achieve almost the same performance compared with standard BB84 even the total number of pulses is as low as 10810^{8}108. We conclude that simplified BB84 can completely replace standard BB84 to be the best choice for QKD in most common practical scenarios.
Subject of study. The effect of different quantum efficiencies of single-photon detectors in quantum key distribution (QKD) systems implementing the BB84 protocol with passive basis choice is investigated. Aim of study. The aim of this study is to determine the secret key generation rate and security of the final key in quantum cryptography systems as functions of detector efficiency mismatch. Additionally, an analytical equation is derived for estimating the secret key generation rate in quantum communication systems with passive basis choice while accounting for different detector efficiencies. Method. The secret key is calculated by dividing the received measurement data according to the measurement bases, followed by performing separate post-processing for each dataset, and finally summing the resulting estimates of the secret key. Main results. An analytical method for estimating the secret key generation rate in systems with passive basis choice and mismatched detector efficiencies is developed. A satellite-to-ground quantum communication channel is modeled using the parameters of the Zvenigorod Observatory in Russia. The proposed theory for accounting for the detector efficiency mismatch is applied to a model of a satellite QKD system, and the determined length of the secret key generated during a single communication session with the satellite is found to be 300 kbit. Practical significance. The obtained results account for the effect of detector efficiency mismatch when estimating the length of the secret key in QKD systems, thereby increasing their reliability in practical implementations.
The Bennett-Brassard 1984 (BB84) protocol is one of the simplest protocols for implementing quantum key distribution (QKD). In the protocol, the sender and the receiver iteratively choose one of two complementary measurement bases. Regarding the basis choice by the receiver, a passive setup has been adopted in a number of its implementations, including satellite QKD and time-bin encoding. However, conventional theoretical techniques to prove the security of the BB84 protocol are not applicable if the receiver chooses their measurement basis passively, rather than actively, with a biased probability, followed by measurement with threshold detectors. Here we present a fully analytical security proof against coherent attacks for such a decoy-state BB84 protocol with the receiver’s passive basis choice and measurement with threshold detectors. Numerical simulations under practical situations show that the difference in secure key rate between the active and passive implementations of the protocol is negligible except for long communication distances.