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International Conference on Quantum Communications, Networking, and Computing (QCNC)

BBM92 Quantum Key Distribution with Non-Maximally Entangled States in Random Scattering Media Using Kraus Operators

Nour Rizk, Angélique Drémeau, Arnaud Coatanhay

Published Apr 6, 2026• 0 citations
Publisher

Abstract

Scattering environments place severe constraints on the security of quantum communication systems, as they induce decoherence mechanisms that degrade entanglement. To quantify these effects, we analyze the performance of the BBM92 protocol in terms of the quantum bit error rate (QBER) and the secret key rate (SKR), considering non-maximally entangled photon pairs transmitted through a quantum channel affected by amplitudedamping and depolarizing effects, described by Kraus operators. We derive closed-form analytical expressions for the QBER and SKR, which establish theoretical security bounds for secure key generation under non-ideal conditions.

DOI
10.1109/qcnc69040.2026.00071

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Scattering environments place severe constraints on the security of quantum communication systems, as they induce decoherence mechanisms that degrade entanglement. To quantify these effects, we analyze the performance of the BBM92 protocol in terms of the quantum bit error rate (QBER) and the secret key rate (SKR), considering non-maximally entangled photon pairs transmitted through a quantum channel affected by amplitudedamping and depolarizing effects, described by Kraus operators. We derive closed-form analytical expressions for the QBER and SKR, which establish theoretical security bounds for secure key generation under non-ideal conditions.
Underwater optical channels pose significant challenges to the security and reliability of quantum communication systems due to absorption and scattering. In this paper, we investigate the BBM92 entanglement-based quantum key distribution (QKD) protocol under realistic underwater channel conditions. Photon pairs are prepared in non-maximally entangled states, and the underwater propagation medium is modeled as a quantum channel incorporating both amplitude-damping and depolarizing effects, described within the Kraus operator formalism. The protocol performance is evaluated in terms of quantum bit error rate (QBER) and secret key rate (SKR), analyzed as functions of the entanglement degree and channel degradation parameters. Closed-form analytical expressions for the QBER and SKR are derived for the proposed channel model and validated through Monte Carlo simulations. The proposed framework is then applied to various realistic underwater scenarios, considering different water types, namely clear ocean, coastal, and turbid water, as well as varying atmospheric conditions.
This study compares the quantum bit error rate (QBER) performance of BB84, SARG04, and the entanglement-based BBM92 protocol in non-turbulent underwater optical channels. Clear, coastal, and turbid water types are considered under different background-illumination and receiver configurations. For BB84 and SARG04, channel attenuation and background-induced detections determine the quantum gain and QBER. For BBM92, photon loss is included through the two arm detection efficiencies, whereas the polarization state conditioned on two-photon detection is described by local depolarizing Kraus channels. The BBM92 QBER is obtained from the correlations in both measurement bases and from an exact partition of recorded events into true and false coincidences. Stochastic numerical estimates provide consistency checks of the analytical implementation. The results show that water turbidity and optical background reduce the distance below the adopted QBER threshold. They also show that a low conditional BBM92 QBER may coexist with a very small coincidence probability; therefore, the QBER-threshold distance alone does not represent a complete operational or secret-key-rate bound.
In the context of secure underwater optical communications, this study examines the performance of the entanglement-based BBM92 protocol when the source generates non-maximally entangled photon pairs. The impact of seawater optical properties, as well as source correlation characteristics, on transmission efficiency is analyzed for three types of nonturbulent channels: clear, coastal, and turbid water. The main performance metrics, including quantum bit error rate (QBER), secret key rate (SKR), and correlation probability, are analytically derived and evaluated as functions of transmission distance between two communicating parties. Monte Carlo simulation results validate our analytical models and quantify performance under different channel conditions.
The current literature on quantum key distribution is limited mainly to transmissions over fiber optic, atmospheric, or satellite links and is not directly applicable to underwater environments with different channel characteristics. In this paper, we analyze the quantum bit error rate (QBER) and secret key rate (SKR) performance of the well-known BB84 protocol in underwater channels. As a path loss model, we consider a modified version of the Beer–Lambert formula, which takes into account the effect of scattering. We derive a closed-form expression for the wave structure function to determine the average power transfer over a turbulent underwater path and use this to obtain an upper bound on QBER as well as a lower bound on SKR. Based on the derived bounds, we present the performance of the BB84 protocol in different water types including clear, coastal, and turbid and under different atmospheric conditions such as clear, hazy, and overcast. We further investigate the effect of system parameters such as aperture size and detector field of view on QBER and SKR performance metrics.
Practical quantum key distribution (QKD) systems operate under noise, but security of most protocols have been analyzed under ideal noiseless scenarios. In this work, we investigated security performance of BB84 protocol under effect of collective rotation noise. Using theoretical quantum information frameworks, we analyzed key security parameters including quantum bit error rate (QBER), mutual information and secret key rate (SKR). Security of protocol is studied under various eavesdropping scenarios based on intercept and resend attacks. Our results show that collective rotation noise has a significant impact on the information shared between the two parties. Particularly, we extended prior treatments by suggesting a noise engineering strategy where we identified a non-zero noise range where information accessed by Eve is minimized while corresponding SKR degradation remains relatively small. This analysis provide insights into robustness of BB84 protocol under realistic noisy channels and may contribute towards development of more resilient QKD systems.
Satellite-based quantum communication for secure key distribution is becoming a more demanding field of research due to its unbreakable security. Prepare and measure protocols such as BB84 consider the satellite as a trusted device, fraught with danger looking at the current trend for satellite-based optical communication. Therefore, entanglement-based protocols must be preferred since, along with overcoming the distance limitation, one can consider the satellite as an untrusted device too. E91 protocol is a good candidate for satellite-based quantum communication; but the key rate is low as most of the measured qubits are utilized to verify a Bell-CHSH inequality to ensure security against Eve. An entanglement-based protocol requires a maximally entangled state for more secure key distribution. The current work discusses the effect of non-maximality on secure key distribution. It establishes a lower bound on the non-maximality condition below which no secure key can be extracted. BBM92 protocol will be more beneficial for key distribution as we found a linear connection between the extent of violation for Bell-CHSH inequality and the quantum bit error rate for a given setup.