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arxiv: 2605.29513 · v1 · pith:ILLH76OJnew · submitted 2026-05-28 · 🪐 quant-ph

Performance Analysis of Underwater Quantum Key Distribution Protocols: BB84, SARG04, and BBM92

Nour Rizk , Ang\'elique Dr\'emeau , Arnaud Coatanhay This is my paper

Pith reviewed 2026-06-29 07:22 UTC · model grok-4.3

classification 🪐 quant-ph
keywords underwater quantum key distributionBB84SARG04BBM92quantum bit error rateKraus operatorsattenuationdepolarization
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The pith

The BBM92 protocol admits an analytical QBER expression derived from Kraus operators that model attenuation and depolarization in underwater channels.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper compares the performance of BB84, SARG04, and BBM92 quantum key distribution protocols in underwater settings. Performance is measured by quantum bit error rate and correlation as functions of distance in clear, coastal, and turbid non-turbulent seawater. For the entanglement-based BBM92, Kraus operators capture the channel effects on entangled photons to derive an analytical QBER from correlation losses. Monte Carlo simulations validate the analytical results for all protocols under various conditions. The analysis identifies conditions for optimal secure underwater quantum communications.

Core claim

For the BBM92 protocol, the quantum channel is modeled using Kraus operators, which characterize the combined effects of attenuation and depolarization on maximally-entangled photons, allowing the derivation of an analytical expression of the QBER, based on the correlation losses between measurements. Monte Carlo simulation results validate the analytical expression of the QBER for all the studied protocols, under various water types, atmospheric conditions, and system parameters.

What carries the argument

Kraus operators for modeling combined attenuation and depolarization effects on maximally-entangled photons in the underwater quantum channel.

Load-bearing premise

The effects of the underwater channel are fully captured by attenuation and depolarization alone in non-turbulent seawater.

What would settle it

A measurement showing that the observed QBER in an underwater quantum channel deviates substantially from the value predicted by the analytical expression based on Kraus operators.

Figures

Figures reproduced from arXiv: 2605.29513 by Ang\'elique Dr\'emeau, Arnaud Coatanhay, Nour Rizk.

Figure 1
Figure 1. Figure 1: Schematic diagram of BBM92 QKD protocol, where the HWP orientation [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: QBERBB84 and QBERSARG04 vs. distance for Scenario 1, with R0(λ) = 10−3 W/m2 , and different pupil diameters in (a) clear water, (b) coastal water, and (c) turbid water. the transmission range increases from 49.78 m to 60.05 m, while for SARG04, it varies from 44.64 m to 53.81 m at d1 = 5 cm and 30 cm, respectively. In [PITH_FULL_IMAGE:figures/full_fig_p025_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: QBERBB84 and QBERSARG04 vs. distance for Scenario 5, with R0(λ) = 500 W/m2 , and different pupil diameters in (a) clear water, (b) coastal water, and (c) turbid water. from 49.78 m to 21.20 m and from 60.05 m to 2.28 m, for BB84 at night and during the day, respectively. For SARG04, it drops to 17.50 m and 1.23 m, respectively. In turbid water, the transmission range becomes extremely short. For BB84, the … view at source ↗
Figure 4
Figure 4. Figure 4: QBERBBM92 vs. distance for different scenarios, with x = L/2, in (a) clear water, (b) coastal water, and (c) turbid water [PITH_FULL_IMAGE:figures/full_fig_p027_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Maximum secure transmission distance where the QBER [PITH_FULL_IMAGE:figures/full_fig_p028_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Correlation ⟨σx ⊗ σx⟩ vs. distance in different types of water (clear, coastal, and turbid) for (a) x = L/2, and (b) x = 0.2L [PITH_FULL_IMAGE:figures/full_fig_p029_6.png] view at source ↗
read the original abstract

Underwater quantum key distribution (UQKD) ensures unconditional communication security based on the fundamental principles of quantum mechanics. This study examines the performance of the BB84, SARG04, and the entanglement-based BBM92 protocols under the impact of both system optical parameters and the physical effects of the underwater channel. The main performance criteria considered are the quantum bit error rate (QBER) and the quantum correlation between the communicating parties as functions of the propagation distance in three types of non-turbulent seawater: clear, coastal, and turbid. For the BBM92 protocol, the quantum channel is modeled using Kraus operators, which characterize the combined effects of attenuation and depolarization on maximally-entangled photons, allowing the derivation of our analytical expression of the QBER, based on the correlation losses between measurements. Monte Carlo simulation results validate the analytical expression of the QBER for all the studied protocols, under various water types, atmospheric conditions, and system parameters. The results determine the conditions under which QKD protocols achieve optimal performance for secure and efficient underwater quantum communications.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The paper examines the performance of BB84, SARG04, and BBM92 quantum key distribution protocols in underwater channels, focusing on QBER and quantum correlation versus propagation distance in clear, coastal, and turbid non-turbulent seawater. For BBM92, it models the channel with Kraus operators combining attenuation and depolarization to derive an analytical QBER expression based on correlation losses, validated by Monte Carlo simulations under various conditions.

Significance. Should the channel model prove adequate, the analytical QBER derivation and simulation results would offer practical guidance for optimizing UQKD systems, particularly in identifying viable distances for secure key distribution in different water types.

major comments (1)
  1. [Abstract] Abstract: The analytical QBER for BBM92 is derived from Kraus operators modeling only attenuation and depolarization. This may not fully capture the effects of multiple scattering in non-turbulent seawater, which can cause polarization-dependent losses and additional decoherence not included in a simple depolarization channel. Since the Monte Carlo validation employs the identical model, it verifies algebraic consistency rather than physical accuracy.
minor comments (1)
  1. The abstract refers to 'system optical parameters' and 'atmospheric conditions' without defining them or indicating how they enter the QBER expressions or simulations.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed review and constructive comment. We address the major point below and will incorporate clarifications in a revised manuscript.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The analytical QBER for BBM92 is derived from Kraus operators modeling only attenuation and depolarization. This may not fully capture the effects of multiple scattering in non-turbulent seawater, which can cause polarization-dependent losses and additional decoherence not included in a simple depolarization channel. Since the Monte Carlo validation employs the identical model, it verifies algebraic consistency rather than physical accuracy.

    Authors: We agree that the Kraus-operator model (attenuation plus depolarization) is an approximation. It is adopted because it permits a closed-form QBER derivation for BBM92 while remaining consistent with standard treatments of non-turbulent scattering channels in the QKD literature. Depolarization is intended to capture the net polarization scrambling induced by multiple scattering events; however, we acknowledge that it does not explicitly include polarization-dependent loss or higher-order decoherence mechanisms. Consequently, the Monte Carlo runs confirm algebraic correctness of the derivation rather than independent physical validation. In the revised manuscript we will (i) state this modeling limitation explicitly in the abstract and channel-model section, (ii) add a short paragraph discussing the scope of the approximation, and (iii) cite representative works that employ more detailed scattering models. No change to the numerical results or analytical expressions is required. revision: partial

Circularity Check

0 steps flagged

QBER derivation is a direct mathematical consequence of the input Kraus model with no reduction to fitted quantities or self-citations

full rationale

The paper states that the BBM92 QBER expression is obtained by applying Kraus operators (modeling attenuation plus depolarization) to the maximally entangled state and extracting correlation losses. Monte Carlo simulation is then run on the identical operator set to confirm algebraic consistency. This is a standard forward derivation from stated channel assumptions to an observable; the result is not obtained by fitting parameters to data and then relabeling the output as a prediction, nor does any load-bearing step rely on a self-citation whose content is itself unverified. No self-definitional, fitted-input, or ansatz-smuggling patterns appear. The derivation chain is therefore self-contained against the paper's own premises.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the channel model is described at a high level without numerical values or unstated assumptions listed.

pith-pipeline@v0.9.1-grok · 5725 in / 1047 out tokens · 26714 ms · 2026-06-29T07:22:25.626336+00:00 · methodology

discussion (0)

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Reference graph

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    He received the Ph.D. degree from the University of Le Havre, Le Havre, France, in 2000, and the HDR degree from the Universit´e de Bretagne Occidentale (UBO), Brest, France, in June 2022. He is currently a Full Professor at the ´Ecole Nationale Sup´erieure de Techniques Avanc´ees (ENSTA, Institut Polytechnique de Paris), Brest. His research interests inc...

    2000