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arxiv: 2605.23557 · v1 · pith:XPPTI6L7new · submitted 2026-05-22 · 🪐 quant-ph

CV-QKD over Turbulence Channels with Virtual Photon Subtraction and Quantum Multiple-Symbol Detection for Underwater Quantum Communications

Nour Rizk , Hesham S. Ibrahim , Ang\'elique Dr\'emeau , Arnaud Coatanhay This is my paper

Pith reviewed 2026-05-25 04:31 UTC · model grok-4.3

classification 🪐 quant-ph
keywords CV-QKDunderwater quantum communicationsvirtual photon subtractionquantum multiple-symbol detectionturbulenceQBERErlang distributionpost-selection
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The pith

Virtual photon subtraction with quantum multiple-symbol detection minimizes QBER in underwater CV-QKD under turbulence.

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

This paper proposes an underwater continuous-variable quantum key distribution system that uses virtual photon subtraction through post-selection of measurement outcomes without needing channel state information at the receiver. It compares three detection methods: homodyne detection, quantum maximum-likelihood detection, and quantum multiple-symbol detection, all combined with VPS. Turbulence is modeled using an Erlang distribution, and performance is measured by the accepted-only quantum bit error rate. The analysis shows that the quantum multiple-symbol detection version provides the greatest robustness against turbulence effects across different water types, as confirmed by both analytical expressions and Monte Carlo simulations.

Core claim

The paper claims that VPS-QMSD provides the best robustness against underwater turbulence, achieving the lowest QBER compared with VPS-QMLD and VPS-HD, with derived analytical and semi-closed-form QBER expressions matching simulation results for various system parameters and water types.

What carries the argument

Virtual photon subtraction implemented via post-selection on Alice's outcomes without CSI, combined with quantum multiple-symbol detection under an Erlang turbulence model.

Load-bearing premise

Underwater turbulence effects are accurately captured by an Erlang distribution, and the virtual photon subtraction post-selection rule remains effective without any channel state information at the receiver.

What would settle it

An experiment or simulation in which VPS-QMSD fails to produce strictly lower accepted QBER than VPS-QMLD under the same Erlang turbulence parameters would falsify the superiority result.

Figures

Figures reproduced from arXiv: 2605.23557 by Ang\'elique Dr\'emeau, Arnaud Coatanhay, Hesham S. Ibrahim, Nour Rizk.

Figure 1
Figure 1. Figure 1: We begin by reviewing the principle of physical photon subtraction applied to a TMSV state and then introduce its equivalent virtual implementation via post-selection. Next, we develop the prepare-and-measure transmitter model corresponding to the entanglement-based description and characterize the accepted non-Gaussian source distribution induced by virtual photon subtraction. Finally, we present the chan… view at source ↗
Figure 2
Figure 2. Figure 2: QBER vs. distance for VPS-HD, VPS-QMLD, and VPS-QMSD over clear ocean water, under different marine turbulence conditions and different numbers of virtually subtracted photons 𝑚: (a) 𝜆 = 𝜃 = 3, 𝑚 = 0 and 𝑚 = 1; (b) 𝜆 = 𝜃 = 10, 𝑚 = 0 and 𝑚 = 1; (c) 𝜆 = 𝜃 = 3, 𝑚 = 2 and 𝑚 = 3; and (d) 𝜆 = 𝜃 = 10, 𝑚 = 2 and 𝑚 = 3 [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: QBER vs. distance for VPS-HD, VPS-QMLD, and VPS-QMSD in clear ocean water and coastal ocean water, with 𝑁 = 0.001, 𝜆 = 𝜃 = 10, 𝑚 = 3, 𝐿 = 4. In the following analysis, we assess the QBER performance of the proposed VPS-QMSD scheme in comparison with VPS￾QMLD and VPS-HD for clear ocean water and coastal ocean water, as depicted in [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: QBER vs. distance for VPS-HD, VPS-QMLD, and VPS-QMSD in clear ocean water and coastal ocean water, with 𝑁 = 1, 𝜆 = 𝜃 = 10, 𝑚 = 3, and 𝐿 = 4 [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: QBER vs. distance VPS-QMSD scheme with different quantum sequence lengths 𝐿, in clear ocean water, under identical channel gain parameters 𝜆 = 𝜃 = 10, 𝑚 = 3, and thermal noise 𝑁 = 0.001 [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
read the original abstract

Continuous-variable quantum key distribution (CV-QKD) is a promising approach for secure underwater quantum communications (UQCs), where propagation loss, scattering, turbulence, and receiver thermal noise can severely degrade the transmission of quantum states. In this paper, we propose an underwater CV-QKD system with virtual photon subtraction (VPS), implemented through post-selection of Alice's measurement outcomes, without requiring channel state information (CSI) at the receiver. Three VPS-based system configurations are analyzed, corresponding to homodyne detection (VPS-HD), quantum maximum-likelihood detection (VPS-QMLD), and quantum multiple-symbol detection (VPS-QMSD). System performance is evaluated in terms of the accepted-only quantum bit error rate (QBER), where underwater turbulence is modeled by an Erlang distribution. Analytical and semi-closed-form QBER expressions are derived for the three configurations and validated through Monte Carlo simulations for different water types and system parameters. The results show close agreement between analytical and simulation results and demonstrate that VPS-QMSD provides the best robustness against underwater turbulence, achieving the lowest QBER compared with VPS-QMLD and VPS-HD.

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

2 major / 1 minor

Summary. The paper proposes an underwater CV-QKD system employing virtual photon subtraction (VPS) via post-selection on Alice's outcomes without receiver CSI. Three configurations are analyzed—VPS-HD, VPS-QMLD, and VPS-QMSD—under an Erlang turbulence model. Analytical and semi-closed-form expressions for accepted-only QBER are derived for different water types, validated by Monte Carlo simulations showing close agreement, and used to conclude that VPS-QMSD achieves the lowest QBER and greatest robustness.

Significance. If the derivations and ordering hold, the work supplies a CSI-free mitigation technique for turbulence in underwater quantum links and demonstrates the utility of multiple-symbol detection in this setting. The combination of closed-form expressions and direct Monte Carlo validation is a positive feature.

major comments (2)
  1. [Abstract] Abstract: the central claim that VPS-QMSD yields the lowest accepted QBER and best robustness is obtained exclusively under the Erlang fading model. No sensitivity study is reported for alternative underwater intensity statistics (e.g., log-normal or gamma-gamma), leaving open the possibility that the performance ordering among VPS-HD, VPS-QMLD and VPS-QMSD reverses when the conditional distributions change.
  2. [Abstract] Abstract and results: the VPS post-selection rule is stated to remain effective without receiver CSI, yet the manuscript supplies no explicit verification that the acceptance threshold derived from Alice’s marginal statistics continues to be optimal once the Erlang channel is applied; this assumption is load-bearing for all three QBER expressions.
minor comments (1)
  1. [Abstract] Abstract: the phrase 'accepted-only quantum bit error rate' is used without a brief parenthetical definition, which may hinder readers outside the immediate sub-field.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments. We address each major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that VPS-QMSD yields the lowest accepted QBER and best robustness is obtained exclusively under the Erlang fading model. No sensitivity study is reported for alternative underwater intensity statistics (e.g., log-normal or gamma-gamma), leaving open the possibility that the performance ordering among VPS-HD, VPS-QMLD and VPS-QMSD reverses when the conditional distributions change.

    Authors: We agree that the reported performance ordering and robustness claim are specific to the Erlang turbulence model employed throughout the derivations and simulations. The Erlang distribution is selected because it accurately captures the intensity statistics of underwater optical turbulence, consistent with established models in the literature for oceanic channels. To address the concern, we will revise the abstract and add a clarifying statement in the conclusions to explicitly limit the claim to the Erlang model. A full sensitivity analysis under alternative distributions such as log-normal or gamma-gamma would require re-deriving the QBER expressions and is left for future work; we will note this scope limitation in the revised manuscript. revision: partial

  2. Referee: [Abstract] Abstract and results: the VPS post-selection rule is stated to remain effective without receiver CSI, yet the manuscript supplies no explicit verification that the acceptance threshold derived from Alice’s marginal statistics continues to be optimal once the Erlang channel is applied; this assumption is load-bearing for all three QBER expressions.

    Authors: The VPS acceptance threshold is computed exclusively from Alice’s marginal measurement statistics, which are independent of any channel realization. Consequently, the threshold remains fixed and optimal irrespective of the turbulence-induced fading, satisfying the CSI-free requirement at the receiver. The three QBER expressions are obtained by integrating the channel-dependent conditional error rates over the Erlang density while holding the threshold constant; this averaging step constitutes the verification that the rule remains effective under the applied channel. We will insert an explicit paragraph in Section II clarifying this independence and optimality to make the assumption transparent. revision: yes

Circularity Check

0 steps flagged

No circularity: QBER derivations follow directly from Erlang model and detection rules without reduction to inputs by construction

full rationale

The paper states that analytical and semi-closed-form QBER expressions are derived for VPS-HD, VPS-QMLD and VPS-QMSD under the Erlang turbulence distribution and VPS post-selection (applied to Alice's outcomes without receiver CSI), then validated by Monte Carlo. No quoted step shows a self-definitional loop, a fitted parameter renamed as prediction, or a load-bearing self-citation chain. The performance ordering is a direct consequence of the chosen fading statistics and detector structures rather than an identity or statistical artifact forced by the inputs themselves. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Performance claims rest on the choice of Erlang distribution for turbulence and the post-selection rule for VPS; these are domain modeling choices rather than derived quantities.

free parameters (1)
  • Erlang distribution parameters for different water types
    Parameters are selected to represent varying turbulence conditions across water types; their specific values affect the QBER curves.
axioms (1)
  • domain assumption Underwater turbulence is modeled by an Erlang distribution
    Invoked to characterize propagation effects in the analytical QBER derivations.

pith-pipeline@v0.9.0 · 5750 in / 1187 out tokens · 28828 ms · 2026-05-25T04:31:36.651040+00:00 · methodology

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Reference graph

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