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arxiv: 1907.04228 · v1 · pith:6B36A6OPnew · submitted 2019-07-09 · 💻 cs.IT · math.IT

Fundamental limits of quantum-secure covert communication over bosonic channels

Michael S. Bullock , Christos N. Gagatsos , Saikat Guha , Boulat A. Bash This is my paper

Pith reviewed 2026-05-25 00:06 UTC · model grok-4.3

classification 💻 cs.IT math.IT
keywords covert communicationbosonic channelsquare root lawquantum securityQPSKthermal noiseHolevo capacity
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The pith

Up to c times sqrt(n) covert bits can be sent reliably over lossy thermal bosonic channels without detection by an unlimited adversary.

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

The paper determines the fundamental scaling limit for quantum-secure covert communication over the lossy thermal-noise bosonic channel, which models many practical optical links. When thermal noise exists that the adversary cannot control, the square-root law applies: at most c sqrt(n) covert bits can be transmitted reliably in n uses, and exceeding this causes certain detection as n grows. The authors derive the explicit value of the constant c and prove that a discrete four-state coherent-state QPSK constellation attains this optimum, matching the performance of a Gaussian prior on amplitude while binary phase-shift keying falls short.

Core claim

Given noise uncontrolled by the adversary, the square root law governs covert communication over the lossy thermal noise bosonic channel: up to c sqrt(n) covert bits can be transmitted reliably in n channel uses. The expression for c is derived, and a QPSK constellation on coherent states achieves the optimal value of c, identical to that obtained with a circularly symmetric complex Gaussian prior; binary phase shift keying is strictly suboptimal for the covert task.

What carries the argument

The square-root law (SRL) constant c for the bosonic channel, obtained from the channel's thermal noise statistics and the adversary's quantum detection capabilities.

If this is right

  • Reliable transmission of up to c sqrt(n) covert bits is possible in n uses.
  • Any attempt to exceed c sqrt(n) bits results in detection probability approaching 1.
  • QPSK modulation achieves the optimal c for the bosonic channel.
  • BPSK modulation achieves strictly less than the optimal c under covert constraints.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Simple discrete constellations suffice to reach the fundamental covert limit, suggesting low-complexity implementations may be viable.
  • The distinction between standard capacity-achieving inputs and covert-optimal inputs may appear in other quantum channels with similar noise.
  • The result separates the covert scaling from the usual Holevo capacity scaling at low SNR.

Load-bearing premise

The analysis requires that some thermal noise remains outside the adversary's control.

What would settle it

A calculation or experiment in which an adversary who can control or correlate with all thermal noise detects any scheme attempting more than c sqrt(n) covert bits with probability approaching 1.

Figures

Figures reproduced from arXiv: 1907.04228 by Boulat A. Bash, Christos N. Gagatsos, Michael S. Bullock, Saikat Guha.

Figure 1
Figure 1. Figure 1: Single-mode bosonic channel E n¯B η modeled by a beamsplitter with transmissivity η and an environment injecting a thermal state ρˆn¯B with mean photon number n¯B. aˆ, eˆ, bˆ, and wˆ label input/output modal annihilation operators. in Section II-A, is parametrized by the power coupling (transmissivity) η between the transmitter Alice and the intended receiver Bob, and the mean photon number n¯B per mode in… view at source ↗
Figure 2
Figure 2. Figure 2: Covert communication over lossy thermal noise bosonic channel. Alice has a lossy thermal noise bosonic channel [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Discrete coherent state constellations used by Alice’s modulator. [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
read the original abstract

We investigate the fundamental limit of quantum-secure covert communication over the lossy thermal noise bosonic channel, the quantum-mechanical model underlying many practical channels. We assume that the adversary has unlimited quantum information processing capabilities as well as access to all transmitted photons that do not reach the legitimate receiver. Given existence of noise that is uncontrolled by the adversary, the square root law (SRL) governs covert communication: up to c*sqrt{n} covert bits can be transmitted reliably in n channel uses. Attempting to surpass this limit results in detection with unity probability as n approaches infinity. Here we present the expression for c, characterizing the SRL for the bosonic channel. We also prove that discrete-valued coherent state quadrature phase shift keying (QPSK) constellation achieves the optimal c, which is the same as that achieved by a circularly-symmetric complex-valued Gaussian prior on coherent state amplitude. Finally, while binary phase shift keying (BPSK) achieves the Holevo capacity for non-covert bosonic channels in the low received signal-to-noise ratio regime, we show that it is strictly sub-optimal for covert communication.

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

0 major / 2 minor

Summary. The manuscript derives the precise constant c governing the square-root law for quantum-secure covert communication over the lossy thermal-noise bosonic channel. Conditioned on the existence of adversary-uncontrolled noise, up to c sqrt(n) covert bits can be sent reliably in n uses; the paper presents an explicit expression for c and proves that a QPSK constellation on coherent states achieves this optimal value (matching the circularly symmetric complex Gaussian prior), while BPSK is strictly suboptimal despite its optimality for non-covert low-SNR bosonic communication.

Significance. If the derivation holds, the result supplies the exact fundamental limit for covert bosonic communication under the standard noise model, with the explicit optimality of discrete QPSK constituting a notable strength that simplifies implementation while preserving the Gaussian rate. The work is a parameter-free derivation from the channel model and the definition of covertness, and the upfront conditioning on uncontrolled noise is clearly stated.

minor comments (2)
  1. [Abstract] The abstract states that an expression for c is presented and that optimality of QPSK is proved, but does not display the expression itself; including the closed-form expression for c in the abstract would improve immediate readability.
  2. Notation for the bosonic channel parameters (e.g., transmissivity, thermal noise mean photon number) should be introduced with a brief reminder of their physical meaning in the first section where the model is defined, to aid readers outside the immediate subfield.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. We are pleased that the derivation of the constant c and the optimality of QPSK were found to be clear and significant.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper derives the constant c characterizing the square-root law for covert communication over the lossy thermal-noise bosonic channel directly from the channel model, the definition of covertness (adversary detection probability), and standard quantum information-theoretic techniques. The optimality of QPSK is shown to match the Gaussian prior via explicit comparison of achievable rates. No equations reduce by construction to fitted inputs, no load-bearing self-citations are invoked for uniqueness theorems, and the central result is explicitly conditioned on the stated noise model. This is a standard conditional information-theoretic derivation with no internal reduction to its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The derivation rests on standard quantum information axioms (bosonic channel model, Holevo information, quantum hypothesis testing for covertness) plus the modeling assumption of uncontrolled thermal noise. No free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption The lossy thermal-noise bosonic channel is the correct quantum-mechanical model for the physical link.
    Invoked in the first sentence of the abstract.
  • domain assumption The adversary has unlimited quantum information processing and receives all photons that miss the legitimate receiver.
    Stated in the abstract.

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Reference graph

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