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arxiv: 2605.08066 · v2 · pith:JOZB35ZRnew · submitted 2026-05-08 · 🪐 quant-ph · cs.IT· math.IT

Covert Signaling for Communication and Sensing over the Bosonic Channels

Tianrui Tan , Evan J.D. Anderson , Michael S. Bullock , Boulat A. Bash This is my paper

Pith reviewed 2026-05-11 02:11 UTC · model grok-4.3

classification 🪐 quant-ph cs.ITmath.IT
keywords covert communicationbosonic channelssparse signalingquantum sensingphoton-number statessquare-root lawthermal-noise channellossy channel
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The pith

The optimal signal state for minimizing detectability in covert communication and sensing over lossy bosonic channels is a mixture of two consecutive photon-number states.

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

This paper studies sparse signaling over lossy thermal-noise bosonic channels that model optical, microwave, and radio-frequency links. The goal is to keep transmissions hard to detect while still supporting communication or active sensing. The central result is that the quantum input state minimizing an eavesdropper's detection probability is a simple probabilistic mixture of just two neighboring photon-number states. In the low-brightness regime this reduces to a mixture of the vacuum state and a single-photon state. The authors also locate the power thresholds at which this choice for covertness begins to conflict with the states that would otherwise maximize communication rate or sensing performance.

Core claim

We characterize the input signal state that minimizes detectability. We find an unintuitive optimal quantum state structure: a mixture of just two consecutive photon-number states. In particular, in the low-brightness regime, the optimal signal state is a mixture of vacuum and a single photon. Since these states are generally suboptimal for both communication and active sensing, we explore the resulting trade-off and identify input-power thresholds for transitions between optimizing for covertness vs. performance in communication and sensing tasks.

What carries the argument

Sparse signaling under the square-root law on the lossy thermal-noise bosonic channel, where the optimal input density operator is the mixture of two consecutive Fock states that minimizes the adversary's detection probability.

Load-bearing premise

The eavesdropper's detection is governed by the standard square-root law applied to the lossy thermal-noise bosonic channel without additional side information or non-standard noise.

What would settle it

Compare the detection error probability achieved by a vacuum-plus-single-photon mixture against other states (such as coherent or thermal states) at identical average photon number over a calibrated lossy thermal channel and check whether the two-state mixture yields the lowest detection rate.

Figures

Figures reproduced from arXiv: 2605.08066 by Boulat A. Bash, Evan J.D. Anderson, Michael S. Bullock, Tianrui Tan.

Figure 1
Figure 1. Figure 1: The lossy thermal-noise bosonic channel E η,n¯B A→BC is characterized by transmissivity η and thermal photon number n¯B, with input subsystem A and output subsystems B and C. Environment is in a thermal state ρˆth(¯nB). The modal annihilation operators aˆ, ˆb, cˆ and eˆ are used to describe the input-output relationships in this channel. in our work, akin to a channel use in classical information theory. S… view at source ↗
Figure 2
Figure 2. Figure 2: Systems models for covert communication and sensing. In both models, adversary Willie has to decide whether Alice [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Covert communication capability vs. mean photon number [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Covert sensing capability vs. mean photon number [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
read the original abstract

Preventing signal detection in communication and active sensing requires careful control of transmission power. In fact, the square-root laws (SRL) for covert classical and quantum communication and sensing prescribe that the average output energy per channel use scales as $1/\sqrt{n}$ for $n$ channel uses. \emph{Diffuse} and \emph{sparse} signaling achieve this. The former transmits signals whose energy decays as $1/\sqrt{n}$ over all $n$ channel uses, which is convenient for mathematical analysis. The latter transmits constant-energy signals only approximately $\propto\sqrt{n}$ times out of $n$ channel uses, remaining silent on the others. This offers significant practical advantages in compatibility with modern digital transmitters. Here, we study sparse signaling over the lossy thermal-noise bosonic channel, which is a quantum model of many practical channels (including optical, microwave, and radio-frequency). We characterize the input signal state that minimizes detectability. We find an unintuitive optimal quantum state structure: a mixture of just two consecutive photon-number states. In particular, in the low-brightness regime, the optimal signal state is a mixture of vacuum and a single photon. Since these states are generally suboptimal for both communication and active sensing, we explore the resulting trade-off and identify input-power thresholds for transitions between optimizing covertness and performance in communication and sensing tasks.

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 / 3 minor

Summary. The manuscript studies sparse signaling over lossy thermal-noise bosonic channels for covert classical and quantum communication and sensing. Under the square-root-law constraint on average output power, it characterizes the input quantum state minimizing eavesdropper detectability (via output-state distinguishability from thermal noise) and shows that the optimum is a mixture of two consecutive Fock states; in the low-brightness regime this reduces to a vacuum-plus-single-photon mixture. The work then examines the resulting trade-offs with communication rate and sensing performance, identifying input-power thresholds at which the optimum switches from covertness to task performance.

Significance. If the central characterization holds, the result supplies an explicit, practical optimal state for sparse covert bosonic signaling that is compatible with modern digital transmitters and directly applicable to optical, microwave, and RF channels. It advances the square-root-law literature by moving beyond diffuse signaling to a concrete quantum-state optimization and quantifies the covertness-performance frontier, providing both theoretical insight and guidance for experimental implementations.

minor comments (3)
  1. Abstract: the statement that the two-Fock mixture is 'unintuitive' would be strengthened by a one-sentence contrast with the classical Poisson or thermal-state expectation that readers might anticipate.
  2. The trade-off section would benefit from an explicit statement of the figure of merit used for the communication and sensing tasks (e.g., Holevo information or Fisher information) so that the power-threshold transitions can be reproduced without ambiguity.
  3. Notation for the bosonic channel parameters (transmissivity, thermal noise photon number) should be introduced once in the model section and used consistently thereafter.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of our manuscript, accurate summary of the results on optimal sparse signaling states for covert bosonic channels, and recommendation for minor revision. We appreciate the recognition of the practical advantages of the two-consecutive-Fock-state structure.

Circularity Check

0 steps flagged

No significant circularity; direct optimization within standard model

full rationale

The paper derives the optimal input state (mixture of two consecutive Fock states, vacuum plus single photon in low brightness) by optimizing over states with fixed mean photon number to minimize output distinguishability from thermal noise on the lossy thermal-noise bosonic channel under the sparse-signaling square-root-law constraint. This is a self-contained mathematical characterization using the photon-number basis natural to the channel model; no step reduces by construction to a fitted parameter renamed as prediction, a self-citation chain, an ansatz smuggled via prior work, or a renamed empirical pattern. The square-root law and channel model are standard external references, and the result is obtained from the optimization rather than presupposed.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the standard bosonic channel model (loss plus thermal noise) and the square-root-law scaling for covertness; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Lossy thermal-noise bosonic channel model governs the physical link
    Invoked to define the channel for which the optimal state is derived.
  • domain assumption Square-root law governs the scaling of average power for covert operation
    Used to set the total power constraint for both diffuse and sparse strategies.

pith-pipeline@v0.9.0 · 5550 in / 1475 out tokens · 39202 ms · 2026-05-11T02:11:07.678836+00:00 · methodology

discussion (0)

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Toward Practical Two-Way Covert Communication

    quant-ph 2026-05 unverdicted novelty 4.0

    Proof-of-concept experiment for two-way covert optical communication with narrowband laser and proposal of correlator receiver for quantum light sources without mode matching.