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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 2026-06-30 22:57 UTC · model grok-4.3

classification 🪐 quant-ph cs.ITmath.IT
keywords covert communicationbosonic channelsparse signalingphoton-number statesthermal noisequantum sensingsquare-root law
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The pith

A mixture of two consecutive photon-number states minimizes detectability for covert sparse signaling over lossy thermal-noise bosonic channels.

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

The paper studies sparse signaling on the lossy thermal-noise bosonic channel to satisfy the square-root law for covert communication and sensing. It shows that the quantum input state minimizing detection probability under this constraint is always a mixture of just two consecutive photon-number states. In the low-brightness regime this mixture reduces to vacuum and a single photon. These states are typically not optimal for maximizing communication rate or sensing accuracy, so the work identifies power thresholds at which the transmitter should switch between covertness and task 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 covertness and performance in communication and sensing tasks.

What carries the argument

The optimal input quantum state, a statistical mixture of two consecutive photon-number states, that satisfies the square-root-law covertness constraint while minimizing detection probability.

If this is right

  • Sparse signaling with these states meets the square-root law covertness requirement while remaining compatible with digital transmitters.
  • Input-power thresholds exist at which the transmitter should switch from the covertness-optimal two-state mixture to states that maximize communication rate or sensing performance.
  • In the low-brightness regime the vacuum-plus-single-photon mixture is the unique optimal state for minimizing detectability.
  • The same two-state structure applies across optical, microwave, and radio-frequency channels modeled by the lossy thermal-noise bosonic channel.

Where Pith is reading between the lines

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

  • Transmitters could be simplified by using on-off keying hardware to realize the low-brightness optimal state.
  • The result suggests similar number-state mixtures may minimize detectability on other bosonic channels with different loss or noise parameters.
  • An optical-table experiment comparing detection error rates for the two-state mixture versus coherent states at equal energy would directly test the prediction.

Load-bearing premise

The analysis assumes sparse signaling with constant-energy pulses sent on a vanishing fraction of channel uses to meet the square-root law covertness constraint.

What would settle it

A calculation or experiment that demonstrates a different photon-number mixture or non-number-state achieves strictly lower detection probability at the same average energy and covertness level would falsify the claimed optimality.

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 examines sparse signaling (constant-energy pulses on a vanishing fraction of uses) over the lossy thermal-noise bosonic channel to satisfy the square-root-law covertness constraint. It characterizes the input quantum state minimizing detectability, reporting that the optimum is a mixture of two consecutive photon-number states; in the low-brightness regime this reduces to a vacuum-plus-single-photon mixture. The work then quantifies the resulting trade-off between covertness and performance for classical communication and active sensing tasks.

Significance. If the optimality claim holds, the result supplies a concrete, low-dimensional state structure that is simultaneously practical for modern transmitters and provably optimal for covertness under the bosonic channel model. The explicit identification of power thresholds separating covertness-optimal from rate- or sensing-optimal regimes is a falsifiable prediction that could guide experimental design in optical and microwave covert systems.

minor comments (3)
  1. §II, Eq. (3): the definition of the sparse-signaling constraint should explicitly state whether the support size is exactly floor(sqrt(n)) or allowed to fluctuate; the subsequent optimization appears to treat it as deterministic.
  2. Fig. 2 caption: the plotted quantity is labeled 'covertness metric' but the axis is the Helstrom error; clarify whether this is 1-P_e or the full error probability.
  3. §IV.B: the transition thresholds between covertness-optimal and communication-optimal regimes are stated numerically; an analytic expression or scaling law would strengthen the claim.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, the recognition of its potential significance for experimental design, and the recommendation of minor revision. No major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper's central result is a characterization of the optimal input state (mixture of two consecutive Fock states) that minimizes detectability under the explicitly stated sparse-signaling model on the lossy thermal-noise bosonic channel. No derivation step reduces by construction to its own inputs, no fitted parameter is relabeled as a prediction, and no load-bearing premise rests on a self-citation chain. The square-root-law covertness constraint and sparse signaling are presented as the modeling scope rather than derived outputs. The abstract and reader's summary contain no equations that exhibit self-definition or renaming of known results.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard square-root law for covertness and the lossy thermal-noise bosonic channel model, both treated as given domain assumptions rather than derived in the abstract.

axioms (2)
  • domain assumption Square-root law for covert classical and quantum communication and sensing (average output energy per channel use scales as 1/sqrt(n))
    Invoked in the abstract as the governing constraint that diffuse and sparse signaling must satisfy.
  • domain assumption Lossy thermal-noise bosonic channel as quantum model of practical optical, microwave, and RF channels
    Stated in the abstract as the channel under study.

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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.

Reference graph

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