Covert Signaling for Communication and Sensing over the Bosonic Channels

This paper investigates sparse signaling over lossy thermal-noise bosonic channels for covert communication and sensing, revealing that a mixture of vacuum and single-photon states minimizes detectability in low-brightness regimes while establishing the trade-offs between covertness and task performance.

The Gist

Imagine you are trying to send a secret message or find a hidden object in a noisy, crowded room. But there's a catch: a very sharp-eared guard (let's call him "Willie") is standing right next to you, listening for any sign that you are doing something. If Willie hears even a whisper that you are active, he raises the alarm, and your mission fails.

This paper explores how to communicate or sense things in this tricky situation, specifically using light waves (like lasers or radio waves) that behave according to the strange rules of quantum physics.

Here is the breakdown of their discovery using simple analogies:

1. The "Square-Root" Rule

The paper starts with a fundamental rule of stealth: To stay hidden while sending information, you can't just shout louder. You have to whisper. But there's a limit to how much you can whisper.

If you try to send a lot of information, you get caught. The math says that if you use a channel $n$ times, you can only send a tiny amount of information (roughly the square root of $n$) without Willie noticing. It's like trying to sneak a few drops of water into a bucket without making a splash; if you add too much, the water overflows, and Willie sees it.

2. Two Ways to Sneak: The "Drip" vs. The "Splash"

The authors compare two strategies for staying under the radar:

• The "Drip" Strategy (Diffuse Signaling): Imagine you have a bucket of water and you want to move it secretly. You could drip a tiny, almost invisible drop into the bucket every single second for a long time. This is mathematically easy to study, but in the real world, it's hard to control a machine to make such tiny, perfect drops every time.
• The "Splash" Strategy (Sparse Signaling): Instead, imagine you stay silent for a long time, and then, very rarely, you drop a normal-sized splash of water. You do this so rarely that Willie thinks it's just background noise. This is much easier to build with real digital machines (like radios).

The paper focuses on the "Splash" (Sparse) strategy because it's more practical for real-world technology.

3. The Big Discovery: The "Two-Step" Dance

The main question the authors asked was: If I decide to use the "Splash" strategy, what exactly should that splash look like to be the hardest for Willie to detect?

You might think the best way to hide is to use a complex, fuzzy cloud of light. Surprisingly, the authors found the opposite. The best "splash" is actually very simple and sharp.

They discovered that the perfect stealth signal is a mixture of only two specific states:

1. Nothing at all (a vacuum).
2. Exactly one particle of light (a single photon).

If you need to send a bit more energy, you mix these two states in a specific ratio. It's like a dancer who only knows two moves: standing perfectly still or taking one single step. By sticking to just these two moves, you create a pattern that blends into the background noise better than any complex dance.

4. Why This Matters for Communication and Sensing

The paper looks at two jobs this stealth signal could do:

• Sending Secrets (Communication): You want to send a message to a friend without Willie knowing you are talking.
• Finding Things (Sensing): You want to shine a light to see if a target is there (like a submarine or a hidden object) without Willie knowing you are looking.

The authors found a trade-off.

• When the signal is very weak (low brightness): The "Two-Step" signal (Vacuum + One Photon) is the absolute best. It minimizes the chance of Willie hearing you. In fact, for sending secrets, this simple signal is just as good as the most complex, theoretical signals scientists have imagined. For sensing, it works just as well as a complex "entangled" signal that usually requires expensive equipment.
• When the signal gets stronger: If you need to send more power, the simple "Two-Step" signal starts to get a bit more noticeable to Willie. At this point, more complex signals (like standard radio waves) might actually be better for the task (sending more data or seeing further), even though they are slightly easier for Willie to detect.

5. The "Crossover" Point

The paper uses graphs to show a "tipping point."

• At low power: The simple "Vacuum + One Photon" signal wins. It's the stealth champion.
• At higher power: The complex signals win on performance (sending more data or seeing better), but they pay a "stealth tax" (Willie is more likely to catch you).

Summary

In a world where a quantum-powered guard is watching you, the best way to stay hidden while doing a job is to be boringly simple. Don't try to be a complex, fuzzy cloud of light. Instead, be a mix of "silence" and "one single flash." This simple "Two-Step" approach is mathematically proven to be the most invisible way to operate in the low-power regime, making it a perfect blueprint for future stealthy quantum radios and sensors.

Technical Summary

Technical Summary: Covert Signaling for Communication and Sensing over the Bosonic Channels

Problem Statement
This paper addresses the fundamental limits of low-probability-of-detection (LPD) or covert signaling over lossy thermal-noise bosonic channels, which model practical optical, microwave, and radio-frequency (RF) systems. The core challenge is to design signaling strategies that allow a transmitter (Alice) to communicate with a receiver (Bob) or perform active sensing (probing a target) while ensuring an adversarial warden (Willie) cannot distinguish the transmission from the innocent background thermal noise.

While the "square-root law" (SRL) establishes that covert communication is limited to $O(\sqrt{n})$ bits over $n$ channel uses, the specific structure of the input quantum state that minimizes detectability under practical constraints remains an open question. Specifically, the paper investigates sparse signaling, where Alice transmits with a small probability $\alpha$ using a fixed mean photon number $\bar{n}_S$, rather than the mathematically convenient but experimentally challenging "diffuse signaling" (transmitting weakly on all modes). The central question is: Under a mean photon number constraint, which active input quantum state minimizes Willie's ability to detect the transmission?

Methodology
The authors employ quantum information theory to analyze the bosonic channel $E_{\eta, \bar{n}_B}^{A \to BC}$, characterized by transmissivity $\eta$ and environmental mean photon number $\bar{n}_B$. The covertness criterion is defined via the Quantum Relative Entropy (QRE) between Willie's received state and the innocent thermal state, ensuring the QRE scales as $O(1)$ (or the trace distance remains small) as $n \to \infty$.

The methodology proceeds through the following steps:

1. Modeling Sparse Signaling: The input ensemble is modeled as a mixture of the vacuum state (innocent) and an active state $\hat{\rho}_A$ with probability $\alpha$.
2. Reduction to Fock-Diagonal States: Using the data-processing inequality for QRE and the phase-covariance of the thermal-loss channel, the authors prove that the optimal input state can be restricted to be diagonal in the Fock (photon-number) basis. This reduces the optimization over general quantum states to an optimization over photon-number distributions.
3. Perturbation Analysis: The authors derive an explicit second-order expansion of the QRE with respect to the activity probability $\alpha$. The leading coefficient, $C_P$, is expressed as a function of the input photon-number distribution.
4. Orthogonal Representation: By utilizing Meixner polynomials, the perturbation coefficient $C_P$ is rewritten as a weighted sum of squares of the input's factorial moments. This representation reveals that minimizing detectability corresponds to suppressing higher-order factorial moments.
5. Convexity Argument: Applying a discrete convexity argument to the factorial moments, the authors demonstrate that the optimal distribution is supported on at most two consecutive Fock states.

Key Contributions

1. Optimal State Structure: The paper characterizes the optimal sparse input state that minimizes Willie's QRE coefficient. The main result (Theorem 1) proves that the optimal state is a mixture of exactly two consecutive photon-number states, $|\kappa\rangle$ and $|\kappa+1\rangle$, where $\kappa = \lfloor \bar{n}_S \rfloor$.
2. Low-Brightness Regime Optimization: In the low-brightness regime ($0 \le \bar{n}_S \le 1$), which is most relevant for covert operations, the optimal state simplifies to a mixture of the vacuum state $|0\rangle$ and the single-photon state $|1\rangle$.
3. Saturation of Converse Bounds: The authors show that this optimal sparse state saturates the converse bound for bosonic covert communication (matching the minimal Willie-side perturbation coefficient derived in prior work [7]).
4. Equivalence to Diffuse Sensing: For covert sensing, the optimal sparse state yields the same leading Willie-side QRE coefficient as the optimal diffuse Two-Mode Squeezed Vacuum (TMSV) scheme under matched mean photon number, demonstrating that the sparse structure does not inherently sacrifice covertness efficiency compared to diffuse strategies in the low-brightness limit.

Results
The paper presents a detailed performance analysis comparing the covertness-optimized sparse Fock-diagonal state against standard signaling schemes (Gaussian coherent states for communication and sparse TMSV for sensing) under the same covertness budget.

• Covert Communication: The performance is measured by the number of reliably transmissible bits per $\sqrt{n}$. The results (Fig. 3) indicate a trade-off: at very low mean photon numbers ($\bar{n}_S$), Gaussian coherent-state modulation offers higher channel capacity (Holevo information) and may outperform the covertness-optimized state. However, as $\bar{n}_S$ increases, the Gaussian scheme incurs a significantly larger covertness penalty (higher QRE coefficient), causing the covertness-optimized Fock-diagonal state to become superior.
• Covert Sensing: The performance is measured by the Chernoff exponent governing the decay of the target-detection error probability. The results (Fig. 4) show that while sparse TMSV schemes benefit from signal-idler correlations to achieve a higher Chernoff exponent at low $\bar{n}_S$, this advantage is offset by a higher covertness penalty. Consequently, the covertness-optimized Fock-diagonal state remains competitive or superior over a broad range of $\bar{n}_S$, with the crossover point occurring at relatively low photon numbers.

Significance
The paper claims that the derived optimal state structure—a mixture of vacuum and single-photon states in the low-brightness regime—provides a practical and theoretically optimal solution for sparse covert signaling. Unlike diffuse signaling, which requires transmitting weak signals on all modes (often impractical due to hardware constraints like finite converter resolution and synchronization), the sparse strategy is compatible with modern digital transmitters.

The significance lies in identifying that suboptimal states for standard communication or sensing tasks (specifically, non-Gaussian mixtures of vacuum and single photons) are actually optimal for covertness. The paper establishes that by strictly adhering to this structure, one can minimize the detectability penalty, thereby maximizing the operational performance (throughput or detection exponent) allowed under a fixed covertness constraint. The work bridges the gap between theoretical covert capacity limits and practical signaling implementations, offering a concrete state design for quantum-secure communication and sensing in noisy environments.