Fundamental Limits of Covert Communication over Classical-Quantum Channels

TL;DR

This work establishes the square-root law for covert communication over memoryless classical-quantum channels with finite input alphabets, showing that at most $L_{ m SRL}\sqrt{n}+o(\sqrt{n})$ covert bits can be reliably transmitted when using product-state inputs, with a channel-dependent covert capacity $L_{ m SRL}$ and a pre-shared secret rate $J_{ m SRL}$. It presents single-letter expressions for $L_{ m SRL}$ and $J_{ m SRL}$ and identifies conditions under which no secret key is needed, while also characterizing non-SRL regimes (constant-rate and $\mathcal{O}(\sqrt{n}\log n)$) and a no-covert-communication scenario. The results rely on quantum-resolvability techniques, pinching maps, and quantum relative-entropy analyses to bound Willie's detection capability and Bob's decoding reliability, with attention to both unrestricted joint measurements and respect to Bob’s measurement constraints. The findings extend the SRL from classical settings to generic classical-quantum channels and lay groundwork for future second-order (finite-block) refinements and entanglement-assisted variants.

Abstract

We investigate covert communication over general memoryless classical-quantum channels with fixed finite-size input alphabets. We show that the square root law (SRL) governs covert communication in this setting when product of $n$ input states is used: $L_{\rm SRL}\sqrt{n}+o(\sqrt{n})$ covert bits (but no more) can be reliably transmitted in $n$ uses of classical-quantum channel, where $L_{\rm SRL}>0$ is a channel-dependent constant that we call covert capacity. We also show that ensuring covertness requires $J_{\rm SRL}\sqrt{n}+o(\sqrt{n})$ bits secret shared by the communicating parties prior to transmission, where $J_{\rm SRL}\geq0$ is a channel-dependent constant. We assume a quantum-powerful adversary that can perform an arbitrary joint (entangling) measurement on all $n$ channel uses. We determine the single-letter expressions for $L_{\rm SRL}$ and $J_{\rm SRL}$, and establish conditions when $J_{\rm SRL}=0$ (i.e., no pre-shared secret is needed). Finally, we evaluate the scenarios where covert communication is not governed by the SRL.

Figures (2)

• Figure 1: Covert communication setting. Alice has a noisy channel to legitimate receiver Bob and adversary Willie. Alice encodes message $W$ with blocklength $n$ code and chooses whether to transmit. Willie observes his channel from Alice to determine whether she is quiet (null hypothesis $H_0$) or not (alternate hypothesis $H_1$). Alice and Bob's coding scheme must ensure that any detector Willie uses is close to ineffective (i.e., a random guess between the hypotheses), while allowing Bob to reliably decode the message (if one is transmitted). Alice and Bob may share a resource (e.g., a secret key exchanged prior to transmission.)
• Figure 2: Covert classical-quantum channel setting. Alice encodes message $m$ drawn from random variable $W$ by using the pre-shared secret key $k$ drawn from random variable $S$ into $\mathbf{x}(m,k) \in \mathcal{X}^n$. She then transmits $\mathbf{x}(m,k)$ in $n$ uses of the classical-quantum channel. Bob uses pre-shared secret key $k$ to select POVM $\left\{\hat{\Lambda}^n_{m,k}\right\}_{m\in \{1,\ldots, M\}}$, and obtain an estimate of the message $\check{W}$ from his received quantum state $\hat{\sigma}_B^n(\mathbf{x}(m,k))$. Willie performs a measurement to determine whether his quantum state $\hat{\rho}_W^n(\mathbf{x}(m,k))$ corresponds to innocent input $\mathbf{0}$ (null hypothesis $H_0$) or not (alternate hypothesis $H_1$).

Theorems & Definitions (36)

• Lemma 1
• Definition 1: Pinching Maps bhatia2013matrix
• Lemma 2
• Lemma 3
• Lemma 4
• Lemma 5
• Lemma 6
• Lemma 7
• Lemma 8
• Lemma 9