Covert Communication and Key Generation Over Quantum State-Dependent Channels

TL;DR

The paper addresses covert communication and covert secret key generation over quantum state-dependent channels with entanglement-assisted CSI at the transmitter. It develops one-shot and asymptotic achievability results for two security settings—covert communication with key generation and covert-secure communication with key generation—using a combination of pinching, channel resolvability, and Gel’fand-Pinsker encoding. The authors derive inner bounds for the CC-CSK and CSC-CSK regions, show reductions to known classical results, and establish optimality in the classical-state regime. They also discuss extensions to causal CSI and stealth variants, outlining important open problems for quantum covert communications.

Abstract

We study covert communication and covert secret key generation with positive rates over quantum state-dependent channels. Specifically, we consider fully quantum state-dependent channels when the transmitter shares an entangled state with the channel. We study this problem setting under two security metrics. For the first security metric, the transmitter aims to communicate covertly with the receiver while simultaneously generating a covert secret key, and for the second security metric, the transmitter aims to transmit a secure message covertly and generate a covert secret key with the receiver simultaneously. Our main results include one-shot and asymptotic achievable positive covert-secret key rate pairs for both security metrics. Our results recover as a special case the best-known results for covert communication over state-dependent classical channels. To the best of our knowledge, our results are the first instance of achieving a positive rate for covert secret key generation and the first instance of achieving a positive covert rate over a quantum channel. Additionally, we show that our results are optimal when the channel is classical and the state is available non-causally at both the transmitter and the receiver.

Figures (3)

• Figure 1: Covert communication over a quantum state-dependent channel. The transmitter generates a secret key $K$ from its share of the CSI $\bar{S}$, encodes the message $M$ along with the key $K$ into a quantum state $A$ by using the CSI $\bar{S}$, and transmits it over a quantum channel $\mathcal{N}_{AS\to BE}$. The goal is for the receiver to reliably decode the message and the secret key upon observing $B$, while the warden, observing $E$, should not be able to decide whether communication is taking place.
• Figure 2: The encoder generates a random codebook $C\triangleq\{U(j,k,m)\}_{(j,k,m)\in\mathcal{J}\times\mathcal{K}\times\mathcal{M}}$, where $m\in\mathcal{M}\triangleq\left[{1:2^R}\right]$ is the message, $k\in\mathcal{K}\triangleq\left[{1:2^{R_K}}\right]$ is the secret key, and $j\in\mathcal{J}\triangleq\left[{1:2^{R_J}}\right]$ represents the local randomness needed to perform Gel'fand-Pinsker encoding and also represents part of the randomness needed to secure the key $k$. Since the message $m$ is not required to be secure, it also serves as part of the local randomness needed to secure the key. To transmit the message $m$, given the CSI $\bar{S}$ and the fixed sub-codebook $\mathcal{C}_m\triangleq\{U(j,k,m)\}_{(j,k)\in\mathcal{J}\times\mathcal{K}}$, the encoder applies the isometry $W^{(\mathcal{C}_m)}_{\bar{S}\to YAJK}$ on its share of the CSI, i.e., $\bar{S}$, which outputs the purification state $Y$, the index $J$, the secret key $K$, and the channel input $A$. Then it transmits the quantum state $A$ over the channel. The decoder applies a set of pinching-based POVM operators on its received state $B$ to recover the message $m$ and the secret key $K$. Using the channel resolvability techniques and properties of the pinching maps, we bound the probability of detection at the warden.
• Figure 3: The orange region illustrates the achievable rate region $\mathcal{R}_{\textup{CC-CSK}}$ in Theorem \ref{['thm:Asymp']} and the blue region illustrates the achievable rate region $\mathcal{R}_{\textup{CSC-CSK}}$ in Theorem \ref{['thm:Asymp_CS']}.

Theorems & Definitions (28)

• Definition 1
• Definition 2: Covert Communication and Covert Secret Key Generation
• Definition 3: Covert Secure Communication and Covert Secret Key Generation
• Definition 4: Covert Communication and Covert Secret Key Generation
• Definition 5: Covert Secure Communication and Covert Secret Key Generation
• Theorem 1: Covert Communication and Covert Secret Key Generation
• proof
• Lemma 1
• proof
• Lemma 2