Fundamental Limits of Covert Communication over Classical-Quantum Channels

TL;DR

This paper establishes the fundamental limits of covert communication over classical-quantum channels, demonstrating the square root law governs the maximum covert bits transmitted and the pre-shared secret requirements, with explicit formulas for channel-dependent constants.

Contribution

It derives single-letter expressions for the covert capacity and secret key requirements in classical-quantum channels, extending the understanding of covert communication limits under quantum adversaries.

Findings

Covert communication follows the square root law with a channel-dependent constant.

Pre-shared secret bits scale with the square root of the number of channel uses.

Conditions are identified when no pre-shared secret is necessary.

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.