Covert Communication over Classical-Quantum Channels

Abstract

The square root law (SRL) is the fundamental limit of covert communication over classical memoryless channels (with a classical adversary) and quantum lossy-noisy bosonic channels (with a quantum-powerful adversary). The SRL states that $\mathcal{O}(\sqrt{n})$ covert bits, but no more, can be reliably transmitted in $n$ channel uses with $\mathcal{O}(\sqrt{n})$ bits of secret pre-shared between the communicating parties. Here we investigate covert communication over general memoryless classical-quantum (cq) channels with fixed finite-size input alphabets, and show that the SRL governs covert communications in typical scenarios. %This demonstrates that the SRL is achievable over any quantum communications channel using a product-state transmission strategy, where the transmitted symbols in every channel use are drawn from a fixed finite-size alphabet. We characterize the optimal constants in front of $\sqrt{n}$ for the reliably communicated covert bits, as well as for the number of the pre-shared secret bits consumed. We assume a quantum-powerful adversary that can perform an arbitrary joint (entangling) measurement on all $n$ channel uses. However, we analyze the legitimate receiver that is able to employ a joint measurement as well as one that is restricted to performing a sequence of measurements on each of $n$ channel uses (product measurement). We also evaluate the scenarios where covert communication is not governed by the SRL