Optimal Throughput for Covert Communication Over a Classical-Quantum Channel

This paper considers the problem of communication over a memoryless classical-quantum wiretap channel subject to the constraint that the eavesdropper on the channel should not be able to learn whether the legitimate parties are using the channel to communicate or not. Specifically, the relative entropy between the output quantum states at the eavesdropper when a codeword is transmitted and when no input is provided must be sufficiently small. Extending earlier works, this paper proves the "square-root law" for a broad class of classical-quantum channels: the maximum amount of information that can be reliably and covertly transmitted over $n$ uses of such a channel scales like $\sqrt{n}$. The scaling constant is also determined.Comment: Corrected version of a paper presented at ITW 2016. In the ITW paper, the denominator in the main formula (10) was incorrect. The current version corrects this mistake and adds an appendix for its derivatio

Covert Communication over Classical-Quantum Channels

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