Fundamental Limits of Covert Communication Over Classical-Quantum Channels

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 a of n input states is used: $L_{\mathrm { SRL}}\sqrt {n}+o(\sqrt {n})$ covert bits (but no more) can be reliably transmitted in n uses of classical-quantum channel, where $L_{\mathrm { SRL}}\gt 0$ is a channel-dependent constant that we call covert capacity. We also show that ensuring covertness requires $J_{\mathrm { SRL}}\sqrt {n}+o(\sqrt {n})$ bits secret key shared by the communicating parties prior to transmission, where $J_{\mathrm { SRL}}\geq 0$ 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 $L_{\mathrm { SRL}}$ and $J_{\mathrm { SRL}}$ , and establish conditions when $J_{\mathrm { SRL}}=0$ (i.e., no pre-shared secret key is needed). Finally, we evaluate scenarios where covert communication is not governed by the SRL.