Covert Signaling for Communication and Sensing over the Bosonic Channels
Preventing signal detection in communication and active sensing requires careful control of transmission power. In fact, the square-root laws (SRL) for covert classical and quantum communication and sensing prescribe that the average output energy per channel use scales as $1/\sqrt{n}$ for $n$ channel uses. \emph{Diffuse} and \emph{sparse} signaling achieve this. The former transmits signals whose energy decays as $1/\sqrt{n}$ over all $n$ channel uses, which is convenient for mathematical analysis. The latter transmits constant-energy signals only approximately $\propto\sqrt{n}$ times out of $n$ channel uses, remaining silent on the others. This offers significant practical advantages in compatibility with modern digital transmitters. Here, we study sparse signaling over the lossy thermal-noise bosonic channel, which is a quantum model of many practical channels (including optical, microwave, and radio-frequency). We characterize the input signal state that minimizes detectability. We find an unintuitive optimal quantum state structure: a mixture of just two consecutive photon-number states. In particular, in the low-brightness regime, the optimal signal state is a mixture of vacuum and a single photon. Since these states are generally suboptimal for both communication and active sensing, we explore the resulting trade-off and identify input-power thresholds for transitions between optimizing covertness and performance in communication and sensing tasks.