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 power per channel use scales as $1/\sqrt{n}$ for $n$ channel uses. Two strategies for achieving this are diffuse and sparse signaling. The former transmits signals with power decaying as $1/\sqrt{n}$ on all $n$ channel uses, which is convenient for mathematical analysis. The latter transmits constant-power signals rarely, on approximately $\sqrt{n}$ out of $n$ channel uses, while remaining silent on the others. This offers significant practical advantages in compatibility with modern digital transmitters. Here, we study sparse signaling over lossy thermal-noise bosonic channels, which describe quantumly 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 for covertness vs. performance in communication and sensing tasks.

翻译：在通信与主动传感中防止信号检测需要精确控制传输功率。事实上，隐蔽经典与量子通信及传感的平方根定律指出，每次信道使用的平均输出功率需随信道使用次数$n$按$1/\sqrt{n}$衰减。实现该目标有两种策略：弥散信令与稀疏信令。前者在所有$n$次信道使用中以$1/\sqrt{n}$的衰减功率传输信号，便于数学分析；后者则仅在约$\sqrt{n}$次信道使用中传输恒定功率信号，其余时间保持静默，在与现代数字发射机的兼容性上具有显著实用优势。本文研究有损热噪声博塞信道上的稀疏信令——该信道量子描述包括光学、微波及射频在内的诸多实际信道。我们刻画了最小化可检测性的输入信号态，发现其最优量子态结构具有反直觉特性：仅由两个相邻光子数态组成的混合态。特别地，在低亮度条件下，最优信号态为真空态与单光子态的混合。由于这些态通常对通信和主动传感均非最优，我们进一步探究其引入的权衡关系，并识别出通信与传感任务中从隐蔽性优化切换至性能优化的输入功率阈值。