We investigate the fundamental limit of quantum-secure covert communication over the lossy thermal noise bosonic channel, the quantum-mechanical model underlying many practical channels. We assume that the adversary has unlimited quantum information processing capabilities as well as access to all transmitted photons that do not reach the legitimate receiver. Given existence of noise that is uncontrolled by the adversary, the square root law (SRL) governs covert communication: up to $c\sqrt {n}$ covert bits can be transmitted reliably in $n$ channel uses. Attempting to surpass this limit results in detection with unity probability as $n\rightarrow \infty $ . Here we present the expression for $c$ , characterizing the SRL for the bosonic channel. We also prove that discrete-valued coherent state quadrature phase shift keying (QPSK) constellation achieves the optimal $c$ , which is the same as that achieved by a circularly-symmetric complex-valued Gaussian prior on coherent state amplitude. Finally, while binary phase shift keying (BPSK) achieves the Holevo capacity for non-covert bosonic channels in the low received signal-to-noise ratio regime, we show that it is strictly sub-optimal for covert communication.

中文翻译：

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波色通道上量子安全隐蔽通信的基本限制

我们研究了有损热噪声玻色子通道上量子安全隐蔽通信的基本限制，这是许多实际通道的量子力学模型。我们假设对手具有无限的量子信息处理能力，并且可以访问所有未到达合法接收器的传输光子。鉴于存在不受对手控制的噪声，平方根定律 (SRL) 管理秘密通信：最多 $c\sqrt {n}$ 隐蔽位可以可靠地传输 $n$ 渠道使用。尝试超过此限制会导致检测到单位概率为 $n\rightarrow \infty $ . 这里我们给出表达式 $c$ ，表征玻色子通道的 SRL。我们还证明了离散值相干状态正交相移键控 (QPSK) 星座实现了最优 $c$ ，这与在相干态振幅上通过圆对称复值高斯先验实现的结果相同。最后，虽然二进制相移键控 (BPSK) 在低接收信噪比机制中实现了非隐蔽玻色子信道的 Holevo 容量，但我们表明它对于隐蔽通信是严格次优的。