Quantum hypothesis testing is one of the most basic tasks in quantum information theory and has fundamental links with quantum communication and estimation theory. In this paper, we establish a formula that characterizes the decay rate of the minimal type-II error probability in a quantum hypothesis test of two Gaussian states given a fixed constraint on the type-I error probability. This formula is a direct function of the mean vectors and covariance matrices of the quantum Gaussian states in question. We give an application to quantum illumination, which is the task of determining whether there is a low-reflectivity object embedded in a target region with a bright thermal-noise bath. For the asymmetric-error setting, we find that a quantum illumination transmitter can achieve an error probability exponent stronger than a coherent-state transmitter of the same mean photon number, and furthermore, that it requires far fewer trials to do so. This occurs when the background thermal noise is either low or bright, which means that a quantum advantage is even easier to witness than in the symmetric-error setting because it occurs for a larger range of parameters. Going forward from here, we expect our formula to have applications in settings well beyond those considered in this paper, especially to quantum communication tasks involving quantum Gaussian channels.

中文翻译：

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高斯假设检验和量子照明

量子假设检验是量子信息理论中最基本的任务之一，与量子通信和估计理论有着根本的联系。在本文中，我们建立了一个公式，该公式描述了在给定I型错误概率的固定约束的情况下，在两个高斯态的量子假设检验中最小II型错误概率的衰减率。该公式是所讨论的量子高斯态的均值向量和协方差矩阵的直接函数。我们将量子照明应用于一个应用程序，该任务是确定在具有明亮热噪声池的目标区域中是否嵌入了低反射率对象。对于非对称错误设置，我们发现，与相同平均光子数的相干态发射器相比，量子照明发射器可以实现更强的错误概率指数，而且，这样做的需要更少的尝试。当背景热噪声较低或较亮时，就会发生这种情况，这意味着与对称误差设置相比，量子优势甚至更容易被证明，因为它出现在较大范围的参数中。从这里开始，我们希望我们的公式能够在远远超出本文所考虑的范围的环境中应用，尤其是涉及量子高斯信道的量子通信任务。这意味着与对称误差设置相比，量子优势甚至更容易证明，因为它出现在更大范围的参数中。从这里开始，我们希望我们的公式可以在远远超出本文所考虑的设置中应用，尤其是涉及量子高斯通道的量子通信任务。这意味着与对称误差设置相比，量子优势甚至更容易证明，因为它出现在更大范围的参数中。从这里开始，我们希望我们的公式能够在远远超出本文所考虑的范围的环境中应用，尤其是涉及量子高斯信道的量子通信任务。