We determine the optimal method of discriminating and comparing quantum states from a certain class of multimode Gaussian states and their mixtures when arbitrary global Gaussian operations and general Gaussian measurements are allowed. We consider the so-called constant-$\widehat{p}$ displaced states, which include mixtures of multimode coherent states arbitrarily displaced along a common axis. We first show that no global or local Gaussian transformations or generalized Gaussian measurements can lead to a better discrimination method than simple homodyne measurements applied to each mode separately and classical postprocessing of the results. This result is applied to binary state comparison problems. We show that homodyne measurements, separately performed on each mode, are the best Gaussian measurement for binary state comparison. We further compare the performance of the optimal Gaussian strategy for binary coherent states comparison with these of non-Gaussian strategies using photon detections.

中文翻译：

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高斯状态判别的高斯运算和度量的极限及其在状态比较中的应用

当允许任意全局高斯运算和一般高斯测量时，我们确定从一类多模高斯态及其混合物中区分和比较量子态的最佳方法。我们认为所谓的常数$\widehat{p}$位移态，包括沿公共轴任意位移的多模相干态的混合。我们首先表明，与分别应用于每种模式的简单零差测量和结果的经典后处理相比，没有全局或局部高斯变换或广义高斯测量可以导致更好的判别方法。此结果适用于二进制状态比较问题。我们表明，分别在每种模式下执行的零差测量是用于二进制状态比较的最佳高斯测量。我们进一步比较了最佳高斯策略的二进制相干态比较与使用光子检测的非高斯策略的性能。