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We determine the optimal method of discriminating and comparing quantumstates from a certain class of multimode Gaussian states and their mixtureswhen arbitrary global Gaussian operations and general Gaussian measurements areallowed. We consider the so-called constant-$\hat{p}$ displaced states whichinclude mixtures of multimode coherent states arbitrarily displaced along acommon axis. We first show that no global or local Gaussian transformations orgeneralized Gaussian measurements can lead to a better discrimination methodthan simple homodyne measurements applied to each mode separately and classicalpostprocessing of the results. This result is applied to binary statecomparison problems. We show that homodyne measurements, separately performedon each mode, are the best Gaussian measurement for binary state comparison. Wefurther compare the performance of the optimal Gaussian strategy for binarycoherent states comparison with these of non-Gaussian strategies using photondetections.

Limit of Gaussian operations and measurements for Gaussian state discrimination, and its application to state comparison