Distinguishing two single-mode Gaussian states by homodyne detection: An information-theoretic approach

It is known that the quantum fidelity, as a measure of the closeness of two quantum states, is operationally equivalent to the minimal overlap of the probability distributions of the two states over all possible POVMs; the POVM realizing the minimum is optimal. We consider the ability of homodyne detection to distinguish two single-mode Gaussian states, and investigate to what extent it is optimal in this information-theoretic sense. We completely identify the conditions under which homodyne detection makes an optimal distinction between two single-mode Gaussian states of the same mean, and show that if the Gaussian states are pure, they are always optimally distinguished.