Distinguishability of Gaussian states in quantum cryptography using postselection

We consider the distinguishability of Gaussian states from the viewpoint of continuous-variable quantum cryptography using postselection. Specifically, we use the probability of error to distinguish between two pure coherent (squeezed) and two particular mixed symmetric coherent (squeezed) states where each mixed state is an incoherent mixture of two pure coherent (squeezed) states with equal and opposite displacements in the conjugate quadrature. We show that the two mixed symmetric Gaussian states (where the various components have the same real part) never give an eavesdropper more information than the two pure Gaussian states. Furthermore, when considering the distinguishability of squeezed states, we show that varying the amount of squeezing leads to a "squeezing" and "antisqueezing" of the net information rates.