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Classical capacity of phase-sensitive Gaussian quantum channels

The full solution of the optimization problem giving the Gaussian capacity of the single-mode fiducial Gaussian quantum channel is provided. Since it was shown that the Gaussian capacity of an arbitrary (phase-sensitive or insensitive) single-mode Gaussian quantum channel is equal to the Gaussian capacity of this fiducial channel, the solution presented in this work can be regarded as universal. The analytical study of this solution, below and above the energy threshold, shows that the dependence of the Gaussian capacity on the environment noise squeezing is not monotonic. In particular, the capacity may have a saddle point, one or two extrema at finite squeezing, or be a monotonically increasing function of the squeezing parameter. The exact dependence is defined by the determinant of the noise covariance matrix and by the transmissivity (or gain) of the fiducial Gaussian channel.