  In the literature the performance of quantum data transmission systems is
usually evaluated in the absence of thermal noise. A more realistic approach
taking into account the thermal noise is intrinsically more difficult because
it requires dealing with Glauber coherent states in an infinite--dimensional
space. In particular, the exact evaluation of the optimal measurement operators
is a very difficult task, and numerical approximation is unavoidable. The paper
faces the problem by approximating the P-representation of the noisy quantum
states with a large but finite number of terms and applying to them the square
root measurement (SRM) approach. Comparisons with the exact solution obtained
with convex semidefinite programming show that the SRM approach gives quite
accurate results. As application, the performance of quadrature amplitude
modulation (QAM) and phase shift keying (PSK) systems is considered. In spite
of the fact that the SRM approach is not optimal and overestimates the error
probability, also in these cases the quantum detection maintains its
superiority with respect to the classical homodyne detection.
