Methods for estimating capacities and rates of Gaussian quantum channels

Optimization methods aimed at estimating the capacities of a general Gaussian channel are developed. Specifically evaluation of classical capacity as maximum of the Holevo information is pursued over all possible Gaussian encodings for the lossy bosonic channel, but extension to other capacities and other Gaussian channels seems feasible. Solutions for both memoryless and memory channels are presented. It is first dealt with single-use (single-mode) channel where the capacity dependence on channel's parameters is analyzed providing the full classification of possible cases. Then, it is dealt with multiple uses (multimode) channel where the capacity dependence on the (multimode) environment state is analyzed when both total environment energy and environment purity are fixed. This allows a fair comparison among different environments, thus understanding the role of memory (intermode correlations) and phenomenon like superadditivity of the capacity. The developed methods are also used for deriving transmission rates with heterodyne and homodyne measurements at the channel output. Classical capacity and transmission rates are presented within a unique framework where the rates can be treated as logarithmic approximations of the capacity.