  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 from channel's parameters is analyzed providing a full
classification of the possible cases. Then it is dealt with multiple uses
(multi-mode) channel where the capacity dependence from the (multi-mode)
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 (inter-mode 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.
