Evaluating capacities of bosonic Gaussian channels

We show how to compute or at least to estimate various capacity-related quantities for bosonic Gaussian channels. Among these are the coherent information, the entanglement-assisted classical capacity, the one-shot classical capacity, and a quantity involving the transpose operation, shown to be a general upper bound on the quantum capacity, even allowing for finite errors. All bounds are explicitly evaluated for the case of a one-mode channel with attenuation or amplification and classical noise.