We give the exact form of the fidelity for general bosonic Gaussian channels in cases where the channel's input are limited to Gaussian pure states. Some examples have been studied and the properties of the fidelity for memory channel are carefully examined.
A noisy Gaussian channel is defined as a channel in which an input field mode is subjected to random Gaussian displacements in phase space. We introduce the quantum fidelity of a Gaussian channel for pure and mixed input states, and we derive a universal scaling law of the fidelity for pure initial states. We also find the maximum fidelity of a Gaussian channel over all input states. Quantum cloning and continuous-variable teleportation are presented as physical examples of Gaussian channels to which the fidelity results can be applied.
As a toy model for the capacity problem in quantum information theory we investigate finite and asymptotic regularizations of the maximum pure-state input-output fidelity F(N) of a general quantum channel N. We show that the asymptotic regularization $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{F}(\mathcal{N})$ is lower bounded by the maximum output $\ensuremath{\infty}$-norm ${\ensuremath{\nu}}_{\ensuremath{\infty}}(\mathcal{N})$ of the channel. For N being a Pauli channel, we find that both quantities are equal.