One-mode Bosonic Gaussian channels: a full weak-degradability classification

A complete weak-degradability analysis of one-mode Gaussian Bosonic channels is performed. A new set of channels which have null quantum capacity is identified. Due to their relatively simple structure Gaussian channels [1] provide an ideal theoretical playground for the study of quantum communication protocols [2]. Bounds and, in some cases, explicit values for the classical [3] or quantum [4] capacities of these channels have been extensively discussed in literature (e.g. see Refs. [1, 5, 6, 7]). Recently weak-degradability was proposed as an useful tool for studying the quantum capacity properties of one-mode Gaussian channels [8]. This suggested the possibility of classifying all one-mode Gaussian channels in terms of a simple canonical form (see Ref. [9]). Moreover, proceeding along similar lines, the exact solution of the quantum capacity of an important subset of those maps was obtained in Ref. [10]. In this manuscript we provide a full weak-degradability classification of one-Gaussian channels and we produce a new set of channels which have null quantum capacity extending a previous result of Ref. [1]. ∗The Leverhulme Visiting Professor at Centre for Quantum Computation, Department of Applied Mathematics and Theoretical Physics, Cambridge University.