On singular Bosonic linear channels

TL;DR

This paper investigates properties of Bosonic linear channels, especially Gaussian channels with degeneracies, revealing their decompositions, reversibility, and classical-quantum subchannel structures.

Contribution

It introduces new decompositions and reversibility insights for Bosonic linear channels with specific degeneracies, enhancing understanding of their structure.

Findings

Degeneracy leads to multiple direct sum decompositions of channels.

Existence of noise-free canonical variables implies reversibility.

Channels can be decomposed into classical-quantum subchannels.

Abstract

Properties of Bosonic linear (quasi-free) channels, in particular, Bosonic Gaussian channels with two types of degeneracy are considered. The first type of degeneracy can be interpreted as existence of noise-free canonical variables (for Gaussian channels it means that $\det\alpha=0$). It is shown that this degeneracy implies existence of (infinitely many) "direct sum decompositions" of Bosonic linear channel, which clarifies reversibility properties of this channel (described in arXiv:1212.2354) and provides explicit construction of reversing channels. The second type of degeneracy consists in rank deficiency of the operator describing transformations of canonical variables. It is shown that this degeneracy implies existence of (infinitely many) decompositions of input space into direct sum of orthogonal subspaces such that the restriction of Bosonic linear channel to each of these subspaces is a discrete classical-quantum channel.