Multi-mode bosonic Gaussian channels

TL;DR

This paper provides a comprehensive analysis of multi-mode bosonic Gaussian channels, clarifying their structure, normal forms, and capacity properties, including the role of auxiliary modes and classical noise, with detailed focus on the two-mode case.

Contribution

It introduces a normal form for multi-mode Gaussian channels, identifies the maximum auxiliary modes needed, and analyzes zero-capacity channels, advancing understanding of their structure and classification.

Findings

Derived a canonical matrix form for noisy n-mode channels

Identified sets of two-mode channels with zero quantum capacity

Simplified weak-degradability classification of Gaussian channels

Abstract

A complete analysis of multi-mode bosonic Gaussian channels is proposed. We clarify the structure of unitary dilations of general Gaussian channels involving any number of bosonic modes and present a normal form. The maximum number of auxiliary modes that is needed is identified, including all rank deficient cases, and the specific role of additive classical noise is highlighted. By using this analysis, we derive a canonical matrix form of the noisy evolution of n-mode bosonic Gaussian channels and of their weak complementary counterparts, based on a recent generalization of the normal mode decomposition for non-symmetric or locality constrained situations. It allows us to simplify the weak-degradability classification. Moreover, we investigate the structure of some singular multi-mode channels, like the additive classical noise channel that can be used to decompose a noisy channel in terms of a less noisy one in order to find new sets of maps with zero quantum capacity. Finally, the two-mode case is analyzed in detail. By exploiting the composition rules of two-mode maps and the fact that anti-degradable channels cannot be used to transfer quantum information, we identify sets of two-mode bosonic channels with zero capacity.