# The semigroup structure of Gaussian channels

## TL;DR

This paper explores the complex semigroup structure of bosonic Gaussian quantum channels, revealing unique properties due to their non-compactness and providing new mathematical representations.

## Contribution

It characterizes divisibility, idempotency, and Markovian properties of Gaussian channels, and introduces new representations for these channels and their semigroup structures.

## Key findings

- Every irreversible Gaussian channel is divisible.
- Non-compactness leads to the absence of generators for some reversible channels.
- New matrix and complex representations for Gaussian channels and semigroups.

## Abstract

We investigate the semigroup structure of bosonic Gaussian quantum channels. Particular focus lies on the sets of channels which are divisible, idempotent or Markovian (in the sense of either belonging to one-parameter semigroups or being infinitesimal divisible). We show that the non-compactness of the set of Gaussian channels allows for remarkable differences when comparing the semigroup structure with that of finite dimensional quantum channels. For instance, every irreversible Gaussian channel is shown to be divisible in spite of the existence of Gaussian channels which are not infinitesimal divisible. A simpler and known consequence of non-compactness is the lack of generators for certain reversible channels. Along the way we provide new representations for classes of Gaussian channels: as matrix semigroup, complex valued positive matrices or in terms of a simple form describing almost all one-parameter semigroups.