# Position representation of single-mode Gaussian channels beyond the Gaussian functional form

## TL;DR

This paper explores the structure of one-mode Gaussian quantum channels beyond the traditional Gaussian functional form, revealing new subsets and characterizing their effects on Gaussian states and master equations.

## Contribution

It introduces the existence of non-Gaussian functional form subsets of Gaussian channels and analyzes their properties and implications.

## Key findings

- Identifies two subsets of Gaussian channels without Gaussian functional form.
- Connects non-Gaussian channels to singular channels and their classification.
- Derives conditions for the existence of master equations for these channels.

## Abstract

We study one-mode Gaussian quantum channels in continuous-variable systems by performing a black-box characterization using complete positivity and trace preserving conditions, and report the existence of two subsets that do not have a functional Gaussian form. Our study covers as particular limit the case of singular channels, thus connecting our results with their known classification scheme based on canonical forms. Our full characterization of Gaussian channels without Gaussian functional form is completed by showing how Gaussian states are transformed under these operations, and by deriving the conditions for the existence of master equations for the non-singular cases. We show that although every functional form can be found in the vicinity of the identity, one of them does not parametrize unitary channels.