Geometry on the manifold of Gaussian quantum channels

TL;DR

This paper develops a geometric framework for analyzing the space of Gaussian quantum channels, enabling statistical predictions about the prevalence of entanglement breaking channels using a metric based on the Choi-Jamio{ }kowski isomorphism and Hilbert-Schmidt distance.

Contribution

It introduces a geometric approach to quantify volumes of Gaussian quantum channels and analytically computes the relative volumes of entanglement breaking and incompatibility breaking channels.

Findings

Analytical expressions for volumes of entanglement breaking channels.

Method to determine incompatibility breaking channels from Choi-Jamio{ }kowski state purities.

A geometric metric based on the Choi-Jamio{ }kowski isomorphism for Gaussian channels.

Abstract

In the space of quantum channels, we establish the geometry that allows us to make statistical predictions about relative volumes of entanglement breaking channels among all the Gaussian quantum channels. The underlying metric is constructed using the Choi-Jamio{\l}kowski isomorphism between the continuous-variable Gaussian states and channels. This construction involves the Hilbert-Schmidt distance in quantum state space. The volume element of the one-mode Gaussian channels can be expressed in terms of local symplectic invariants. We analytically compute the relative volumes of the one-mode Gaussian entanglement breaking and incompatibility breaking channels. Finally, we show that, when given the purities of the Choi-Jamio{\l}kowski state of the channel, one can determine whether or not such channel is incompatibility breaking.