Gaussian unsteerable channels and computable quantifications of Gaussian steering

TL;DR

This paper advances the resource theory of Gaussian steering by characterizing Gaussian unsteerable channels, introducing computable steering quantifiers based on covariance matrices, and analyzing their properties and implications in quantum systems.

Contribution

It introduces the class of Gaussian unsteerable channels and maximal channels, completing the resource theory for Gaussian steering and providing efficient quantification methods.

Findings

Gaussian unsteerable channels are characterized and used as free operations.

The quantifiers $\\mathcal{J}_j$ are computationally simple and rely only on covariance matrices.

Gaussian steering can decay rapidly in Markovian environments.

Abstract

The current quantum resource theory for Gaussian steering for continuous-variable systems is flawed and incomplete. Its primary shortcoming stems from an inadequate comprehension of the architecture of Gaussian channels transforming Gaussian unsteerable states into Gaussian unsteerable states, resulting in a restricted selection of free operations. In the present paper, we explore in depth the structure of such $(m+n)$-mode Gaussian channels, and introduce the class of the Gaussian unsteerable channels and the class of maximal Gaussian unsteerable channels, both of them may be chosen as the free operations, which completes the resource theory for Gaussian steering from $A$ to $B$ by Alice's Gaussian measurements. We also propose two quantifications $\mathcal{J}_{j}$ $(j=1,2)$ of $(m+n)$-mode Gaussian steering from $A$ to $B$. The computation of the value of $\mathcal{J}_{j}$ is straightforward and efficient, as it solely relies on the covariance matrices of Gaussian states, eliminating the need for any optimization procedures. Though $\mathcal{J}_{j}$s are not genuine Gaussian steering measures, they have some nice properties such as non-increasing under certain Gaussian unsteerable channels. Additionally, we compare ${\mathcal J}_2$ with the Gaussian steering measure $\mathcal N_3$, which is based on the Uhlmann fidelity, revealing that ${\mathcal J}_2$ is an upper bound of $\mathcal N_3$ at certain class of $(1+1)$-mode Gaussian pure states. As an illustration, we apply $\mathcal J_2$ to discuss the behaviour of Gaussian steering for a special class of $(1+1)$-mode Gaussian states in Markovian environments, which uncovers the intriguing phenomenon of rapid decay in quantum steering.