Gaussian unsteerable channels and computable quantifications of Gaussian steering

The current quantum resource theory for Gaussian steering in 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 a class of Gaussian unsteerable channels and a class of maximal Gaussian unsteerable channels, both of which may be chosen as free operations, which completes the resource theory for Gaussian steering from $A$ to $B$ by using 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 relies solely on the covariance matrices of Gaussian states, eliminating the need for any optimization procedures. Though the ${\mathcal{J}}_{j}$ are not genuine Gaussian steering measures, they have some nice properties such as being nonincreasing 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 ${\mathcal{N}}_{3}$ for a certain class of $(1+1)$-mode Gaussian pure states. As an illustration, we apply ${\mathcal{J}}_{2}$ to discuss the behavior 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.