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 J_j (j=1,2) of (m+n)-mode Gaussian steering from A to B. The computation of the value of 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 J_js are not genuine Gaussian steering measures, they have some nice properties such as non-increasing under certain Gaussian unsteerable channels. Additionally, we compare J₂ with the Gaussian steering measure N₃, which is based on the Uhlmann fidelity, revealing that J₂ is an upper bound of N₃ at certain class of (1+1)-mode Gaussian pure states. As an illustration, we apply J₂ 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.