Evolution of quantum steering in a Gaussian noisy channel

We describe the behaviour of the Gaussian quantum steering in a system of two bosonic modes evolving in a Gaussian noisy channel. The evolution based on completely positive quantum dynamical semigroups assures that the Gaussian form of the initial state of the subsystem is preserved in time. We study the quantum steering using the symplectic formalism and we find out that the steerability in the considered open system interacting with a squeezed thermal reservoir is destroyed by thermal noise and dissipation. The Gaussian quantum steerability manifests a sudden suppression, namely it decays at a finite moment of time. This behaviour is similar to the one of the quantum entanglement, compared to the Gaussian quantum discord, which is only asymptotically decreasing in time.