Quantum steering with Gaussian states: A tutorial

Quantum steering refers to the apparent possibility of exploiting quantum correlations to remotely influence the quantum state of a subsystem, by measuring local degrees of freedom. In continuousvariable (CV) quantum information, this notion is strongly linked to the possibility of demonstrating the EPR paradox, whence the name EPR steering. Recently, another type of steering with CV Gaussian states was proposed under the name of nonclassical steering, stemming from the idea of remotely generating Glauber P-nonclassicality by conditional Gaussian measurements on two-mode Gaussian states. In this tutorial, we thoroughly illustrate these phenomena, firstly introducing quantum steering in its most general setting, and then focusing on Gaussian states and the connection with P-nonclassicality. We discuss the strong and weak forms of nonclassical steering, their relation with entanglement and how to formulate them in an invariant form with respect to local Gaussian unitary operations. For two-mode squeezed thermal states (TMST), we show that EPR steering coincides with nonclassical steering, and in particular this implies that a single type of Gaussian measurements is sufficient to check steerability for this class, unlike for the more general situation which usually requires the choice of distinct measurements.