Gaussian–Rényi-2 correlations hierarchy in optomechanics

Abstract

In a two-mode Gaussian state [Formula: see text], we report on stationary evolution of three measures of correlations defined via the Rényi-2 entropy, i.e. quantum mutual information (QMI) [Formula: see text], the Gaussian–Rényi-2 entanglement (GR2E) [Formula: see text] and Gaussian quantum steering (GQS) [Formula: see text]. We evaluate analytical expression of the covariance matrix fully describing the state [Formula: see text]. Further, we study, under influences of parameters characterizing the state at hand and its environment, the behavior of the three considered measures. We find that quantum steering [Formula: see text] is always upper bounded by (GR2E) [Formula: see text], which in turn is found always upper bounded by half of the QMI [Formula: see text]. This therefore satisfies the hierarchical relation [Formula: see text] established in [L. Lami, C. Hirche, G. Adesso and A. Winter, Phys. Rev. Lett. 117 (2016) 220502]. Importantly, we find that both GR2E [Formula: see text] and GQS [Formula: see text] are strongly affected by the thermal effects. Remarkably, when the GR2E [Formula: see text] thoroughly vanishes, the GMI [Formula: see text] exhibits a freezing behavior, and seems to be captured within a wide range of temperature.