Gaussian–Rényi-2 correlations hierarchy in optomechanics.

In a two-mode Gaussian state ϱ̂AB, we report on stationary evolution of three measures of correlations defined via the Rényi-2 entropy, i.e. quantum mutual information (QMI) ℐR,2, the Gaussian–Rényi-2 entanglement (GR2E) ℰR,2 and Gaussian quantum steering (GQS) 풢. We evaluate analytical expression of the covariance matrix fully describing the state ϱ̂AB. 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 풢 is always upper bounded by (GR2E) ℰR,2, which in turn is found always upper bounded by half of the QMI ℐR,2. This therefore satisfies the hierarchical relation 12ℐR,2≥ℰR,2≥풢 established in [L. Lami, C. Hirche, G. Adesso and A. Winter, <italic>Phys. Rev. Lett.</italic><bold>117</bold> (2016) 220502]. Importantly, we find that both GR2E ℰR,2 and GQS 풢 are strongly affected by the thermal effects. Remarkably, when the GR2E ℰR,2 thoroughly vanishes, the GMI ℐR,2 exhibits a <italic>freezing behavior</italic>, and seems to be captured within a wide range of temperature. [ABSTRACT FROM AUTHOR]