A holographic realization of correlation and mutual information

Abstract

The status of the inequality existing between mutual information and (normalized) thermal two-point connected correlation function, namely, $I(A:B)\ge \frac{{({〈O_A{O}_B〉}_{\beta }-{〈O_A〉}_{\beta }{〈O_B〉}_{\beta })}^2}{2{〈O_A^2〉}_{\beta }{〈O_B^2〉}_{\beta }}$ has been explicitly probed by using the gauge/gravity correspondence. In the holographic analysis, the geodesic approximation for heavy operators ($\Delta \sim mR$) has been used. We observe that the study leads to some non-trivial insights depending upon the method of calculating the thermal object ${〈O^2〉}_{\beta }$. For a particular computed result of ${〈O^2〉}_{\beta }$ we propose that all of the existing quantum mechanical dependencies (correlations) and classical correlations between the subsystems $A$ and $B$ vanishes at two different separation lengths, namely, $sT|c$ and ${sT|}_I$ where ${sT|}_I{>sT|}_c$.