A holographic realization of correlation and mutual information

The status of the inequality existing between mutual information and (normalized) thermal two-point connected correlation function, namely, $I(A:B)\ge \frac{(\text{expval}{{\mathcal{O}}_A{\mathcal{O}}_B}_{\beta }-\text{expval}{{\mathcal{O}}_A}_{\beta }\text{expval}{{\mathcal{O}}_B}_{\beta }{)}^2}{2\text{expval}{{\mathcal{O}}_A^2}_{\beta }\text{expval}{{\mathcal{O}}_B^2}_{\beta }}$ has been explicitly probed by using the gauge/gravity correspondence. In the holographic analysis, the geodesic approximation for heavy operators ($\mathrm{\Delta }\sim mR$) has been used. We observe that the study leads to some non-trivial insights depending upon the method of calculating the thermal object $\text{expval}{{\mathcal{O}}^2}_{\beta }$. For a particular computed result of $\text{expval}{{\mathcal{O}}^2}_{\beta }$ 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, $sT\mathrm{\mid }c$ and $sT{\mathrm{\mid }}_I$ where $sT{\mathrm{\mid }}_I>sT{\mathrm{\mid }}_c$.