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A holographic realization of correlation and mutual information

Ashis Saha, Anirban Roy Chowdhury, Sunandan Gangopadhyay

TL;DR

The paper investigates a finite-temperature, holographic realization of the information-theoretic bound $I(A:B) \ge \frac{(\langle O_A O_B\rangle_\beta - \langle O_A\rangle_\beta \langle O_B\rangle_\beta)^2}{2 \langle O_A^2\rangle_\beta \langle O_B^2\rangle_\beta}$ by embedding a large-$N$ QFT with Lifshitz dual into Lifshitz black hole geometries. Using geodesic approximations for heavy operators, it computes the thermal two-point function, thermal one-point function, and the composite operator norm $\langle O^2\rangle_\beta$ in various temperature regimes, and evaluates holographic mutual information for disjoint subsystems. The analysis reveals two characteristic separation scales, $sT|_c$ and $sT|_I$, separating quantum and classical correlation contributions, with $sT|_I > sT|_c$, and shows that the bound can be saturated or violated depending on how $\langle O^2\rangle_\beta$ is determined. The work highlights the sensitivity of information-theoretic inequalities to finite-temperature holographic data and underscores the role of thermal operator norms in diagnosing quantum versus classical correlations in strongly coupled systems.

Abstract

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

A holographic realization of correlation and mutual information

TL;DR

The paper investigates a finite-temperature, holographic realization of the information-theoretic bound by embedding a large- QFT with Lifshitz dual into Lifshitz black hole geometries. Using geodesic approximations for heavy operators, it computes the thermal two-point function, thermal one-point function, and the composite operator norm in various temperature regimes, and evaluates holographic mutual information for disjoint subsystems. The analysis reveals two characteristic separation scales, and , separating quantum and classical correlation contributions, with , and shows that the bound can be saturated or violated depending on how is determined. The work highlights the sensitivity of information-theoretic inequalities to finite-temperature holographic data and underscores the role of thermal operator norms in diagnosing quantum versus classical correlations in strongly coupled systems.

Abstract

The status of the inequality existing between mutual information and (normalized) thermal two-point connected correlation function, namely, has been explicitly probed by using the gauge/gravity correspondence. In the holographic analysis, the geodesic approximation for heavy operators () has been used. We observe that the study leads to some non-trivial insights depending upon the method of calculating the thermal object . For a particular computed result of we propose that all of the existing quantum mechanical dependencies (correlations) and classical correlations between the subsystems and vanishes at two different separation lengths, namely, and where .
Paper Structure (13 sections, 69 equations, 1 figure, 1 table)

This paper contains 13 sections, 69 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Holographic setup for the geometric computations
  3. Computation of the thermal two-point correlator
  4. Computation of the thermal one-point function
  5. Computation of $\langle\mathcal{O}^2\rangle_{\beta}$
  6. Computation of mutual information $I(A:B)$
  7. Checking the inequality and results
  8. Case 1: For $\langle\mathcal{O}^2\rangle_{\beta}=2^{4\Delta (\frac{1}{\xi}-\frac{1}{d+\xi-1})}[\frac{\pi T}{d+\xi-1}]^{\frac{2\Delta}{\xi}}$
  9. Case 2: For $\langle\mathcal{O}^2\rangle_{\beta}=b_dT^{\frac{2\Delta}{\xi}}$
  10. Case 3: For $\langle\mathcal{O}^2\rangle_{\beta}=\Delta a_d T^{\frac{2\Delta}{\xi}}$
  11. Conclusion
  12. Appendix 1: Classical derivation of the inequality involving mutual information and connected thermal two-point correlator, $I(X:Y)\geq \frac{\sigma(X,Y)^2}{2||X||^2~||Y||^2}$
  13. Appendix 2: Explicit expressions of various constants

Figures (1)

  • Figure 1: Plot of holographic mutual information $I(A:B)$ and two-point connected correlator $C$. In the left panel, we plot $I(A:B)$ and $C$ with respect to $\alpha=\frac{sT}{lT}$, for $lT=0.4$. In the right panel, we plot $I(A:B)$ and $C$ with respect to $sT$, for $lT=10$. In these plots, we set $d=3$ and $\Delta=10$.