A Holographic Proof of R\'enyi Entropic Inequalities

TL;DR

This paper establishes holographic Renyi entropic inequalities by reformulating them as thermodynamic principles, providing new insights into quantum entanglement and modular Hamiltonian fluctuations in holography.

Contribution

It introduces a holographic proof of Renyi entropic inequalities using a thermodynamic analogy, and derives a formula for quantum fluctuations of the modular Hamiltonian.

Findings

Proof of Renyi entropic inequalities in holography

Holographic formula for quantum fluctuation of modular Hamiltonian

Analysis of capacity of entanglement examples

Abstract

We prove R\'enyi entropic inequalities in a holographic setup based on the recent proposal for the holographic formula of R\'enyi entropies when the bulk is stable against any perturbation. Regarding the R\'enyi parameter as an inverse temperature, we reformulate the entropies in analogy with statistical mechanics, which provides us a concise interpretation of the inequalities as the positivities of entropy, energy and heat capacity. This analogy also makes clear a thermodynamic structure in deriving the holographic formula. As a by-product of the proof we obtain a holographic formula to calculate the quantum fluctuation of the modular Hamiltonian. A few examples of the capacity of entanglement are examined in detail.