A Holographic Proof of R\'enyi Entropic Inequalities

Understand

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.