Maximin is Not Enough

TL;DR

This paper explores the limitations of the maximin method for proving inequalities in holographic entanglement entropy in dynamical spacetimes and proposes a new, weaker inequality based on Shannon entropy properties.

Contribution

It demonstrates the current limitations of the maximin approach for certain inequalities and introduces a new inequality for covariant holographic entanglement entropy based on Shannon entropy properties.

Findings

Maximin method has limited utility for inequalities involving five or fewer regions.

Proposed a new inequality for covariant holographic entanglement entropy.

The new inequality is easier to prove and supports the role of holographic entanglement in cosmology.

Abstract

The RT formula for static spacetimes arising in the AdS/CFT correspondence satisfies inequalities that are not yet proven in the case of the HRT formula, which applies to general dynamical spacetimes. Wall's maximin construction is the only known technique for extending inequalities of holographic entanglement entropy from the static to dynamical case. We show that this method currently has no further utility when dealing with inequalities for five or fewer regions. Despite this negative result, we propose the validity of one new inequality for covariant holographic entanglement entropy for five regions. This inequality, while not maximin provable, is much weaker than many of the inequalities satisfied by the RT formula and should therefore be easier to prove. If it is valid, then there is strong evidence that holographic entanglement entropy plays a role in general spacetimes including those that arise in cosmology. Our new inequality is obtained by the assumption that the HRT formula satisfies every known balanced inequality obeyed by the Shannon entropies of classical probability distributions. This is a property that the RT formula has been shown to possess and which has been previously conjectured to hold for quantum mechanics in general.