Positivity, entanglement entropy, and minimal surfaces

TL;DR

This paper investigates the boundary representation of entanglement entropy in quantum field theory and holography, deriving inequalities for minimal surfaces and entropies, and analyzing their validity through examples and analytic tools.

Contribution

It introduces a boundary insertion approach for entanglement entropy at first order in n-1 and explores the resulting positivity inequalities in QFT and holographic contexts.

Findings

Positivity inequalities hold in known QFT examples.

Counterexamples found in complex geometries.

Analytic tools show local positivity of correlation functions.

Abstract

The path integral representation for the Renyi entanglement entropies of integer index n implies these information measures define operator correlation functions in QFT. We analyze whether the limit $n\rightarrow 1$, corresponding to the entanglement entropy, can also be represented in terms of a path integral with insertions on the region's boundary, at first order in $n-1$. This conjecture has been used in the literature in several occasions, and specially in an attempt to prove the Ryu-Takayanagi holographic entanglement entropy formula. We show it leads to conditional positivity of the entropy correlation matrices, which is equivalent to an infinite series of polynomial inequalities for the entropies in QFT or the areas of minimal surfaces representing the entanglement entropy in the AdS-CFT context. We check these inequalities in several examples. No counterexample is found in the few known exact results for the entanglement entropy in QFT. The inequalities are also remarkable satisfied for several classes of minimal surfaces but we find counterexamples corresponding to more complicated geometries. We develop some analytic tools to test the inequalities, and as a byproduct, we show that positivity for the correlation functions is a local property when supplemented with analyticity. We also review general aspects of positivity for large N theories and Wilson loops in AdS-CFT.