Entropy inequalities from reflection positivity

TL;DR

This paper derives a new set of entropy inequalities in quantum field theory using reflection positivity, linking correlator positivity to entropy inequalities, and proves an infinite sequence of such inequalities for Renyi entropies.

Contribution

It introduces a generalized reflection positivity framework to establish entropy inequalities and proves an infinite sequence of inequalities for Renyi entropies in quantum field theory.

Findings

Derived an infinite sequence of entropy inequalities for Renyi entropies.

Connected correlator positivity to entropy inequalities in quantum field theory.

Provided geometrical interpretations and examples of the inequalities.

Abstract

We investigate the question of whether the entropy and the Renyi entropies of the vacuum state reduced to a region of the space can be represented in terms of correlators in quantum field theory. In this case, the positivity relations for the correlators are mapped into inequalities for the entropies. We write them using a real time version of reflection positivity, which can be generalized to general quantum systems. Using this generalization we can prove an infinite sequence of inequalities which are obeyed by the Renyi entropies of integer index. There is one independent inequality involving any number of different subsystems. In quantum field theory the inequalities acquire a simple geometrical form and are consistent with the integer index Renyi entropies being given by vacuum expectation values of twisting operators in the Euclidean formulation. Several possible generalizations and specific examples are analyzed.