R\'enyi mutual information inequalities from Rindler positivity

TL;DR

This paper derives inequalities for Rènyi mutual information in relativistic quantum field theories using Rindler positivity, revealing conditions like complete monotonicity and applying them to conformal theories and OPE coefficient constraints.

Contribution

It introduces new inequalities for Rènyi mutual information based on Rindler positivity, including conditions like complete monotonicity and applications to conformal field theories.

Findings

Rindler positivity implies an infinite set of inequalities for Rènyi mutual information.

In 1+1D CFTs, conformal invariance leads to stronger monotonicity conditions.

These inequalities constrain OPE coefficients of twist operators.

Abstract

Rindler positivity is a property that holds in any relativistic Quantum Field Theory and implies an infinite set of inequalities involving the exponential of the R\'enyi mutual information $I_n(A_i,\bar{A}_j)$ between $A_i$ and $\bar{A}_j$, where $A_i$ is a spacelike region in the right Rindler wedge and $\bar{A}_j$ is the wedge reflection of $A_j$. We explore these inequalities in order to get local inequalities for $I_n(A,\bar{A})$ as a function of the distance between $A$ and its mirror region $\bar{A}$. We show that the assumption, based on the cluster property of the vacuum, that $I_n$ goes to zero when the distance goes to infinity, implies the more stringent and simple condition that $F_n\equiv{e}^{(n-1)I_n}$ should be a completely monotonic function of the distance, meaning that all the even (odd) derivatives are non-negative (non-positive). In the case of a CFT in 1+1 dimensions, we show that conformal invariance implies stronger conditions, including a sort of monotonicity of the R\'enyi mutual information for pairs of intervals. An application of these inequalities to obtain constraints for the OPE coefficients of the $4-$point function of certain twist operators is also discussed.