Rényi mutual information inequalities from Rindler positivity

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 Rényi 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 Rényi 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.

16 pages, 4 figures

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