Rényi mutual information inequalities from Rindler positivity

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 Renyi mutual information In (Ai,$$ \overline{A} $$j) between Ai and $$ \overline{A} $$j, where Ai is a spacelike region in the right Rindler wedge and $$ \overline{A} $$j is the wedge reflection of Aj. We explore these inequalities in order to get local inequalities for In (A,$$ \overline{A} $$) as a function of the distance between A and its mirror region $$ \overline{A} $$. We show that the assumption, based on the cluster property of the vacuum, that In goes to zero when the distance goes to infinity, implies the more stringent and simple condition that Fn≡ e(n−1)In 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, we show that conformal invariance implies stronger conditions, including a sort of monotonicity of the Renyi mutual information for pairs of balls. An application of these inequalities to obtain constraints for the OPE coefficients of the 4-point function of certain twist operators is also discussed.