Rényi mutual information inequalities from Rindler positivity

Abstract

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 , A ¯ $$ \overline{A} $$ j ) between A i and A ¯ $$ \overline{A} $$ j , where A i is a spacelike region in the right Rindler wedge and A ¯ $$ \overline{A} $$ j is the wedge reflection of A j . We explore these inequalities in order to get local inequalities for I n (A, A ¯ $$ \overline{A} $$ ) as a function of the distance between A and its mirror region A ¯ $$ \overline{A} $$ . 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 ≡ 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, we show that conformal invariance implies stronger conditions, including a sort of monotonicity of the Rényi 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.

Keywords

• Conformal Field Theory
• Field Theories in Lower Dimensions