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 ,Ā j ) between A i andĀ j , where A i is a spacelike region in the right Rindler wedge andĀ j is the wedge reflection of A j . We explore these inequalities in order to get local inequalities for I n (A,Ā) as a function of the distance between A and its mirror regionĀ. 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)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 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.

Is the EMI model a QFT? An inquiry on the space of allowed entropy functions

“…The Rényi entropies of integer order n ≥ 2, as they are produced by expectation values of twist operators, also satisfy this infinite set of inequalities. In real time, these operator inequalities -equivalent to the reflection positive inequalities in imaginary time -are called wedge reflection positivity, Rindler positivity, or CRT positivity (for chargereflection-time inversion) [10,54,55].…”

Global aspects of conformal symmetry and the ANEC in dS and AdS

“…Following a similar approach as in refs. [48,49] it would be interesting to explore the consequences of these properties regarding entanglement entropy inequalities.…”

Interplay between reflection positivity and crossing symmetry in the bootstrap approach to CFT

“…This change of x by 1 − x implies that, on the rhs, these inequalities are not automatically fulfilled, and at least the OPE coefficients are restricted. In order to understand this, let us briefly describe the connection between RP and some inequalities for a function of a single variable used by us in [7].…”

“…Organization of the paper. In section 2, we recall the general form of RP inequalities and the material about positive definite functions of single variables with the one-parameter arrange of points used in [7]. In section 3 we will see the implications of RP in the case of CFT and check some positivity conditions in the conformal blocks.…”

“…Although we have not used test functions, the choice of each sequence correspond formally to test functions sharp at the t i α . We want to convert the positive definiteness of this quadratic form into a positive definiteness of a function of a single variable, as we did in [7]. Let us make a brief summary of it below.…”