Mutual Information of Measurement in Conformal Field Theories

Abstract

The mutual information of measurement of a local operator between two spatial regions characterizes the reduction in the uncertainty of measurement outcomes in one region given the outcome in the other. We study the Rényi-Shannon mutual information of measurement in conformal field theories (CFTs). Explicit calculations for scalar primaries and descendants in large $N$ CFTs are performed in the two extreme limits of the intervals being adjacent or largely separated. The results depend on only the cross ratio and the scaling dimension $\Delta$ of the operator being measured, with striking similarities to the mutual information of entanglement. We find a surprising quadratic dependence as $(\Delta -1/2{)}^2$ for adjacent intervals. This dependence superficially seems to violate the information processing inequality, which we discuss and resolve. We also offer a holographic interpretation of the mutual information of measurement.