On Phi-entropic Dependence Measures and Non-local Correlations

Abstract

We say that a measure of dependence between two random variables $X$ and $Y$, denoted as $\ddot{I}(X;Y)$, satisfies the data processing property if $\ddot{I}(X;Y)\ge \ddot{I}(X^{\prime };Y^{\prime })$ for every $X^{\prime }\to X\to Y\to Y^{\prime }$, and satisfies the tensorization property if $\ddot{I}(X_1{X}_2;Y_1{Y}_2)=\max \{\ddot{I}(X_1;Y_1),\ddot{I}(X_2;Y_2)\}$ when $(X_1,Y_1)$ is independent of $(X_2,Y_2)$. It is known that measures of dependence defined based on $\hat{I}¦$-entropy satisfy these properties. These measures are important because they generalize R{é}nyi's maximal correlation and the hypercontractivity ribbon. The data processing and tensorization properties are special cases of monotonicity under wirings of non-local boxes. We show that ribbons defined using $\hat{I}¦$-entropic measures of dependence are monotone under wiring of non-local no-signaling boxes, generalizing an earlier result. In addition, we also discuss the evaluation of $\hat{I}¦$-strong data processing inequality constant for joint distributions obtained from a $Z$-channel.