On Phi-entropic Dependence Measures and Non-local Correlations

TL;DR

The work presents a unified framework for dependence measures based on $\Phi$-entropy, generalizing both maximal correlation and the hypercontractivity ribbon. It defines the $\Phi$-ribbon $\mathfrak{R}_{\Phi}$ and proves that these ribbons satisfy data-processing and tensorization, aligning with monotonicity under wirings of no-signaling boxes. A key contribution is showing monotonicity of $\mathfrak{R}_{\Phi}$ under arbitrary box wirings, extending known results for MC and HC ribbons to the full family of $\Phi$-ribbons. Additionally, the paper analyzes the $\Phi$-SDPI constant $\eta_{\Phi}$ for Z-channel sources, reducing the optimization to two function classes and identifying conditions under which the maximizer occurs at $f_X(1)=0$ for broad $\mathscr{F}$, with explicit constants for standard $\Phi$ choices. Overall, these results provide a coherent approach to nonlocal correlations and information-processing inequalities in both classical and no-signaling quantum settings, with potential implications for characterizing nonlocality via ribbon-based monotones.

Abstract

We say that a measure of dependence between two random variables $X$ and $Y$, denoted as $ρ(X;Y)$, satisfies the data processing property if $ρ(X;Y)\geq ρ(X';Y')$ for every $X'\rightarrow X\rightarrow Y\rightarrow Y'$, and satisfies the tensorization property if $ρ(X_1X_2;Y_1Y_2)=\max\{ρ(X_1;Y_1),ρ(X_2;Y_2)\}$ when $(X_1,Y_1)$ is independent of $(X_2,Y_2)$. It is known that measures of dependence defined based on $Φ$-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 $Φ$-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 $Φ$-strong data processing inequality constant for joint distributions obtained from a $Z$-channel.

Figures (2)

• Figure 1: Consider a scenario where two parties each possess subsystems of a bipartite physical system, which can exhibit correlations. Each party can perform a measurement on their respective subsystem by adjusting the measurement device using a specific parameter, and subsequently observe the outcome. Let the measurement settings be denoted by $x$ and $y$, and the corresponding outcomes by $a$ and $b$. In the general case, the outcomes $a$ and $b$ resulting from the measurements $x$ and $y$ occur with a conditional probability $p_{AB|XY}(ab|xy)$. This setup can be conceptualized as a “box” divided into two parts, where each part has an input and an output. For a given pair of inputs $x$ and $y$, the corresponding outputs $a$ and $b$ are produced according to the probability distribution $p_{AB|XY}(ab|xy)$.
• Figure 2: Due to no-signaling property, Alice and Bob can choose boxes in different orders. For example, Alice can use the order $2,3,1$ to simulate the result given $x'$. Alice simulates the box $2$ with input $x_2$ determined by $x'$ and get the result $a_2$. She then uses the box $3$ with input $x_3$ determined by the former output $a_2$ and get the result $a_3$. Then she used the box $1$ with input $x_1$ determined by $a_3$ and get the result $a_1$. Finally she simulates the result $a'$ by applying a stochastic map. Bob can use the order of boxes as $3,1,2$ given $y'$ and follow the same steps.

Theorems & Definitions (2)

• proof
• proof : Proof of \ref{['tensorization-of-two']}