Tsirelson's bound from a Generalised Data Processing Inequality

TL;DR

This paper links Tsirelson's bound on quantum correlations to the data processing inequality in generalized probabilistic theories, showing that certain entropic properties imply the bound.

Contribution

It demonstrates that the validity of the data processing inequality with a generalized entropy measure necessarily enforces Tsirelson's bound in any convex probabilistic theory.

Findings

Data processing inequality implies Tsirelson's bound

Generalized entropy measures can be used to derive quantum bounds

Not all entropic relations are necessary for Tsirelson's bound

Abstract

The strength of quantum correlations is bounded from above by Tsirelson's bound. We establish a connection between this bound and the fact that correlations between two systems cannot increase under local operations, a property known as the data processing inequality. More specifically, we consider arbitrary convex probabilistic theories. These can be equipped with an entropy measure that naturally generalizes the von Neumann entropy, as shown recently in [Short and Wehner, Barnum et al.]. We prove that if the data processing inequality holds with respect to this generalized entropy measure then the underlying theory necessarily respects Tsirelson's bound. We moreover generalise this statement to any entropy measure satisfying certain minimal requirements. A consequence of our result is that not all of the entropic relations used to derive Tsirelson's bound via information causality in [Pawlowski et al.] are necessary.