Geometry of the set of quantum correlations

TL;DR

This paper explores the geometric structure of quantum correlations using convex geometry, revealing counter-intuitive features in simple Bell scenarios and discussing implications for self-testing.

Contribution

It provides the first detailed geometric analysis of the quantum set of correlations, highlighting complex features and limitations in self-testing.

Findings

Counter-intuitive geometric features in simple Bell scenarios

More complex features in scenarios with more inputs or parties

Limitations on self-testing imposed by quantum set geometry

Abstract

It is well known that correlations predicted by quantum mechanics cannot be explained by any classical (local-realistic) theory. The relative strength of quantum and classical correlations is usually studied in the context of Bell inequalities, but this tells us little about the geometry of the quantum set of correlations. In other words, we do not have good intuition about what the quantum set actually looks like. In this paper we study the geometry of the quantum set using standard tools from convex geometry. We find explicit examples of rather counter-intuitive features in the simplest non-trivial Bell scenario (two parties, two inputs and two outputs) and illustrate them using 2-dimensional slice plots. We also show that even more complex features appear in Bell scenarios with more inputs or more parties. Finally, we discuss the limitations that the geometry of the quantum set imposes on the task of self-testing.