Classifying the simplest Bell inequalities beyond qubits and their applications towards self-testing

TL;DR

This paper characterizes all Bell inequalities in the (2,2,3) scenario derived from sum-of-squares decompositions, enabling self-testing of maximally entangled states of local dimension three and associated measurements.

Contribution

It provides a complete characterization of Bell inequalities beyond qubits in the (2,2,3) scenario and applies them to self-test maximally entangled states.

Findings

All Bell inequalities from the sum-of-squares decomposition in (2,2,3) scenario identified.

Bell inequalities used for self-testing maximally entangled states of dimension three.

Enhanced understanding of nonlocal correlations beyond the simplest qubit case.

Abstract

Bell inequalities reveal the fundamentally nonlocal character of quantum mechanics. In this regard, one of the interesting problems is to explore all possible Bell inequalities that demonstrate a gap between local and nonlocal quantum behaviour. This is useful for the geometric characterisation of the set of nonlocal correlations achievable within quantum theory. Moreover, it provides a systematic way to construct Bell inequalities that are tailored to specific quantum information processing tasks. This characterisation is well understood in the simplest $(2,2,2)$ scenario, namely two parties performing two binary outcome measurements. However, beyond this setting, relatively few Bell inequalities are known, and the situation becomes particularly scarce in scenarios involving a greater number of outcomes. Here, we consider the $(2,2,3)$ scenario, or two parties performing two three-outcome measurements, and characterise all Bell inequalities that can arise from the simplest sum-of-squares decomposition and are maximally violated by the maximally entangled state of local dimension three. We then utilise them to self-test this state, along with a class of three-outcome measurements.