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

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.

💡 Research Summary

The paper addresses the largely unexplored territory of Bell inequalities in the (2,2,3) scenario, where two distant parties each have two measurement settings with three possible outcomes. While the (2,2,2) case (two parties, two settings, binary outcomes) is completely characterized, the landscape for higher‑outcome scenarios remains sparse. The authors set out to classify all Bell inequalities that arise from the simplest sum‑of‑squares (SOS) decomposition and that are maximally violated by the maximally entangled state of local dimension three (the qutrit maximally entangled state).

The authors first define a general Bell operator \