Quantum value for a family of $I_{3322}$-like Bell functionals

TL;DR

This paper introduces a three-parameter family of Bell functionals extending previous work, analyzes their quantum values, and characterizes optimal quantum realizations, including self-testing properties and numerical studies of the $I_{3322}$-like functionals.

Contribution

It extends Bell functionals with a new marginal term, provides a complete characterization of quantum values for one branch, and numerically studies the other, including the $I_{3322}$ functional.

Findings

Quantum value for the first branch is a simple function of parameters.

Complete characterization of optimal realizations for the first branch.

Numerical identification of quantum regions and optimal states for the second branch.

Abstract

We introduce a three-parameter family of Bell functionals that extends those studied in reference [Phys. Rev. Research 2, 033420 (2020)] by including a marginal contribution. An analysis of their quantum value naturally splits the family into two branches, and for the first of them we show that this value is given by a simple function of the parameters defining the functionals. In this case we completely characterise the realisations attaining the optimal value and show that these functionals can be used to self-test any partially entangled state of two qubits. The optimal measurements, however, are not unique and form a one-parameter family of qubit measurements. The second branch, which includes the well-known $I_{3322}$ functional, is studied numerically. We identify the region in the parameter space where the quantum value can be attained, with two-dimensional systems and characterise the state and measurements attaining this value. Finally, we show that the set of realisations introduced in reference [Phys. Rev. A 82, 022116 (2010)] to obtain the maximal violation of the $I_{3322}$ inequality succeeds in approaching the optimal value for a large subset of the functionals in this branch. In these cases we analyse and discuss the main features of the optimal realisations.