Quantum value for a family of I-3322 -like Bell functionals

We introduce a three-parameter family of Bell functionals that extends those studied previously [Kaniewski, Phys. Rev. Res. 2, 033420 (2020)] by including a marginal contribution. An analysis of their largest value achievable by quantum realizations 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 characterize the realizations 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 optimal value can be attained with two-dimensional quantum systems and characterize the state and measurements attaining this value. Finally, we show that the set of realizations introduced by Pal and Vertesi [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 analyze and discuss the main features of the optimal realizations.