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We explore quantum nonlocality in one of the simplest bipartite scenarios.Several new facet-defining Bell inequalities for the {[3 3 3] [3 3 3]} scenarioare obtained with their quantum violations analyzed in details. Surprisingly,all these inequalities involving only genuine ternary-outcome measurements canbe violated maximally by some two-qubit entangled states, such as the maximallyentangled two-qubit state. This gives further evidence that in analyzing thequantum violation of Bell inequalities, or in the application of the latter todevice-independent quantum information processing tasks, the commonly-heldwisdom of equating the local Hilbert space dimension of the optimal state withthe number of measurement outcomes is not necessarily justifiable. In addition,when restricted to the minimal qubit subspace, it can be shown that one ofthese Bell inequalities requires non-projective measurements to attain maximalquantum violation, thereby giving the first example of a facet-defining Bellinequality where a genuine positive-operator-valued measure is relevant. Weexperimentally demonstrate the quantum violation of this and two other Bellinequalities for this scenario using energy-time entangled photon pairs. Usingthe obtained measurement statistics, we demonstrate how characterization of theunderlying resource in the spirit of device-independence, but supplemented withauxiliary assumptions, can be achieved. In particular, we discuss how one mayget around the fact that, due to finite-size effects, raw measurementstatistics typically violate the non-signaling condition.Comment: 16 pages, 2 tables and 6 figure

Bipartite Bell inequalities with three ternary-outcome measurements—from theory to experiments