  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]} scenario
are obtained with their quantum violations analyzed in details. Surprisingly,
all these inequalities involving only genuine ternary-outcome measurements can
be violated maximally by some two-qubit entangled states, such as the maximally
entangled two-qubit state. This gives further evidence that in analyzing the
quantum violation of Bell inequalities, or in the application of the latter to
device-independent quantum information processing tasks, the commonly-held
wisdom of equating the local Hilbert space dimension of the optimal state with
the number of measurement outcomes is not necessarily justifiable. In addition,
when restricted to the minimal qubit subspace, it can be shown that one of
these Bell inequalities requires non-projective measurements to attain maximal
quantum violation, thereby giving the first example of a facet-defining Bell
inequality where a genuine positive-operator-valued measure is relevant. We
experimentally demonstrate the quantum violation of this and two other Bell
inequalities for this scenario using energy-time entangled photon pairs. Using
the obtained measurement statistics, we demonstrate how characterization of the
underlying resource in the spirit of device-independence, but supplemented with
auxiliary assumptions, can be achieved. In particular, we discuss how one may
get around the fact that, due to finite-size effects, raw measurement
statistics typically violate the non-signaling condition.
