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Noncontextuality inequalities are usually derived from the distinguishability properties of quantum states, i.e. their orthogonality. Here, we show that antidistinguishability can also be used to derive noncontextuality inequalities. The Yu-Oh 13 ray noncontextuality inequality can be re-derived and generalized as an instance of our antidistinguishability method. For some sets of states, the antidistinguishability method gives tighter bounds on noncontextual models than just considering orthogonality, and the Hadamard states provide an example of this. We also derive noncontextuality inequalities based on mutually unbiased bases and symmetric informationally complete POVMs. Antidistinguishability based inequalities were initially discovered as overlap bounds for the reality of the quantum state. Our main contribution here is to show that they are also noncontextuality inequalities., Comments are welcome!

In an ideal contextuality scenario it is assumed that, whenever two contexts overlap, the marginal distribution obtained for the intersection of these contexts must be the same. Although this nondisturbance idealization has been extremely useful in investigating the limitations of physical theories, in this paper, we will work out a more robust and possibly more experimentally friendly framework for contextuality. Simply put, we will restate the extended definition of noncontextuality provided by the noncontextuality by default and show that in this extended perspective it is possible to avoid weakening the nondisturbance hypothesis. By doing so, we also show how standard tools for characterizing and quantifying contextuality can also be applied to this extended scenario. In particular, we came up with a linear program to check whether or not a behavior is contextual in the extended sense. The connection with the usual definition of contextuality is also established because whenever nondisturbance is fully satisfied the extended conditions reduce to the original ones.

Bell's theorem shows that local measurements on entangled states give rise to correlations incompatible with local hidden variable models. The degree of quantum nonlocality is not maximal though, as there are even more nonlocal theories beyond quantum theory still compatible with the nonsignalling principle. In spite of decades of research, we still have a very fragmented picture of the whole geometry of these different sets of correlations. Here we employ both analytical and numerical tools to ameliorate that. First, we identify two different classes of Bell scenarios where the nonsignalling correlations can behave very differently: in one case, the correlations are generically quantum and nonlocal while on the other quite the opposite happens as the correlations are generically classical and local. Second, by randomly sampling over nonsignalling correlations, we compute the distribution of a nonlocality quantifier based on the trace distance to the local set. With that, we conclude that the nonlocal correlations can show concentration phenomena: their distribution is peaked at a distance from the local set that increases both with the number of parts or measurements being performed., Comment: Minor modifications. Still with 13 pages, 8 figures, 5 tables. Comments are welcome!