Concentration Phenomena in the Geometry of Bell Correlations

Cris Duarte(Chapman University), Raphael Drumond (Federal University of Minas Gerais), Roberto Oliveira (Brazilian Institute for Pure and Applied Mathematics)

For any finite number of parts, measurements and outcomes in a Bell scenario we estimate the probability of random N-qudit pure states to substantially violate any Bell inequality with uniformly bounded coefficients. We prove that under some conditions on the local dimension the probability to find any significant amount of violation goes to zero exponentially fast as the number of parts goes to infinity. In addition, we also prove that if the number of parts is at least 3, this probability also goes to zero as the local Hilbert space dimension goes to infinity. In addition to that, shedding light upon the fragmented picture we have for the set of quantum correlations, we employ both analytical and numerical tools to ameliorate that. First, we identify two different classes of 
Bell scenarios where the nonsignaling correlations can behave very differently: In one case, the correlations are generically quantum and nonlocal while in the other quite the opposite happens as the correlations are generically classical and local. Second, by randomly sampling over nonsignaling 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 a concentration phenomenon, as their distribution is peaked at a distance from the local set that increases both with the number of parts or measurements being performed.