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Bell showed 50 years ago that quantum theory is nonlocal via his celebratedinequalities, turning the issue of quantum nonlocality from a matter of tasteinto a matter of test. Years later, Hardy proposed a test for nonlocalitywithout inequality, which is a kind of "something-versus-nothing" argument.Hardy's test for $n$ particles induces an $n$-partite Bell's inequality withtwo dichotomic local measurements for each observer, which has been shown to beviolated by all entangled pure states. Our first result is to show that theBell-Hardy inequality arising form Hardy's nonlocality test is tight for anarbitrary number of parties, i.e., it defines a facet of the Bell polytope inthe given scenario. On the other hand quantum theory is not that nonlocal sinceit forbids signaling and even not as nonlocal as allowed by non-signalingconditions, i.e., quantum mechanical predictions form a strict subset of the socalled non-signaling polytope. In the scenario of each observer measuring twodichotomic observables, Fritz established a duality between the Bell polytopeand the non-signaling polytope: tight Bell's inequalities, the facets of theBell polytope, are in a one-to-one correspondence with extremal non-signalingboxes, the vertices of the non-signaling polytope. Our second result is toprovide an alternative and more direct formula for this duality. As an example,the tight Bell-Hardy inequality gives rise to an extremal non-signaling boxthat serves as a natural multipartite generalization of Popescu-Rohrlich box.

Bell's inequality and extremal nonlocal box from Hardy's test for nonlocality