A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations

Abstract

We are interested in the problem of characterizing the correlations that arise when performing local measurements on separate quantum systems. In a previous work [Phys. Rev. Lett. 98, 010401 (2007)], we introduced an infinite hierarchy of conditions necessarily satisfied by any set of quantum correlations. Each of these conditions could be tested using semidefinite programming. We present here new results concerning this hierarchy. We prove in particular that it is complete, in the sense that any set of correlations satisfying every condition in the hierarchy has a quantum representation in terms of commuting measurements. Although our tests are conceived to rule out nonquantum correlations, and can in principle certify that a set of correlations is quantum only in the asymptotic limit where all tests are satisfied, we show that in some cases it is possible to conclude that a given set of correlations is quantum after performing only a finite number of tests. We provide a criterion to detect when such a situation arises, and we explain how to reconstruct the quantum states and measurement operators reproducing the given correlations. Finally, we present several applications of our approach. We use it in particular to bound the quantum violation of various Bell inequalities.

Nonlocality in many-body quantum systems detected with two-body correlators

“…Unlike the local set, Q is not a polytope and hence its boundary is more difficult to characterize. Still, it can be well approximated [57]. Non-signalling correlations.…”

“…These techniques can be applied to any (n, m, d) scenario with K-body correlators when the symmetry group G is S n and they will be studied elsewhere. These approximations are in the spirit of the NavascuesPironio-Acín (NPA) hierarchy [57], which bounds the set of quantum correlations Q appearing in (8) from above by providing a hierarchy of sets Q ⊆ . .…”

Maximal violation of a bipartite three-setting, two-outcome Bell inequality using infinite-dimensional quantum systems

“…The notion of levels and partial levels, and their notation is explained in Ref. [11], and also in Ref [13]. This 19 Bell inequalities for which the tightness of the quantum bound has not been proven.…”

“…On the other hand, the best upper bounds are based on the NPA method [10] and at level three it yields the significantly higher upper bound, 0.250 875 56 [11,16]. We could even go above level three to an intermediate level in [13], and presently we have got the upper bound 0.250 875 38 at level four.…”

More nonlocality with less entanglement

“…This would not only have very interesting consequences for our understanding of Bell inequalities, but also for the optimization of polynomials with noncommutative variables. In particular, it would imply that the SDP hierarchies suggested in [25,32,33] only converge in the limit of infinitely many levels, which is an open problem even outside the realm of quantum information.…”