ON THE COMPLEXITY OF SEMIDEFINITE PROGRAMS ARISING IN POLYNOMIAL OPTIMIZATION

In this paper we investigate matrix inequalities which hold irrespective of the size of the matrices involved, and explain how the search for such inequalities can be implemented as a semidefinite program (SDP). We provide a comprehensive discussion of the time complexity of these SDPs

Probabilistic models on contextuality scenarios

We introduce a framework to describe probabilistic models in Bell experiments, and more generally in contextuality scenarios. Such a scenario is a hypergraph whose vertices represent elementary events and hyperedges correspond to measurements. A probabilistic model on such a scenario associates to each event a probability, in such a way that events in a given measurement have a total probability equal to one. We discuss the advantages of this framework, like the unification of the notions of contexuality and nonlocality, and give a short overview of results obtained elsewhere.Comment: In Proceedings QPL 2013, arXiv:1412.791

Generalizations of Boxworld

Boxworld is a toy theory that can generate extremal nonlocal correlations known as PR boxes. These have been well established as an important tool to examine general nonlocal correlations, even beyond the correlations that are possible in quantum theory. We modify boxworld to include new features. The first modification affects the construction of joint systems such that the new theory allows entangled measurements as well as entangled states in contrast to the standard version of boxworld. The extension to multipartite systems and the consequences for entanglement swapping are analysed. Another modification provides continuous transitions between classical probability theory and boxworld, including the algebraic expression for the maximal CHSH violation as a function of the transition parameters.Comment: In Proceedings QPL 2011, arXiv:1210.029

ON THE COMPLEXITY OF SEMIDEFINITE PROGRAMS ARISING IN POLYNOMIAL OPTIMIZATION

In this paper we investigate matrix inequalities which hold irrespective of the size of the matrices involved, and explain how the search for such inequalities can be implemented as a semidefinite program (SDP). We provide a comprehensive discussion of the time complexity of these SDPs

More non-locality with less entanglement

We provide an explicit example of a Bell inequality with 3 settings and 2 outcomes per site for which the largest violation is not obtained by the maximally entangled state, even if its dimension is allowed to be arbitrarily large. This complements recent results by Junge and Palazuelos (arXiv:1007.3042) who show, employing tools from operator space theory, that such inequalities do exist. Our elementary example provides arguably the simplest setting in which it can be demonstrated that even an infinite supply of EPR pairs is not the strongest possible nonlocal resource.Comment: 9 pages; Added reference to arXiv:1012.151

The convex Positivstellensatz in a free algebra

Given a monic linear pencil L in g variables let D_L be its positivity domain, i.e., the set of all g-tuples X of symmetric matrices of all sizes making L(X) positive semidefinite. Because L is a monic linear pencil, D_L is convex with interior, and conversely it is known that convex bounded noncommutative semialgebraic sets with interior are all of the form D_L. The main result of this paper establishes a perfect noncommutative Nichtnegativstellensatz on a convex semialgebraic set. Namely, a noncommutative polynomial p is positive semidefinite on D_L if and only if it has a weighted sum of squares representation with optimal degree bounds: p = s^* s + \sum_j f_j^* L f_j, where s, f_j are vectors of noncommutative polynomials of degree no greater than 1/2 deg(p). This noncommutative result contrasts sharply with the commutative setting, where there is no control on the degrees of s, f_j and assuming only p nonnegative, as opposed to p strictly positive, yields a clean Positivstellensatz so seldom that such cases are noteworthy.Comment: 22 page