A Maximum Entropy Approach to the Realizability of Spin

Paolo Dai Pra , Michele Pavon * and Neeraja SahasrabudheDepartment of Mathematics, University of Padova, via Trieste 63, 35121 Padova, Italy;E-Mails: daipra@math.unipd.it (P.D.P.); neeraja@math.unipd.it (N.S.)* Author to whom correspondence should be addressed; E-Mail: pavon@math.unipd.it;Tel.: +39-049-827-1341; Fax: +39-049-827-1428.Received: 26 February 2013; in revised form: 10 June 2013 / Accepted: 15 June 2013 /Published: 21 June 2013Abstract: Deriving the form of the optimal solution of a maximum entropy problem, weobtain an infinite family of linear inequalities characterizing the polytope of spin correlationmatrices. For n6, the facet description of such a polytope is provided through a minimalsystem of Bell-type inequalities.Keywords: correlation matrix; spin system; maximum entropy; Bell’s inequalities;moment problem1. IntroductionMoment problems are fairly common in many areas of applied mathematics, statistics and probability,economics, engineering, physics and operations research. Historically, moment problems came intofocus with Stieltjes in 1894 [1], in the context of studying the analytic behavior of continued fractions.The term “moment” was borrowed from mechanics: the moments could represent the total mass ofan unknown mass density, the torque necessary to support the mass on a beam, etc. Over time,however, moment problems took the shape of an important field in their own right. A deep connectionwith convex geometry was discovered by Krein in the mid 1930s and developed by the Russianschool; see, e.g., [2,3]. Another fundamental connection with the work of Caratheodory, Toeplitz,Schur, Nevanlinna and Pick on analytic interpolation was investigated in the first half of the twentiethcentury [4]. This led to important developments in operator theory; see, e.g., [5,6]. In more recenttimes, a rather impressive application and generalization of this mathematics has been developed by the