Abstract
Deriving the form of the optimal solution of a maximum entropy problem, we obtain an infinite family of linear inequalities characterizing the polytope of spin correlation matrices. For n ≤ 6, the facet description of such a polytope is provided through a minimal system of Bell-type inequalities.
Highlights
Moment problems are fairly common in many areas of applied mathematics, statistics and probability, economics, engineering, physics and operations research
The term “moment” was borrowed from mechanics: the moments could represent the total mass of an unknown mass density, the torque necessary to support the mass on a beam, etc
One seeks a multivariate, stationary stochastic process as the input of a bank of rational filters whose output covariance has been estimated. This turns into a Nevanlinna-Pick interpolation problem with a bounded degree [7,8]. The latter can be viewed as a generalized moment problem, which is advantageously cast in the frame of various convex optimization problems, often featuring entropic-type criteria
Summary
Moment problems are fairly common in many areas of applied mathematics, statistics and probability, economics, engineering, physics and operations research. A relevant problem for applications considers the situation in which only some entries of the covariance matrix are given, for example, those that have been estimated from the data In this context, one aims at characterizing all possible completions of the partially given covariance matrix or completions that possess certain desirable properties; see, e.g., [29,31,32,33,34,35,36,37] and references therein. One aims at characterizing all possible completions of the partially given covariance matrix or completions that possess certain desirable properties; see, e.g., [29,31,32,33,34,35,36,37] and references therein Another more theoretical problem investigates the geometry of correlation matrices, namely, covariances of standardized random variables. We obtain necessary and sufficient conditions for the existence of a covariance completion, as well as a “canonical” (maximum entropy) probability realizing the given covariances
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