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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

Moment problems are fairly common in many areas of applied mathematics, statistics and probability, economics, engineering, physics and operations research. Historically, moment problems came into focus with Stieltjes in 1894 [

A general moment problem can be stated as follows. Suppose we are given a measurable space,

Among the various instances of this inverse problem, the covariance realization/completion problem has raised wide interest, in part because of its important applications in mathematical statistics [

The geometry of correlation matrices may change dramatically if one adds constraints on the values of the random vector realizing the covariance matrix. In this paper, we consider the case in which the components of the random vector are required to be

A more delicate problem is to characterize covariance matrices of spin systems by a system of linear inequalities. This problem was tackled in [

The aim of this paper is two-fold.

We derive an infinite family of linear inequalities characterizing covariances of spin systems, via the solution of a maximum entropy problem. Besides its intrinsic interest, this method has the advantage of describing, in terms of certain Lagrangian multipliers, an explicit probability realizing the covariances, whenever they are realizable. The search for the Lagrange multipliers is an interesting computational problem, which will be addressed in a forthcoming paper.

Via a computer-aided proof, we determine the facets of the polytope of covariance matrices of spin systems for

Let us define a spin system first. Let

Suppose that we are given the spin-spin correlations,

Under what conditions does a distribution with those correlations exist?

If one such distribution exists, that is, if the given correlations are realizable, then how does one characterize the maximum entropy probability measure?

The spin correlation/covariance matrices form a convex polytope whose description in terms of vertices is known. Let us denote the convex polytope of the spin correlation matrices of the order,

These

It is interesting to note that the description of extremals of spin correlation matrices is rather simple when compared with the description of extremals of the convex set of correlation matrices in general (see [

The necessity of these inequalities is easy to show:

We have seen that the spin correlation matrices form a convex polytope with extreme points given by Theorem 1. Every convex polytope has two representations: one as the convex hull of finitely many extreme points (known as the

The program, cdd+ (cdd, respectively), is a C++ (ANSI C) program that performs both tasks. Given the equations of faces of the polytope, it returns the set of vertices and extreme rays and

We executed the cdd+ program to find the necessary and sufficient condition for

These are the simplest cases, already covered in [

Bell’s inequalities imply positivity of the matrix;

Bell’s inequalities correspond to the facets of the polytope of spin correlation matrices in dimension three and four; in particular, they provide the “minimal” description in terms of linear inequalities.

The polytope of spin-correlation matrices has 56 facets. Forty of these are given by the Bell’s inequalities, corresponding to

The remaining 16 facets correspond to the following inequalities: for every

There are 368 facets. We can group the corresponding inequalities into three groups.

We have the

For

For

We will see in the next section that inequalities of the types above hold for spin-correlation matrices also in higher dimensions, where, however, facets of different types appear.

Our aim now is to find an explicit measure that realizes the given covariances. One of the most natural and popular approach in these kind of problems is to use the maximum entropy method. The rationale underline this approach has been discussed over the years by various “deep thinkers" such as Jaynes [

We want to find a probability measure that realizes the given covariances and that also maximizes the entropy of the system. In other words, we want to solve the following optimization problem:

Consider the

Also note that this last formula simply specifies a class of probability measures on

Λ is a critical point for

A critical point exists if

It is also clear that the following set of inequalities ensures the properness of

Let us denote by

We first show that

Thus, for

Now, to show

Its main consequence is that it guarantees that whenever

Positivity: let

Bell’s inequalities: let

Generalizations of Bell’s inequalities: Let us consider

Many other variants of the Bell’s inequalities could be obtained with other choices of the

We have observed that the maximum entropy method allows us to reduce the problem of realizing a given spin correlation matrix to finding the minimum of the function,

Let

By elementary computations, we can compute the gradient,

Step 1:

Step 2:

Step 3:

Step 4:

Step 5:

It should be observed that the computation of

We are grateful to an anonymous reviewer for pointing out to us some literature in discrete geometry that is relevant for the present paper. The work of M. Pavon was partially supported by the QuantumFuture research grant of the University of Padova and by an Alexander von Humboldt Foundation fellowship at the Institut für Angewandte Mathematik, Universität Heidelberg, Germany.

The authors declare no conflict of interest.

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