Bell's celebrated inequality gives a condition which a correlated set of measurements must satisfy to be explainable in terms of some hidden variables model. Its interest lies in the fact that quantum mechanics can make predictions violating this condition. In such cases, experimental examination of the situation can conclusively demonstrate a real world situation inexplicable by any classical model, while at the same time providing positive (but less conclusive) evidence for the validity of the quantum model.

We consider a situation in which 2k measurements L_{1},...,L_{k} and R_{1},...,R_{k} are made repeatedly on an ensemble of systems. The L measurements interfere with each other and the R_{1},...,R_{k} measurements interfere with each other, but the L and R measurements do not interfere, so in each experiment we can make one L and one R measurement. Repeated experiments can then generate statistics for the correlations corr(L_{i},R_{j}) of all possible L and R experiments, which can be summarized in a correlation matrix C_{i,j}. The *Bell figure* for this class of experiment is the set of matrices C_{i,j} compatible with some hidden variables model of the situation.

Such a model would explain the L_{j} outcome of each experiment as the value of some unknown function f_{j}(x) of one or more hidden variables x being drawn uncontrollably from a space of such variables, but subject to some probability distribution D(x). Similarly the R _{j} outcome 1s the value of some unknown function g_{j}(x), and the correlation matrix C_{i,j} is _{i,j} = ∫ f_{i}(x)g_{j}(x) D(x)dx/(∫ (f_{i}(x))^{2}D(x)dx)^{1/2}(∫ (g_{j}(x))^{2}D(x)dx)^{1/2}

Since by assumption we know nothing about the variables f or the functions f and g, all we can say about the matrix C_{i,j} is that its elements are dot products l_{i}•r_{j} of unknown unit vectors l_{i} and r_{j}.

The simplest case to consider (typified by the measurement of spins in a spin-0 configuration of two entangled particles) is that in which the diagonal elements of C_{i,j} are all -1. In this case each vector r_{j} must be the negative of the corresponding l_{j}, so C must be the negative of the positive definite symmetric matrix _{i,j} = r_{i} • r_{j}

Any positive definite matrix H with unit diagonal elements must have this form, since it has a positive definite hermitian square root h, and then _{i}^{j} = ∑_{k} h_{i}^{k}h_{j}^{k},_{i} to have components h_{i}^{k}. In this case the Bell figure B is just the set of positive definite matrices P with unit diagonal. This is a convex set, which can be defined by the family of inequalities

If k = 2 the applicable inequalities tell us nothing useful, since the one off-diagonal element in _{i,j} = r_{i} • r_{j}_{1} • r_{2}_{1} and r_{2}. To understand the less trivial case k = 3, we can parametrize the space of all our 3 by 3 positive definite matrices by standardizing and then parametrizing the space of all configurations of 3 unit vectors in 3 space. The first of these three points r_{1}, r_{2}, r_{3} on the unit sphere in 3-space can be put at the north pole and the second on the xz plane, at an angle a from r_{1}. (Working with angles here is more convenient than working with dot products; the angle between two vectors is the arccos of the dot product. We will soon see that in this angle-based representation the Bell figure is polyhedral.) Suppose that the angular distances of r_{2} and r_{3} from r_{1} are a and b respectively. Then there are 3 cases to consider: (i) a and b both less than 90^{o}; (ii) a less than 90^{o}; and b greater; (iii) both greater than 90^{o}.

Let L be the great circle thru r_{1} and r_{2}, M_{1} be the semicircle of L starting at r_{1} and containing r_{2}, and M_{2} the semicircle opposite to M_{1}.

This is shown in the second figure, which also shows the surface of values predicted by quantum mechanics for 3 correlation measurements of spin in a two-particle entangled spin 0 experimental situation like the Stern-Gerlach experiment.

For completeness sake we also consider the general case in which the diagonal elements of C_{i,j} need not be -1, and specifically its subclass k = 2. Here we are interested in characterizing the 2 by 2 matrices which can be written in terms of 4 unit vectors l_{1}, l_{2}, r_{1}, r_{2} as l_{i}•r_{j}. Again we work by standardizing the position of these points on the unit sphere (this time in 4 dimensions) and in terms of angles. Put r_{1} at the north pole, let z be the angle between it and r_{1}, and let x,y,u,v be the great circle distances r_{1}l_{1}, r_{1}l_{2}, r_{2}l_{1}, r_{2}l_{2} respectively. We need to find the range of variation of x,y,u,v; z is merely an auxiliary.

Any three distances of which none is greater than the sum of the other two define a possible spherical triangle. So for given z, i.e. for given r_{1}l_{2}, r_{2}l_{1}, the point l_{1} can be placed at the specified angular distances x,y from
r_{1} and r_{2} if and only if _{1} can be placed if and only if z ≤ u + v, |u - v| ≤ z.
The angles x,y,u,v are therefore possible if and only if these two intervals overlap, which is to say max(|x - y|,|u - v|) ≤ min(x + y,u + v). The 4-dimensional polyhedron defined by this piecewise linear inequality is the Bell figure for this case.