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 L₁,...,L_k and R₁,...,R_k are made repeatedly on an ensemble of systems. The L measurements interfere with each other and the R₁,...,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 C_i,j = ∫ f_i(x)g_j(x) D(x)dx/(∫ (f_i(x))²D(x)dx)^1/2(∫ (g_j(x))²D(x)dx)^1/2. So the Bell figure is simply the set of all matrices C which can have this form.

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 c_i,j = r_i • r_j, whose diagonal elements must be 1.

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 H_i^j = ∑_k h_i^kh_j^k, so we can take the k vectors r_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 Pv•v ≥ 0, where v ranges over all unit vectors. But a better way of testing a given experimental P to see if it belongs to B is to calculate the k eigenvalues of c numerically and verify that all are non-negative.

If k = 2 the applicable inequalities tell us nothing useful, since the one off-diagonal element in c_i,j = r_i • r_j must lie between -1 and + 1 in any case, while conversely any number in the range has the form r₁ • r₂ for some unit vectors r₁ and r₂. 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 r₁, r₂, r₃ 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 r₁. (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 r₂ and r₃ from r₁ 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 r₁ and r₂, M₁ be the semicircle of L starting at r₁ and containing r₂, and M₂ the semicircle opposite to M₁. Then it is easily seen by inspection of the attached diagram that in all three cases r₃ is closest to r2 when it lies on M₁, and furthest when it lies on M₂. It follows that the angle c between r₂ and r₃ ranges from a minimum of |a - b| to a maximum of min(a + b, 360^o - a - b). So in terms of angles the Bell figure for the case under consideration is the polyhedron 0^o ≤ a ≤ 180^o, 0^o ≤ b ≤ 180^o, |a - b| ≤ c ≤ min(a + b, 360^o - a - b).

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 2 measurement angles X and X + Y relative to a specified 'up' direction, the correlations predicted by quantum theory, and verified experimentally, these are cos²(X/2),cos²((X + Y)/2), and cos²(Y/2).) The parts of the surface outside the polyhedral Bell figure are painted in. The corresponding correlations are inconsistent with a hidden variables explanation of the data.

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 l₁, l₂, r₁, r₂ 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 r₁ at the north pole, let z be the angle between it and r₁, and let x,y,u,v be the great circle distances r₁l₁, r₁l₂, r₂l₁, r₂l₂ 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 r₁l₂, r₂l₁, the point l₁ can be placed at the specified angular distances x,y from r₁ and r₂ if and only if z ≤ x + y, |x - y| ≤ z. Similarly l₁ 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.