Bell's inequality and Bell figures

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,...,Lk and R1,...,Rk are made repeatedly on an ensemble of systems. The L measurements interfere with each other and the R1,...,Rk 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(Li,Rj) of all possible L and R experiments, which can be summarized in a correlation matrix Ci,j. The Bell figure for this class of experiment is the set of matrices Ci,j compatible with some hidden variables model of the situation.

Such a model would explain the Lj outcome of each experiment as the value of some unknown function fj(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 gj(x), and the correlation matrix Ci,j is Ci,j = ∫ fi(x)gj(x) D(x)dx/(∫ (fi(x))2D(x)dx)1/2(∫ (gj(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 Ci,j is that its elements are dot products li•rj of unknown unit vectors li and rj.

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 Ci,j are all -1. In this case each vector rj must be the negative of the corresponding lj, so C must be the negative of the positive definite symmetric matrix ci,j = ri • rj, 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 Hij = ∑k hikhjk, so we can take the k vectors ri to have components hik. 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 ci,j = ri • rj 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 90o; (ii) a less than 90o; and b greater; (iii) both greater than 90o.

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, 360o - a - b). So in terms of angles the Bell figure for the case under consideration is the polyhedron 0o ≤ a ≤ 180o, 0o ≤ b ≤ 180o, |a - b| ≤ c ≤ min(a + b, 360o - 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 Ci,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 li•rj. 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.