1         Abstract

The violation of Bell-type inequalities by quantum probabilities represents an important aspect of quantum mechanics which is linked to the mind-boggling quantum features of nonlocality, complementarity and contextuality. Formally, such bounds on the classical probabilities from consistency arguments have already been investigated by Boole in the middle of the nineteenth century. Boole called them "conditions of possible experience." We introduce CddIF, a Mathematica package to compute all the Boole-Bell-type inequalities associated with an arbitrary physical setup. We have also computed with Mathematica the tractable special cases of two particles with up to three possible detection angles per particle, and the three-particle/two directions (Greenberger-Horne-Zeilinger) case.

2	Boole-Bell Type Inequalities

Would you believe someone telling you that the chances of rain in Vienna and Budapest are 0.1 in each one of the cities alone, and the joint probability of rainfall in both cities is 0.99? I believe you would not. It just does not make any intuitive sense to claim that it rains almost never in one of the cities but almost always in both of them. The nagging question appears, though: which numbers would you believe in? Of course one could argue that the joint probability should not exceed any single probability. This certainly appears to be a necessary condition, but is it a sufficient one?  In the middle of the 19th century George Boole, in response to such queries, formulated a theory of "conditions of possible experience" [1,2,3,4,5]. These conditions on the (joint) probabilities of logically connected events are expressed by certain equations or inequalities relating those probabilities. More recently, similar inequalities for a particular setup which are relevant in the quantum mechanical context have been discussed by Clauser and  Horne and others [6,7,8]. Pitowsky has given a geometrical interpretation in terms of correlation polytopes [9, 4, 10, 5].

2.1	Geometrical interpretation [4,5,9,10]

Consider the truth table (Table 1) of the above example, where (Subscript[a, 1]) and (Subscript[b, 1]) represent the statements that "it rains in Vienna" and "it rains in Budapest'',  and (Subscript[a, 1])(Subscript[b, 1]) represents the proposition "it rains in Vienna and in Budapest ,''  respectively; i.e.,

TruthTable[2, 1]

In the weather example, there are two places with one observable each; hence the arguments 2,1. The third "component bit" of the vector is a function of the first components.  Since we are dealing with the classical logical "and" operation here, this function is a multiplication.  This amounts to interpreting the rows of the truth table as extremal points; the entries of the rows being the components.  The vectors have the following meaning: {0,0,0} stands for the case of neither rain in Vienna nor in Budapest,  {1,0,0} stands for the case of  rain in Vienna but not in Budapest, and so on. Any probability distribution characterized by the three numbers  {a1, b1, a1b1} should be a weighted linear average over all these (weather) conditions. 

Interpret the set of all numbers {a1, b1, a1b1} as a set of points in a three-dimensional real space.  As pointed out by Pitowsky [4], this set is the polytope formed by a weighted linear average over all extremal points of Table 1 representing the possible (weather) conditions. More precisely, the  surface of the convex polytope with vertices {0,0,0}, {1,0,0}, {0,1,0} and {1,1,1} (cf. Figure 1) represents all conceivable classical probability distributions. The faces of the polytope represent the conditions by which classical probabilities are bound; and these "inside-outside" conditions are expressed as inequalities. These inequalities are called Boole-Bell inequalities.

2.2	Minkowski-Weyl representation theorem

Generally, according to the Minkoswki-Weyl representation theorem (eg.,[10,p.29]), every convex polytope in an Euclidean space has a dual description: either as the convex hull of its extreme points (vertices), or as the intersection of a finite number of half-spaces, each one given by a linear inequality. The problem to obtain all inequalities from the vertices of a convex polytope is known as the hull problem. For the Boole-Bell type problems at hand, the Minkowski-Weyl representation theorem represents a direct, provable method to obtain a complete set of optimal inequalities for classical probability distributions characterizing any physical setup or arrangement.

3	Calculating Inequalities using CddIF

This package acts as an interface to the cdd-program by Komei Fukuda [11] for a transformation from a vertex to a half-space representation and vice versa, making use of the Minkowski-Weyl representation theorem by means of the Double Description Method [12], which we shall use but not review here. A complete description can be found in [13] and the package can be downloaded from http://tph.tuwien.ac.at/~svozil/cdd/.  Available Mathematica functions are for example TruthTable[x,y] to build the corresponding truth table for the configuration x-particles/y-measurements, ConvToHRep to construct the half-spaces representation or PlotInequalities to plot the calculated inequalities.

3.1	Computation of inequalities corresponding to {{0,0,0}, {1,0,0}, {0,1,0} and {1,1,1}}

The appropriate inequalities for the probabilities of rain in Vienna (a1), rain in Budapest (b1) and  rain in Vienna and in Budapest (a1b1) from the example above can easily be calculated with the use of CddIF:

InequToRead[
      GetInequFromHRep[
ConvToHRep[2, 1]]]

Here ConvToHRep[2,1] calculates the half-space representation for this particular configuration. Again, as in the weather example discussed above, there are two places with one observable each; hence the arguments 2, 1. The functions InequToRead[GetInequFromHRep[...] ] display the inequalities in a human-readable form.

3.2	Quantum mechanical context

In the quantum mechanical case elementary events and their probabilities refer to clicks in particle detectors. Quantum probabilities have to be calculated using the formalism of quantum mechanics based on Hilbert space entities (i.e., Gleason's theorem) rather than on Boolean algebras. There exist complementary observables which cannot be measured simultaneously. And unlike classical Boolean algebras, Quantum Hilbert lattices are non-boolean (non-distributive). It is by no means trivial that quantum probabilities should satisfy the consistency conditions for classical probabilities; and indeed Bell has discovered physical configurations where Boole's "conditions of possible experience" are violated by quantum probabilities.

As an example, consider a source that produces pairs of spin-1/2 particles in a singlet-state, meaning that, no matter in what direction the spins of the two particles are measured, they will always be opposed to each other. The particles fly apart along the z axis, and after the particles have separated, measurements on spin components along one out of two directions are made. If, for simplicity, the measurements are made in the x-y plane perpendicular to the trajectory of the particles, the direction of the measurement can be given by angles measured from the vertical x axis ((Subscript[α, 1]) and (Subscript[α, 2]) on one side, (Subscript[β, 1]) and (Subscript[β, 2]) on the other side). On each side the measurement angle is chosen randomly for each pair of incoming particles and each measurement can yield two results - in ℏ/2 units: "+1'' for spin up and "-1'' for spin down (cf. Figure 2).

Deploying this configuration we get probabilities to find a particle measured along the axis specified by the angles (Subscript[α, 1]), (Subscript[α, 2]),(Subscript[β, 1]) and (Subscript[β, 2]) either in spin up or in spin down state  denoted as a1, a2, b1, b2 - and we also take the joint event of finding a particle on one side at the angle (Subscript[α, 1]) ((Subscript[α, 2])) in a specific spin state and the other particle on the other side along the vector (Subscript[β, 1]) ((Subscript[β, 2])) in a specific spin state, denoted by a1b1, a2b1, a1b2 and a2b2.

GetInequFromHRep[ConvToHRep[2, 2]] // InequToRead

The  first inequalities correspond to the well known CHSH inequalities, which are thereby obtained in a very natural way.

3.3	Two particles - three measurement directions

In the case of two particles (a and b) with three properties (whereas the properties are three different angles of the detectors for each particle) denoted by (Subscript[a, 1]), (Subscript[a, 2]), (Subscript[a, 3]), (Subscript[b, 1]), (Subscript[b, 2]), (Subscript[b, 3]) 15 different events can be found: {(Subscript[a, 1]), (Subscript[a, 2]),  ...   , (Subscript[b, 3]), (Subscript[a, 1])⋀(Subscript[b, 1]), 
        (Subscript[a, 1])⋀(Subscript[b, 2]),  ...   , (Subscript[a, 3])⋀(Subscript[b, 3])}. Using

TruthTable[2, 3]

all vertices of the corresponding correlation polytope can be found -  we get a dimension of 15 and 64 vertices as result. 
A representation in terms of inequalities of the polytope described by the truth table above can be created by

h = ConvToHRep[2, 3]

This results in 684 hyper-planes or 684 inequalities, respectively, from the 64 vertices limiting the polytope. All inequalities can be displayed by

GetInequFromHRep[h] // InequToRead

To show all inequalities that are violated at the specific measurement directions (Subscript[a, 1]) = (Subscript[b, 1]) = 0, (Subscript[a, 2]) = (Subscript[b, 2]) = 2  π/3, (Subscript[a, 3]) = (Subscript[b, 3]) = 4  π/3 with the functions P[{x_}]:=1/2 and P[{x_,y_}]:=1/2 (Sin^2)[(x-y)/2] yielding the quantum probabilities to find two spin-1/2 particles (in a singlet state) both either in spin up or both in spin down, we execute

P[{x_}] := 1/2; 
P[{x_, y_}] := Sin[(x - y)/2]^2/2;
GetViolInequalities[
        h, {{0, 2  π/3, 4  π/3}, {0, 2  π/3, 4  π/3}}, P, 
        All] // TableForm

A two-dimensional graphical representation of all inequalities (cf. Figure 3) can be generated for example by

PlotInequalities[
          h, {x, 0, π}, {{0, x, 4  π/3}, {0, x, 4  π/3}}, P];

A very similar procedure involving  TruthTable[3,2] yields all Boole-Bell inequalities for the three-particle and two measurement directions (Greenberger-Horne-Zeilinger) case.

4	Summary

In summary we have implemented a systematic method to compute all optimal Boole-Bell type inequalities satisfied by classical probabilities, given a physical setup. By optimal we mean that there is no condition such as equality or inequality which is satisfied by classical probabilities and which is a less stringent bound thereof. We have used this method to compute well known as well as new types of bounds on probabilities in configurations relevant to quantum physics. We believe that this represents a powerful new tool for the investigation of the nonclassical features of quantum mechanics.

5	References

[1] Boole, G.   An investigation of the laws of thought, New York, 1958, Dover Edition.
[2] Boole, G.   On the theory of probabilities, in Phil. Trans. R. Soc. London,  152:225-252, 1862.
[3] Hailperin, T.    Boole's logic and probability (Studies in logic and the foundations of mathematics; 85), Amsterdam, 1976, North-Holland.   
[4] Pitowsky, I.   Quantum Probability - Quantum Logic, Berlin, 1989, Springer.
[5] Pitowsky, I.   George Boole's 'conditions od possible experience' and the quantum puzzle, in Brit. J. Phil. Sci, 45:95-125, 1994.
[6] Clauser, J. F. and Horne, M. A.   Experimental consequences of objective local theories, in Phys. Rev D, 10: 526-535, 1974.
[7] Clauser, J. F. et al.   Proposed experiment to test local hidden-variables theories, in Phys. Rev. Lett., 23:880-884, 1969.
[8] Clauser, J. F. and Shimony, A.   Bell's theorem: experimental tests and implications, in Rep. Prog. Phys., 41:1881-1926, 1978.
[9] Pitowsky, I.   From George Boole to John Bell: The origin of Bell's inequality, in Bell's Theorem, Quantum Theory and the Conception of the Universe, Dordrecht, 1989 (ed. Kafatos, M.), Kluwer.
[10] Ziegler, G.   Lectures on Polytopes, New York, 1994, Springer.
[11] Komei Fukuda, http://www.cs.mcgill.ca/~fukuda/soft/cdd_home/cdd.html.
[12] Motzkin, T. S. et al.   The Double Description Method, in Contributions to theory of games, Vol. 2, New Jersey, Princeton, 1953, Princeton University Press.
[13] Filipp, S. and Svozil, K.   Boole-Bell-type inequalities in Mathematica, e-print: quant-ph/0105083.