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Inria / Raweb 2004
Project: Coprin

Project : coprin

Section: New Results


Keywords : quantum mechanic, interval analysis.

Quantum mechanic

Participant : Jean-Pierre Merlet.

At the beginning of year 2004 we have been contacted to help solving a difficult problem in quantum mechanics related to the Kochen-Specker theorem that shows that the hidden variables hypothesis was not valid. For that purpose it is necessary to examine sets of 4D unit vectors (denoted 1,2,...,9,A,B,...), called KS vectors, that will be used in a diagram constituted of groups of four 4D vectors (although any finite dimension larger than 2 may be used for the vectors space). An example of such diagram is

FGHI,BCDE,789A,3456,1256,2ADE,49BC,18HI,37FG,CEGI

with groups FGHI, BCDE, .... All vectors in a group must be orthogonal to all the other vectors in the group and no vectors should be collinear or opposite. We may try to assign a state 0 or 1 to each vector so that one vector in each group has the state 0 while the three others have the state 1. Kochen was able to exhibit a system with 117 3D vectors for which such assignment was not possible i.e. at least one vector belong to one group that impose that its state is 0 while the same vector belong to another group in which its state should be 1.

Diagram with vectors having real valued components are called Kochen-Specker systems and the existence of such diagram is a key point in the proof of the Kochen-Specker theorem. But finding Kochen-Specker systems is also important for experiments as the vectors describe measurements that can be carried out on a finite dimensional quantum system for verifying the theorem. Note that different diagrams may correspond to the same measurement arrangements: they will be called isomorphic systems.

So far known Kochen-Specker systems have been found using approaches that rely on human ingenuity and the complexity of finding such system grows exponentially with increasing numbers of vectors and dimension for the vectors space.

A team constituted of Mladen Pavičić, University of Zagreb, Brendan McKay, Australian National University, Norman Megill, Boston Information Group and COPRIN has proposed another approach. A software was designed to provide all non-isomorphic systems with a given number a of vectors and a given number b of groups. Another software is then used on the input to eliminate diagrams that admit a 0-1 state assignment for the vectors. The remaining diagrams are KS systems candidate but it remains to show that the components of the vectors admit at least one real value so that the orthogonality, non collinearity and unitary constraints are satisfied. Note that it is not necessary to determine all possible solution vectors: one set is sufficient.

The problem is not well posed: if a set of solutions vectors {S1, S2, ..., Sa} is found, then the set {RS1, RS2, ..., RSa}, where R is an arbitrary rotation matrix, is also a solution. This may be avoided by assuming that a group is an orthonormal basis of R4. Hence remains 4(a-4) unknowns (the components of the vectors) with the constraints that the vectors are unitary (a-4 constraints) and that the vectors in a group are mutually orthogonal: this induces 6 constraint equations for a group and a total of 6(b-1) equations. We end up with a system of 4(a-4) unknowns for a-4 + 6(b-1) equations. Classical algebraic methods were initially used for the solving but in general without success due to the size of the system. It appears that interval analysis was a good candidate for the solving: indeed as the unknowns are components of an unit vector their values must lie in the range [-1,1]. Furthermore if S is a solution vector, so is -S: this allow to restrict the range of one component of each vector to [0,1]. Finally looking only for one solution is convenient for interval analysis-based solving method.

Preliminary trials with ALIAS have shown that indeed the approach was working well for reasonable values of a, b. But the exhaustive generation was not appropriate for large values as the number of generated diagrams grows exponentially with the number of vectors. For example for a = 18, b = 12 the exhaustive generation will generate more than 2.9 1016 diagrams and would require more than 30 million years on a 2 GHz CPU. To solve this problem the generation program has been modified to create diagrams incrementally i.e. when a sequence of n groups has been created all the diagrams which share this n groups are generated before changing the n-th group. Then we have implemented within the generation program a filter based on constraint programming that in many cases allows to show that for the current sequence the orthogonality, non-collinearity and unitary constraints cannot be satisfied. Eliminating sequences early in the generation process allows to drastically improve the generation time. For example for a = 18, b = 12 the software generates only 100220 systems in less than 30 minutes among which only 26800 cannot have a 0-1 state assignment and are submitted to the solver.

Using this approach all 4D KS vector systems with up to 24 vectors and all 3D system with up to 30 vectors were generated. This exhaustive approach has lead us to solve approximately 200 106 non-linear systems (having between 30 and 200 equations), which constitute probably a world record.

Among the various results that we have found we may mention that Cabello's system 1234,4567,789A,ABCD,DEFG,GHI1,35CE,29BI,68FH with a = 18, b = 9 is the smallest 4-dim Kochen-Specker real system. Further results can be found inĀ [39].

For larger values of a we have encountered in some cases solving difficulties. For example we have not been able to solve a case with a = 40, b = 20 that involves 108 unknowns and 150 equations. A theoretical work has been started to better understand the structure of the equations. At the same time we have started a collaboration with the projects OASIS, APACHE and PARIS to determine if a grid-computing approach was able to solve the problem.

The presented algorithms can easily be generalized beyond the Kochen-Specker theorem. One can use the diagrams to generate Hilbert lattice counterexamples, partial Boolean algebras, and general quantum algebras which could eventually serve as an algebra for quantum computers.


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