# 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 {S_{1}, S_{2}, ..., S_{a}} is found, then the set
{RS_{1}, RS_{2}, ..., RS_{a}}, 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 10^{16} 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 10^{6}
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.