Graph imaging of quantum detections
M. Pavicic,
We present a generalised and exhaustive method of
finding the directions of the quantisation axes of
the measured eigenstates within experiments which have no classical
counterparts. The method relies on a constructive and exhaustive definition of
sets of such directions (which we call Kochen-Specker
vectors) in a Hilbert space of any dimension as well as of all the remaining
vectors of the space. Kochen-Specker vectors are
elements of any set of orthonormal states, i.e.,
vectors in n-dim Hilbert space, n > 2 to which it is impossible to assign 1s
and 0s in such a way that no two mutually orthogonal vectors from the set are
both assigned 1 and that not all mutually orthogonal vectors are assigned 0.
Our constructive definition of such Kochen-Specker
vectors is based on imaging of nonlinear equations that define the geometry of
the vectors to linear graphs, the so-called MMP diagrams. Thus we substitute
solving nonlinear equations by checking conditions imposed on the corresponding
graphs. In doing so we reduce the exponentially complex task
of solving nonlinear equations to a polynomially complex
task of generating and sorting the graph images of the equations. The
latter procedure invokes a
2-dimensional meta-imaging - a representation of graphs as figures on which we
can define states and find final solutions by rejecting all those ones that allow
them. The algorithms are limited neither by the number of dimensions nor by the
number of vectors and can also be used as a general method for solving particular
nonlinear equations in any other imaginable application. We obtained thousands
of new Kochen-Specker vectors in practically no time.
While solving systems of nonlinear equations by brute force would take ages and
ages of the Universe, generation and elimination of graphs take minutes and
hours.
Reference:
M. Pavicic,
J.-P. Merlet, B. D. McKay and N. D. Megill, Kochen-Specker Vectors,
J. Phys. A, 38, 497-503 (2005); Corrigendum,
J. Phys. A, 38, 3709 (2005).