510 research outputs found
A Kochen-Specker system has at least 22 vectors (extended abstract)
At the heart of the Conway-Kochen Free Will theorem and Kochen and Specker's
argument against non-contextual hidden variable theories is the existence of a
Kochen-Specker (KS) system: a set of points on the sphere that has no
0,1-coloring such that at most one of two orthogonal points are colored 1 and
of three pairwise orthogonal points exactly one is colored 1. In public
lectures, Conway encouraged the search for small KS systems. At the time of
writing, the smallest known KS system has 31 vectors. Arends, Ouaknine and
Wampler have shown that a KS system has at least 18 vectors, by reducing the
problem to the existence of graphs with a topological embeddability and
non-colorability property. The bottleneck in their search proved to be the
sheer number of graphs on more than 17 vertices and deciding embeddability.
Continuing their effort, we prove a restriction on the class of graphs we
need to consider and develop a more practical decision procedure for
embeddability to improve the lower bound to 22.Comment: In Proceedings QPL 2014, arXiv:1412.810
Noncontextuality, Finite Precision Measurement and the Kochen-Specker Theorem
Meyer recently queried whether non-contextual hidden variable models can,
despite the Kochen-Specker theorem, simulate the predictions of quantum
mechanics to within any fixed finite experimental precision. Clifton and Kent
have presented constructions of non-contextual hidden variable theories which,
they argued, indeed simulate quantum mechanics in this way. These arguments
have evoked some controversy. One aim of this paper is to respond to and rebut
criticisms of the MCK papers. We thus elaborate in a little more detail how the
CK models can reproduce the predictions of quantum mechanics to arbitrary
precision. We analyse in more detail the relationship between classicality,
finite precision measurement and contextuality, and defend the claims that the
CK models are both essentially classical and non-contextual. We also examine in
more detail the senses in which a theory can be said to be contextual or
non-contextual, and in which an experiment can be said to provide evidence on
the point. In particular, we criticise the suggestion that a decisive
experimental verification of contextuality is possible, arguing that the idea
rests on a conceptual confusion.Comment: 27 pages; published version; minor changes from previous versio
Kochen-Specker Vectors
We give a constructive and exhaustive definition of Kochen-Specker (KS)
vectors in a Hilbert space of any dimension as well as of all the remaining
vectors of the space. KS vectors are elements of any set of orthonormal states,
i.e., vectors in n-dim Hilbert space, H^n, n>3 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 KS vectors is based on
algorithms that generate MMP diagrams corresponding to blocks of orthogonal
vectors in R^n, on algorithms that single out those diagrams on which algebraic
0-1 states cannot be defined, and on algorithms that solve nonlinear equations
describing the orthogonalities of the vectors by means of statistically
polynomially complex interval analysis and self-teaching programs. The
algorithms are limited neither by the number of dimensions nor by the number of
vectors. To demonstrate the power of the algorithms, all 4-dim KS vector
systems containing up to 24 vectors were generated and described, all 3-dim
vector systems containing up to 30 vectors were scanned, and several general
properties of KS vectors were found.Comment: 19 pages, 6 figures, title changed, introduction thoroughly
rewritten, n-dim rotation of KS vectors defined, original Kochen-Specker 192
(117) vector system translated into MMP diagram notation with a new graphical
representation, results on Tkadlec's dual diagrams added, several other new
results added, journal version: to be published in J. Phys. A, 38 (2005). Web
page: http://m3k.grad.hr/pavici
Finite precision measurement nullifies the Kochen-Specker theorem
Only finite precision measurements are experimentally reasonable, and they
cannot distinguish a dense subset from its closure. We show that the rational
vectors, which are dense in S^2, can be colored so that the contradiction with
hidden variable theories provided by Kochen-Specker constructions does not
obtain. Thus, in contrast to violation of the Bell inequalities, no
quantum-over-classical advantage for information processing can be derived from
the Kochen-Specker theorem alone.Comment: 7 pages, plain TeX; minor corrections, interpretation clarified,
references update
New Examples of Kochen-Specker Type Configurations on Three Qubits
A new example of a saturated Kochen-Specker (KS) type configuration of 64
rays in 8-dimensional space (the Hilbert space of a triple of qubits) is
constructed. It is proven that this configuration has a tropical dimension 6
and that it contains a critical subconfiguration of 36 rays. A natural
multicolored generalisation of the Kochen-Specker theory is given based on a
concept of an entropy of a saturated configuration of rays.Comment: 24 page
Kochen-Specker Sets and Generalized Orthoarguesian Equations
Every set (finite or infinite) of quantum vectors (states) satisfies
generalized orthoarguesian equations (OA). We consider two 3-dim
Kochen-Specker (KS) sets of vectors and show how each of them should be
represented by means of a Hasse diagram---a lattice, an algebra of subspaces of
a Hilbert space--that contains rays and planes determined by the vectors so as
to satisfy OA. That also shows why they cannot be represented by a special
kind of Hasse diagram called a Greechie diagram, as has been erroneously done
in the literature. One of the KS sets (Peres') is an example of a lattice in
which 6OA pass and 7OA fails, and that closes an open question of whether the
7oa class of lattices properly contains the 6oa class. This result is important
because it provides additional evidence that our previously given proof of noa
=< (n+1)oa can be extended to proper inclusion noa < (n+1)oa and that nOA form
an infinite sequence of successively stronger equations.Comment: 16 pages and 5 figure
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