TY - JOUR

T1 - Kochen-Specker vectors

AU - Pavičić, Mladen

AU - Merlet, Jean Pierre

AU - McKay, Brendan

AU - Megill, Norman D.

PY - 2005/2/18

Y1 - 2005/2/18

N2 - We give a constructive and exhaustive definition of Kochen-Specker (KS) vectors in a Hubert 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 an n-dimensional Hilbert space, Hn,n ≥ 3, to which it is impossible to assign Is 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", 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 selfteaching 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 four-dimensional KS vector systems containing up to 24 vectors were generated and described, all three-dimensional vector systems containing up to 30 vectors were scanned, and several general properties of KS vectors were found.

AB - We give a constructive and exhaustive definition of Kochen-Specker (KS) vectors in a Hubert 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 an n-dimensional Hilbert space, Hn,n ≥ 3, to which it is impossible to assign Is 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", 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 selfteaching 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 four-dimensional KS vector systems containing up to 24 vectors were generated and described, all three-dimensional vector systems containing up to 30 vectors were scanned, and several general properties of KS vectors were found.

UR - http://www.scopus.com/inward/record.url?scp=14544276338&partnerID=8YFLogxK

U2 - 10.1088/0305-4470/38/7/013

DO - 10.1088/0305-4470/38/7/013

M3 - Article

SN - 0305-4470

VL - 38

SP - 1577

EP - 1592

JO - Journal of Physics A: Mathematical and General

JF - Journal of Physics A: Mathematical and General

IS - 7

ER -