Kochen–Specker vectors

Abstract

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.

Arbitrarily exhaustive hypergraph generation of 4-, 6-, 8-, 16-, and 32-dimensional quantum contextual sets

“…Program states01 carries out an exhaustive search according to a backtracking algorithm. This is a much less demanding task than a constructive upward generation we used previously in [33] and although we have to deal with a huge volume of hypergraphs, on computing clusters we can carry out generations successfully, as we show in subsequent sections below.…”

“…When dealing with such ASCII line encoding of MMP hypergraphs we call them MMP hypergraphs lines or strings when needed. The order of the strings and characters is irrelevant; gaps in characters are allowed and its number is not limited; tens of thousands of them are not a problem for our programs shortd.c, mmpstrip.c, subgraph.c, vectorfind.c, states01.c and others [9,27,31,[35][36][37].…”

“…In this section we shall make use of the results about the 24-24 class of 4-dim KS sets we obtained in [31,36,[39][40][41]] to introduce the main steps and strategy we shall undertake to obtain the results in the subsequent sections.…”

“…This can be compared with less than a dozen of KS sets discovered by several researchers between 1967 and the end of the 20th century [29,[42][43][44][45][46][47], mostly exploring highly symmetrical geometrical structures defined by mutually orthogonal vectors. In 2004 Pavičić, McKay, Merlet, and Megill [36,40] generated non-isomorphic MMP hypergraphs and filtered them by means of a program which was written for our algorithm of assigning 0s and 1s to their vertices and another algorithm for assigning vector components to vertices. The generation and assignments are exponentially complex tasks in general but applied to our KS MMP hypergraphs they turned out to be polynomially complex for the great majority of jobs.…”

“…After we generated the first few hundred KS sets in [36,40] and started to draw their hypergraphs we visually recognised-see Fig. 1-that they were all subgraphs of the hypergraph we drew for Peres' 24-24 set.…”