Generalised Kochen–Specker Theorem in Three Dimensions

Abstract

We show that there is no non-constant assignment of zeros and ones to points of a unit sphere in $$\mathbb{R}^3$$ R 3 such that for every three pairwisely orthogonal vectors, an odd number of them is assigned 1. This is a new strengthening of the Bell–Kochen–Specker theorem, which proves the non-existence of hidden variables in quantum theories.

Automated Generation of Arbitrarily Many Kochen-Specker and Other Contextual Sets in Odd Dimensional Hilbert Spaces

“…For instance with the help of {0, ±1, 2, 3} (Peres') components we obtain the master 81-52 which contains just one single critical set-Peres' 57-40; {0, ±1, ±2, 5} yield the master 97-64 which generates 20 critical KS MMPHs from 49-36 to 55-40; in contrast, {0, ±1, 2, ±2, ±3, 5} yield the master 301-184 which generates 81 critical KS MMPHs from 49-36 to 92-66; {0, ±ω, 2ω, ±ω We did not find simple real vector components which would yield KS MMPHs smaller than 49-36, although we are able to generate many smaller KS NBMMPHs down to 19-13 shown in Fig. 2(b), or 39-27 shown in the Supplemental Material (SM), some of which might possess vectors similar to those obtained by Voráček and Navara [14] for the 21-13 BMMPH shown in Fig. 2(a).…”

“…The path taken in [14] is intractable for hundreds of small KS NBMMPHs we checked on our supercomputer since the number of free variables is too high.…”

“…2. (a) BMMPH from[14]; (b) KS MMPH which does not have a real coordinatization and most probably also not a complex one; (c) One of nine 51-37 KS MMPHs we generated; it is a 22-gon non-isomorphic to Conway-Kochen's 51-37.…”