New Kochen-Specker Sets in Four Dimensions

TL;DR

This paper systematically derives all 4-dimensional Kochen-Specker sets with 18 to 24 vectors from a single known set, revealing new configurations and analyzing their critical subsets.

Contribution

It introduces algorithms to generate and analyze 4D Kochen-Specker sets, expanding the known configurations and identifying critical subsets within them.

Findings

All 388 KS sets with 18-23 vectors and 844 with 24 vectors can be derived from Peres' set.

Discovered additional KS sets with 22-24 vectors beyond the derived ones.

Identified that Peres' 24-24 KS set contains six critical KS subsets.

Abstract

We show that all possible 388 4-dim Kochen-Specker (KS) (vector) sets (of yes-no questions) with 18 through 23 vectors and 844 sets with 24 vectors all with component values from \{-1,0,1\} can be obtained by stripping vectors off a single system provided by Peres 20 years ago. In addition to them, we have found a number of other KS sets with 22 through 24 vectors. We present the algorithms we used and features we found, such as, for instance, that Peres' 24-24 KS set has altogether six critical KS subsets.