On Simplest Kochen-Specker Sets

TL;DR

This paper critically examines the simplest known Kochen-Specker sets, revealing prior generation, challenging their fundamentality, and demonstrating that automatic methods can efficiently generate all minimal contextual sets across dimensions.

Contribution

It shows the previously claimed simplest 3D KS set was generated earlier, argues against its fundamental status, and proves that automatic generation captures all minimal contextual sets in any dimension.

Findings

The 3D KS set was previously generated in 2023.

Automatic generation methods efficiently produce all minimal contextual sets.

Many smaller 3D contextual sets exist, questioning the set's fundamental status.

Abstract

In Phys. Rev. Lett. 135, 190203 (2025) a discovery of the simplest 3D contextual set with 33 vertices, 50 bases, and 14 complete bases is claimed. In this paper, we show that it was previously generated in Quantum 7, 953 (2023) and analyze the meaning, origin, and significance of the simplest contextual sets in any dimension. In particular, we prove that there is no ground to consider the aforementioned set as fundamental since there are many 3D contextual sets with a smaller number of complete bases. We also show that automatic generation of contextual sets from basic vector components automatically yields all known minimal contextual sets of any kind in any dimension and therefore also the aforementioned set in no CPU-time. In the end, we discuss varieties of contextual sets, in particular Kochen-Specker (KS), extended KS, and non-KS sets as well as ambiguities in their definitions.