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On Simplest Kochen-Specker Sets

Mladen Pavicic

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.

On Simplest Kochen-Specker Sets

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.

Paper Structure

This paper contains 1 section, 4 figures.

Table of Contents

  1. Appendix. Coordinatization of the 69-50 in Fig. \ref{['fig:CK']}(c)

Figures (4)

  • Figure 1: (a) The Yu-Oh set 13-16 yu-oh-12 in an MMPH representation; (b) the Yu-Oh set filled with grey vertices of multiplicity 1---the 25-16 MMPH; (c) the 69-50 MMPH from pavicic-quantum-23 redrawn so as to flash colored vertices from cabello-25b; its variety with grey vertices of multiplicity 1 dropped---33-50---is isomorphic to cabello-25b and to Fig. \ref{['fig:yuoh']} below; its coordinatization is given in the Appendix.
  • Figure 2: A 3D representation of the 33-50 set; snapshots from two different angles are taken from a Blender output obtained in pavicic-ravlic-2025 which the reader can interactively rotate at will; (a) top view; (b) side view.
  • Figure 3: (a) Partially "extended" Yu-Oh set which is still contextual and has thirteen complete bases; (b) contextual subset of the 33-50 contextual set; (c) contextual sub-hypergraph of the 33-50 contextual set with seven complete bases.
  • Figure 4: (a) The 18-9 contextual 4D set cabell-est-96a, pmmm04c, cabello-08; (b) the 17-9---a contextual critical subset of 18-9 obtained obtained by means of a weak deletion of a vertex voloshin-09; in Cabello's notation it is a KS set while it is a non-KS in the notation of ours; (c) the 19-9---non-contextual weakly extended set of 17-9.