The simplest Kochen-Specker set

TL;DR

This paper introduces a highly symmetrical Kochen-Specker set in three dimensions with fewer bases, simplifying proofs and refuting a previous conjecture, thereby advancing understanding of quantum contextuality.

Contribution

It presents the simplest and most symmetrical Kochen-Specker set in dimension three, reducing the number of bases and inputs needed, and refutes a prior conjecture in quantum theory.

Findings

New KS set with 14 bases, the minimum known

Refutes Conjecture 2 in Phys. Rev. Lett. 134, 010201 (2025)

Establishes fundamental quantum properties of the set

Abstract

Kochen-Specker (KS) sets are fundamental in physics. Every time nature produces bipartite correlations attaining the nonsignaling limit, or two parties always win a nonlocal game impossible to always win classically, is because the parties are measuring a KS set. The simplest quantum system in which all these phenomena occur is a pair of three-level systems. However, the simplest KS sets in dimension three known are asymmetrical and require a large number of bases (the current minimum is 16, set by Peres and Penrose). Here we present a KS set that is much more symmetrical and easier to prove than any previous example. It sets a new record for minimum number of bases, 14, and enables us to refute Conjecture 2 in Phys. Rev. Lett. 134, 010201 (2025), setting a new record for qutrit-qutrit perfect strategies with a minimum number of inputs: 5-9. We establish the fundamental nature of this set in quantum theory.