The simplest Kochen-Specker 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.

Figures (2)

• Figure 1: Relations of orthogonality between the 33 vectors of our KS set. Adjacent nodes represent orthogonal vectors. The three red (green, blue, yellow) nodes are mutually orthogonal. $1\omega\bar{\omega^2}$ represents $\tfrac{1}{\sqrt{3}}(1,\omega,-\omega^2)$, where $\omega = e^{2 \pi i /3}$.
• Figure 2: Majorana representation of the KS set. Each of the $33$ vectors corresponds to a unique pair of points, but some points are shared among different vectors.