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arXiv:2508.07335 · Physical Review Letters

Simplest Kochen-Specker Set

Adán Cabello

Published Nov 6, 2025• 4 citations
arXivHTMLPDF

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, it 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 known in dimension three 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 Cabello Simplest bipartite perfect quantum strategies, Phys. Rev. Lett. 134, 010201 (2025)PRLTAO0031-900710.1103/PhysRevLett.134.010201, 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.

DOI
10.1103/fx9d-488z

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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, it 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 known in dimension three 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 Cabello Simplest bipartite perfect quantum strategies, Phys. Rev. Lett. 134, 010201 (2025)PRLTAO0031-900710.1103/PhysRevLett.134.010201, 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.
Bipartite perfect quantum strategies (BPQSs) allow two players isolated from each other to win every trial of a nonlocal game. BPQSs have crucial roles in recent developments in quantum information and quantum computation. However, only few BPQSs with a small number of inputs are known and only one of them has been experimentally tested. It has recently been shown that every BPQS has an associated Kochen-Specker (KS) set. Here, we first prove that any BPQS of minimum input cardinality that can be obtained from a generalized KS set can also be obtained from a KS set of pure states. We address the problem of finding BPQSs of small input cardinality starting from KS sets. We introduce an algorithm that identifies the BPQS with the minimum number of settings for any given KS set. We apply it to many well-known KS sets of small cardinality in dimensions 3, 4, 5, 6, 7, and 8. In each dimension, the algorithm either recovers the best BPQS known or finds one with fewer inputs.
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.
A bipartite perfect quantum strategy (BPQS) allows two players who cannot communicate with each other to always win a nonlocal game. BPQSs are rare but fundamental in light of some recent results in quantum information, computation, and foundations. A more than 40-year-old open problem is how many inputs (measurement settings) a BPQS requires. A related problem is how many inputs are needed if, in addition, the quantum system has minimum dimension. A third, apparently unrelated, problem is what is the connection between BPQSs and state-independent contextuality, which inspired the first BPQSs. Here, we solve the third problem: we prove that every BPQS defines a Kochen-Specker set. We use this result to identify the BPQS with the smallest number of inputs, both in the general case and in the case of minimum dimension, and solve some related problems. We conjecture that the BPQSs presented here are the solutions to the first two problems.
A perfect strategy is one that allows the mutually in-communicated players of a nonlocal game to win every trial of the game. Perfect strategies are basic tools for some fundamental results in quantum computation and crucial resources for some applications in quantum information. Here, we address the problem of producing qudit-qudit perfect quantum strategies with a small number of settings. For that, we exploit a recent result showing that any perfect quantum strategy induces a Kochen-Specker set. We identify a family of KS sets in even dimension d≥6d \ge 6d≥6 that, for many dimensions, require the smallest number of orthogonal bases known: d+1d+1d+1. This family was only defined for some ddd. We first extend the family to infinitely many more dimensions. Then, we show the optimal way to use each of these sets to produce a bipartite perfect strategy with minimum input cardinality. As a result, we present a family of perfect quantum strategies in any (2,d−1,d)(2,d-1,d)(2,d−1,d) Bell scenario, with d=2kpmd = 2^kp^md=2kpm for ppp prime, m≥k≥0m \geq k \geq 0m≥k≥0 (excluding m=k=0m=k=0m=k=0), d=8pd = 8pd=8p for p≥19p \geq 19p≥19, d=kpd=kpd=kp for p>((k−2)2k−2)2p > ((k-2)2^{k-2})^2p>((k−2)2k−2)2 whenever there exists a Hadamard matrix of order kkk, other sporadic examples, as well as a recursive construction that produces perfect quantum strategies for infinitely many dimensions ddd from any dimension d′d'd′ with a perfect quantum strategy. We identify their associated Bell inequalities and prove that they are not tight, which provides a second counterexample to a conjecture of 2007.
Kochen—Specker (KS) theorem denies the possibility for the noncontextual hidden variable theories to reproduce the predictions of quantum mechanics. A set of projection operators (projectors) and bases used to show the impossibility of noncontextual definite values assignment is named as the KS set. Since one KS set with a mixture of 16 rank-1 projectors and 14 rank-2 projectors was proposed in 1995 [Kernaghan M and Peres A Phys. Lett. A 198 (1995) 1 ] for a three-qubit system, there have been plenty of the same type KS sets and we propose a systematic way to produce them. We also propose a probabilistic state-dependent proof of the KS theorem that mainly focuses on the values assignment for all the rank-2 projectors.
The Kochen-Specker (KS) theorem is a central result in quantum theory and has applications in quantum information. Its proof requires several yes-no tests that can be grouped in contexts or subsets of jointly measurable tests. Arguably, the best measure of simplicity of a KS set is the number of contexts. The smaller this number is, the smaller the number of experiments needed to reveal the conflict between quantum theory and noncontextual theories and to get a quantum vs classical outperformance. The original KS set had 132 contexts. Here we introduce a KS set with seven contexts and prove that this is the simplest KS set that admits a symmetric parity proof.