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arXiv:1308.6012 · Physical Review A

Kochen-Specker set with seven contexts

Petr Lisoněk, Piotr Badziąg, José R. Portillo, Adán Cabello

Published Apr 1, 2014• 52 citations
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Abstract

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.

DOI
10.1103/physreva.89.042101

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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.
A Kochen-Specker (KS) set is a specific set of projectors and measurement contexts that prove the Bell-Kochen-Specker contextuality theorem. The simplest known KS sets in Hilbert space dimensions d=3,4,5,6,8d=3,4,5,6,8d=3,4,5,6,8 are reproduced, and several methods by which a new KS set can be constructed using one or more known KS sets in lower dimensions are reviewed and improved. These KS sets and improved methods enable the construction of explicitly critical new KS sets in all dimensions, where critical refers to the irreducibility of the set of contexts. The simplest known critical KS sets are derived in all even dimensions $d\ensuremath{\ge}10$ with at most nine contexts and 30 projectors, and in all odd dimensions $d\ensuremath{\ge}7$ with at most 13 contexts and 39 projectors. These results show that neither the number of contexts nor the number of projectors in a minimal KS set scales with dimension ddd.
Kochen-Specker (KS) sets are key tools for proving some fundamental results in quantum theory and also have potential applications in quantum information processing. However, so far, their intrinsic complexity has prevented experimentalists from using them for any application. The KS set requiring the smallest number of contexts has been recently found. Relying on this simple KS set, here we report an input state-independent experimental technique to certify whether a set of measurements is actually accessing a preestablished quantum six-dimensional space encoded in the transverse momentum of single photons.
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.
We present a number of observables-based proofs of the Kochen-Specker (KS) theorem based on the NNN-qubit Pauli group for $N\ensuremath{\ge}4$, thus adding to the proofs that have been presented earlier for the 2- and 3-qubit groups. These proofs have the attractive feature that they can be presented in the form of diagrams from which they are obvious by inspection. They are also irreducible in the sense that they cannot be reduced to smaller proofs by ignoring some subset of qubits and/or observables in them. A simple algorithm is given for transforming any observables-based KS proof into a large number of projectors-based KS proofs; if the observables-based proof has OOO observables, with each observable occurring in exactly two commuting sets and any two commuting sets having at most one observable in common, the number of associated projectors-based parity proofs is 2O{}^{O}O. We introduce symbols for the observables- and projectors-based KS proofs that capture their important features and also convey a feeling for the enormous variety of both these types of proofs within the NNN-qubit Pauli group. We discuss an infinite family of observables-based proofs, whose members include all numbers of qubits from 2 up, and show how it can be used to generate projectors-based KS proofs involving only 9 bases (or experimental contexts) in any dimension of the form 2N{2}^{N}2N for $N\ensuremath{\ge}2$. Some implications of our results are discussed.
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.
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.