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arXiv:1702.05215 · Physical Review A

Minimal complexity of Kochen-Specker sets does not scale with dimension

Mordecai Waegell, P. K. Aravind

Published May 2, 2017• 9 citations
arXivHTMLPDF

Abstract

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 $d=3,4,5,6,8$ 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 $d$.

DOI
10.1103/physreva.95.050101

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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≥10d\ensuremath{\ge}10d\ensuremath≥10 with at most nine contexts and 30 projectors, and in all odd dimensions d\ensuremath≥7d\ensuremath{\ge}7d\ensuremath≥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.
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.
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.
Recently, handling of contextual sets, in particular Kochen-Specker (KS) sets, in higher dimensions has been given an increasing attention, both theoretically and experimentally. However, methods of their generation are diverse, not generally applicable in every dimension, and of exponential complexity. Therefore, we design a dimensional upscaling method, whose complexity does not scale with dimension. As a proof of principle we generate manageable-sized KS master sets in up to 27 dimensional spaces and show that well over 32 dimensions can be reached. From these master sets we obtain an ample number of smaller KS sets. We discuss three kinds of applications that work with KS sets in higher dimensions. We anticipate other applications of KS sets for quantum information processing that make use of large families of nonisomorphic KS sets.
We look at generalisations of sets of vectors proving the Kochen-Specker theorem in 3 and 4 dimensions. It has been shown that two such sets, although unitarily inequivalent, are part of a larger 3-parameter family of vectors that share the same orthogonality graph. We find that these sets are unusual, in that the vectors in all other Kochen-Specker sets investigated here are fully determined by orthog-onality conditions and thus admit no free parameters. The Kochen-Specker (KS) theorem [1] is a statement about the fundamen-tal nature of quantum mechanics: no non-contextual hidden variable theory can reliably reproduce quantum mechanical predictions. The KS theorem states that in a Hilbert space with dimension n ≥ 3 it is impossible to assign definite values v ∈ {0, 1} to m projection operators Pm so that for every commuting set of projection operators satisfying Pm = I, the correspond-ing definite values obey v (Pm) = 1. The KS theorem is often visualised by means of a set of "uncolourable" vectors, where projection operators are represented by their constituent rays in Hilbert space and rays can be simplified to vectors as only directions are relevant here. The vectors are coloured according to their pre-assigned val-ues and the summation conditions above translate into colouring rules. This allows us to construct orthogonality graphs that, when uncolourable, pro-vide a contradiction within non-contextual hidden variable models and thus a proof of the KS theorem. Sets of vectors exemplifying the KS theorem have been discovered by nu-merous authors, with an aim to reduce the number of vectors required to reach a contradiction. Most notably, in a 3-dimensional Hilbert space, the original KS proof of 117 vectors has been replicated with sets using between 31 and 33 vectors [2, 3, 4, 5] and in a 4-dimensional Hilbert space, sets of between 24 and the lower limit of 18 vectors have been realised [2, 6, 7]. Additionally, an extensive computer search has recently revealed over one thousand of such KS sets in R4 [8]. In a publication by E. Gould and P. K. Aravind [10], two of these sets in 3 dimensions have been shown to share the same orthogonality graph. Specifically, the sets of 33 vectors found by Penrose [3] and Peres [2] have
Recent results show that Kochen-Specker (KS) sets of observables are fundamental to quantum information, computation, and foundations beyond previous expectations. Among KS sets, those that are unique up to unitary transformations (i.e., ``rigid'') are especially important. The problem is that we do not know any rigid KS set in C3{\mathbb{C}}^{3}C3, the smallest quantum system that allows for KS sets. Moreover, none of the existing methods for constructing KS sets leads to rigid KS sets in C3{\mathbb{C}}^{3}C3. Here, we show that two fundamental structures of quantum theory define two rigid KS sets. One of these structures is the supersymmetric informationally complete positive-operator-valued measure. The other is the minimal state-independent contextuality set. The second construction provides a clue to solve the minimal KS problem, the most important open problem in this field. We prove that there is no KS set of 30 elements that can be obtained from the minimal state-independent contextuality set by completing bases and adding elements that are orthogonal to two previous elements. We conjecture that 31 is the solution to the minimal KS set problem.
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.