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Keywords = MMP hypergraphs

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28 pages, 2541 KB  
Article
Engineering and Applying Quantum Contextuality
by Mladen Pavičić
Entropy 2026, 28(4), 446; https://doi.org/10.3390/e28040446 - 14 Apr 2026
Viewed by 744
Abstract
The endeavor to refute hidden variable theories underlying quantum theory has yielded the discipline of contextual sets. A plethora of various kinds of sets of arbitrary structure in any dimension have been developed, alongside extensive experimental validation. These advancements incited us to investigate [...] Read more.
The endeavor to refute hidden variable theories underlying quantum theory has yielded the discipline of contextual sets. A plethora of various kinds of sets of arbitrary structure in any dimension have been developed, alongside extensive experimental validation. These advancements incited us to investigate to what extent we might move beyond hidden variables, engineer contextual sets, and find their applications within quantum theory itself, without any reference to hidden variable models. To this end, we consider possible applications of contextual sets in quantum computation, cryptography, pseudo-telepathy, and nonlocality, as well as generating them from error-correction protocols, complex Hadamard gates, and simple quantum gates. We found that the results in this field are still scarce, and we therefore investigated the directions in which future research might be carried out and the potential obstacles to realizing such undertakings in the past. Full article
(This article belongs to the Special Issue Quantum Information and Probability: From Foundations to Engineering IV)
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35 pages, 1317 KB  
Article
Quantum Contextual Hypergraphs, Operators, Inequalities, and Applications in Higher Dimensions
by Mladen Pavičić
Entropy 2025, 27(1), 54; https://doi.org/10.3390/e27010054 - 9 Jan 2025
Cited by 2 | Viewed by 2369
Abstract
Quantum contextuality plays a significant role in supporting quantum computation and quantum information theory. The key tools for this are the Kochen–Specker and non-Kochen–Specker contextual sets. Traditionally, their representation has been predominantly operator-based, mainly focusing on specific constructs in dimensions ranging from three [...] Read more.
Quantum contextuality plays a significant role in supporting quantum computation and quantum information theory. The key tools for this are the Kochen–Specker and non-Kochen–Specker contextual sets. Traditionally, their representation has been predominantly operator-based, mainly focusing on specific constructs in dimensions ranging from three to eight. However, nearly all of these constructs can be represented as low-dimensional hypergraphs. This study demonstrates how to generate contextual hypergraphs in any dimension using various methods, particularly those that do not scale in complexity with increasing dimensions. Furthermore, we introduce innovative examples of hypergraphs extending to dimension 32. Our methodology reveals the intricate structural properties of hypergraphs, enabling precise quantifications of contextuality. Additionally, we investigate several promising applications of hypergraphs in quantum communication and quantum computation, paving the way for future breakthroughs in the field. Full article
(This article belongs to the Special Issue Quantum Probability and Randomness V)
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21 pages, 1029 KB  
Article
Non-Kochen–Specker Contextuality
by Mladen Pavičić
Entropy 2023, 25(8), 1117; https://doi.org/10.3390/e25081117 - 26 Jul 2023
Cited by 3 | Viewed by 3271 | Correction
Abstract
Quantum contextuality supports quantum computation and communication. One of its main vehicles is hypergraphs. The most elaborated are the Kochen–Specker ones, but there is also another class of contextual sets that are not of this kind. Their representation has been mostly operator-based and [...] Read more.
Quantum contextuality supports quantum computation and communication. One of its main vehicles is hypergraphs. The most elaborated are the Kochen–Specker ones, but there is also another class of contextual sets that are not of this kind. Their representation has been mostly operator-based and limited to special constructs in three- to six-dim spaces, a notable example of which is the Yu-Oh set. Previously, we showed that hypergraphs underlie all of them, and in this paper, we give general methods—whose complexity does not scale up with the dimension—for generating such non-Kochen–Specker hypergraphs in any dimension and give examples in up to 16-dim spaces. Our automated generation is probabilistic and random, but the statistics of accumulated data enable one to filter out sets with the required size and structure. Full article
(This article belongs to the Special Issue Quantum Probability and Randomness IV)
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20 pages, 410 KB  
Article
Hypergraph Contextuality
by Mladen Pavičić
Entropy 2019, 21(11), 1107; https://doi.org/10.3390/e21111107 - 12 Nov 2019
Cited by 10 | Viewed by 4105
Abstract
Quantum contextuality is a source of quantum computational power and a theoretical delimiter between classical and quantum structures. It has been substantiated by numerous experiments and prompted generation of state independent contextual sets, that is, sets of quantum observables capable of revealing quantum [...] Read more.
Quantum contextuality is a source of quantum computational power and a theoretical delimiter between classical and quantum structures. It has been substantiated by numerous experiments and prompted generation of state independent contextual sets, that is, sets of quantum observables capable of revealing quantum contextuality for any quantum state of a given dimension. There are two major classes of state-independent contextual sets—the Kochen-Specker ones and the operator-based ones. In this paper, we present a third, hypergraph-based class of contextual sets. Hypergraph inequalities serve as a measure of contextuality. We limit ourselves to qutrits and obtain thousands of 3-dim contextual sets. The simplest of them involves only 5 quantum observables, thus enabling a straightforward implementation. They also enable establishing new entropic contextualities. Full article
(This article belongs to the Special Issue Entropy in Foundations of Quantum Physics)
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12 pages, 571 KB  
Article
Vector Generation of Quantum Contextual Sets in Even Dimensional Hilbert Spaces
by Mladen Pavičić and Norman D. Megill
Entropy 2018, 20(12), 928; https://doi.org/10.3390/e20120928 - 5 Dec 2018
Cited by 16 | Viewed by 4221
Abstract
Recently, quantum contextuality has been proved to be the source of quantum computation’s power. That, together with multiple recent contextual experiments, prompts improving the methods of generation of contextual sets and finding their features. The most elaborated contextual sets, which offer blueprints for [...] Read more.
Recently, quantum contextuality has been proved to be the source of quantum computation’s power. That, together with multiple recent contextual experiments, prompts improving the methods of generation of contextual sets and finding their features. The most elaborated contextual sets, which offer blueprints for contextual experiments and computational gates, are the Kochen–Specker (KS) sets. In this paper, we show a method of vector generation that supersedes previous methods. It is implemented by means of algorithms and programs that generate hypergraphs embodying the Kochen–Specker property and that are designed to be carried out on supercomputers. We show that vector component generation of KS hypergraphs exhausts all possible vectors that can be constructed from chosen vector components, in contrast to previous studies that used incomplete lists of vectors and therefore missed a majority of hypergraphs. Consequently, this unified method is far more efficient for generations of KS sets and their implementation in quantum computation and quantum communication. Several new KS classes and their features have been found and are elaborated on in the paper. Greechie diagrams are discussed. Full article
(This article belongs to the Special Issue Quantum Probability and Randomness)
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