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21 pages, 1173 KB

Open AccessArticle

Quantum Correlations in Classical Systems

by Ghenadie N. Mardari

Quantum Rep. 2026, 8(2), 35; https://doi.org/10.3390/quantum8020035 - 18 Apr 2026

Viewed by 2190

A classical fluid splitter produces the same patterns of energy redistribution as a Stern–Gerlach quantum device, with rotationally invariant coefficients of correlation between molecular paths. Alternative settings express a cosine squared relationship, leading to Tsirelson-type Bell violations with outcome independence. This result confirms [...] Read more.

A classical fluid splitter produces the same patterns of energy redistribution as a Stern–Gerlach quantum device, with rotationally invariant coefficients of correlation between molecular paths. Alternative settings express a cosine squared relationship, leading to Tsirelson-type Bell violations with outcome independence. This result confirms the Correspondence Principle of quantum mechanics, where individual detection events express system-level properties according to Born’s Rule. Kochen–Specker contextuality and Bell Locality are not formally contradicted, but their interpretation is in question. Current definitions of “Local Realism” are limited to intrinsic particle properties. In contrast, quantum-like correlations require the acknowledgement of ensemble effects on dynamically inseparable entities, even when those entities are observed one at a time. Full article

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28 pages, 2541 KB

Open AccessFeature PaperArticle

Engineering and Applying Quantum Contextuality

by Mladen Pavičić

Entropy 2026, 28(4), 446; https://doi.org/10.3390/e28040446 - 14 Apr 2026

Viewed by 723

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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13 pages, 216 KB

Open AccessArticle

Reassessing the Strength of a Class of Wigner’s Friend No-Go Theorems

by Elias Okon

Entropy 2025, 27(6), 563; https://doi.org/10.3390/e27060563 - 27 May 2025

Cited by 1 | Viewed by 2105

Abstract

Two recent, prominent theorems—the “no-go theorem for observer-independent facts” and the “Local Friendliness no-go theorem”—employ so-called extended Wigner’s friend scenarios to try to impose novel, non-trivial constraints on the possible nature of physical reality. While the former is argued to entail that there [...] Read more.

Two recent, prominent theorems—the “no-go theorem for observer-independent facts” and the “Local Friendliness no-go theorem”—employ so-called extended Wigner’s friend scenarios to try to impose novel, non-trivial constraints on the possible nature of physical reality. While the former is argued to entail that there can be no theory in which the results of Wigner and his friend can both be considered objective, the latter is said to place on reality stronger constraints than the Bell and Kochen–Specker theorems. Here, I conduct a thorough analysis of these theorems and show that they suffer from a list of shortcomings that question their validity and limit their strength. I conclude that the “no-go theorem for observer-independent facts” and the “Local Friendliness no-go theorem” fail to impose significant constraints on the nature of physical reality. Full article

(This article belongs to the Section Quantum Information)

9 pages, 261 KB

Open AccessArticle

Chromatic Quantum Contextuality

by Karl Svozil

Entropy 2025, 27(4), 387; https://doi.org/10.3390/e27040387 - 5 Apr 2025

Cited by 3 | Viewed by 1619

Abstract

Chromatic quantum contextuality is a criterion of quantum nonclassicality based on (hyper)graph coloring constraints. If a quantum hypergraph requires more colors than the number of outcomes per maximal observable (context), it lacks a classical realization with n-uniform outcomes per context. Consequently, it [...] Read more.

Chromatic quantum contextuality is a criterion of quantum nonclassicality based on (hyper)graph coloring constraints. If a quantum hypergraph requires more colors than the number of outcomes per maximal observable (context), it lacks a classical realization with n-uniform outcomes per context. Consequently, it cannot represent a “completable” noncontextual set of coexisting n-ary outcomes per maximal observable. This result serves as a chromatic analogue of the Kochen-Specker theorem. We present an explicit example of a four-colorable quantum logic in dimension three. Furthermore, chromatic contextuality suggests a novel restriction on classical truth values, thereby excluding two-valued measures that cannot be extended to n-ary colorings. Using this framework, we establish new bounds for the house, pentagon, and pentagram hypergraphs, refining previous constraints. Full article

(This article belongs to the Special Issue Quantum Probability and Randomness V)

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35 pages, 1317 KB

Open AccessFeature PaperArticle

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 2354

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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17 pages, 377 KB

Open AccessArticle

Hidden Variables in Quantum Mechanics from the Perspective of Boltzmannian Statistical Mechanics

by Dustin Lazarovici

Quantum Rep. 2024, 6(3), 465-481; https://doi.org/10.3390/quantum6030031 - 6 Sep 2024

Cited by 1 | Viewed by 4751

Abstract

This paper examines no-hidden-variables theorems in quantum mechanics from the point of view of statistical mechanics. It presents a general analysis of the measurement process in the Boltzmannian framework that leads to a characterization of (in)compatible measurements and reproduces several features of quantum [...] Read more.

This paper examines no-hidden-variables theorems in quantum mechanics from the point of view of statistical mechanics. It presents a general analysis of the measurement process in the Boltzmannian framework that leads to a characterization of (in)compatible measurements and reproduces several features of quantum probabilities often described as “non-classical”. The analysis is applied to versions of the Kochen–Specker and Bell theorems to shed more light on their implications. It is shown how, once the measurement device and the active role of the measurement process are taken into account, contextuality appears as a natural feature of random variables. This corroborates Bell’s criticism that no-go results of the Kochen–Specker type are based on gratuitous assumptions. In contrast, Bell-type theorems are much more profound, but should be understood as nonlocality theorems rather than no-hidden-variables theorems. Finally, the paper addresses misunderstandings and misleading terminology that have confused the debate about hidden variables in quantum mechanics. Full article

(This article belongs to the Special Issue Exclusive Feature Papers of Quantum Reports in 2024–2025)

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1 pages, 458 KB

Open AccessCorrection

Correction: Pavičić, M. Non-Kochen–Specker Contextuality. Entropy 2023, 25, 1117

by Mladen Pavičić

Entropy 2024, 26(2), 100; https://doi.org/10.3390/e26020100 - 24 Jan 2024

Viewed by 1383

Abstract

In the original publication [...] Full article

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21 pages, 465 KB

Open AccessEditor’s ChoiceArticle

Generalized Bell Scenarios: Disturbing Consequences on Local-Hidden-Variable Models

by André Mazzari, Gabriel Ruffolo, Carlos Vieira, Tassius Temistocles, Rafael Rabelo and Marcelo Terra Cunha

Entropy 2023, 25(9), 1276; https://doi.org/10.3390/e25091276 - 30 Aug 2023

Cited by 1 | Viewed by 2365

Abstract

Bell nonlocality and Kochen–Specker contextuality are among the main topics in the foundations of quantum theory. Both of them are related to stronger-than-classical correlations, with the former usually referring to spatially separated systems, while the latter considers a single system. In recent works, [...] Read more.

Bell nonlocality and Kochen–Specker contextuality are among the main topics in the foundations of quantum theory. Both of them are related to stronger-than-classical correlations, with the former usually referring to spatially separated systems, while the latter considers a single system. In recent works, a unified framework for these phenomena was presented. This article reviews, expands, and obtains new results regarding this framework. Contextual and disturbing features inside the local models are explored, which allows for the definition of different local sets with a non-trivial relation among them. The relations between the set of quantum correlations and these local sets are also considered, and post-quantum local behaviours are found. Moreover, examples of correlations that are both local and non-contextual but such that these two classical features cannot be expressed by the same hidden variable model are shown. Extensions of the Fine–Abramsky–Brandenburger theorem are also discussed. Full article

(This article belongs to the Special Issue Quantum Correlations, Contextuality, and Quantum Nonlocality)

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21 pages, 1029 KB

Open AccessArticle

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 3257 | 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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11 pages, 543 KB

Open AccessArticle

Bypassing the Kochen–Specker Theorem: An Explicit Non-Contextual Statistical Model for the Qutrit

by David H. Oaknin

Axioms 2023, 12(1), 90; https://doi.org/10.3390/axioms12010090 - 15 Jan 2023

Cited by 2 | Viewed by 2648

Abstract

We describe an explicitly non-contextual statistical model of hidden variables for the qutrit, which fully reproduces the predictions of quantum mechanics, and thus, bypasses the constraints imposed by the Kochen–Specker theorem and its subsequent reformulations. We notice that these renowned theorems crucially rely [...] Read more.

We describe an explicitly non-contextual statistical model of hidden variables for the qutrit, which fully reproduces the predictions of quantum mechanics, and thus, bypasses the constraints imposed by the Kochen–Specker theorem and its subsequent reformulations. We notice that these renowned theorems crucially rely on the implicitly assumed existence of an absolute frame of reference with respect to which physically indistinguishable tests related by spurious gauge transformations can supposedly be assigned well-defined distinct identities. We observe that the existence of such an absolute frame of reference is not required by fundamental physical principles, and hence, assuming it is an unnecessarily restrictive demand. Full article

(This article belongs to the Special Issue Advances in Stochastic Modelling)

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12 pages, 319 KB

Open AccessFeature PaperArticle

Extending Kolmogorov’s Axioms for a Generalized Probability Theory on Collections of Contexts

by Karl Svozil

Entropy 2022, 24(9), 1285; https://doi.org/10.3390/e24091285 - 12 Sep 2022

Cited by 5 | Viewed by 3069

Abstract

Kolmogorov’s axioms of probability theory are extended to conditional probabilities among distinct (and sometimes intertwining) contexts. Formally, this amounts to row stochastic matrices whose entries characterize the conditional probability to find some observable (postselection) in one context, given an observable (preselection) in another [...] Read more.

Kolmogorov’s axioms of probability theory are extended to conditional probabilities among distinct (and sometimes intertwining) contexts. Formally, this amounts to row stochastic matrices whose entries characterize the conditional probability to find some observable (postselection) in one context, given an observable (preselection) in another context. As the respective probabilities need not (but, depending on the physical/model realization, can) be of the Born rule type, this generalizes approaches to quantum probabilities by Aufféves and Grangier, which in turn are inspired by Gleason’s theorem. Full article

(This article belongs to the Special Issue Quantum Information and Probability: From Foundations to Engineering)

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15 pages, 506 KB

Open AccessArticle

Contextuality-by-Default Description of Bell Tests: Contextuality as the Rule and Not as an Exception

by Marian Kupczynski

Entropy 2021, 23(9), 1104; https://doi.org/10.3390/e23091104 - 25 Aug 2021

Cited by 18 | Viewed by 4576

Abstract

Contextuality and entanglement are valuable resources for quantum computing and quantum information. Bell inequalities are used to certify entanglement; thus, it is important to understand why and how they are violated. Quantum mechanics and behavioural sciences teach us that random variables ‘measuring’ the [...] Read more.

Contextuality and entanglement are valuable resources for quantum computing and quantum information. Bell inequalities are used to certify entanglement; thus, it is important to understand why and how they are violated. Quantum mechanics and behavioural sciences teach us that random variables ‘measuring’ the same content (the answer to the same Yes or No question) may vary, if ‘measured’ jointly with other random variables. Alice’s and BoB′s raw data confirm Einsteinian non-signaling, but setting dependent experimental protocols are used to create samples of coupled pairs of distant ±1 outcomes and to estimate correlations. Marginal expectations, estimated using these final samples, depend on distant settings. Therefore, a system of random variables ‘measured’ in Bell tests is inconsistently connected and it should be analyzed using a Contextuality-by-Default approach, what is done for the first time in this paper. The violation of Bell inequalities and inconsistent connectedness may be explained using a contextual locally causal probabilistic model in which setting dependent variables describing measuring instruments are correctly incorporated. We prove that this model does not restrict experimenters’ freedom of choice which is a prerequisite of science. Contextuality seems to be the rule and not an exception; thus, it should be carefully tested. Full article

(This article belongs to the Special Issue Quantum Probability and Randomness III)

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21 pages, 413 KB

Open AccessArticle

Quantum Randomness is Chimeric

by Karl Svozil

Entropy 2021, 23(5), 519; https://doi.org/10.3390/e23050519 - 24 Apr 2021

Cited by 4 | Viewed by 5785

Abstract

If quantum mechanics is taken for granted, the randomness derived from it may be vacuous or even delusional, yet sufficient for many practical purposes. “Random” quantum events are intimately related to the emergence of both space-time as well as the identification of physical [...] Read more.

If quantum mechanics is taken for granted, the randomness derived from it may be vacuous or even delusional, yet sufficient for many practical purposes. “Random” quantum events are intimately related to the emergence of both space-time as well as the identification of physical properties through which so-called objects are aggregated. We also present a brief review of the metaphysics of indeterminism. Full article

(This article belongs to the Special Issue Noisy Intermediate-Scale Quantum Technologies (NISQ))

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6 pages, 217 KB

Open AccessArticle

Quantum Probability’s Algebraic Origin

by Gerd Niestegge

Entropy 2020, 22(11), 1196; https://doi.org/10.3390/e22111196 - 23 Oct 2020

Cited by 4 | Viewed by 5283

Abstract

Max Born’s statistical interpretation made probabilities play a major role in quantum theory. Here we show that these quantum probabilities and the classical probabilities have very different origins. Although the latter always result from an assumed probability measure, the first include transition probabilities [...] Read more.

Max Born’s statistical interpretation made probabilities play a major role in quantum theory. Here we show that these quantum probabilities and the classical probabilities have very different origins. Although the latter always result from an assumed probability measure, the first include transition probabilities with a purely algebraic origin. Moreover, the general definition of transition probability introduced here comprises not only the well-known quantum mechanical transition probabilities between pure states or wave functions, but further physically meaningful and experimentally verifiable novel cases. A transition probability that differs from 0 and 1 manifests the typical quantum indeterminacy in a similar way as Heisenberg’s and others’ uncertainty relations and, furthermore, rules out deterministic states in the same way as the Bell-Kochen-Specker theorem. However, the transition probability defined here achieves a lot more beyond that: it demonstrates that the algebraic structure of the Hilbert space quantum logic dictates the precise values of certain probabilities and it provides an unexpected access to these quantum probabilities that does not rely on states or wave functions. Full article

(This article belongs to the Special Issue Quantum Probability and Randomness II)

24 pages, 2851 KB

Open AccessArticle

A Knot Theoretic Extension of the Bloch Sphere Representation for Qubits in Hilbert Space and Its Application to Contextuality and Many-Worlds Theories

by Stefan Heusler, Paul Schlummer and Malte S. Ubben

Symmetry 2020, 12(7), 1135; https://doi.org/10.3390/sym12071135 - 7 Jul 2020

Cited by 3 | Viewed by 5101

Abstract

We argue that the usual Bloch sphere is insufficient in various aspects for the representation of qubits in quantum information theory. For example, spin flip operations with the quaternions

I J K = e^{\frac{2 π i}{2}} = - 1

and [...] Read more.

We argue that the usual Bloch sphere is insufficient in various aspects for the representation of qubits in quantum information theory. For example, spin flip operations with the quaternions

I J K = e^{\frac{2 π i}{2}} = - 1

and

J I K = + 1

cannot be distinguished on the Bloch sphere. We show that a simple knot theoretic extension of the Bloch sphere representation is sufficient to track all unitary operations for single qubits. Next, we extend the Bloch sphere representation to entangled states using knot theory. As applications, we first discuss contextuality in quantum physics—in particular the Kochen-Specker theorem. Finally, we discuss some arguments against many-worlds theories within our knot theoretic model of entanglement. The key ingredients of our approach are symmetries and geometric properties of the unitary group. Full article

(This article belongs to the Special Issue The Importance of Being Symmetrical)

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