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Quantum Probability and Randomness V

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A special issue of Entropy (ISSN 1099-4300). This special issue belongs to the section "Quantum Information".

Deadline for manuscript submissions: 30 May 2025 | Viewed by 11419

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Prof. Dr. Andrei Khrennikov

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International Center for Mathematical Modeling in Physics and Cognitive Sciences, Linnaeus University, SE-351 95 Växjö, Sweden
Interests: quantum foundations; information; probability; contextuality; applications of the mathematical formalism of quantum theory outside of physics: cognition, psychology, decision making, economics, finances, and social and political sciences; p-adic numbers; p-adic and ultrametric analysis; dynamical systems; p-adic theoretical physics; utrametric models of cognition and psychological behavior; p-adic models in geophysics and petroleum research
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Prof. Dr. Karl Svozil

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Guest Editor

Institute for Theoretical Physics, Vienna University of Technology Wiedner Hauptstrasse 8-10/136, A-1040 Vienna, Austria
Interests: quantum logic; automaton logic; conventionality in relativity theory; intrinsic embedded observers; physical (in)determinism; physical random number generators; generalized probability theory
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

This is the fifth Special Issue devoted to the theme of “Quantum Probability and Randomness”; for the first four issues, visit the following links:

The previous Special Issues collected a sample of high-quality papers, both theoretical and experiment-related, written by experts in this area, which attracted considerable interest (including numerous downloads). This is why we have decided to proceed once again with this hot topic by considering structuring this theme into a regular series based on the Entropy journal.

The last few years have been characterized by tremendous developments in quantum information and probability and their applications, including quantum computing, quantum cryptography, and quantum random generators. Despite the successful development of quantum technology, its foundational basis is still not concrete and contains a few sandy and shaky slices. Quantum random generators are one of the most promising outputs of the recent quantum information revolution. Therefore, it is very important to reconsider the foundational basis of this project, starting with the notion of irreducible quantum randomness.

Quantum probabilities present a powerful tool to model uncertainty. Interpretations of quantum probability and foundational meanings of its basic tools, starting with the Born rule, are among the topics which will be covered in this Special Issue.

Recently, quantum probability has started to play an important role in a few areas of research outside quantum physics—in particular, the quantum probabilistic treatment of problems of the theory of decision making under uncertainty. Such studies are also among the topics addressed in this Special Issue.

The areas covered include the following:

Foundations of quantum information theory and quantum probability;
Quantum versus classical randomness and quantum random generators;
Generalized probabilistic models;
Quantum contextuality and generalized contextual models;
Bell’s inequality, entanglement, and randomness;
Quantum-like probabilistic modeling of the process of decision making under uncertainty;
Quantum probabilistic models of cognition and AI;
Quantum probability and foundational questions of quantum technologies (computing and cryptography);
Quantum probability and information in biology.

Of course, possible topics need not be restricted to the list above; any contribution directed to the improvement of quantum foundations and the development of quantum probability and randomness is welcome.

Prof. Dr. Andrei Khrennikov
Prof. Dr. Karl Svozil
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Entropy is an international peer-reviewed open access monthly journal published by MDPI.

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Keywords

quantum foundation
quantum vs. classical probability and randomness
quantum information
Bell inequality
entanglement
contextuality
random generators
generalized probabilistic models

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Related Special Issues

Quantum Probability and Randomness in Entropy (14 articles)
Quantum Probability and Randomness II in Entropy (13 articles)
Quantum Probability and Randomness III in Entropy (12 articles)
Quantum Probability and Randomness IV in Entropy (15 articles)

Published Papers (13 papers)

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Research

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15 pages, 429 KiB

Open AccessArticle

A Note on the Relativistic Transformation Properties of Quantum Stochastic Calculus

by John E. Gough

Entropy 2025, 27(5), 529; https://doi.org/10.3390/e27050529 - 15 May 2025

Viewed by 66

Abstract

We present a simple argument to derive the transformation of the quantum stochastic calculus formalism between inertial observers and derive the quantum open system dynamics for a system moving in a vacuum (or, more generally, a coherent) quantum field under the usual Markov approximation. We argue, however, that, for uniformly accelerated open systems, the formalism must break down as we move from a Fock representation over the algebra of field observables over all of Minkowski space to the restriction regarding the algebra of observables over a Rindler wedge. This leads to quantum noise having a unitarily inequivalent non-Fock representation: in particular, the latter is a thermal representation at the Unruh temperature. The unitary inequivalence is ultimately a consequence of the underlying flat noise spectrum approximation for the fundamental quantum stochastic processes. We derive the quantum stochastic limit for a uniformly accelerated (two-level) detector and establish an open system description of the relaxation to thermal equilibrium at the Unruh temperature. Full article

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

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9 pages, 261 KiB

Open AccessArticle

Chromatic Quantum Contextuality

by Karl Svozil

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

Cited by 1 | Viewed by 240

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 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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18 pages, 277 KiB

Open AccessFeature PaperArticle

Fitting Copulas with Maximal Entropy

by Milan Bubák and Mirko Navara

Entropy 2025, 27(1), 87; https://doi.org/10.3390/e27010087 - 18 Jan 2025

Viewed by 587

Abstract

We deal with two-dimensional copulas from the perspective of their differential entropy. We formulate a problem of finding a copula with maximum differential entropy when some copula values are given. As expected, the solution is a copula with a piecewise constant density (a checkerboard copula). This allows us to simplify the optimization of the continuous objective function, the differential entropy, to an optimization of finitely many density values. We present several ideas to simplify this problem. It has a feasible numerical solution. We also present several instances that admit closed-form solutions. Full article

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

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35 pages, 1317 KiB

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

Viewed by 725

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 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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12 pages, 285 KiB

Open AccessArticle

Problem of Existence of Joint Distribution on Quantum Logic

by Oľga Nánásiová, Karla Čipková and Michal Zákopčan

Entropy 2024, 26(12), 1121; https://doi.org/10.3390/e26121121 - 21 Dec 2024

Viewed by 621

Abstract

This paper deals with the topics of modeling joint distributions on a generalized probability space. An algebraic structure known as quantum logic is taken as the basic model. There is a brief summary of some earlier published findings concerning a function s-map, which is a mathematical tool suitable for constructing virtual joint probabilities of even non-compatible propositions. The paper completes conclusions published in 2020 and extends the results for three or more random variables if the marginal distributions are known. The existence of a

(n + 1)

-variate joint distribution is shown in special cases when the quantum logic consists of at most n blocks of Boolean algebras. Full article

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

33 pages, 855 KiB

Open AccessArticle

Statistical Testing of Random Number Generators and Their Improvement Using Randomness Extraction

by Cameron Foreman, Richie Yeung and Florian J. Curchod

Entropy 2024, 26(12), 1053; https://doi.org/10.3390/e26121053 - 4 Dec 2024

Cited by 3 | Viewed by 1382

Abstract

Random number generators (RNGs) are notoriously challenging to build and test, especially for cryptographic applications. While statistical tests cannot definitively guarantee an RNG’s output quality, they are a powerful verification tool and the only universally applicable testing method. In this work, we design, implement, and present various post-processing methods, using randomness extractors, to improve the RNG output quality and compare them through statistical testing. We begin by performing intensive tests on three RNGs—the 32-bit linear feedback shift register (LFSR), Intel’s ‘RDSEED,’ and IDQuantique’s ‘Quantis’—and compare their performance. Next, we apply the different post-processing methods to each RNG and conduct further intensive testing on the processed output. To facilitate this, we introduce a comprehensive statistical testing environment, based on existing test suites, that can be parametrised for lightweight (fast) to intensive testing. Full article

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

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14 pages, 292 KiB

Open AccessArticle

Effects of the Quantum Vacuum at a Cosmic Scale and of Dark Energy

by Emilio Santos

Entropy 2024, 26(12), 1042; https://doi.org/10.3390/e26121042 - 30 Nov 2024

Viewed by 970

Abstract

The Einstein equation in a semiclassical approximation is applied to a spherical region of the universe, with the stress-energy tensor consisting of the mass density and pressure of the

Λ C D M

cosmological model plus an additional contribution due to the quantum [...] Read more.

The Einstein equation in a semiclassical approximation is applied to a spherical region of the universe, with the stress-energy tensor consisting of the mass density and pressure of the

Λ C D M

cosmological model plus an additional contribution due to the quantum vacuum. Expanding the equation in powers of Newton constant G, the vacuum contributes to second order. The result is that at least a part of the acceleration in the expansion of the universe may be due to the quantum vacuum fluctuations. Full article

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

12 pages, 359 KiB

Open AccessArticle

Statistical Properties of Superpositions of Coherent Phase States with Opposite Arguments

by Miguel Citeli de Freitas and Viktor V. Dodonov

Entropy 2024, 26(11), 977; https://doi.org/10.3390/e26110977 - 15 Nov 2024

Viewed by 740

Abstract

We calculate the second-order moments, the Robertson–Schrödinger uncertainty product, and the Mandel factor for various superpositions of coherent phase states with opposite arguments, comparing the results with similar superpositions of the usual (Klauder–Glauber–Sudarshan) coherent states. We discover that the coordinate variance in the analog of even coherent states can show the most strong squeezing effect, close to the maximal possible squeezing for the given mean photon number. On the other hand, the Robertson–Schrödinger (RS) uncertainty product in superpositions of coherent phase states increases much slower (as function of the mean photon number) than in superpositions of the usual coherent states. A nontrivial behavior of the Mandel factor for small mean photon numbers is discovered in superpositions with unequal weights of two components. An exceptional nature of the even and odd superpositions is demonstrated. Full article

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

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14 pages, 321 KiB

Open AccessArticle

On the Negative Result Experiments in Quantum Mechanics

by Kenichi Konishi

Entropy 2024, 26(11), 958; https://doi.org/10.3390/e26110958 - 7 Nov 2024

Cited by 2 | Viewed by 923

Abstract

We comment on the so-called negative result experiments (also known as null measurements, interaction-free measurements, and so on) in quantum mechanics (QM), in the light of the new general understanding of the quantum-measurement processes, proposed recently. All experiments of this kind (null measurements) can be understood as improper measurements with an intentionally biased detector set up, which introduces exclusion or selection of certain events. The prediction on the state of a microscopic system under study based on a null measurement is sometimes dramatically described as “wave-function collapse without any microsystem-detector interactions”. Though certainly correct, such a prediction is just a consequence of the standard QM laws, not different from the situation in the so-called state-preparation procedure. Another closely related concept is the (first-class or) repeatable measurements. The verification of the prediction made by a null measurement requires eventually a standard unbiased measurement involving the microsystem-macroscopic detector interactions, which are nonadiabatic, irreversible processes of signal amplification. Full article

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

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28 pages, 347 KiB

Open AccessArticle

Conditional Values in Quantum Mechanics

by Leon Cohen

Entropy 2024, 26(10), 838; https://doi.org/10.3390/e26100838 - 30 Sep 2024

Viewed by 1008

Abstract

We consider the local value of an operator for a given position or momentum and, more generally on the value of another arbitrary observable. We develop a general approach that is based on breaking up

A ψ (x)

as [...] Read more.

A ψ (x)

\frac{A ψ (x)}{ψ (x)} = {(\frac{A ψ (x)}{ψ (x)})}_{R} + i {(\frac{A ψ (x)}{ψ (x)})}_{I}

where

A

is the operator whose local value we seek and

ψ (x)

is the position wave function. We show that the real part is related to the conditional value for a given position and the imaginary part is related to the standard deviation of the conditional value. We show that the uncertainty of an operator can be expressed in two parts that depend on the real and imaginary parts. In the case of the position representation, the expression for the uncertainty of an operator shows that there are two fundamental contributions, one due to the amplitude of the wave function and the other due to the phase. We obtain the equation of motion for the conditional values, and in particular, we generalize the Ehrenfest theorem by deriving a local version of the theorem. We give a number of examples, including the local value of momentum, kinetic energy, and Hamiltonian. We also discuss other approaches for obtaining a conditional value in quantum mechanics including using quasi-probability distributions and the characteristic function approach, among others. Full article

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

23 pages, 388 KiB

Open AccessArticle

Statistical Signatures of Quantum Contextuality

by Holger F. Hofmann

Entropy 2024, 26(9), 725; https://doi.org/10.3390/e26090725 - 26 Aug 2024

Cited by 1 | Viewed by 852

Abstract

Quantum contextuality describes situations where the statistics observed in different measurement contexts cannot be explained by a measurement of the independent reality of the system. The most simple case is observed in a three-dimensional Hilbert space, with five different measurement contexts related to each other by shared measurement outcomes. The quantum formalism defines the relations between these contexts in terms of well-defined relations between operators, and these relations can be used to reconstruct an unknown quantum state from a finite set of measurement results. Here, I introduce a reconstruction method based on the relations between the five measurement contexts that can violate the bounds of non-contextual statistics. A complete description of an arbitrary quantum state requires only five of the eight elements of a Kirkwood–Dirac quasiprobability, but only an overcomplete set of eleven elements provides an unbiased description of all five contexts. A set of five fundamental relations between the eleven elements reveals a deterministic structure that links the five contexts. As illustrated by a number of examples, these relations provide a consistent description of contextual realities for the measurement outcomes of all five contexts. Full article

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

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19 pages, 731 KiB

Open AccessArticle

Correlations in the EPR State Observables

by Daniel F. Orsini, Luna R. N. Oliveira and Marcos G. E. da Luz

Entropy 2024, 26(6), 476; https://doi.org/10.3390/e26060476 - 30 May 2024

Viewed by 1284

Abstract

The identification and physical interpretation of arbitrary quantum correlations are not always effortless. Two features that can significantly influence the dispersion of the joint observable outcomes in a quantum bipartite system composed of systems I and II are: (a) All possible pairs of observables describing the composite are equally probable upon measurement, and (b) The absence of concurrence (positive reinforcement) between any of the observables within a particular system; implying that their associated operators do not commute. The so-called EPR states are known to observe (a). Here, we demonstrate in very general (but straightforward) terms that they also satisfy condition (b), a relevant technical fact often overlooked. As an illustration, we work out in detail the three-level systems, i.e., qutrits. Furthermore, given the special characteristics of EPR states (such as maximal entanglement, among others), one might intuitively expect the CHSH correlation, computed exclusively for the observables of qubit EPR states, to yield values greater than two, thereby violating Bell’s inequality. We show such a prediction does not hold true. In fact, the combined properties of (a) and (b) lead to a more limited range of values for the CHSH measure, not surpassing the nonlocality threshold of two. The present constitutes an instructive example of the subtleties of quantum correlations. Full article

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

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Review

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21 pages, 384 KiB

Open AccessReview

The Group-Algebraic Formalism of Quantum Probability and Its Applications in Quantum Statistical Mechanics

by Yan Gu and Jiao Wang

Entropy 2025, 27(1), 59; https://doi.org/10.3390/e27010059 - 10 Jan 2025

Viewed by 773

Abstract

We show that the theory of quantum statistical mechanics is a special model in the framework of the quantum probability theory developed by mathematicians, by extending the characteristic function in the classical probability theory to the quantum probability theory. As dynamical variables of a quantum system must respect certain commutation relations, we take the group generated by a Lie algebra constructed with these commutation relations as the bridge, so that the classical characteristic function defined in a Euclidean space is transformed to a normalized, non-negative definite function defined in this group. Indeed, on the quantum side, this group-theoretical characteristic function is equivalent to the density matrix; hence, it can be adopted to represent the state of a quantum ensemble. It is also found that this new representation may have significant advantages in applications. As two examples, we show its effectiveness and convenience in solving the quantum-optical master equation for a harmonic oscillator coupled with its thermal environment, and in simulating the quantum cat map, a paradigmatic model for quantum chaos. Other related issues are reviewed and discussed as well. Full article

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

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