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Quantum Probability and Randomness III

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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: closed (31 December 2021) | Viewed by 30700

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Special Issue Editors

Prof. Dr. Andrei Khrennikov

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Guest Editor

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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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 third Special Issue devoted to the theme: “Quantum Probability and Randomness”; for the first two issues, see

https://www.mdpi.com/si/entropy/Probability_Randomness

https://www.mdpi.com/si/entropy/Probability_Randomness_ii

The previous issues collected a sample of good papers, both theoretical and experiment-related, written by experts in this area, and it attracted a lot of 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 Entropy journal.

The last few years have been characterized by a tremendous development of 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 meaning of its basic tools, starting with the Born rule, are among the topics which will be covered by this issue.

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

The areas covered include:

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 probabilistic modeling of the process of decision making under uncertainty.

Of course, possible topics need not be restricted to the list above; any contribution directed to the improvement of quantum foundations, 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.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Related Special Issues

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Quantum Probability and Randomness II in Entropy (13 articles)

Published Papers (12 papers)

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Research

32 pages, 369 KiB

Open AccessArticle

A Toss without a Coin: Information, Discontinuity, and Mathematics in Quantum Theory

by Arkady Plotnitsky

Entropy 2022, 24(4), 532; https://doi.org/10.3390/e24040532 - 10 Apr 2022

Cited by 4 | Viewed by 1314

Abstract

The article argues that—at least in certain interpretations, such as the one assumed in this article under the heading of “reality without realism”—the quantum-theoretical situation appears as follows: While—in terms of probabilistic predictions—connected to and connecting the information obtained in quantum phenomena, the mathematics of quantum theory (QM or QFT), which is continuous, does not represent and is discontinuous with both the emergence of quantum phenomena and the physics of these phenomena, phenomena that are physically discontinuous with each other as well. These phenomena, and thus this information, are described by classical physics. All actually available information (in the mathematical sense of information theory) is classical: it is composed of units, such as bits, that are—or are contained in—entities described by classical physics. On the other hand, classical physics cannot predict this information when it is created, as manifested in measuring instruments, in quantum experiments, while quantum theory can. In this epistemological sense, this information is quantum. The article designates the discontinuity between quantum theory and the emergence of quantum phenomena the “Heisenberg discontinuity”, because it was introduced by W. Heisenberg along with QM, and the discontinuity between QM or QFT and the classical physics of quantum phenomena, the “Bohr discontinuity”, because it was introduced as part of Bohr’s interpretation of quantum phenomena and QM, under the assumption of Heisenberg discontinuity. Combining both discontinuities precludes QM or QFT from being connected to either physical reality, that ultimately responsible for quantum phenomena or that of these phenomena themselves, other than by means of probabilistic predictions concerning the information, classical in character, contained in quantum phenomena. The nature of quantum information is, in this view, defined by this situation. A major implication, discussed in the Conclusion, is the existence and arguably the necessity of two—classical and quantum—or with relativity, three and possibly more essentially different theories in fundamental physics. Full article

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

9 pages, 299 KiB

Open AccessArticle

Generalized Householder Transformations

by Karl Svozil

Entropy 2022, 24(3), 429; https://doi.org/10.3390/e24030429 - 19 Mar 2022

Cited by 1 | Viewed by 1666

Abstract

The Householder transformation, allowing a rewrite of probabilities into expectations of dichotomic observables, is generalized in terms of its spectral decomposition. The dichotomy is modulated by allowing more than one negative eigenvalue or by abandoning binaries altogether, yielding generalized operator-valued arguments for contextuality. [...] Read more.

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

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20 pages, 420 KiB

Open AccessArticle

Quantifying and Interpreting Connection Strength in Macro- and Microscopic Systems: Lessons from Bell’s Approach

by Christoph Gallus, Pawel Blasiak and Emmanuel M. Pothos

Entropy 2022, 24(3), 364; https://doi.org/10.3390/e24030364 - 03 Mar 2022

Cited by 2 | Viewed by 2306

Abstract

Bell inequalities were created with the goal of improving the understanding of foundational questions in quantum mechanics. To this end, they are typically applied to measurement results generated from entangled systems of particles. They can, however, also be used as a statistical tool for macroscopic systems, where they can describe the connection strength between two components of a system under a causal model. We show that, in principle, data from macroscopic observations analyzed with Bell’ s approach can invalidate certain causal models. To illustrate this use, we describe a macroscopic game setting, without a quantum mechanical measurement process, and analyze it using the framework of Bell experiments. In the macroscopic game, violations of the inequalities can be created by cheating with classically defined strategies. In the physical context, the meaning of violations is less clear and is still vigorously debated. We discuss two measures for optimal strategies to generate a given statistic that violates the inequalities. We show their mathematical equivalence and how they can be computed from CHSH-quantities alone, if non-signaling applies. As a macroscopic example from the financial world, we show how the unfair use of insider knowledge could be picked up using Bell statistics. Finally, in the discussion of realist interpretations of quantum mechanical Bell experiments, cheating strategies are often expressed through the ideas of free choice and locality. In this regard, violations of free choice and locality can be interpreted as two sides of the same coin, which underscores the view that the meaning these terms are given in Bell’s approach should not be confused with their everyday use. In general, we conclude that Bell’s approach also carries lessons for understanding macroscopic systems of which the connectedness conforms to different causal structures. Full article

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

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15 pages, 909 KiB

Open AccessArticle

Quantum Bitcoin Mining

by Robert Benkoczi, Daya Gaur, Naya Nagy, Marius Nagy and Shahadat Hossain

Entropy 2022, 24(3), 323; https://doi.org/10.3390/e24030323 - 24 Feb 2022

Cited by 4 | Viewed by 7371

Abstract

This paper studies the effect of quantum computers on Bitcoin mining. The shift in computational paradigm towards quantum computation allows the entire search space of the golden nonce to be queried at once by exploiting quantum superpositions and entanglement. Using Grover’s algorithm, a [...] Read more.

O (\sqrt{2^{256} / t})

, where t is the target value for the nonce. This is better using a square root over the classical search algorithm that requires

O (2^{256} / t)

tries. If sufficiently large quantum computers are available for the public, mining activity in the classical sense becomes obsolete, as quantum computers always win. Without considering quantum noise, the size of the quantum computer needs to be

\approx

10^{4}

qubits. Full article

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

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13 pages, 368 KiB

Open AccessArticle

Plasma-like Description for Elementary and Composite Quantum Particles

by Andrey Akhmeteli

Entropy 2022, 24(2), 261; https://doi.org/10.3390/e24020261 - 10 Feb 2022

Cited by 1 | Viewed by 2010

Abstract

Schrödinger noticed in 1952 that a scalar complex wave function can be made real by a gauge transformation. The author showed recently that one real function is also enough to describe matter in the Dirac equation in an arbitrary electromagnetic or Yang–Mills field. This suggests some “symmetry” between positive and negative frequencies and, therefore, particles and antiparticles, so the author previously considered a description of one-particle wave functions as plasma-like collections of a large number of particles and antiparticles. The description has some similarities with Bohmian mechanics. This work offers a criterion for approximation of continuous charge density distributions by discrete ones with quantized charge based on the equality of partial Fourier sums, and an example of such approximation is computed using the homotopy continuation method. An example mathematical model of the description is proposed. The description is also extended to composite particles, such as nucleons or large molecules, regarded as collections including a composite particle and a large number of pairs of elementary particles and antiparticles. While it is not clear if this is a correct description of the reality, it can become a basis of an interesting model or useful picture of quantum mechanics. Full article

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

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20 pages, 337 KiB

Open AccessArticle

The Classical-Quantum Dichotomy from the Perspective of the Process Algebra

by William Sulis

Entropy 2022, 24(2), 184; https://doi.org/10.3390/e24020184 - 26 Jan 2022

Cited by 2 | Viewed by 1750

Abstract

The classical-quantum dichotomy is analyzed from the perspective of the Process Algebra approach, which views fundamental phenomena through the lens of complex systems theory and Whitehead’s process theory. Broadly, the dichotomy can be framed in terms of differences in ontology (phenomena and their behavior) and differences in epistemology (theoretical languages used in their description). The Process Algebra posits a reality, generated by processes, whose fundamental characteristics include becoming, generativity, transience, locality, and contextuality. From this perspective, the classical-quantum dichotomy appears to be a false dichotomy—it arises because of stereotyped, strawman-like depictions of what it means to be classical or quantum. A more careful examination reveals that reality is unitary, that whether a system behaves in a quantum or classical manner depends upon its particularities, in particular, whether it is complex or not, and how information flows govern its dynamics. Full article

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

12 pages, 2438 KiB

Open AccessArticle

Measurement of the Temperature Using the Tomographic Representation of Thermal States for Quadratic Hamiltonians

by Julio A. López-Saldívar, Margarita A. Man’ko and Vladimir I. Man’ko

Entropy 2021, 23(11), 1445; https://doi.org/10.3390/e23111445 - 31 Oct 2021

Cited by 8 | Viewed by 1383

Abstract

The Wigner and tomographic representations of thermal Gibbs states for one- and two-mode quantum systems described by a quadratic Hamiltonian are obtained. This is done by using the covariance matrix of the mentioned states. The area of the Wigner function and the width of the tomogram of quantum systems are proposed to define a temperature scale for this type of states. This proposal is then confirmed for the general one-dimensional case and for a system of two coupled harmonic oscillators. The use of these properties as measures for the temperature of quantum systems is mentioned. Full article

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

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21 pages, 1670 KiB

Open AccessArticle

Conceiving Particles as Undulating Granular Systems Allows Fundamentally Realist Interpretation of Quantum Mechanics

by Stéphane Avner

Entropy 2021, 23(10), 1338; https://doi.org/10.3390/e23101338 - 14 Oct 2021

Cited by 3 | Viewed by 2486

Abstract

The strange behavior of subatomic particles is described by quantum theory, whose standard interpretation rejected some fundamental principles of classical physics such as causality, objectivity, locality, realism and determinism. Recently, a granular relativistic electrodynamical model of the electron could capture the measured values of its observables and predict its mass from the stability of its substructure. The model involves numerous subparticles that constitute some tight nucleus and loosely bound envelope allegedly forming real waves. The present study examines whether such a substructure and associated dynamics allow fundamentally realist interpretations of emblematic quantum phenomena, properties and principles, such as wave-particle duality, loss of objectivity, quantization, simultaneous multipath exploration, collapse of wavepacket, measurement problem, and entanglement. Drawing inspiration from non-linear dynamical systems, subparticles would involve realist hidden variables while high-level observables would not generally be determined, as particles would generally be in unstable states before measurements. Quantum mechanics would constitute a high-level probabilistic description emerging from an underlying causal, objective, local, albeit contextual and unpredictable reality. Altogether, by conceiving particles as granular systems composed of numerous extremely sensitive fluctuating subcorpuscles, this study proposes the possible existence of a local fundamentally realist interpretation of quantum mechanics. Full article

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

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15 pages, 506 KiB

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 16 | Viewed by 2293

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 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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13 pages, 282 KiB

Open AccessArticle

Contextuality in Classical Physics and Its Impact on the Foundations of Quantum Mechanics

by Fritiof Wallentin

Entropy 2021, 23(8), 968; https://doi.org/10.3390/e23080968 - 27 Jul 2021

Cited by 2 | Viewed by 1714

Abstract

It is shown that the hallmark quantum phenomenon of contextuality is present in classical statistical mechanics (CSM). It is first shown that the occurrence of contextuality is equivalent to there being observables that can differentiate between pure and mixed states. CSM is formulated in the formalism of quantum mechanics (FQM), a formulation commonly known as the Koopman–von Neumann formulation (KvN). In KvN, one can then show that such a differentiation between mixed and pure states is possible. As contextuality is a probabilistic phenomenon and as it is exhibited in both classical physics and ordinary quantum mechanics (OQM), it is concluded that the foundational issues regarding quantum mechanics are really issues regarding the foundations of probability. Full article

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

17 pages, 348 KiB

Open AccessEditor’s ChoiceArticle

The Violation of Bell-CHSH Inequalities Leads to Different Conclusions Depending on the Description Used

by Aldo F. G. Solis-Labastida, Melina Gastelum and Jorge G. Hirsch

Entropy 2021, 23(7), 872; https://doi.org/10.3390/e23070872 - 08 Jul 2021

Cited by 2 | Viewed by 2362

Abstract

Since the experimental observation of the violation of the Bell-CHSH inequalities, much has been said about the non-local and contextual character of the underlying system. However, the hypothesis from which Bell’s inequalities are derived differ according to the probability space used to write them. The violation of Bell’s inequalities can, alternatively, be explained by assuming that the hidden variables do not exist at all, that they exist but their values cannot be simultaneously assigned, that the values can be assigned but joint probabilities cannot be properly defined, or that averages taken in different contexts cannot be combined. All of the above are valid options, selected by different communities to provide support to their particular research program. Full article

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

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16 pages, 273 KiB

Open AccessArticle

Is the Devil in h?

by Andrei Khrennikov

Entropy 2021, 23(5), 632; https://doi.org/10.3390/e23050632 - 19 May 2021

Cited by 17 | Viewed by 2255

Abstract

This note is a part of my effort to rid quantum mechanics (QM) nonlocality. Quantum nonlocality is a two faced Janus: one face is a genuine quantum mechanical nonlocality (defined by the Lüders’ projection postulate). Another face is the nonlocality of the hidden variables model that was invented by Bell. This paper is devoted the deconstruction of the latter. The main casualty of Bell’s model is that it straightforwardly contradicts Heisenberg’s uncertainty and Bohr’s complementarity principles generally. Thus, we do not criticize the derivation or interpretation of the Bell inequality (as was done by numerous authors). Our critique is directed against the model as such. The original Einstein-Podolsky-Rosen (EPR) argument assumed the Heisenberg’s principle without questioning its validity. Hence, the arguments of EPR and Bell differ crucially, and it is necessary to establish the physical ground of the aforementioned principles. This is the quantum postulate: the existence of an indivisible quantum of action given by the Planck constant. Bell’s approach with hidden variables implicitly implies rejection of the quantum postulate, since the latter is the basis of the reference principles. Full article

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

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