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Quantum Probability and Randomness II

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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 (18 December 2020) | Viewed by 39601

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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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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 second Special Issue devoted to the theme: “Quantum Probability and Randomness”; for the first issue, see https://www.mdpi.com/journal/entropy/special_issues/Probability_Randomness

The first issue 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). That is why we have decided to proceed once again with this hot topic.

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. In spite of 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 at the improvement of quantum foundations, development of quantum probability and randomness is welcome.

Prof. 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

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

Published Papers (13 papers)

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Research

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20 pages, 3411 KiB

Open AccessArticle

Extending Quantum Probability from Real Axis to Complex Plane

by Ciann-Dong Yang and Shiang-Yi Han

Entropy 2021, 23(2), 210; https://doi.org/10.3390/e23020210 - 08 Feb 2021

Cited by 7 | Viewed by 3507

Abstract

Probability is an important question in the ontological interpretation of quantum mechanics. It has been discussed in some trajectory interpretations such as Bohmian mechanics and stochastic mechanics. New questions arise when the probability domain extends to the complex space, including the generation of complex trajectory, the definition of the complex probability, and the relation of the complex probability to the quantum probability. The complex treatment proposed in this article applies the optimal quantum guidance law to derive the stochastic differential equation governing a particle’s random motion in the complex plane. The probability distribution

ρ_{c} (t, x, y)

of the particle’s position over the complex plane

z = x + i y

is formed by an ensemble of the complex quantum random trajectories, which are solved from the complex stochastic differential equation. Meanwhile, the probability distribution

ρ_{c} (t, x, y)

is verified by the solution of the complex Fokker–Planck equation. It is shown that quantum probability

{| Ψ |}^{2}

and classical probability can be integrated under the framework of complex probability

ρ_{c} (t, x, y)

, such that they can both be derived from

ρ_{c} (t, x, y)

by different statistical ways of collecting spatial points. Full article

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

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20 pages, 360 KiB

Open AccessArticle

Kolmogorovian versus Non-Kolmogorovian Probabilities in Contextual Theories

by Claudio Garola

Entropy 2021, 23(1), 121; https://doi.org/10.3390/e23010121 - 18 Jan 2021

Cited by 1 | Viewed by 1580

Abstract

Most scholars maintain that quantum mechanics (QM) is a contextual theory and that quantum probability does not allow for an epistemic (ignorance) interpretation. By inquiring possible connections between contextuality and non-classical probabilities we show that a class

T^{μ MP}

of theories can [...] Read more.

T^{μ MP}

of theories can be selected in which probabilities are introduced as classical averages of Kolmogorovian probabilities over sets of (microscopic) contexts, which endows them with an epistemic interpretation. The conditions characterizing

T^{μ MP}

are compatible with classical mechanics (CM), statistical mechanics (SM), and QM, hence we assume that these theories belong to

T^{μ MP}

. In the case of CM and SM, this assumption is irrelevant, as all of the notions introduced in them as members of

T^{μ MP}

reduce to standard notions. In the case of QM, it leads to interpret quantum probability as a derived notion in a Kolmogorovian framework, explains why it is non-Kolmogorovian, and provides it with an epistemic interpretation. These results were anticipated in a previous paper, but they are obtained here in a general framework without referring to individual objects, which shows that they hold, even if only a minimal (statistical) interpretation of QM is adopted in order to avoid the problems following from the standard quantum theory of measurement. Full article

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

26 pages, 3224 KiB

Open AccessArticle

Emerging Complexity in Distributed Intelligent Systems

by Valentina Guleva, Egor Shikov, Klavdiya Bochenina, Sergey Kovalchuk, Alexander Alodjants and Alexander Boukhanovsky

Entropy 2020, 22(12), 1437; https://doi.org/10.3390/e22121437 - 19 Dec 2020

Cited by 13 | Viewed by 3347

Abstract

Distributed intelligent systems (DIS) appear where natural intelligence agents (humans) and artificial intelligence agents (algorithms) interact, exchanging data and decisions and learning how to evolve toward a better quality of solutions. The networked dynamics of distributed natural and artificial intelligence agents leads to emerging complexity different from the ones observed before. In this study, we review and systematize different approaches in the distributed intelligence field, including the quantum domain. A definition and mathematical model of DIS (as a new class of systems) and its components, including a general model of DIS dynamics, are introduced. In particular, the suggested new model of DIS contains both natural (humans) and artificial (computer programs, chatbots, etc.) intelligence agents, which take into account their interactions and communications. We present the case study of domain-oriented DIS based on different agents’ classes and show that DIS dynamics shows complexity effects observed in other well-studied complex systems. We examine our model by means of the platform of personal self-adaptive educational assistants (avatars), especially designed in our University. Avatars interact with each other and with their owners. Our experiment allows finding an answer to the vital question: How quickly will DIS adapt to owners’ preferences so that they are satisfied? We introduce and examine in detail learning time as a function of network topology. We have shown that DIS has an intrinsic source of complexity that needs to be addressed while developing predictable and trustworthy systems of natural and artificial intelligence agents. Remarkably, our research and findings promoted the improvement of the educational process at our university in the presence of COVID-19 pandemic conditions. Full article

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

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15 pages, 1253 KiB

Open AccessArticle

From Quantum Probabilities to Quantum Amplitudes

by Sofia Martínez-Garaot, Marisa Pons and Dmitri Sokolovski

Entropy 2020, 22(12), 1389; https://doi.org/10.3390/e22121389 - 08 Dec 2020

Cited by 2 | Viewed by 2015

Abstract

The task of reconstructing the system’s state from the measurements results, known as the Pauli problem, usually requires repetition of two successive steps. Preparation in an initial state to be determined is followed by an accurate measurement of one of the several chosen operators in order to provide the necessary “Pauli data”. We consider a similar yet more general problem of recovering Feynman’s transition (path) amplitudes from the results of at least three consecutive measurements. The three-step histories of a pre- and post-selected quantum system are subjected to a type of interference not available to their two-step counterparts. We show that this interference can be exploited, and if the intermediate measurement is “fuzzy”, the path amplitudes can be successfully recovered. The simplest case of a two-level system is analysed in detail. The “weak measurement” limit and the usefulness of the path amplitudes are also discussed. Full article

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

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6 pages, 217 KiB

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 3 | Viewed by 3287

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 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, 1378 KiB

Open AccessArticle

Quantum Measurements with, and Yet without an Observer

by Dmitri Sokolovski

Entropy 2020, 22(10), 1185; https://doi.org/10.3390/e22101185 - 21 Oct 2020

Cited by 6 | Viewed by 2487

Abstract

It is argued that Feynman’s rules for evaluating probabilities, combined with von Neumann’s principle of psycho-physical parallelism, help avoid inconsistencies, often associated with quantum theory. The former allows one to assign probabilities to entire sequences of hypothetical Observers’ experiences, without mentioning the problem of wave function collapse. The latter limits the Observer’s (e.g., Wigner’s friend’s) participation in a measurement to the changes produced in material objects, thus leaving his/her consciousness outside the picture. Full article

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

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30 pages, 363 KiB

Open AccessArticle

Reality, Indeterminacy, Probability, and Information in Quantum Theory

by Arkady Plotnitsky

Entropy 2020, 22(7), 747; https://doi.org/10.3390/e22070747 - 07 Jul 2020

Cited by 6 | Viewed by 2376

Abstract

Following the view of several leading quantum-information theorists, this paper argues that quantum phenomena, including those exhibiting quantum correlations (one of their most enigmatic features), and quantum mechanics may be best understood in quantum-informational terms. It also argues that this understanding is implicit already in the work of some among the founding figures of quantum mechanics, in particular W. Heisenberg and N. Bohr, half a century before quantum information theory emerged and confirmed, and gave a deeper meaning to, to their insights. These insights, I further argue, still help this understanding, which is the main reason for considering them here. My argument is grounded in a particular interpretation of quantum phenomena and quantum mechanics, in part arising from these insights as well. This interpretation is based on the concept of reality without realism, RWR (which places the reality considered beyond representation or even conception), introduced by this author previously, in turn, following Heisenberg and Bohr, and in response to quantum information theory. Full article

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

18 pages, 946 KiB

Open AccessArticle

A Gaussian-Distributed Quantum Random Number Generator Using Vacuum Shot Noise

by Min Huang, Ziyang Chen, Yichen Zhang and Hong Guo

Entropy 2020, 22(6), 618; https://doi.org/10.3390/e22060618 - 02 Jun 2020

Cited by 10 | Viewed by 4372

Abstract

Among all the methods of extracting randomness, quantum random number generators are promising for their genuine randomness. However, existing quantum random number generator schemes aim at generating sequences with a uniform distribution, which may not meet the requirements of specific applications such as a continuous-variable quantum key distribution system. In this paper, we demonstrate a practical quantum random number generation scheme directly generating Gaussian distributed random sequences based on measuring vacuum shot noise. Particularly, the impact of the sampling device in the practical system is analyzed. Furthermore, a related post-processing method, which maintains the fine distribution and autocorrelation properties of raw data, is exploited to extend the precision of generated Gaussian distributed random numbers to over 20 bits, making the sequences possible to be utilized by the following system with requiring high precision numbers. Finally, the results of normality and randomness tests prove that the generated sequences satisfy Gaussian distribution and can pass the randomness testing well. Full article

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

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25 pages, 494 KiB

Open AccessEditor’s ChoiceArticle

Differential Parametric Formalism for the Evolution of Gaussian States: Nonunitary Evolution and Invariant States

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

Entropy 2020, 22(5), 586; https://doi.org/10.3390/e22050586 - 23 May 2020

Cited by 12 | Viewed by 2761

Abstract

In the differential approach elaborated, we study the evolution of the parameters of Gaussian, mixed, continuous variable density matrices, whose dynamics are given by Hermitian Hamiltonians expressed as quadratic forms of the position and momentum operators or quadrature components. Specifically, we obtain in generic form the differential equations for the covariance matrix, the mean values, and the density matrix parameters of a multipartite Gaussian state, unitarily evolving according to a Hamiltonian

\hat{H}

. We also present the corresponding differential equations, which describe the nonunitary evolution of the subsystems. The resulting nonlinear equations are used to solve the dynamics of the system instead of the Schrödinger equation. The formalism elaborated allows us to define new specific invariant and quasi-invariant states, as well as states with invariant covariance matrices, i.e., states were only the mean values evolve according to the classical Hamilton equations. By using density matrices in the position and in the tomographic-probability representations, we study examples of these properties. As examples, we present novel invariant states for the two-mode frequency converter and quasi-invariant states for the bipartite parametric amplifier. Full article

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

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21 pages, 376 KiB

Open AccessArticle

Specifying the Unitary Evolution of a Qudit for a General Nonstationary Hamiltonian via the Generalized Gell-Mann Representation

by Elena R. Loubenets and Christian Käding

Entropy 2020, 22(5), 521; https://doi.org/10.3390/e22050521 - 03 May 2020

Cited by 5 | Viewed by 2573

Abstract

Optimal realizations of quantum technology tasks lead to the necessity of a detailed analytical study of the behavior of a d-level quantum system (qudit) under a time-dependent Hamiltonian. In the present article, we introduce a new general formalism describing the unitary evolution [...] Read more.

(d \geq 2)

in terms of the Bloch-like vector space and specify how, in a general case, this formalism is related to finding time-dependent parameters in the exponential representation of the evolution operator under an arbitrary time-dependent Hamiltonian. Applying this new general formalism to a qubit case

(d = 2)

, we specify the unitary evolution of a qubit via the evolution of a unit vector in

R^{4}

, and this allows us to derive the precise analytical expression of the qubit unitary evolution operator for a wide class of nonstationary Hamiltonians. This new analytical expression includes the qubit solutions known in the literature only as particular cases. Full article

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

17 pages, 302 KiB

Open AccessArticle

Higher-Dimensional Quantum Walk in Terms of Quantum Bernoulli Noises

by Ce Wang and Caishi Wang

Entropy 2020, 22(5), 504; https://doi.org/10.3390/e22050504 - 28 Apr 2020

Cited by 2 | Viewed by 1975

Abstract

As a discrete-time quantum walk model on the one-dimensional integer lattice

Z

As a discrete-time quantum walk model on the one-dimensional integer lattice

Z

, the quantum walk recently constructed by Wang and Ye [Caishi Wang and Xiaojuan Ye, Quantum walk in terms of quantum Bernoulli noises, Quantum Information Processing 15 (2016), 1897–1908] exhibits quite different features. In this paper, we extend this walk to a higher dimensional case. More precisely, for a general positive integer

d \geq 2

, by using quantum Bernoulli noises we introduce a model of discrete-time quantum walk on the d-dimensional integer lattice

Z^{d}

, which we call the d-dimensional QBN walk. The d-dimensional QBN walk shares the same coin space with the quantum walk constructed by Wang and Ye, although it is a higher dimensional extension of the latter. Moreover we prove that, for a range of choices of its initial state, the d-dimensional QBN walk has a limit probability distribution of d-dimensional standard Gauss type, which is in sharp contrast with the case of the usual higher dimensional quantum walks. Some other results are also obtained. Full article

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

17 pages, 538 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Two Faced Janus of Quantum Nonlocality

by Andrei Khrennikov

Entropy 2020, 22(3), 303; https://doi.org/10.3390/e22030303 - 06 Mar 2020

Cited by 31 | Viewed by 3320

Abstract

This paper is a new step towards understanding why “quantum nonlocality” is a misleading concept. Metaphorically speaking, “quantum nonlocality” is Janus faced. One face is an apparent nonlocality of the Lüders projection and another face is Bell nonlocality (a wrong conclusion that the violation of Bell type inequalities implies the existence of mysterious instantaneous influences between distant physical systems). According to the Lüders projection postulate, a quantum measurement performed on one of the two distant entangled physical systems modifies their compound quantum state instantaneously. Therefore, if the quantum state is considered to be an attribute of the individual physical system and if one assumes that experimental outcomes are produced in a perfectly random way, one quickly arrives at the contradiction. It is a primary source of speculations about a spooky action at a distance. Bell nonlocality as defined above was explained and rejected by several authors; thus, we concentrate in this paper on the apparent nonlocality of the Lüders projection. As already pointed out by Einstein, the quantum paradoxes disappear if one adopts the purely statistical interpretation of quantum mechanics (QM). In the statistical interpretation of QM, if probabilities are considered to be objective properties of random experiments we show that the Lüders projection corresponds to the passage from joint probabilities describing all set of data to some marginal conditional probabilities describing some particular subsets of data. If one adopts a subjective interpretation of probabilities, such as QBism, then the Lüders projection corresponds to standard Bayesian updating of the probabilities. The latter represents degrees of beliefs of local agents about outcomes of individual measurements which are placed or which will be placed at distant locations. In both approaches, probability-transformation does not happen in the physical space, but only in the information space. Thus, all speculations about spooky interactions or spooky predictions at a distance are simply misleading. Coming back to Bell nonlocality, we recall that in a recent paper we demonstrated, using exclusively the quantum formalism, that CHSH inequalities may be violated for some quantum states only because of the incompatibility of quantum observables and Bohr’s complementarity. Finally, we explain that our criticism of quantum nonlocality is in the spirit of Hertz-Boltzmann methodology of scientific theories. Full article

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

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Review

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44 pages, 628 KiB

Open AccessEditor’s ChoiceReview

What Is So Special about Quantum Clicks?

by Karl Svozil

Entropy 2020, 22(6), 602; https://doi.org/10.3390/e22060602 - 28 May 2020

Cited by 24 | Viewed by 4150

Abstract

This is an elaboration of the “extra” advantage of the performance of quantized physical systems over classical ones, both in terms of single outcomes as well as probabilistic predictions. From a formal point of view, it is based on entities related to (dual) vectors in (dual) Hilbert spaces, as compared to the Boolean algebra of subsets of a set and the additive measures they support. Full article

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

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