Special Issue "Quantum Probability and Randomness II"

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

Deadline for manuscript submissions: closed (30 April 2020).

Special Issue Editors

Prof. Dr. Andrei Khrennikov
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Guest Editor
International Center Math Modeling: Physics, Engineering, Economics, and Cognitive Science, Linnaeus University, Växjö, Sweden
Interests: Foundations of Probability Theory, Quantization of Systems; Classical Random Field Model of Quantum Mechanics, Quantum-Like Models: Molecular Biology, Cognition, Psychology, Behavioral Economics, Social Science
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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 and Collections 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 papers will be 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 1600 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.

Published Papers (5 papers)

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Research

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Open AccessArticle
Differential Parametric Formalism for the Evolution of Gaussian States: Nonunitary Evolution and Invariant States
Entropy 2020, 22(5), 586; https://doi.org/10.3390/e22050586 - 23 May 2020
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 [...] Read more.
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 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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Open AccessArticle
Specifying the Unitary Evolution of a Qudit for a General Nonstationary Hamiltonian via the Generalized Gell-Mann Representation
Entropy 2020, 22(5), 521; https://doi.org/10.3390/e22050521 - 03 May 2020
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.
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 of a qudit ( d 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)
Open AccessArticle
Higher-Dimensional Quantum Walk in Terms of Quantum Bernoulli Noises
Entropy 2020, 22(5), 504; https://doi.org/10.3390/e22050504 - 28 Apr 2020
Abstract
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 [...] Read more.
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 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)
Open AccessFeature PaperArticle
Two Faced Janus of Quantum Nonlocality
Entropy 2020, 22(3), 303; https://doi.org/10.3390/e22030303 - 06 Mar 2020
Cited by 4
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 [...] Read more.
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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Open AccessReview
What Is so Special about Quantum Clicks?
Entropy 2020, 22(6), 602; https://doi.org/10.3390/e22060602 - 28 May 2020
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) [...] Read more.
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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