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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 August 2018) | Viewed by 57824

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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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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,

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 to the improvement of quantum foundations, development of quantum probability and randomness is welcome.

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

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Related Special Issues

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)
Quantum Probability and Randomness V in Entropy (5 articles)

Published Papers (14 papers)

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Editorial

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5 pages, 217 KiB

Open AccessEditorial

Quantum Probability and Randomness

by Andrei Khrennikov and Karl Svozil

Entropy 2019, 21(1), 35; https://doi.org/10.3390/e21010035 - 7 Jan 2019

Cited by 6 | Viewed by 4125

Abstract

The recent quantum information revolution has stimulated interest in the quantum foundations by perceiving and re-evaluating the theory from a novel information-theoretical viewpoint [...] Full article

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

Research

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12 pages, 571 KiB

Open AccessArticle

Vector Generation of Quantum Contextual Sets in Even Dimensional Hilbert Spaces

by Mladen Pavičić and Norman D. Megill

Entropy 2018, 20(12), 928; https://doi.org/10.3390/e20120928 - 5 Dec 2018

Cited by 14 | Viewed by 3299

Abstract

Recently, quantum contextuality has been proved to be the source of quantum computation’s power. That, together with multiple recent contextual experiments, prompts improving the methods of generation of contextual sets and finding their features. The most elaborated contextual sets, which offer blueprints for contextual experiments and computational gates, are the Kochen–Specker (KS) sets. In this paper, we show a method of vector generation that supersedes previous methods. It is implemented by means of algorithms and programs that generate hypergraphs embodying the Kochen–Specker property and that are designed to be carried out on supercomputers. We show that vector component generation of KS hypergraphs exhausts all possible vectors that can be constructed from chosen vector components, in contrast to previous studies that used incomplete lists of vectors and therefore missed a majority of hypergraphs. Consequently, this unified method is far more efficient for generations of KS sets and their implementation in quantum computation and quantum communication. Several new KS classes and their features have been found and are elaborated on in the paper. Greechie diagrams are discussed. Full article

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

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13 pages, 615 KiB

Open AccessArticle

Advanced Statistical Testing of Quantum Random Number Generators

by Aldo C. Martínez, Aldo Solis, Rafael Díaz Hernández Rojas, Alfred B. U'Ren, Jorge G. Hirsch and Isaac Pérez Castillo

Entropy 2018, 20(11), 886; https://doi.org/10.3390/e20110886 - 17 Nov 2018

Cited by 14 | Viewed by 4646

Abstract

Pseudo-random number generators are widely used in many branches of science, mainly in applications related to Monte Carlo methods, although they are deterministic in design and, therefore, unsuitable for tackling fundamental problems in security and cryptography. The natural laws of the microscopic realm provide a fairly simple method to generate non-deterministic sequences of random numbers, based on measurements of quantum states. In practice, however, the experimental devices on which quantum random number generators are based are often unable to pass some tests of randomness. In this review, we briefly discuss two such tests, point out the challenges that we have encountered in experimental implementations and finally present a fairly simple method that successfully generates non-deterministic maximally random sequences. Full article

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

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22 pages, 861 KiB

Open AccessArticle

Probabilities and Epistemic Operations in the Logics of Quantum Computation

by Maria Luisa Dalla Chiara, Hector Freytes, Roberto Giuntini, Roberto Leporini and Giuseppe Sergioli

Entropy 2018, 20(11), 837; https://doi.org/10.3390/e20110837 - 31 Oct 2018

Cited by 3 | Viewed by 2431

Abstract

Quantum computation theory has inspired new forms of quantum logic, called quantum computational logics, where formulas are supposed to denote pieces of quantum information, while logical connectives are interpreted as special examples of quantum logical gates. The most natural semantics for these logics is a form of holistic semantics, where meanings behave in a contextual way. In this framework, the concept of quantum probability can assume different forms. We distinguish an absolute concept of probability, based on the idea of quantum truth, from a relative concept of probability (a form of transition-probability, connected with the notion of fidelity between quantum states). Quantum information has brought about some intriguing epistemic situations. A typical example is represented by teleportation-experiments. In some previous works we have studied a quantum version of the epistemic operations “to know”, “to believe”, “to understand”. In this article, we investigate another epistemic operation (which is informally used in a number of interesting quantum situations): the operation “being probabilistically informed”. Full article

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

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13 pages, 3461 KiB

Open AccessArticle

Enhancing Extractable Quantum Entropy in Vacuum-Based Quantum Random Number Generator

by Xiaomin Guo, Ripeng Liu, Pu Li, Chen Cheng, Mingchuan Wu and Yanqiang Guo

Entropy 2018, 20(11), 819; https://doi.org/10.3390/e20110819 - 24 Oct 2018

Cited by 22 | Viewed by 4755

Abstract

Information-theoretically provable unique true random numbers, which cannot be correlated or controlled by an attacker, can be generated based on quantum measurement of vacuum state and universal-hashing randomness extraction. Quantum entropy in the measurements decides the quality and security of the random number generator (RNG). At the same time, it directly determines the extraction ratio of true randomness from the raw data, in other words, it obviously affects quantum random bits generating rate. In this work, we commit to enhancing quantum entropy content in the vacuum noise based quantum RNG. We have taken into account main factors in this proposal to establish the theoretical model of quantum entropy content, including the effects of classical noise, the optimum dynamical analog-digital convertor (ADC) range, the local gain and the electronic gain of the homodyne system. We demonstrate that by amplifying the vacuum quantum noise, abundant quantum entropy is extractable in the step of post-processing even classical noise excursion, which may be deliberately induced by an eavesdropper, is large. Based on the discussion and the fact that the bandwidth of quantum vacuum noise is infinite, we propose large dynamical range and moderate TIA gain to pursue higher local oscillator (LO) amplification of vacuum quadrature and broader detection bandwidth in homodyne system. High true randomness extraction ratio together with high sampling rate is attainable. Experimentally, an extraction ratio of true randomness of 85.3% is achieved by finite enhancement of the laser power of the LO when classical noise excursions of the raw data is obvious. Full article

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

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19 pages, 8857 KiB

Open AccessArticle

Entanglement of Three-Qubit Random Pure States

by Marco Enríquez, Francisco Delgado and Karol Życzkowski

Entropy 2018, 20(10), 745; https://doi.org/10.3390/e20100745 - 29 Sep 2018

Cited by 31 | Viewed by 5215

Abstract

We study entanglement properties of generic three-qubit pure states. First, we obtain the distributions of both the coefficients and the only phase in the five-term decomposition of Acín et al. for an ensemble of random pure states generated by the Haar measure on [...] Read more.

U (8)

. Furthermore, we analyze the probability distributions of two sets of polynomial invariants. One of these sets allows us to classify three-qubit pure states into four classes. Entanglement in each class is characterized using the minimal Rényi-Ingarden-Urbanik entropy. Besides, the fidelity of a three-qubit random state with the closest state in each entanglement class is investigated. We also present a characterization of these classes in terms of the corresponding entanglement polytope. The entanglement classes related to stochastic local operations and classical communication (SLOCC) are analyzed as well from this geometric perspective. The numerical findings suggest some conjectures relating some of those invariants with entanglement properties to be ground in future analytical work. Full article

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

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17 pages, 843 KiB

Open AccessArticle

New Entropic Inequalities and Hidden Correlations in Quantum Suprematism Picture of Qudit States

by Margarita A. Man’ko and Vladimir I. Man’ko

Entropy 2018, 20(9), 692; https://doi.org/10.3390/e20090692 - 11 Sep 2018

Cited by 25 | Viewed by 3316

Abstract

We study an analog of Bayes’ formula and the nonnegativity property of mutual information for systems with one random variable. For single-qudit states, we present new entropic inequalities in the form of the subadditivity and condition corresponding to hidden correlations in quantum systems. We present qubit states in the quantum suprematism picture, where these states are identified with three probability distributions, describing the states of three classical coins, and illustrate the states by Triada of Malevich’s squares with areas satisfying the quantum constraints. We consider arbitrary quantum states belonging to N-dimensional Hilbert space as

(N^{2} - 1)

fair probability distributions describing the states of

(N^{2} - 1)

classical coins. We illustrate the geometrical properties of the qudit states by a set of Triadas of Malevich’s squares. We obtain new entropic inequalities for matrix elements of an arbitrary density N×N-matrix of qudit systems using the constructed maps of the density matrix on a set of the probability distributions. In addition, to construct the bijective map of the qudit state onto the set of probabilities describing the positions of classical coins, we show that there exists a bijective map of any quantum observable onto the set of dihotomic classical random variables with statistics determined by the above classical probabilities. Finally, we discuss the physical meaning and possibility to check derived inequalities in the experiments with superconducting circuits based on Josephson junction devices. Full article

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

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46 pages, 425 KiB

Open AccessArticle

“The Heisenberg Method”: Geometry, Algebra, and Probability in Quantum Theory

by Arkady Plotnitsky

Entropy 2018, 20(9), 656; https://doi.org/10.3390/e20090656 - 30 Aug 2018

Cited by 9 | Viewed by 4856

Abstract

The article reconsiders quantum theory in terms of the following principle, which can be symbolically represented as QUANTUMNESS → PROBABILITY → ALGEBRA and will be referred to as the QPA principle. The principle states that the quantumness of physical phenomena, that is, the specific character of physical phenomena known as quantum, implies that our predictions concerning them are irreducibly probabilistic, even in dealing with quantum phenomena resulting from the elementary individual quantum behavior (such as that of elementary particles), which in turn implies that our theories concerning these phenomena are fundamentally algebraic, in contrast to more geometrical classical or relativistic theories, although these theories, too, have an algebraic component to them. It follows that one needs to find an algebraic scheme able make these predictions in a given quantum regime. Heisenberg was first to accomplish this in the case of quantum mechanics, as matrix mechanics, whose matrix character testified to his algebraic method, as Einstein characterized it. The article explores the implications of the Heisenberg method and of the QPA principle for quantum theory, and for the relationships between mathematics and physics there, from a nonrealist or, in terms of this article, “reality-without-realism” or RWR perspective, defining the RWR principle, thus joined to the QPA principle. Full article

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

41 pages, 3102 KiB

Open AccessArticle

SU(2) Decomposition for the Quantum Information Dynamics in 2d-Partite Two-Level Quantum Systems

by Francisco Delgado

Entropy 2018, 20(8), 610; https://doi.org/10.3390/e20080610 - 17 Aug 2018

Cited by 5 | Viewed by 4121

Abstract

The gate array version of quantum computation uses logical gates adopting convenient forms for computational algorithms based on the algorithms classical computation. Two-level quantum systems are the basic elements connecting the binary nature of classical computation with the settlement of quantum processing. Despite this, their design depends on specific quantum systems and the physical interactions involved, thus complicating the dynamics analysis. Predictable and controllable manipulation should be addressed in order to control the quantum states in terms of the physical control parameters. Resources are restricted to limitations imposed by the physical settlement. This work presents a formalism to decompose the quantum information dynamics in

S U (2^{2 d})

for

2 d

-partite two-level systems into

2^{2 d - 1}

S U (2)

quantum subsystems. It generates an easier and more direct physical implementation of quantum processing developments for qubits. Easy and traditional operations proposed by quantum computation are recovered for larger and more complex systems. Alternating the parameters of local and non-local interactions, the procedure states a universal exchange semantics on the basis of generalized Bell states. Although the main procedure could still be settled on other interaction architectures by the proper selection of the basis as natural grammar, the procedure can be understood as a momentary splitting of the

2 d

information channels into

2^{2 d - 1}

pairs of 2 level quantum information subsystems. Additionally, it is a settlement of the quantum information manipulation that is free of the restrictions imposed by the underlying physical system. Thus, the motivation of decomposition is to set control procedures easily in order to generate large entangled states and to design specialized dedicated quantum gates. They are potential applications that properly bypass the general induced superposition generated by physical dynamics. Full article

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

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16 pages, 394 KiB

Open AccessArticle

An Information-Theoretic Perspective on the Quantum Bit Commitment Impossibility Theorem

by Marius Nagy and Naya Nagy

Entropy 2018, 20(3), 193; https://doi.org/10.3390/e20030193 - 13 Mar 2018

Cited by 6 | Viewed by 3890

Abstract

This paper proposes a different approach to pinpoint the causes for which an unconditionally secure quantum bit commitment protocol cannot be realized, beyond the technical details on which the proof of Mayers’ no-go theorem is constructed. We have adopted the tools of quantum entropy analysis to investigate the conditions under which the security properties of quantum bit commitment can be circumvented. Our study has revealed that cheating the binding property requires the quantum system acting as the safe to harbor the same amount of uncertainty with respect to both observers (Alice and Bob) as well as the use of entanglement. Our analysis also suggests that the ability to cheat one of the two fundamental properties of bit commitment by any of the two participants depends on how much information is leaked from one side of the system to the other and how much remains hidden from the other participant. Full article

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

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1393 KiB

Open AccessArticle

Paths of Cultural Systems

by Paul Ballonoff

Entropy 2018, 20(1), 8; https://doi.org/10.3390/e20010008 - 25 Dec 2017

Cited by 2 | Viewed by 3729

Abstract

A theory of cultural structures predicts the objects observed by anthropologists. We here define those which use kinship relationships to define systems. A finite structure we call a partially defined quasigroup (or pdq, as stated by Definition 1 below) on a dictionary (called a natural language) allows prediction of certain anthropological descriptions, using homomorphisms of pdqs onto finite groups. A viable history (defined using pdqs) states how an individual in a population following such history may perform culturally allowed associations, which allows a viable history to continue to survive. The vector states on sets of viable histories identify demographic observables on descent sequences. Paths of vector states on sets of viable histories may determine which histories can exist empirically. Full article

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

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Review

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19 pages, 310 KiB

Open AccessReview

Developments in Quantum Probability and the Copenhagen Approach

by Gregg Jaeger

Entropy 2018, 20(6), 420; https://doi.org/10.3390/e20060420 - 31 May 2018

Cited by 8 | Viewed by 3914

Abstract

In the Copenhagen approach to quantum mechanics as characterized by Heisenberg, probabilities relate to the statistics of measurement outcomes on ensembles of systems and to individual measurement events via the actualization of quantum potentiality. Here, brief summaries are given of a series of key results of different sorts that have been obtained since the final elements of the Copenhagen interpretation were offered and it was explicitly named so by Heisenberg—in particular, results from the investigation of the behavior of quantum probability since that time, the mid-1950s. This review shows that these developments have increased the value to physics of notions characterizing the approach which were previously either less precise or mainly symbolic in character, including complementarity, indeterminism, and unsharpness. Full article

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

Other

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24 pages, 413 KiB

Open AccessConcept Paper

On Interpretational Questions for Quantum-Like Modeling of Social Lasing

by Andrei Khrennikov, Alexander Alodjants, Anastasiia Trofimova and Dmitry Tsarev

Entropy 2018, 20(12), 921; https://doi.org/10.3390/e20120921 - 2 Dec 2018

Cited by 15 | Viewed by 4566

Abstract

The recent years were characterized by increasing interest to applications of the quantum formalism outside physics, e.g., in psychology, decision-making, socio-political studies. To distinguish such approach from quantum physics, it is called quantum-like. It is applied to modeling socio-political processes on the basis of the social laser model describing stimulated amplification of social actions. The main aim of this paper is establishing the socio-psychological interpretations of the quantum notions playing the basic role in lasing modeling. By using the Copenhagen interpretation and the operational approach to the quantum formalism, we analyze the notion of the social energy. Quantum formalizations of such notions as a social atom, s-atom, and an information field are presented. The operational approach based on the creation and annihilation operators is used. We also introduce the notion of the social color of information excitations representing characteristics linked to lasing coherence of the type of collimation. The Bose–Einstein statistics of excitations is coupled with the bandwagon effect, one of the basic effects of social psychology. By using the operational interpretation of the social energy, we present the thermodynamical derivation of this quantum statistics. The crucial role of information overload generated by the modern mass-media is emphasized. In physics laser’s resonator, the optical cavity, plays the crucial role in amplification. We model the functioning of social laser’s resonator by “distilling” the physical scheme from connection with optics. As the mathematical basis, we use the master equation for the density operator for the quantum information field. Full article

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

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5 pages, 229 KiB

Open AccessLetter

Dimensional Lifting through the Generalized Gram–Schmidt Process

by Hans Havlicek and Karl Svozil

Entropy 2018, 20(4), 284; https://doi.org/10.3390/e20040284 - 14 Apr 2018

Cited by 7 | Viewed by 3746

Abstract

A new way of orthogonalizing ensembles of vectors by “lifting” them to higher dimensions is introduced. This method can potentially be utilized for solving quantum decision and computing problems. Full article

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