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Quantum Probability and Randomness Quantum Probability and Randomness Edited by Andrei Khrennikov and Karl Svozil Printed Edition of the Special Issue Published in Entropy www.mdpi.com/journal/entropy Quantum Probability and Randomness Quantum Probability and Randomness Special Issue Editors Andrei Khrennikov Karl Svozil MDPI • Basel • Beijing • Wuhan • Barcelona • Belgrade Special Issue Editors Andrei Khrennikov Karl Svozil Linnaeus University Institute for Theoretical Physics of the Sweden Vienna Technical University Austria Editorial Office MDPI St. Alban-Anlage 66 4052 Basel, Switzerland This is a reprint of articles from the Special Issue published online in the open access journal Entropy (ISSN 1099-4300) from 2018 to 2019 (available at: https://www.mdpi.com/journal/entropy/special issues/Probability Randomness) For citation purposes, cite each article independently as indicated on the article page online and as indicated below: LastName, A.A.; LastName, B.B.; LastName, C.C. Article Title. Journal Name Year, Article Number, Page Range. ISBN 978-3-03897-714-8 (Pbk) ISBN 978-3-03897-715-5 (PDF) Cover image courtesy of R.C.-Z. Quehenberger. c 2019 by the authors. Articles in this book are Open Access and distributed under the Creative Commons Attribution (CC BY) license, which allows users to download, copy and build upon published articles, as long as the author and publisher are properly credited, which ensures maximum dissemination and a wider impact of our publications. The book as a whole is distributed by MDPI under the terms and conditions of the Creative Commons license CC BY-NC-ND. Contents About the Special Issue Editors ..................................... vii Andrei Khrennikov and Karl Svozil Quantum Probability and Randomness Reprinted from: Entropy 2019, 21, 35, doi:10.3390/e21010035 ..................... 1 Mladen Paviˇci´c and Norman D. Megill Vector Generation of Quantum Contextual Sets in Even Dimensional Hilbert Spaces Reprinted from: Entropy 2018, 20, 928, doi:10.3390/e20120928 ..................... 6 Aldo C. Mart´ınez, Aldo Sol´ıs, Rafael D´ıaz Hern´andez Rojas, Alfred B. U’Ren, Jorge G. Hirsch and Isaac P´erez Castillo Advanced Statistical Testing of Quantum Random Number Generators Reprinted from: Entropy 2018, 20, 886, doi:10.3390/e20110886 ..................... 18 Maria Luisa Dalla Chiara, Hector Freytes, Roberto Giuntini, Roberto Leporini and Giuseppe Sergioli Probabilities and Epistemic Operations in the Logics of Quantum Computation Reprinted from: Entropy 2018, 20, 837, doi:10.3390/e20110837 ..................... 31 Xiaomin Guo, Ripeng Liu, Pu Li, Chen Cheng, Mingchuan Wu and Yanqiang Guo Enhancing Extractable Quantum Entropy in Vacuum-Based Quantum Random Number Generato Reprinted from: Entropy 2018, 20, 819, doi:10.3390/e20110819 ..................... 53 Marco Enr´ıquez, Francisco Delgado and Karol Zyczkowski˙ Entanglement of Three-Qubit Random Pure States Reprinted from: Entropy 2018, 20, 745, doi:10.3390/e20100745 ..................... 66 Margarita A. Man’ko and Vladimir I. Man’ko New Entropic Inequalities and Hidden Correlations in Quantum Suprematism Picture of Qudit States Reprinted from: Entropy 2018, 20, 692, doi:10.3390/e20090692 ..................... 85 Arkady Plotnitsky “The Heisenberg Method”: Geometry, Algebra, and Probability in Quantum Theory Reprinted from: Entropy 2018, 20, 656, doi:10.3390/e20090656 .....................102 Francisco Delgado SU(2) Decomposition for the Quantum Information Dynamics in 2d-Partite Two-Level Quantum Systems Reprinted from: Entropy 2018, 20, 610, doi:10.3390/e20080610 .....................148 Marius Nagy and Naya Nagy An Information-Theoretic Perspective on the Quantum Bit Commitment Impossibility Theorem Reprinted from: Entropy 2018, 20, 193, doi:10.3390/e20030193 .....................189 Gregg Jaeger Developments in Quantum Probability and the Copenhagen Approach Reprinted from: Entropy 2018, 20, 420, doi:10.3390/e20060420 .....................205 v Hans Havlicek and Karl Svozil Dimensional Lifting through the Generalized Gram–Schmidt Process Reprinted from: Entropy 2018, 20, 284, doi:10.3390/e20040284 .....................224 Andrei Khrennikov, Alexander Alodjants, Anastasiia Trofimova and Dmitry Tsarev On Interpretational Questions for Quantum-Like Modeling of Social Lasing Reprinted from: Entropy 2018, 20, 921, doi:10.3390/e20120921 .....................229 Paul Ballonoff Paths of Cultural Systems Reprinted from: Entropy 2018, 20, 8, doi:10.3390/e20010008 ......................253 vi About the Special Issue Editors Andrei Khrennikov was born in 1958 in Volgorad and spent his childhood in the town of Bratsk, in Siberia, north from the lake Baikal. In the period between 1975–1980, he studied at Moscow State University, department of Mechanics and Mathematics, and in 1983, he received his PhD in mathematical physics (quantum field theory) at the same department. In 1990, he became full professor at Moscow University for Electronic Engineering. Since 1997, he has been a professor of applied mathematics at Linnaueus University, South-East Sweden, and since 2002, the director of the multidisciplinary research center at this university, as well as the International Center for Mathematical Modeling in Physics, Engineering, Economics, and Cognitive Science. His research interests are multidisciplinary, e.g., foundations of quantum physics, information, and probability, cognitive modeling, ultrametric (non-Archimedean) mathematics, dynamical systems, infinite-dimensional analysis, quantum-like models in psychology, and economics and finances. He is the author of approcimately 500 papers and 20 monographs in mathematics, physics, biology, psychology, cognitive science, economics, and finances. Karl Svozil is a professor of theoretical physics at Vienna’s University of Technology. He earned a Dr. Phil. while studying philosophy and sciences in the old, “Humboldtian” tradition in Heidelberg and Vienna, emphasizing the unity of knowledge. After attending the Lawrence Berkeley Laboratory and UC Berkeley, he worked as a physicist in Vienna, with many shorter stays abroad—among them, the Lomonosov Moscow State University, Lebedev Physical Institute and ICPT Trieste. He recently held an honorary position at the University of Auckland and served as president of the International Quantum Structures Association. Svozil’s main interests include quantum logic, issues related to (in)determininsism in physics, and “relativizing” relativity theory in the spirit of Alexandrov’s theorem of incidence geometry. vii entropy Editorial Quantum Probability and Randomness ∗ Andrei Khrennikov 1, and Karl Svozil 2 1 International Center for Mathematical Modeling in Physics, Engineering, Economics, and Cognitive Science, Linnaeus University, 351 95 Växjö, Sweden 2 Institute for Theoretical Physics, Vienna University of Technology, Wiedner Hauptstrasse 8-10/136, 1040 Vienna, Austria; [email protected] * Correspondence: [email protected] Received: 2 January 2019; Accepted: 3 January 2019; Published: 7 January 2019 Keywords: quantum foundations; probability; irreducible randomness; random number generators; quantum technology; entanglement; quantum-like models for social stochasticity; contextuality 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 [1–5]. Quantum probability and randomness play the crucial role in foundations of quantum mechanics. It might not be totally unreasonable to claim that, already starting from some of the earliest (in hindsight) indications of quanta in the 1902 Rutherford–Soddy exponential decay law and the small aberrations predicted by Schweidler [6], the tide of indeterminism [7,8] was rolling against chartered territories of fin de siécle mechanistic determinism. Riding the waves were researchers like Exner, who already in his 1908 inaugural lecture as rector magnificus [9] postulated that irreducible randomness is, and probability theory therefore needs to be, at the heart of all sciences; natural as well as social. Exner [10] was forgotten but cited in Schrödinger’s alike “Zürcher Antrittsvorlesung” of 1922 [11]. Not much later Born expressed his inclinations to give up determinism in the world of the atoms [12], thereby denying the existence of some inner properties of the quanta which condition a definite outcome for, say, the scattering after collisions. Von Neumann [13] was among the first who emphasized this new feature which was very different from the “in principle knowable unknowns” grounded in epistemology alone. Quantum randomness was treated as individual randomness; that is, as if single electrons or photons are sometimes capable of behaving acausally and irreducibly randomly. Such randomness cannot be reduced to a variability of properties of systems in some ensemble. Therefore, quantum randomness is often considered as irreducible randomness. Von Neumann understood well that it is difficult, if not outright impossible in general, to check empirically the randomness for individual systems, say for electrons or photons. In particular, he proceeded with the statistical interpretation of probability based on the mathematical model of von Mises [14,15] based upon relative frequencies after admissible place selections. At the same time, it is just and fair to note that the aforementioned tendencies to ground physics, and by reductionism, all of science, in ontological indeterminism, have been strongly contested and fiercely denied by eminent
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