Randomness: quantum versus classical

Recent tremendous development of quantum information theory led to a number of quantum technological projects, e.g., quantum random generators. This development stimulates a new wave of interest in quantum foundations. One of the most intriguing problems of quantum foundations is elaboration of a consistent and commonly accepted interpretation of quantum state. Closely related problem is clarification of the notion of quantum randomness and its interrelation with classical randomness. In this short review we shall discuss basics of classical theory of randomness (which by itself is very complex and characterized by diversity of approaches) and compare it with irreducible quantum randomness. The second part of this review is devoted to the information interpretation of quantum mechanics (QM) in the spirit of Zeilinger and Brukner (and QBism of Fuchs et al.) and physics in general (e.g., Wheeler's "it from bit") as well as digital philosophy of Chaitin (with historical coupling to ideas of Leibnitz). Finally, we continue discussion on interrelation of quantum and classical randomness and information interpretation of QM.

[1]  J. S. BELLt.  Einstein-Podolsky-Rosen Paradox , 2018 .

[2]  A. Zeilinger,et al.  Speakable and Unspeakable in Quantum Mechanics , 1989 .

[3]  Vulovic,et al.  Randomness of a true coin toss. , 1986, Physical review. A, General physics.

[4]  Ray J. Solomonoff,et al.  A Formal Theory of Inductive Inference. Part II , 1964, Inf. Control..

[5]  A. Shiryayev.  On Tables of Random Numbers , 1993 .

[6]  K. Michielsen,et al.  A local realist model for correlations of the singlet state , 2006 .

[7]  Jeff Dean,et al.  Time Series , 2009, Encyclopedia of Database Systems.

[8]  S. Wehner,et al.  Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres , 2015, Nature.

[9]  P. Martin-Lof,et al.  Complexity Oscillations in Infinite Binary Sequences , 2004 .

[10]  Ray J. Solomonoff,et al.  A Formal Theory of Inductive Inference. Part I , 1964, Inf. Control..

[11]  Karl Hess,et al.  Bell’s theorem: Critique of proofs with and without inequalities , 2005 .

[12]  Tsuyoshi Mizuguchi,et al.  Dynamics of Coin Tossing , 2006 .

[13]  Armin W. Schulz,et al.  Interpretations of probability , 2003 .

[14]  Anton Zeilinger,et al.  Dance of the Photons: From Einstein to Quantum Teleportation , 2010 .

[15]  Persi Diaconis,et al.  c ○ 2007 Society for Industrial and Applied Mathematics Dynamical Bias in the Coin Toss ∗ , 2022 .

[16]  A. Robinson.  I. Introduction , 1991 .

[17]  Johannes Kofler,et al.  Quantum Information and Randomness , 2010, European Review.

[18]  D. A. Edwards.  The mathematical foundations of quantum mechanics , 1979, Synthese.

[19]  Per Martin-Löf,et al.  The Definition of Random Sequences , 1966, Inf. Control..

[20]  C. Fuchs,et al.  Quantum probabilities as Bayesian probabilities , 2001, quant-ph/0106133.

[21]  Andrei N. Kolmogorov,et al.  Logical basis for information theory and probability theory , 1968, IEEE Trans. Inf. Theory.

[22]  A. A..  Probability, Statistics and Truth , 1940, Nature.

[23]  A. Zeilinger,et al.  Significant-Loophole-Free Test of Bell's Theorem with Entangled Photons. , 2015, Physical review letters.

[24]  H. Stowell.  The emperor's new mind R. Penrose, Oxford University Press, New York (1989) 466 pp. $24.95 , 1990, Neuroscience.

[25]  J. Bell.  On the impossible pilot wave , 1982 .

[26]  N. David Mermin,et al.  An introduction to QBism with an application to the locality of quantum mechanics , 2013, 1311.5253.

[27]  Andrei Khrennikov,et al.  Ubiquitous Quantum Structure: From Psychology to Finance , 2010 .

[28]  T. Nieuwenhuizen,et al.  Where Bell went wrong , 2008, 0812.3058.

[29]  C. Fuchs,et al.  A Quantum-Bayesian Route to Quantum-State Space , 2009, 0912.4252.

[30]  Caslav Brukner,et al.  Information Invariance and Quantum Probabilities , 2009, 0905.0653.

[31]  A. Kolmogorov.  Three approaches to the quantitative definition of information , 1968 .

[32]  Yoshiharu Tanaka,et al.  Quantum Adaptivity in Biology: From Genetics to Cognition , 2015, Springer Netherlands.

[33]  Marian Kupczynski.  Time Series, Stochastic Processes and Completeness of Quantum Theory , 2011 .

[34]  Przemyslaw Perlikowski,et al.  Understanding Coin-Tossing , 2010 .

[35]  Louis de Broglie,et al.  The current interpretation of wave mechanics : a critical study , 1964 .

[36]  J. Linnett,et al.  Quantum mechanics , 1975, Nature.

[37]  A J Leggett,et al.  The Quantum Measurement Problem , 2005, Science.

[38]  A. Zeilinger.  A Foundational Principle for Quantum Mechanics , 1999, Synthese Library.

[39]  Gregory J. Chaitin,et al.  On the Simplicity and Speed of Programs for Computing Infinite Sets of Natural Numbers , 1969, J. ACM.

[40]  J. Neumann.  Mathematical Foundations of Quantum Mechanics , 1955 .

[41]  Caslav Brukner,et al.  OPERATIONALLY INVARIANT INFORMATION IN QUANTUM MEASUREMENTS , 1999 .

[42]  Erhard Tornier,et al.  Grundlagen der Wahrscheinlichkeitsrechnung , 1933 .

[43]  J. Wheeler.  Information, physics, quantum: the search for links , 1999 .

[44]  C. Schnorr.  Zufälligkeit und Wahrscheinlichkeit , 1971 .

[45]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1951 .

[46]  Harald Bergstriim.  Mathematical Theory of Probability and Statistics , 1966 .

[47]  Andrei Khrennikov,et al.  Ubiquitous Quantum Structure , 2010 .

[48]  Andrei N. Kolmogorov,et al.  On Tables of Random Numbers (Reprinted from "Sankhya: The Indian Journal of Statistics", Series A, Vol. 25 Part 4, 1963) , 1998, Theor. Comput. Sci..

[49]  Karl Hess,et al.  Boole and Bell inequality , 2011 .

[50]  R. Solomonoff.  A PRELIMINARY REPORT ON A GENERAL THEORY OF INDUCTIVE INFERENCE , 2001 .

[51]  Aaron D. O’Connell.  Dance of the Photons: From Einstein to Quantum Teleportation , 2011 .

[52]  Emanuel Knill,et al.  Asymptotically optimal data analysis for rejecting local realism , 2011 .

[53]  M. Born,et al.  Statistical Interpretation of Quantum Mechanics. , 1955, Science.

[54]  Marian Kupczynski,et al.  EPR Paradox, Locality and Completeness of Quantum Theory , 2007, 0710.3510.

[55]  Stefano Pironio,et al.  Random numbers certified by Bell’s theorem , 2009, Nature.

[56]  A. Church.  On the concept of a random sequence , 1940 .

[57]  Andrei Khrennikov,et al.  On the equivalence of the Clauser–Horne and Eberhard inequality based tests , 2014, 1403.2811.

[58]  C. Fuchs.  Quantum Mechanics as Quantum Information (and only a little more) , 2002, quant-ph/0205039.

[59]  Christopher A. Fuchs.  Delirium Quantum Or, where I will take quantum mechanics if it will let me , 2007 .

[60]  E. Knill,et al.  A strong loophole-free test of local realism , 2015, 2016 Conference on Lasers and Electro-Optics (CLEO).

[61]  Andrei Khrennikov,et al.  Contextual Approach to Quantum Formalism , 2009 .

[62]  Yoshiharu Tanaka,et al.  Quantum Information Biology: From Information Interpretation of Quantum Mechanics to Applications in Molecular Biology and Cognitive Psychology , 2015, Foundations of Physics.