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Quantum Information and Probability: From Foundations to Engineering
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Quantum Information and Probability: From Foundations to Engineering

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

Deadline for manuscript submissions: closed (31 March 2023) | Viewed by 22364

Special Issue Editors

Prof. Dr. Francesco Buscemi

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Guest Editor
Department of Mathematical Informatics, Nagoya University, Nagoya 464-8601, Japan
Interests: quantum information theory; quantum thermodynamics; quantum statistical mechanics; quantum measurement theory; quantum statistics; quantum foundations
Prof. Dr. Andrei Khrennikov

E-Mail Website
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
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

The quantum information revolution has large, foundational impacts on theories and experiments, and recently also on engineering. We invite all kinds of contributions devoted to quantum theory, experiments, and engineering, especially to foundational questions with coupling to the quantum information, probability, and measurement theories.

Prof. Dr. Francesco Buscemi
Prof. Dr. Andrei Khrennikov
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.

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Published Papers (14 papers)

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Research

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31 pages, 402 KiB  
Article
Process and Time
by William Sulis
Entropy 2023, 25(5), 803; https://doi.org/10.3390/e25050803 - 15 May 2023
Cited by 2 | Viewed by 1616
Abstract
In regards to the nature of time, it has become commonplace to hear physicists state that time does not exist and that the perception of time passing and of events occurring in time is an illusion. In this paper, I argue that physics [...] Read more.
In regards to the nature of time, it has become commonplace to hear physicists state that time does not exist and that the perception of time passing and of events occurring in time is an illusion. In this paper, I argue that physics is actually agnostic on the question of the nature of time. The standard arguments against its existence all suffer from implicit biases and hidden assumptions, rendering many of them circular in nature. An alternative viewpoint to that of Newtonian materialism is the process view of Whitehead. I will show that the process perspective supports the reality of becoming, of happening, and of change. At the fundamental level, time is an expression of the action of process generating the elements of reality. Metrical space–time is an emergent aspect of relations between process-generated entities. Such a view is compatible with existing physics. The situation of time in physics is reminiscent of that of the continuum hypothesis in mathematical logic. It may be an independent assumption, not provable within physics proper (though it may someday be amenable to experimental exploration). Full article
(This article belongs to the Special Issue Quantum Information and Probability: From Foundations to Engineering)
12 pages, 299 KiB  
Article
On Transmitted Complexity Based on Modified Compound States
by Noboru Watanabe
Entropy 2023, 25(3), 455; https://doi.org/10.3390/e25030455 - 05 Mar 2023
Viewed by 699
Abstract
Based on the classical dynamical entropy, the channel coding theorem is investigated. Attempts to extend the dynamical entropy to quantum systems have been made by several researchers In 1999, Kossakowski, Ohya and I introduced the quantum dynamical entropy (KOW entropy) for completely positive [...] Read more.
Based on the classical dynamical entropy, the channel coding theorem is investigated. Attempts to extend the dynamical entropy to quantum systems have been made by several researchers In 1999, Kossakowski, Ohya and I introduced the quantum dynamical entropy (KOW entropy) for completely positive maps containing an automorphism describing the time evolution. Its formulation used transition expectations and lifting in the sense of Accardi and Ohya and was studied as a measure of the complexity of quantum mechanical systems. This KOW entropy allowed the extension of generalized AF (Alicki and Fannes) entropy and generalized AOW (Accardi, Ohya and Watanabe) entropy. In addition, the S-Mixing entropy and S-mixing mutual-entropy were formulated by Ohya in 1985. Compound states are an important tool for formulating mutual entropy, and the complexity was constructed by the generalized AOW entropy. In this paper, the complexity associated with the entangled compound states in the C* dynamical system based on the generalized AOW entropy based on the KOW entropy is investigated to lay the foundation for the proof of the theorem of channel coding for quantum systems. We show that the fundamental inequalities of the mutual entropy are satisfied when the initial state is transmitted over the channel changes with time. Full article
(This article belongs to the Special Issue Quantum Information and Probability: From Foundations to Engineering)
8 pages, 239 KiB  
Article
Environment-Assisted Invariance Does Not Necessitate Born’s Rule for Quantum Measurement
by Lotte Mertens and Jasper van Wezel
Entropy 2023, 25(3), 435; https://doi.org/10.3390/e25030435 - 01 Mar 2023
Cited by 2 | Viewed by 1127
Abstract
The argument of environment-assisted invariance (known as envariance) implying Born’s rule is widely used in models for quantum measurement to reason that they must yield the correct statistics, specifically for linear models. However, it has recently been shown that linear collapse models can [...] Read more.
The argument of environment-assisted invariance (known as envariance) implying Born’s rule is widely used in models for quantum measurement to reason that they must yield the correct statistics, specifically for linear models. However, it has recently been shown that linear collapse models can never give rise to Born’s rule. Here, we address this apparent contradiction and point out an inconsistency in the assumptions underlying the arguments based on envariance. We use a construction in which the role of the measurement machine is made explicit and shows that the presence of envariance does not imply that every measurement will behave according to Born’s rule. Rather, it implies that every quantum state allows a measurement machine to be constructed, which yields Born’s rule when measuring that particular state. This resolves the paradox and is in agreement with the recent result of objective collapse models necessarily being nonlinear. Full article
(This article belongs to the Special Issue Quantum Information and Probability: From Foundations to Engineering)
8 pages, 578 KiB  
Article
Experimental Counterexample to Bell’s Locality Criterion
by Ghenadie N. Mardari
Entropy 2022, 24(12), 1742; https://doi.org/10.3390/e24121742 - 29 Nov 2022
Cited by 1 | Viewed by 1479
Abstract
The EPR paradox was caused by the provision that quantum variables must have pre-existing values. This type of “hidden property realism” was later falsified by Bell’s Theorem. Accordingly, the physical basis for action-at-a-distance between entangled quanta was removed. Yet, modern interpretations present Bell’s [...] Read more.
The EPR paradox was caused by the provision that quantum variables must have pre-existing values. This type of “hidden property realism” was later falsified by Bell’s Theorem. Accordingly, the physical basis for action-at-a-distance between entangled quanta was removed. Yet, modern interpretations present Bell’s inequality as a Locality Criterion, as if Bell violations can only happen at the quantum level, and only with remote interactions. This is a questionable practice, considering that classical joint measurements also violate such inequalities for mutually exclusive wave properties. In particular, consecutive measurements of polarization produce the same coefficients of correlation as parallel measurements with entangled quanta, yet they are explicitly local. Furthermore, it is possible to combine parallel and consecutive measurements of Type I polarization-entangled photons in a single experiment, conclusively showing that quantum Bell violations can be local. Surprisingly, classical phenomena also require nonlocal interpretations if pre-existing properties are taken for granted. Hence, the solution is to reject the models with pre-existing properties for both classical and quantum wave-like phenomena. Full article
(This article belongs to the Special Issue Quantum Information and Probability: From Foundations to Engineering)
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11 pages, 267 KiB  
Article
Schrödinger’s Cat Meets Occam’s Razor
by Richard D. Gill
Entropy 2022, 24(11), 1586; https://doi.org/10.3390/e24111586 - 01 Nov 2022
Cited by 3 | Viewed by 2236
Abstract
We discuss V.P. Belavkin’s approach to the Schrödinger cat problem and show its close relation to ideas based on superselection and interaction with the environment developed by N.P. Landsman. The purpose of the paper is to explain these ideas in the most simple [...] Read more.
We discuss V.P. Belavkin’s approach to the Schrödinger cat problem and show its close relation to ideas based on superselection and interaction with the environment developed by N.P. Landsman. The purpose of the paper is to explain these ideas in the most simple possible context, namely: discrete time and separable Hilbert spaces, in order to make them accessible to those coming from the philosophy of science and not too happy with idiosyncratic notation and terminology and sophisticated mathematical tools. Conventional elementary mathematical descriptions of quantum mechanics take “measurement” to be a primitive concept. Paradoxes arise when we choose to consider smaller or larger systems as measurement devices in their own right, by making different and apparently arbitrary choices of location of the “Heisenberg cut”. Various quantum interpretations have different resolutions of the paradox. In Belavkin’s approach, the classical world around us does really exist, and it evolves stochastically and dynamically in time according to probability laws following from successive applications of the Born law. It is a collapse theory. The quantum/classical distinction is determined by the arrow of time. The underlying unitary evolution of the wave-function of the universe enables the designation of a collection of beables which grows as time evolves, and which therefore can be assigned random, classical trajectories. In a slogan: the past is particles, the future is a wave. We, living in the now, are located on the cutting edge between past and future. Full article
(This article belongs to the Special Issue Quantum Information and Probability: From Foundations to Engineering)
6 pages, 225 KiB  
Article
Unitarity of Decoherence Implies Possibility of Decoherence-like Dynamics towards Macroscopic Superpositions
by Stanislav Filatov and Marcis Auzinsh
Entropy 2022, 24(11), 1546; https://doi.org/10.3390/e24111546 - 28 Oct 2022
Viewed by 930
Abstract
Quantum decoherence is crucial to understanding the emergence of the classical world from the underlying quantum reality. Decoherence dynamics are unitary, although they superselect a preferred eigenbasis. Decoherence dynamics result in stable macroscopic, localized, classical-like states. We show that the above-mentioned facts imply [...] Read more.
Quantum decoherence is crucial to understanding the emergence of the classical world from the underlying quantum reality. Decoherence dynamics are unitary, although they superselect a preferred eigenbasis. Decoherence dynamics result in stable macroscopic, localized, classical-like states. We show that the above-mentioned facts imply the possibility of the existence of decoherence-like dynamics that result in stable macroscopic non-localized non-classical-like states. Being rooted in the fabric of the decoherence theory itself, this property implies environments that steer the decoherence towards, for example, spatial superpositions of macroscopic objects. To demonstrate this, we provide thought-experimental, mathematical and philosophical arguments. Full article
(This article belongs to the Special Issue Quantum Information and Probability: From Foundations to Engineering)
17 pages, 327 KiB  
Article
New Challenges for Classical and Quantum Probability
by Luigi Accardi
Entropy 2022, 24(10), 1502; https://doi.org/10.3390/e24101502 - 21 Oct 2022
Cited by 1 | Viewed by 1297
Abstract
The discovery that any classical random variable with all moments gives rise to a full quantum theory (that in the Gaussian and Poisson cases coincides with the usual one) implies that a quantum–type formalism will enter into practically all applications of classical probability [...] Read more.
The discovery that any classical random variable with all moments gives rise to a full quantum theory (that in the Gaussian and Poisson cases coincides with the usual one) implies that a quantum–type formalism will enter into practically all applications of classical probability and statistics. The new challenge consists in finding the classical interpretation, for different types of classical contexts, of typical quantum notions such as entanglement, normal order, equilibrium states, etc. As an example, every classical symmetric random variable has a canonically associated conjugate momentum. In usual quantum mechanics (associated with Gaussian or Poisson classical random variables), the interpretation of the momentum operator was already clear to Heisenberg. How should we interpret the conjugate momentum operator associated with classical random variables outside the Gauss–Poisson class? The Introduction is intended to place in historical perspective the recent developments that are the main object of the present exposition. Full article
(This article belongs to the Special Issue Quantum Information and Probability: From Foundations to Engineering)
29 pages, 670 KiB  
Article
“Yet Once More”: The Double-Slit Experiment and Quantum Discontinuity
by Arkady Plotnitsky
Entropy 2022, 24(10), 1455; https://doi.org/10.3390/e24101455 - 12 Oct 2022
Cited by 4 | Viewed by 1587
Abstract
This article reconsiders the double-slit experiment from the nonrealist or, in terms of this article, “reality-without-realism” (RWR) perspective, grounded in the combination of three forms of quantum discontinuity: (1) “Heisenberg discontinuity”, defined by the impossibility of a representation or even conception of how [...] Read more.
This article reconsiders the double-slit experiment from the nonrealist or, in terms of this article, “reality-without-realism” (RWR) perspective, grounded in the combination of three forms of quantum discontinuity: (1) “Heisenberg discontinuity”, defined by the impossibility of a representation or even conception of how quantum phenomena come about, even though quantum theory (such as quantum mechanics or quantum field theory) predicts the data in question strictly in accord with what is observed in quantum experiments); (2) “Bohr discontinuity”, defined, under the assumption of Heisenberg discontinuity, by the view that quantum phenomena and the data observed therein are described by classical and not quantum theory, even though classical physics cannot predict them; and (3) “Dirac discontinuity” (not considered by Dirac himself, but suggested by his equation), according to which the concept of a quantum object, such as a photon or electron, is an idealization only applicable at the time of observation and not to something that exists independently in nature. Dirac discontinuity is of particular importance for the article’s foundational argument and its analysis of the double-slit experiment. Full article
(This article belongs to the Special Issue Quantum Information and Probability: From Foundations to Engineering)
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13 pages, 487 KiB  
Article
Electron Spin Correlations: Probabilistic Description and Geometric Representation
by Ana María Cetto
Entropy 2022, 24(10), 1439; https://doi.org/10.3390/e24101439 - 09 Oct 2022
Cited by 1 | Viewed by 1486
Abstract
The electron spin correlation is shown to be expressible in terms of a bona fide probability distribution function with an associated geometric representation. With this aim, an analysis is presented of the probabilistic features of the spin correlation within the quantum formalism, which [...] Read more.
The electron spin correlation is shown to be expressible in terms of a bona fide probability distribution function with an associated geometric representation. With this aim, an analysis is presented of the probabilistic features of the spin correlation within the quantum formalism, which helps clarify the concepts of contextuality and measurement dependence. The dependence of the spin correlation on conditional probabilities allows for a clear separation between system state and measurement context; the latter determines how the probability space should be partitioned in calculating the correlation. A probability distribution function ρ(ϕ) is then proposed, which reproduces the quantum correlation for a pair of single-particle spin projections and is amenable to a simple geometric representation that gives meaning to the variable ϕ. The same procedure is shown to be applicable to the bipartite system in the singlet spin state. This endows the spin correlation with a clear probabilistic meaning and leaves the door open for a possible physical picture of the electron spin, as discussed at the end of the paper. Full article
(This article belongs to the Special Issue Quantum Information and Probability: From Foundations to Engineering)
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12 pages, 319 KiB  
Article
Extending Kolmogorov’s Axioms for a Generalized Probability Theory on Collections of Contexts
by Karl Svozil
Entropy 2022, 24(9), 1285; https://doi.org/10.3390/e24091285 - 12 Sep 2022
Cited by 4 | Viewed by 1315
Abstract
Kolmogorov’s axioms of probability theory are extended to conditional probabilities among distinct (and sometimes intertwining) contexts. Formally, this amounts to row stochastic matrices whose entries characterize the conditional probability to find some observable (postselection) in one context, given an observable (preselection) in another [...] Read more.
Kolmogorov’s axioms of probability theory are extended to conditional probabilities among distinct (and sometimes intertwining) contexts. Formally, this amounts to row stochastic matrices whose entries characterize the conditional probability to find some observable (postselection) in one context, given an observable (preselection) in another context. As the respective probabilities need not (but, depending on the physical/model realization, can) be of the Born rule type, this generalizes approaches to quantum probabilities by Aufféves and Grangier, which in turn are inspired by Gleason’s theorem. Full article
(This article belongs to the Special Issue Quantum Information and Probability: From Foundations to Engineering)
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19 pages, 384 KiB  
Article
Interpretation of Quantum Theory: The Quantum “Grue-Bleen” Problem
by Benjamin Schumacher and Michael D. Westmoreland
Entropy 2022, 24(9), 1268; https://doi.org/10.3390/e24091268 - 09 Sep 2022
Viewed by 1312
Abstract
We present a critique of the many-world interpretation of quantum mechanics, based on different “pictures” that describe the time evolution of an isolated quantum system. Without an externally imposed frame to restrict these possible pictures, the theory cannot yield non-trivial interpretational statements. This [...] Read more.
We present a critique of the many-world interpretation of quantum mechanics, based on different “pictures” that describe the time evolution of an isolated quantum system. Without an externally imposed frame to restrict these possible pictures, the theory cannot yield non-trivial interpretational statements. This is analogous to Goodman’s famous “grue-bleen” problem of language and induction. Using a general framework applicable to many kinds of dynamical theories, we try to identify the kind of additional structure (if any) required for the meaningful interpretation of a theory. We find that the “grue-bleen” problem is not restricted to quantum mechanics, but also affects other theories including classical Hamiltonian mechanics. For all such theories, absent external frame information, an isolated system has no interpretation. Full article
(This article belongs to the Special Issue Quantum Information and Probability: From Foundations to Engineering)
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18 pages, 383 KiB  
Article
Law of Total Probability in Quantum Theory and Its Application in Wigner’s Friend Scenario
by Jianhao M. Yang
Entropy 2022, 24(7), 903; https://doi.org/10.3390/e24070903 - 29 Jun 2022
Cited by 3 | Viewed by 1357
Abstract
It is well-known that the law of total probability does not generally hold in quantum theory. However, recent arguments on some of the fundamental assumptions in quantum theory based on the extended Wigner’s friend scenario show a need to clarify how the law [...] Read more.
It is well-known that the law of total probability does not generally hold in quantum theory. However, recent arguments on some of the fundamental assumptions in quantum theory based on the extended Wigner’s friend scenario show a need to clarify how the law of total probability should be formulated in quantum theory and under what conditions it still holds. In this work, the definition of conditional probability in quantum theory is extended to POVM measurements. A rule to assign two-time conditional probability is proposed for incompatible POVM operators, which leads to a more general and precise formulation of the law of total probability. Sufficient conditions under which the law of total probability holds are identified. Applying the theory developed here to analyze several quantum no-go theorems related to the extended Wigner’s friend scenario reveals logical loopholes in these no-go theorems. The loopholes exist as a consequence of taking for granted the validity of the law of total probability without verifying the sufficient conditions. Consequently, the contradictions in these no-go theorems only reconfirm the invalidity of the law of total probability in quantum theory rather than invalidating the physical statements that the no-go theorems attempt to refute. Full article
(This article belongs to the Special Issue Quantum Information and Probability: From Foundations to Engineering)
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Review

Jump to: Research

13 pages, 341 KiB  
Review
Contextuality or Nonlocality: What Would John Bell Choose Today?
by Marian Kupczynski
Entropy 2023, 25(2), 280; https://doi.org/10.3390/e25020280 - 02 Feb 2023
Cited by 10 | Viewed by 2021
Abstract
A violation of Bell-CHSH inequalities does not justify speculations about quantum non-locality, conspiracy and retro-causation. Such speculations are rooted in a belief that setting dependence of hidden variables in a probabilistic model (called a violation of measurement independence (MI)) would mean a violation [...] Read more.
A violation of Bell-CHSH inequalities does not justify speculations about quantum non-locality, conspiracy and retro-causation. Such speculations are rooted in a belief that setting dependence of hidden variables in a probabilistic model (called a violation of measurement independence (MI)) would mean a violation of experimenters’ freedom of choice. This belief is unfounded because it is based on a questionable use of Bayes Theorem and on incorrect causal interpretation of conditional probabilities. In Bell-local realistic model, hidden variables describe only photonic beams created by a source, thus they cannot depend on randomly chosen experimental settings. However, if hidden variables describing measuring instruments are correctly incorporated into a contextual probabilistic model a violation of inequalities and an apparent violation of no-signaling reported in Bell tests can be explained without evoking quantum non-locality. Therefore, for us, a violation of Bell-CHSH inequalities proves only that hidden variables have to depend on settings confirming contextual character of quantum observables and an active role played by measuring instruments. Bell thought that he had to choose between non-locality and the violation of experimenters’ freedom of choice. From two bad choices he chose non-locality. Today he would probably choose the violation of MI understood as contextuality. Full article
(This article belongs to the Special Issue Quantum Information and Probability: From Foundations to Engineering)
29 pages, 426 KiB  
Review
Contextuality, Complementarity, Signaling, and Bell Tests
by Andrei Khrennikov
Entropy 2022, 24(10), 1380; https://doi.org/10.3390/e24101380 - 28 Sep 2022
Cited by 15 | Viewed by 2408
Abstract
This is a review devoted to the complementarity–contextuality interplay with connection to the Bell inequalities. Starting the discussion with complementarity, I point to contextuality as its seed. Bohr contextuality is the dependence of an observable’s outcome on the experimental context; on the system–apparatus [...] Read more.
This is a review devoted to the complementarity–contextuality interplay with connection to the Bell inequalities. Starting the discussion with complementarity, I point to contextuality as its seed. Bohr contextuality is the dependence of an observable’s outcome on the experimental context; on the system–apparatus interaction. Probabilistically, complementarity means that the joint probability distribution (JPD) does not exist. Instead of the JPD, one has to operate with contextual probabilities. The Bell inequalities are interpreted as the statistical tests of contextuality, and hence, incompatibility. For context-dependent probabilities, these inequalities may be violated. I stress that contextuality tested by the Bell inequalities is the so-called joint measurement contextuality (JMC), the special case of Bohr’s contextuality. Then, I examine the role of signaling (marginal inconsistency). In QM, signaling can be considered as an experimental artifact. However, often, experimental data have signaling patterns. I discuss possible sources of signaling—for example, dependence of the state preparation on measurement settings. In principle, one can extract the measure of “pure contextuality” from data shadowed by signaling. This theory is known as contextuality by default (CbD). It leads to inequalities with an additional term quantifying signaling: Bell–Dzhafarov–Kujala inequalities. Full article
(This article belongs to the Special Issue Quantum Information and Probability: From Foundations to Engineering)
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