Entropy

Research

9 pages, 1125 KiB

Open AccessFeature PaperArticle

VIP-2 —High-Sensitivity Tests on the Pauli Exclusion Principle for Electrons

by Kristian Piscicchia, Johann Marton, Sergio Bartalucci, Massimiliano Bazzi, Sergio Bertolucci, Mario Bragadireanu, Michael Cargnelli, Alberto Clozza, Raffaele Del Grande, Luca De Paolis, Carlo Fiorini, Carlo Guaraldo, Mihail Iliescu, Matthias Laubenstein, Marco Miliucci, Edoardo Milotti, Fabrizio Napolitano, Andreas Pichler, Alessandro Scordo, Hexi Shi, Diana Laura Sirghi, Florin Sirghi, Laura Sperandio, Oton Vazquez Doce, Johann Zmeskal and Catalina Curceanu Show full author list Hide full author list

Entropy 2020, 22(11), 1195; https://doi.org/10.3390/e22111195 - 22 Oct 2020

Cited by 12 | Viewed by 3035

Abstract

The VIP collaboration is performing high sensitivity tests of the Pauli Exclusion Principle for electrons in the extremely low cosmic background environment of the underground Gran Sasso National Laboratory INFN (Italy). In particular, the VIP-2 Open Systems experiment was conceived to put strong [...] Read more.

The VIP collaboration is performing high sensitivity tests of the Pauli Exclusion Principle for electrons in the extremely low cosmic background environment of the underground Gran Sasso National Laboratory INFN (Italy). In particular, the VIP-2 Open Systems experiment was conceived to put strong constraints on those Pauli Exclusion Principle violation models which respect the so-called Messiah–Greenberg superselection rule. The experimental technique consists of introducing a direct current in a copper conductor, and searching for the X-rays emission coming from a forbidden atomic transition from the L shell to the K shell of copper when the K shell is already occupied by two electrons. The analysis of the first three months of collected data (in 2018) is presented. The obtained result represents the best bound on the Pauli Exclusion Principle violation probability which fulfills the Messiah–Greenberg rule. Full article

(This article belongs to the Special Issue Quantum Information Revolution: Impact to Foundations)

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24 pages, 329 KiB

Open AccessArticle

A Note on Complexities by Means of Quantum Compound Systems

by Noboru Watanabe

Entropy 2020, 22(3), 298; https://doi.org/10.3390/e22030298 - 5 Mar 2020

Cited by 3 | Viewed by 2007

Abstract

It has been shown that joint probability distributions of quantum systems generally do not exist, and the key to solving this concern is the compound state invented by Ohya. The Ohya compound state constructed by the Schatten decomposition (i.e., one-dimensional orthogonal projection) of [...] Read more.

It has been shown that joint probability distributions of quantum systems generally do not exist, and the key to solving this concern is the compound state invented by Ohya. The Ohya compound state constructed by the Schatten decomposition (i.e., one-dimensional orthogonal projection) of the input state shows the correlation between the states of the input and output systems. In 1983, Ohya formulated the quantum mutual entropy by applying this compound state. Since this mutual entropy satisfies the fundamental inequality, one may say that it represents the amount of information correctly transmitted from the input system through the channel to the output system, and it may play an important role in discussing the efficiency of information transfer in quantum systems. Since the Ohya compound state is separable state, it is important that we must look more carefully into the entangled compound state. This paper is intended as an investigation of the construction of the entangled compound state, and the hybrid entangled compound state is introduced. The purpose of this paper is to consider the validity of the compound states constructing the quantum mutual entropy type complexity. It seems reasonable to suppose that the quantum mutual entropy type complexity defined by using the entangled compound state is not useful to discuss the efficiency of information transmission from the initial system to the final system. Full article

(This article belongs to the Special Issue Quantum Information Revolution: Impact to Foundations)

10 pages, 2525 KiB

Open AccessArticle

Applying the Bell’s Test to Chinese Texts

by Igor A. Bessmertny, Xiaoxi Huang, Aleksei V. Platonov, Chuqiao Yu and Julia A. Koroleva

Entropy 2020, 22(3), 275; https://doi.org/10.3390/e22030275 - 28 Feb 2020

Cited by 3 | Viewed by 2641

Abstract

Search engines are able to find documents containing patterns from a query. This approach can be used for alphabetic languages such as English. However, Chinese is highly dependent on context. The significant problem of Chinese text processing is the missing blanks between words, [...] Read more.

Search engines are able to find documents containing patterns from a query. This approach can be used for alphabetic languages such as English. However, Chinese is highly dependent on context. The significant problem of Chinese text processing is the missing blanks between words, so it is necessary to segment the text to words before any other action. Algorithms for Chinese text segmentation should consider context; that is, the word segmentation process depends on other ideograms. As the existing segmentation algorithms are imperfect, we have considered an approach to build the context from all possible n-grams surrounding the query words. This paper proposes a quantum-inspired approach to rank Chinese text documents by their relevancy to the query. Particularly, this approach uses Bell’s test, which measures the quantum entanglement of two words within the context. The contexts of words are built using the hyperspace analogue to language (HAL) algorithm. Experiments fulfilled in three domains demonstrated that the proposed approach provides acceptable results. Full article

(This article belongs to the Special Issue Quantum Information Revolution: Impact to Foundations)

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12 pages, 1921 KiB

Open AccessEditor’s ChoiceArticle

On the Irrationality of Being in Two Minds

by Shahram Dehdashti, Lauren Fell and Peter Bruza

Entropy 2020, 22(2), 174; https://doi.org/10.3390/e22020174 - 4 Feb 2020

Cited by 4 | Viewed by 4363

Abstract

This article presents a general framework that allows irrational decision making to be theoretically investigated and simulated. Rationality in human decision making under uncertainty is normatively prescribed by the axioms of probability theory in order to maximize utility. However, substantial literature from psychology [...] Read more.

This article presents a general framework that allows irrational decision making to be theoretically investigated and simulated. Rationality in human decision making under uncertainty is normatively prescribed by the axioms of probability theory in order to maximize utility. However, substantial literature from psychology and cognitive science shows that human decisions regularly deviate from these axioms. Bistable probabilities are proposed as a principled and straight forward means for modeling (ir)rational decision making, which occurs when a decision maker is in “two minds”. We show that bistable probabilities can be formalized by positive-operator-valued projections in quantum mechanics. We found that (1) irrational decision making necessarily involves a wider spectrum of causal relationships than rational decision making, (2) the accessible information turns out to be greater in irrational decision making when compared to rational decision making, and (3) irrational decision making is quantum-like because it violates the Bell–Wigner polytope. Full article

(This article belongs to the Special Issue Quantum Information Revolution: Impact to Foundations)

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Graphical abstract

Graphical abstract
Full article ">Figure 1
A schematic setup for a bistable model structure. Full article ">Figure 2
Alternative causal models based on Reichenbach’s principle. Full article ">Figure 3
Polytopes for different values of <math display="inline"><semantics> <mrow> <mi>k</mi> <mo>∈</mo> <mo>{</mo> <mn>1.0</mn> <mo>,</mo> <mn>0.9</mn> <mo>,</mo> <mn>0.8</mn> <mo>,</mo> <mn>0.7</mn> <mo>,</mo> <mn>0.6</mn> <mo>,</mo> <mn>0.5</mn> <mo>}</mo> </mrow> </semantics></math> are respectively shown in plots (a)–(f). Full article ">Figure 4
Pure irrational information volume (PIIV), i.e., <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </semantics></math> as a function of the bistable parameter k. Full article ">Figure 5
Bell–Wigner polytopes for different values of <math display="inline"><semantics> <mrow> <msub> <mi>k</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>0.9</mn> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <mn>0.5</mn> </mrow> </semantics></math> are illustrated in plots (a), (b), ⋯, (f). Colors differentiate different values of <math display="inline"><semantics> <msub> <mi>k</mi> <mn>1</mn> </msub> </semantics></math>, with blue signifying <math display="inline"><semantics> <mrow> <msub> <mi>k</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math> and green signifying <math display="inline"><semantics> <mrow> <msub> <mi>k</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>. Paramater <math display="inline"><semantics> <msub> <mi>k</mi> <mn>2</mn> </msub> </semantics></math> varies from 0.5 to 1. Bars below the <math display="inline"><semantics> <mrow> <msub> <mi>k</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>k</mi> <mn>2</mn> </msub> </mrow> </semantics></math> plane signify negative probabilities. Full article ">

26 pages, 2751 KiB

Open AccessArticle

Balanced Quantum-Like Bayesian Networks

by Andreas Wichert, Catarina Moreira and Peter Bruza

Entropy 2020, 22(2), 170; https://doi.org/10.3390/e22020170 - 2 Feb 2020

Cited by 14 | Viewed by 3949

Abstract

Empirical findings from cognitive psychology indicate that, in scenarios under high levels of uncertainty, many people tend to make irrational decisions. To address this problem, models based on quantum probability theory, such as the quantum-like Bayesian networks, have been proposed. However, this model [...] Read more.

Empirical findings from cognitive psychology indicate that, in scenarios under high levels of uncertainty, many people tend to make irrational decisions. To address this problem, models based on quantum probability theory, such as the quantum-like Bayesian networks, have been proposed. However, this model makes use of a Bayes normalisation factor during probabilistic inference to convert the likelihoods that result from quantum interference effects into probability values. The interpretation of this operation is not clear and leads to extremely skewed intensity waves that make the task of prediction of these irrational decisions challenging. This article proposes the law of balance, a novel mathematical formalism for probabilistic inferences in quantum-like Bayesian networks, based on the notion of balanced intensity waves. The general idea is to balance the intensity waves resulting from quantum interference in such a way that, during Bayes normalisation, they cancel each other. With this representation, we also propose the law of maximum uncertainty, which is a method to predict these paradoxes by selecting the amplitudes of the wave with the highest entropy. Empirical results show that the law of balance together with the law of maximum uncertainty were able to accurately predict different experiments from cognitive psychology showing paradoxical or irrational decisions, namely in the Prisoner’s Dilemma game and the Two-Stage Gambling Game. Full article

(This article belongs to the Special Issue Quantum Information Revolution: Impact to Foundations)

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20 pages, 855 KiB

Open AccessEditor’s ChoiceArticle

Adapting Logic to Physics: The Quantum-Like Eigenlogic Program

by Zeno Toffano and François Dubois

Entropy 2020, 22(2), 139; https://doi.org/10.3390/e22020139 - 24 Jan 2020

Cited by 11 | Viewed by 4061

Abstract

Considering links between logic and physics is important because of the fast development of quantum information technologies in our everyday life. This paper discusses a new method in logic inspired from quantum theory using operators, named Eigenlogic. It expresses logical propositions using linear [...] Read more.

Considering links between logic and physics is important because of the fast development of quantum information technologies in our everyday life. This paper discusses a new method in logic inspired from quantum theory using operators, named Eigenlogic. It expresses logical propositions using linear algebra. Logical functions are represented by operators and logical truth tables correspond to the eigenvalue structure. It extends the possibilities of classical logic by changing the semantics from the Boolean binary alphabet

{0, 1}

using projection operators to the binary alphabet

{+ 1, ? 1}

employing reversible involution operators. Also, many-valued logical operators are synthesized, for whatever alphabet, using operator methods based on Lagrange interpolation and on the Cayley–Hamilton theorem. Considering a superposition of logical input states one gets a fuzzy logic representation where the fuzzy membership function is the quantum probability given by the Born rule. Historical parallels from Boole, Post, Poincaré and Combinatory Logic are presented in relation to probability theory, non-commutative quaternion algebra and Turing machines. An extension to first order logic is proposed inspired by Grover’s algorithm. Eigenlogic is essentially a logic of operators and its truth-table logical semantics is provided by the eigenvalue structure which is shown to be related to the universality of logical quantum gates, a fundamental role being played by non-commutativity and entanglement. Full article

(This article belongs to the Special Issue Quantum Information Revolution: Impact to Foundations)

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16 pages, 299 KiB

Open AccessArticle

An Information Ontology for the Process Algebra Model of Non-Relativistic Quantum Mechanics

by William Sulis

Entropy 2020, 22(2), 136; https://doi.org/10.3390/e22020136 - 23 Jan 2020

Cited by 8 | Viewed by 2679

Abstract

The process algebra model has been suggested as an alternative mathematical framework for non-relativistic quantum mechanics (NRQM). It appears to reproduce the wave functions of non-relativistic quantum mechanics to a high degree of accuracy. It posits a fundamental level of finite, discrete events [...] Read more.

The process algebra model has been suggested as an alternative mathematical framework for non-relativistic quantum mechanics (NRQM). It appears to reproduce the wave functions of non-relativistic quantum mechanics to a high degree of accuracy. It posits a fundamental level of finite, discrete events upon which the usual entities of NRQM supervene. It has been suggested that the process algebra model provides a true completion of NRQM, free of divergences and paradoxes, with causally local information propagation, contextuality, and realism. Arguments in support of these claims have been mathematical. Missing has been an ontology of this fundamental level from which the formalism naturally emerges. In this paper, it is argued that information and information flow provides this ontology. Higher level constructs such as energy, momentum, mass, spacetime, are all emergent from this fundamental level. Full article

(This article belongs to the Special Issue Quantum Information Revolution: Impact to Foundations)

14 pages, 269 KiB

Open AccessArticle

Application of Theory of Quantum Instruments to Psychology: Combination of Question Order Effect with Response Replicability Effect

by Masanao Ozawa and Andrei Khrennikov

Entropy 2020, 22(1), 37; https://doi.org/10.3390/e22010037 - 26 Dec 2019

Cited by 33 | Viewed by 4590

Abstract

Recently, quantum formalism started to be actively used outside of quantum physics: in psychology, decision-making, economics, finances, and social science. Human psychological behavior is characterized by a few basic effects; one of them is the question order effect (QOE). This effect was successfully [...] Read more.

Recently, quantum formalism started to be actively used outside of quantum physics: in psychology, decision-making, economics, finances, and social science. Human psychological behavior is characterized by a few basic effects; one of them is the question order effect (QOE). This effect was successfully modeled (Busemeyer–Wang) by representing questions A and B by Hermitian observables and mental-state transformations (back action of answering) by orthogonal projectors. However, then it was demonstrated that such representation cannot be combined with another psychological effect, known as the response replicability effect (RRE). Later, this no-go result was generalized to representation of questions and state transformations by quantum instruments of the atomic type. In light of these results, the possibility of using quantum formalism in psychology was questioned. In this paper, we show that, nevertheless, the combination of the QOE and RRE can be modeled within quantum formalism, in the framework of theory of non-atomic quantum instruments. Full article

(This article belongs to the Special Issue Quantum Information Revolution: Impact to Foundations)

13 pages, 535 KiB

Open AccessArticle

Quantum Games with Unawareness with Duopoly Problems in View

by Piotr Frąckiewicz and Jakub Bilski

Entropy 2019, 21(11), 1097; https://doi.org/10.3390/e21111097 - 10 Nov 2019

Cited by 5 | Viewed by 2405

Abstract

Playing the Cournot duopoly in the quantum domain can lead to the optimal strategy profile in the case of maximally correlated actions of the players. However, that result can be obtained if the fact that the players play the quantum game is common [...] Read more.

Playing the Cournot duopoly in the quantum domain can lead to the optimal strategy profile in the case of maximally correlated actions of the players. However, that result can be obtained if the fact that the players play the quantum game is common knowledge among the players. Our purpose is to determine reasonable game outcomes when players’ perceptions about what game is actually played are limited. To this end, we consider a collection consisting of the classical and quantum games that specifies how each player views the game and how each player views the other players’ perceptions of the game. We show that a slight change in how the players perceive the game may considerably affect the result of the game and, in the case of maximally correlated strategies, may vary from the inefficient Nash equilibrium outcome in the classical Cournot duopoly to the Pareto optimal outcome. We complete our work by investigating in the same way the Bertrand duopoly model. Full article

(This article belongs to the Special Issue Quantum Information Revolution: Impact to Foundations)

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19 pages, 14074 KiB

Open AccessArticle

Communication Enhancement through Quantum Coherent Control of N Channels in an Indefinite Causal-Order Scenario

by Lorenzo M. Procopio, Francisco Delgado, Marco Enríquez, Nadia Belabas and Juan Ariel Levenson

Entropy 2019, 21(10), 1012; https://doi.org/10.3390/e21101012 - 18 Oct 2019

Cited by 74 | Viewed by 5060

Abstract

In quantum Shannon theory, transmission of information is enhanced by quantum features. Up to very recently, the trajectories of transmission remained fully classical. Recently, a new paradigm was proposed by playing quantum tricks on two completely depolarizing quantum channels i.e., using coherent control [...] Read more.

In quantum Shannon theory, transmission of information is enhanced by quantum features. Up to very recently, the trajectories of transmission remained fully classical. Recently, a new paradigm was proposed by playing quantum tricks on two completely depolarizing quantum channels i.e., using coherent control in space or time of the two quantum channels. We extend here this control to the transmission of information through a network of an arbitrary number N of channels with arbitrary individual capacity i.e., information preservation characteristics in the case of indefinite causal order. We propose a formalism to assess information transmission in the most general case of N channels in an indefinite causal order scenario yielding the output of such transmission. Then, we explicitly derive the quantum switch output and the associated Holevo limit of the information transmission for

N = 2

,

N = 3

as a function of all involved parameters. We find in the case

N = 3

that the transmission of information for three channels is twice that of transmission of the two-channel case when a full superposition of all possible causal orders is used. Full article

(This article belongs to the Special Issue Quantum Information Revolution: Impact to Foundations)

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13 pages, 331 KiB

Open AccessArticle

A Generic Model for Quantum Measurements

by Alexia Auffèves and Philippe Grangier

Entropy 2019, 21(9), 904; https://doi.org/10.3390/e21090904 - 17 Sep 2019

Cited by 20 | Viewed by 3715

Abstract

In previous articles, we presented a derivation of Born’s rule and unitary transforms in Quantum Mechanics (QM), from a simple set of axioms built upon a physical phenomenology of quantization—physically, the structure of QM results of an interplay between the quantized number of [...] Read more.

In previous articles, we presented a derivation of Born’s rule and unitary transforms in Quantum Mechanics (QM), from a simple set of axioms built upon a physical phenomenology of quantization—physically, the structure of QM results of an interplay between the quantized number of “modalities” accessible to a quantum system, and the continuum of “contexts” required to define these modalities. In the present article, we provide a unified picture of quantum measurements within our approach, and justify further the role of the system–context dichotomy, and of quantum interferences. We also discuss links with stochastic quantum thermodynamics, and with algebraic quantum theory. Full article

(This article belongs to the Special Issue Quantum Information Revolution: Impact to Foundations)

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Figure 1

Figure 1
Two possible ways to change the context and come back. An irreversible change of context is signaled by the realization of one modality <math display="inline"><semantics> <msub> <mi>v</mi> <mi>j</mi> </msub> </semantics></math> in the new context. This phenomenon eliminates all other possible new modalities <math display="inline"><semantics> <msub> <mi>v</mi> <mrow> <mi>i</mi> <mo>≠</mo> <mi>j</mi> </mrow> </msub> </semantics></math> because modalities are mutually exclusive in a given context. If the outcomes are not read, one has to sum probabilities over the N possibilities, and one gets <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>(</mo> <msub> <mi>u</mi> <mi>k</mi> </msub> <mo stretchy="false">|</mo> <msub> <mi>u</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> </semantics></math> given by Equation (<a href="#FD2-entropy-21-00904" class="html-disp-formula">2</a>). As the opposite extreme case, a reversible change of context does not give rise to the realization of any new modality in the context <math display="inline"><semantics> <mrow> <mo>{</mo> <msub> <mi>v</mi> <mi>j</mi> </msub> <mo>}</mo> </mrow> </semantics></math>. In that case, one has <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mrow> <msub> <mi>u</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">|</mo> </mrow> <msub> <mi>u</mi> <mi>i</mi> </msub> </mrow> </msub> <mo>=</mo> <msub> <mi>δ</mi> <mrow> <mi>k</mi> <mi>i</mi> </mrow> </msub> </mrow> </semantics></math>, corresponding to a sum of probability amplitudes (see text). The above figure in the reversible case can be seen as an interferometer, where the <math display="inline"><semantics> <mrow> <mo>{</mo> <msub> <mi>v</mi> <mi>j</mi> </msub> <mo>}</mo> </mrow> </semantics></math> modalities correspond to “which-path” results. Full article ">Figure 2
General way to change the context and come back, by entangling the initial system with another system, considered as the meter. To recover the previous cases, the system–meter interaction must be a QND interaction in the context <math display="inline"><semantics> <mrow> <mo>{</mo> <msub> <mi>v</mi> <mi>j</mi> </msub> <mo>}</mo> </mrow> </semantics></math> (see text). Depending on the strength of this interaction, one can recover real, virtual, and “weak” measurements. Full article ">

13 pages, 277 KiB

Open AccessArticle

Probability Theory as a Physical Theory Points to Superdeterminism

by Louis Vervoort

Entropy 2019, 21(9), 848; https://doi.org/10.3390/e21090848 - 30 Aug 2019

Cited by 10 | Viewed by 5735

Abstract

Probability theory as a physical theory is, in a sense, the most general physics theory available, more encompassing than relativity theory and quantum mechanics, which comply with probability theory. Taking this simple fact seriously, I argue that probability theory points towards superdeterminism, a [...] Read more.

Probability theory as a physical theory is, in a sense, the most general physics theory available, more encompassing than relativity theory and quantum mechanics, which comply with probability theory. Taking this simple fact seriously, I argue that probability theory points towards superdeterminism, a principle that underlies, notably, ‘t Hooft’s Cellular Automaton Interpretation of quantum mechanics. Specifically, I argue that superdeterminism offers a solution for: (1) Kolmogorov’s problem of probabilistic dependence; (2) the interpretation of the Central Limit Theorem; and (3) Bell’s theorem. Superdeterminism’s competitor, indeterminism (“no hidden variables”), remains entirely silent regarding (1) and (2), and leaves (3) as an obstacle rather than a solution for the unification of quantum mechanics and general relativity. This suggests that, if one wishes to stick to the standard position in physics and adopt the principles with the highest explanatory power, one should adopt superdeterminism and reject indeterminism. Throughout the article precise questions to mathematicians are formulated to advance this research. Full article

(This article belongs to the Special Issue Quantum Information Revolution: Impact to Foundations)

15 pages, 298 KiB

Open AccessArticle

Get Rid of Nonlocality from Quantum Physics

by Andrei Khrennikov

Entropy 2019, 21(8), 806; https://doi.org/10.3390/e21080806 - 18 Aug 2019

Cited by 48 | Viewed by 5081

Abstract

This paper is aimed to dissociate nonlocality from quantum theory. We demonstrate that the tests on violation of the Bell type inequalities are simply statistical tests of local incompatibility of observables. In fact, these are tests on violation of the Bohr complementarity principle. [...] Read more.

This paper is aimed to dissociate nonlocality from quantum theory. We demonstrate that the tests on violation of the Bell type inequalities are simply statistical tests of local incompatibility of observables. In fact, these are tests on violation of the Bohr complementarity principle. Thus, the attempts to couple experimental violations of the Bell type inequalities with “quantum nonlocality” is really misleading. These violations are explained in the quantum theory as exhibitions of incompatibility of observables for a single quantum system, e.g., the spin projections for a single electron or the polarization projections for a single photon. Of course, one can go beyond quantum theory with the hidden variables models (as was suggested by Bell) and then discuss their possible nonlocal features. However, conventional quantum theory is local. Full article

(This article belongs to the Special Issue Quantum Information Revolution: Impact to Foundations)

21 pages, 544 KiB

Open AccessEditor’s ChoiceArticle

PT Symmetry, Non-Gaussian Path Integrals, and the Quantum Black–Scholes Equation

by Will Hicks

Entropy 2019, 21(2), 105; https://doi.org/10.3390/e21020105 - 23 Jan 2019

Cited by 6 | Viewed by 4448

Abstract

The Accardi–Boukas quantum Black–Scholes framework, provides a means by which one can apply the Hudson–Parthasarathy quantum stochastic calculus to problems in finance. Solutions to these equations can be modelled using nonlocal diffusion processes, via a Kramers–Moyal expansion, and this provides useful tools to [...] Read more.

The Accardi–Boukas quantum Black–Scholes framework, provides a means by which one can apply the Hudson–Parthasarathy quantum stochastic calculus to problems in finance. Solutions to these equations can be modelled using nonlocal diffusion processes, via a Kramers–Moyal expansion, and this provides useful tools to understand their behaviour. In this paper we develop further links between quantum stochastic processes, and nonlocal diffusions, by inverting the question, and showing how certain nonlocal diffusions can be written as quantum stochastic processes. We then go on to show how one can use path integral formalism, and

PT

symmetric quantum mechanics, to build a non-Gaussian kernel function for the Accardi–Boukas quantum Black–Scholes. Behaviours observed in the real market are a natural model output, rather than something that must be deliberately included. Full article

(This article belongs to the Special Issue Quantum Information Revolution: Impact to Foundations)

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13 pages, 259 KiB

Open AccessArticle

Logical Structures Underlying Quantum Computing

by Federico Holik, Giuseppe Sergioli, Hector Freytes and Angel Plastino

Entropy 2019, 21(1), 77; https://doi.org/10.3390/e21010077 - 16 Jan 2019

Cited by 6 | Viewed by 4271

Abstract

In this work we advance a generalization of quantum computational logics capable of dealing with some important examples of quantum algorithms. We outline an algebraic axiomatization of these structures. Full article

(This article belongs to the Special Issue Quantum Information Revolution: Impact to Foundations)

Review

Jump to: Research, Other

11 pages, 1108 KiB

Open AccessFeature PaperEditor’s ChoiceReview

Some Notes on Counterfactuals in Quantum Mechanics

by Avshalom C. Elitzur and Eliahu Cohen

Entropy 2020, 22(3), 266; https://doi.org/10.3390/e22030266 - 26 Feb 2020

Cited by 1 | Viewed by 3998

Abstract

Counterfactuals, i.e., events that could have occurred but eventually did not, play a unique role in quantum mechanics in that they exert causal effects despite their non-occurrence. They are therefore vital for a better understanding of quantum mechanics (QM) and possibly the universe [...] Read more.

Counterfactuals, i.e., events that could have occurred but eventually did not, play a unique role in quantum mechanics in that they exert causal effects despite their non-occurrence. They are therefore vital for a better understanding of quantum mechanics (QM) and possibly the universe as a whole. In earlier works, we have studied counterfactuals both conceptually and experimentally. A fruitful framework termed quantum oblivion has emerged, referring to situations where one particle seems to "forget" its interaction with other particles despite the latter being visibly affected. This framework proved to have significant explanatory power, which we now extend to tackle additional riddles. The time-symmetric causality employed by the Two State-Vector Formalism (TSVF) reveals a subtle realm ruled by “weak values,” already demonstrated by numerous experiments. They offer a realistic, simple and intuitively appealing explanation to the unique role of quantum non-events, as well as to the foundations of QM. In this spirit, we performed a weak value analysis of quantum oblivion and suggest some new avenues for further research. Full article

(This article belongs to the Special Issue Quantum Information Revolution: Impact to Foundations)

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Figure 1

Figure 1
Possible electron–positron interactions and their outcomes. (a) The setting. (b,c) Annihilation. (d) Oblivion. Full article ">Figure 2
The two vectors taking place in the nested MZI and their joint prediction. (a) Forward state-vector (blue lines) with full and “empty” paths. (b) Backwards state-vector (red), again with full and “empty” paths. (c) The overlap between the pre- and post-selected states gives rise to an odd trajectory (purple) which harbors a short-lived particle in the middle of the “empty” path. Full article ">Figure 2 Cont.
The two vectors taking place in the nested MZI and their joint prediction. (a) Forward state-vector (blue lines) with full and “empty” paths. (b) Backwards state-vector (red), again with full and “empty” paths. (c) The overlap between the pre- and post-selected states gives rise to an odd trajectory (purple) which harbors a short-lived particle in the middle of the “empty” path. Full article ">Figure 3
The oblivion experiment analyzed by the Two State-Vector Formalism (TSVF). (a) Weak values corresponding to the presence of the electron and positron at several positions are denoted in boldface for <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="bold-italic">t</mi> <mn mathvariant="bold">0</mn> </msub> <mo><</mo> <mi mathvariant="bold-italic">t</mi> <mo><</mo> <msub> <mi mathvariant="bold-italic">t</mi> <mn mathvariant="bold">1</mn> </msub> </mrow> </semantics></math> (b) Similarly, for the times before post-selection, in both cases anomalous weak values emerge which correspond to the whereabouts of the particles. Full article ">

Other

Jump to: Research, Review

21 pages, 313 KiB

Open AccessFeature PaperDiscussion

Does Geometric Algebra Provide a Loophole to Bell’s Theorem?

by Richard David Gill

Entropy 2020, 22(1), 61; https://doi.org/10.3390/e22010061 - 31 Dec 2019

Cited by 18 | Viewed by 5646 | Correction

Abstract

In 2007, and in a series of later papers, Joy Christian claimed to refute Bell’s theorem, presenting an alleged local realistic model of the singlet correlations using techniques from geometric algebra (GA). Several authors published papers refuting his claims, and Christian’s ideas did [...] Read more.

In 2007, and in a series of later papers, Joy Christian claimed to refute Bell’s theorem, presenting an alleged local realistic model of the singlet correlations using techniques from geometric algebra (GA). Several authors published papers refuting his claims, and Christian’s ideas did not gain acceptance. However, he recently succeeded in publishing yet more ambitious and complex versions of his theory in fairly mainstream journals. How could this be? The mathematics and logic of Bell’s theorem is simple and transparent and has been intensely studied and debated for over 50 years. Christian claims to have a mathematical counterexample to a purely mathematical theorem. Each new version of Christian’s model used new devices to circumvent Bell’s theorem or depended on a new way to misunderstand Bell’s work. These devices and misinterpretations are in common use by other Bell critics, so it useful to identify and name them. I hope that this paper can serve as a useful resource to those who need to evaluate new “disproofs of Bell’s theorem”. Christian’s fundamental idea is simple and quite original: he gives a probabilistic interpretation of the fundamental GA equation

a \cdot b = (a b + b a) / 2

. After that, ambiguous notation and technical complexity allows sign errors to be hidden from sight, and new mathematical errors can be introduced. Full article

(This article belongs to the Special Issue Quantum Information Revolution: Impact to Foundations)

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