Editorial

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3 pages, 176 KiB

Open AccessEditorial

Special Issue “Quantum Foundations: 90 Years of Uncertainty”

by Gustavo M. Bosyk, Sebastian Fortin, Pedro W. Lamberti and Federico Holik

Entropy 2019, 21(2), 159; https://doi.org/10.3390/e21020159 - 08 Feb 2019

Cited by 1 | Viewed by 2109

Abstract

The VII Conference on Quantum Foundations: 90 years of uncertainty (https://sites [...] Full article

(This article belongs to the Special Issue Quantum Foundations: 90 Years of Uncertainty)

Research

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15 pages, 289 KiB

Open AccessArticle

Evaluating the Maximal Violation of the Original Bell Inequality by Two-Qudit States Exhibiting Perfect Correlations/Anticorrelations

by Andrei Y. Khrennikov and Elena R. Loubenets

Entropy 2018, 20(11), 829; https://doi.org/10.3390/e20110829 - 29 Oct 2018

Cited by 9 | Viewed by 2291

Abstract

We introduce the general class of symmetric two-qubit states guaranteeing the perfect correlation or anticorrelation of Alice and Bob outcomes whenever some spin observable is measured at both sites. We prove that, for all states from this class, the maximal violation of the [...] Read more.

We introduce the general class of symmetric two-qubit states guaranteeing the perfect correlation or anticorrelation of Alice and Bob outcomes whenever some spin observable is measured at both sites. We prove that, for all states from this class, the maximal violation of the original Bell inequality is upper bounded by

\frac{3}{2}

and specify the two-qubit states where this quantum upper bound is attained. The case of two-qutrit states is more complicated. Here, for all two-qutrit states, we obtain the same upper bound

\frac{3}{2}

for violation of the original Bell inequality under Alice and Bob spin measurements, but we have not yet been able to show that this quantum upper bound is the least one. We discuss experimental consequences of our mathematical study. Full article

(This article belongs to the Special Issue Quantum Foundations: 90 Years of Uncertainty)

19 pages, 957 KiB

Open AccessArticle

Revisiting Entanglement within the Bohmian Approach to Quantum Mechanics

by Claudia Zander and Angel Ricardo Plastino

Entropy 2018, 20(6), 473; https://doi.org/10.3390/e20060473 - 18 Jun 2018

Cited by 12 | Viewed by 4049

Abstract

We revisit the concept of entanglement within the Bohmian approach to quantum mechanics. Inspired by Bohmian dynamics, we introduce two partial measures for the amount of entanglement corresponding to a pure state of a pair of quantum particles. One of these measures is [...] Read more.

We revisit the concept of entanglement within the Bohmian approach to quantum mechanics. Inspired by Bohmian dynamics, we introduce two partial measures for the amount of entanglement corresponding to a pure state of a pair of quantum particles. One of these measures is associated with the statistical correlations exhibited by the joint probability density of the two Bohmian particles in configuration space. The other partial measure corresponds to the correlations associated with the phase of the joint wave function, and describes the non-separability of the Bohmian velocity field. The sum of these two components is equal to the total entanglement of the joint quantum state, as measured by the linear entropy of the single-particle reduced density matrix. Full article

(This article belongs to the Special Issue Quantum Foundations: 90 Years of Uncertainty)

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22 pages, 615 KiB

Open AccessArticle

New Forms of Quantum Value Indefiniteness Suggest That Incompatible Views on Contexts Are Epistemic

by Karl Svozil

Entropy 2018, 20(6), 406; https://doi.org/10.3390/e20060406 - 24 May 2018

Cited by 11 | Viewed by 3633

Abstract

Extensions of the Kochen–Specker theorem use quantum logics whose classical interpretation suggests a true-implies-value indefiniteness property. This can be interpreted as an indication that any view of a quantum state beyond a single context is epistemic. A remark by Gleason about the ad [...] Read more.

Extensions of the Kochen–Specker theorem use quantum logics whose classical interpretation suggests a true-implies-value indefiniteness property. This can be interpreted as an indication that any view of a quantum state beyond a single context is epistemic. A remark by Gleason about the ad hoc construction of probability measures in Hilbert spaces as a result of the Pythagorean property of vector components is interpreted platonically. Unless there is a total match between preparation and measurement contexts, information about the former from the latter is not ontic, but epistemic. This is corroborated by configurations of observables and contexts with a truth-implies-value indefiniteness property. Full article

(This article belongs to the Special Issue Quantum Foundations: 90 Years of Uncertainty)

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28 pages, 1603 KiB

Open AccessArticle

Adiabatic Quantum Computation Applied to Deep Learning Networks

by Jeremy Liu, Federico M. Spedalieri, Ke-Thia Yao, Thomas E. Potok, Catherine Schuman, Steven Young, Robert Patton, Garrett S. Rose and Gangotree Chamka

Entropy 2018, 20(5), 380; https://doi.org/10.3390/e20050380 - 18 May 2018

Cited by 16 | Viewed by 9727

Abstract

Training deep learning networks is a difficult task due to computational complexity, and this is traditionally handled by simplifying network topology to enable parallel computation on graphical processing units (GPUs). However, the emergence of quantum devices allows reconsideration of complex topologies. We illustrate [...] Read more.

Training deep learning networks is a difficult task due to computational complexity, and this is traditionally handled by simplifying network topology to enable parallel computation on graphical processing units (GPUs). However, the emergence of quantum devices allows reconsideration of complex topologies. We illustrate a particular network topology that can be trained to classify MNIST data (an image dataset of handwritten digits) and neutrino detection data using a restricted form of adiabatic quantum computation known as quantum annealing performed by a D-Wave processor. We provide a brief description of the hardware and how it solves Ising models, how we translate our data into the corresponding Ising models, and how we use available expanded topology options to explore potential performance improvements. Although we focus on the application of quantum annealing in this article, the work discussed here is just one of three approaches we explored as part of a larger project that considers alternative means for training deep learning networks. The other approaches involve using a high performance computing (HPC) environment to automatically find network topologies with good performance and using neuromorphic computing to find a low-power solution for training deep learning networks. Our results show that our quantum approach can find good network parameters in a reasonable time despite increased network topology complexity; that HPC can find good parameters for traditional, simplified network topologies; and that neuromorphic computers can use low power memristive hardware to represent complex topologies and parameters derived from other architecture choices. Full article

(This article belongs to the Special Issue Quantum Foundations: 90 Years of Uncertainty)

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15 pages, 307 KiB

Open AccessArticle

Entropic Uncertainty Relations for Successive Measurements in the Presence of a Minimal Length

by Alexey E. Rastegin

Entropy 2018, 20(5), 354; https://doi.org/10.3390/e20050354 - 09 May 2018

Cited by 6 | Viewed by 2992

Abstract

We address the generalized uncertainty principle in scenarios of successive measurements. Uncertainties are characterized by means of generalized entropies of both the Rényi and Tsallis types. Here, specific features of measurements of observables with continuous spectra should be taken into account. First, we [...] Read more.

We address the generalized uncertainty principle in scenarios of successive measurements. Uncertainties are characterized by means of generalized entropies of both the Rényi and Tsallis types. Here, specific features of measurements of observables with continuous spectra should be taken into account. First, we formulated uncertainty relations in terms of Shannon entropies. Since such relations involve a state-dependent correction term, they generally differ from preparation uncertainty relations. This difference is revealed when the position is measured by the first. In contrast, state-independent uncertainty relations in terms of Rényi and Tsallis entropies are obtained with the same lower bounds as in the preparation scenario. These bounds are explicitly dependent on the acceptance function of apparatuses in momentum measurements. Entropic uncertainty relations with binning are discussed as well. Full article

(This article belongs to the Special Issue Quantum Foundations: 90 Years of Uncertainty)

22 pages, 7322 KiB

Open AccessArticle

Quantization and Bifurcation beyond Square-Integrable Wavefunctions

by Ciann-Dong Yang and Chung-Hsuan Kuo

Entropy 2018, 20(5), 327; https://doi.org/10.3390/e20050327 - 29 Apr 2018

Cited by 1 | Viewed by 4374

Abstract

Probability interpretation is the cornerstone of standard quantum mechanics. To ensure the validity of the probability interpretation, wavefunctions have to satisfy the square-integrable (SI) condition, which gives rise to the well-known phenomenon of energy quantization in confined quantum systems. On the other hand, [...] Read more.

Probability interpretation is the cornerstone of standard quantum mechanics. To ensure the validity of the probability interpretation, wavefunctions have to satisfy the square-integrable (SI) condition, which gives rise to the well-known phenomenon of energy quantization in confined quantum systems. On the other hand, nonsquare-integrable (NSI) solutions to the Schrödinger equation are usually ruled out and have long been believed to be irrelevant to energy quantization. This paper proposes a quantum-trajectory approach to energy quantization by releasing the SI condition and considering both SI and NSI solutions to the Schrödinger equation. Contrary to our common belief, we find that both SI and NSI wavefunctions contribute to energy quantization. SI wavefunctions help to locate the bifurcation points at which energy has a step jump, while NSI wavefunctions form the flat parts of the stair-like distribution of the quantized energies. The consideration of NSI wavefunctions furthermore reveals a new quantum phenomenon regarding the synchronicity between the energy quantization process and the center-saddle bifurcation process. Full article

(This article belongs to the Special Issue Quantum Foundations: 90 Years of Uncertainty)

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10 pages, 254 KiB

Open AccessArticle

Gudder’s Theorem and the Born Rule

by Francisco De Zela

Entropy 2018, 20(3), 158; https://doi.org/10.3390/e20030158 - 02 Mar 2018

Cited by 4 | Viewed by 3207

Abstract

We derive the Born probability rule from Gudder’s theorem—a theorem that addresses orthogonally-additive functions. These functions are shown to be tightly connected to the functions that enter the definition of a signed measure. By imposing some additional requirements besides orthogonal additivity, the addressed [...] Read more.

We derive the Born probability rule from Gudder’s theorem—a theorem that addresses orthogonally-additive functions. These functions are shown to be tightly connected to the functions that enter the definition of a signed measure. By imposing some additional requirements besides orthogonal additivity, the addressed functions are proved to be linear, so they can be given in terms of an inner product. By further restricting them to act on projectors, Gudder’s functions are proved to act as probability measures obeying Born’s rule. The procedure does not invoke any property that fully lies within the quantum framework, so Born’s rule is shown to apply within both the classical and the quantum domains. Full article

(This article belongs to the Special Issue Quantum Foundations: 90 Years of Uncertainty)

9 pages, 426 KiB

Open AccessArticle

Uncertainty Relation Based on Wigner–Yanase–Dyson Skew Information with Quantum Memory

by Jun Li and Shao-Ming Fei

Entropy 2018, 20(2), 132; https://doi.org/10.3390/e20020132 - 20 Feb 2018

Cited by 8 | Viewed by 4108

Abstract

We present uncertainty relations based on Wigner–Yanase–Dyson skew information with quantum memory. Uncertainty inequalities both in product and summation forms are derived. It is shown that the lower bounds contain two terms: one characterizes the degree of compatibility of two measurements, and the [...] Read more.

We present uncertainty relations based on Wigner–Yanase–Dyson skew information with quantum memory. Uncertainty inequalities both in product and summation forms are derived. It is shown that the lower bounds contain two terms: one characterizes the degree of compatibility of two measurements, and the other is the quantum correlation between the measured system and the quantum memory. Detailed examples are given for product, separable and entangled states. Full article

(This article belongs to the Special Issue Quantum Foundations: 90 Years of Uncertainty)

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Review

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36 pages, 1391 KiB

Open AccessReview

Uncertainty Relations for Coarse–Grained Measurements: An Overview

by Fabricio Toscano, Daniel S. Tasca, Łukasz Rudnicki and Stephen P. Walborn

Entropy 2018, 20(6), 454; https://doi.org/10.3390/e20060454 - 10 Jun 2018

Cited by 18 | Viewed by 5797

Abstract

Uncertainty relations involving incompatible observables are one of the cornerstones of quantum mechanics. Aside from their fundamental significance, they play an important role in practical applications, such as detection of quantum correlations and security requirements in quantum cryptography. In continuous variable systems, the [...] Read more.

Uncertainty relations involving incompatible observables are one of the cornerstones of quantum mechanics. Aside from their fundamental significance, they play an important role in practical applications, such as detection of quantum correlations and security requirements in quantum cryptography. In continuous variable systems, the spectra of the relevant observables form a continuum and this necessitates the coarse graining of measurements. However, these coarse-grained observables do not necessarily obey the same uncertainty relations as the original ones, a fact that can lead to false results when considering applications. That is, one cannot naively replace the original observables in the uncertainty relation for the coarse-grained observables and expect consistent results. As such, several uncertainty relations that are specifically designed for coarse-grained observables have been developed. In recognition of the 90th anniversary of the seminal Heisenberg uncertainty relation, celebrated last year, and all the subsequent work since then, here we give a review of the state of the art of coarse-grained uncertainty relations in continuous variable quantum systems, as well as their applications to fundamental quantum physics and quantum information tasks. Our review is meant to be balanced in its content, since both theoretical considerations and experimental perspectives are put on an equal footing. Full article

(This article belongs to the Special Issue Quantum Foundations: 90 Years of Uncertainty)

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