Research

1152 KiB

Open AccessArticle

Recall Performance for Content-Addressable Memory Using Adiabatic Quantum Optimization

by Jonathan Schrock, Alex J. McCaskey, Kathleen E. Hamilton, Travis S. Humble and Neena Imam

Entropy 2017, 19(9), 500; https://doi.org/10.3390/e19090500 - 15 Sep 2017

Cited by 6 | Viewed by 3860

A content-addressable memory (CAM) stores key-value associations such that the key is recalled by providing its associated value. While CAM recall is traditionally performed using recurrent neural network models, we show how to solve this problem using adiabatic quantum optimization. Our approach maps [...] Read more.

A content-addressable memory (CAM) stores key-value associations such that the key is recalled by providing its associated value. While CAM recall is traditionally performed using recurrent neural network models, we show how to solve this problem using adiabatic quantum optimization. Our approach maps the recurrent neural network to a commercially available quantum processing unit by taking advantage of the common underlying Ising spin model. We then assess the accuracy of the quantum processor to store key-value associations by quantifying recall performance against an ensemble of problem sets. We observe that different learning rules from the neural network community influence recall accuracy but performance appears to be limited by potential noise in the processor. The strong connection established between quantum processors and neural network problems supports the growing intersection of these two ideas. Full article

(This article belongs to the Special Issue Foundations of Quantum Mechanics)

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489 KiB

Open AccessFeature PaperArticle

On Generalized Stam Inequalities and Fisher–Rényi Complexity Measures

by Steeve Zozor, David Puertas-Centeno and Jesús S. Dehesa

Entropy 2017, 19(9), 493; https://doi.org/10.3390/e19090493 - 14 Sep 2017

Cited by 9 | Viewed by 3462

Abstract

Information-theoretic inequalities play a fundamental role in numerous scientific and technological areas (e.g., estimation and communication theories, signal and information processing, quantum physics, …) as they generally express the impossibility to have a complete description of a system via a finite number of [...] Read more.

Information-theoretic inequalities play a fundamental role in numerous scientific and technological areas (e.g., estimation and communication theories, signal and information processing, quantum physics, …) as they generally express the impossibility to have a complete description of a system via a finite number of information measures. In particular, they gave rise to the design of various quantifiers (statistical complexity measures) of the internal complexity of a (quantum) system. In this paper, we introduce a three-parametric Fisher–Rényi complexity, named

(p, β, λ)

-Fisher–Rényi complexity, based on both a two-parametic extension of the Fisher information and the Rényi entropies of a probability density function

ρ

characteristic of the system. This complexity measure quantifies the combined balance of the spreading and the gradient contents of

ρ

, and has the three main properties of a statistical complexity: the invariance under translation and scaling transformations, and a universal bounding from below. The latter is proved by generalizing the Stam inequality, which lowerbounds the product of the Shannon entropy power and the Fisher information of a probability density function. An extension of this inequality was already proposed by Bercher and Lutwak, a particular case of the general one, where the three parameters are linked, allowing to determine the sharp lower bound and the associated probability density with minimal complexity. Using the notion of differential-escort deformation, we are able to determine the sharp bound of the complexity measure even when the three parameters are decoupled (in a certain range). We determine as well the distribution that saturates the inequality: the

(p, β, λ)

-Gaussian distribution, which involves an inverse incomplete beta function. Finally, the complexity measure is calculated for various quantum-mechanical states of the harmonic and hydrogenic systems, which are the two main prototypes of physical systems subject to a central potential. Full article

(This article belongs to the Special Issue Foundations of Quantum Mechanics)

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314 KiB

Open AccessArticle

Contextuality and Indistinguishability

by José Acacio De Barros, Federico Holik and Décio Krause

Entropy 2017, 19(9), 435; https://doi.org/10.3390/e19090435 - 23 Aug 2017

Cited by 14 | Viewed by 3619

Abstract

It is well known that in quantum mechanics we cannot always define consistently properties that are context independent. Many approaches exist to describe contextual properties, such as Contextuality by Default (CbD), sheaf theory, topos theory, and non-standard or signed probabilities. In this paper, [...] Read more.

It is well known that in quantum mechanics we cannot always define consistently properties that are context independent. Many approaches exist to describe contextual properties, such as Contextuality by Default (CbD), sheaf theory, topos theory, and non-standard or signed probabilities. In this paper, we propose a treatment of contextual properties that is specific to quantum mechanics, as it relies on the relationship between contextuality and indistinguishability. In particular, we propose that if we assume the ontological thesis that quantum particles or properties can be indistinguishable yet different, no contradiction arising from a Kochen–Specker-type argument appears: when we repeat an experiment, we are in reality performing an experiment measuring a property that is indistinguishable from the first, but not the same. We will discuss how the consequences of this move may help us understand quantum contextuality. Full article

(This article belongs to the Special Issue Foundations of Quantum Mechanics)

1002 KiB

Open AccessArticle

A Quantum Description of the Stern–Gerlach Experiment

by Håkan Wennerström and Per-Olof Westlund

Entropy 2017, 19(5), 186; https://doi.org/10.3390/e19050186 - 25 Apr 2017

Cited by 8 | Viewed by 11019

Abstract

A detailed analysis of the classic Stern–Gerlach experiment is presented. An analytical simple solution is presented for the quantum description of the translational and spin dynamics of a silver atom in a magnetic field with a gradient along a single z-direction. This [...] Read more.

A detailed analysis of the classic Stern–Gerlach experiment is presented. An analytical simple solution is presented for the quantum description of the translational and spin dynamics of a silver atom in a magnetic field with a gradient along a single z-direction. This description is then used to obtain an approximate quantum description of the more realistic case with a magnetic field gradient also in a second y-direction. An explicit relation is derived for how an initial off center deviation in the y-direction affects the final result observed at the detector. This shows that the “mouth shape” pattern at the detector observed in the original Stern–Gerlach experiment is a generic consequence of the gradient in the y-direction. This is followed by a discussion of the spin dynamics during the entry of the silver atom into the magnet. An analytical relation is derived for a simplified case of a field only along the z-direction. A central question for the conceptual understanding of the Stern–Gerlach experiment has been how an initially unpolarized spin ends up in a polarized state at the detector. It is argued that this can be understood with the use of the adiabatic approximation. When the atoms first experience the magnetic field outside the magnet, there is in general a change in the spin state, which transforms from a degenerate eigenstate in the absence of a field into one of two possible non-degenerate states in the field. If the direction of the field changes during the passage through the device, there is a corresponding adiabatic change of the spin state. It is shown that an application of the adiabatic approximation in this way is consistent with the previously derived exact relations. Full article

(This article belongs to the Special Issue Foundations of Quantum Mechanics)

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1780 KiB

Open AccessArticle

Using Measured Values in Bell’s Inequalities Entails at Least One Hypothesis in Addition to Local Realism

by Alejandro Andrés Hnilo

Entropy 2017, 19(4), 180; https://doi.org/10.3390/e19040180 - 22 Apr 2017

Cited by 3 | Viewed by 4499

Abstract

The recent loophole-free experiments have confirmed the violation of Bell’s inequalities in nature. Yet, in order to insert measured values in Bell’s inequalities, it is unavoidable to make a hypothesis similar to “ergodicity at the hidden variables level”. This possibility opens a promising [...] Read more.

The recent loophole-free experiments have confirmed the violation of Bell’s inequalities in nature. Yet, in order to insert measured values in Bell’s inequalities, it is unavoidable to make a hypothesis similar to “ergodicity at the hidden variables level”. This possibility opens a promising way out from the old controversy between quantum mechanics and local realism. Here, I review the reason why such a hypothesis (actually, it is one of a set of related hypotheses) in addition to local realism is necessary, and present a simple example, related to Bell’s inequalities, where the hypothesis is violated. This example shows that the violation of the additional hypothesis is necessary, but not sufficient, to violate Bell’s inequalities without violating local realism. The example also provides some clues that may reveal the violation of the additional hypothesis in an experiment. Full article

(This article belongs to the Special Issue Foundations of Quantum Mechanics)

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329 KiB

Open AccessArticle

Heisenberg and Entropic Uncertainty Measures for Large-Dimensional Harmonic Systems

by David Puertas-Centeno, Irene V. Toranzo and Jesús S. Dehesa

Entropy 2017, 19(4), 164; https://doi.org/10.3390/e19040164 - 09 Apr 2017

Cited by 16 | Viewed by 4035

Abstract

The D-dimensional harmonic system (i.e., a particle moving under the action of a quadratic potential) is, together with the hydrogenic system, the main prototype of the physics of multidimensional quantum systems. In this work, we rigorously determine the leading term of the [...] Read more.

The D-dimensional harmonic system (i.e., a particle moving under the action of a quadratic potential) is, together with the hydrogenic system, the main prototype of the physics of multidimensional quantum systems. In this work, we rigorously determine the leading term of the Heisenberg-like and entropy-like uncertainty measures of this system as given by the radial expectation values and the Rényi entropies, respectively, at the limit of large D. The associated multidimensional position-momentum uncertainty relations are discussed, showing that they saturate the corresponding general ones. A conjecture about the Shannon-like uncertainty relation is given, and an interesting phenomenon is observed: the Heisenberg-like and Rényi-entropy-based equality-type uncertainty relations for all of the D-dimensional harmonic oscillator states in the pseudoclassical (

D \to \infty

) limit are the same as the corresponding ones for the hydrogenic systems, despite the so different character of the oscillator and Coulomb potentials. Full article

(This article belongs to the Special Issue Foundations of Quantum Mechanics)

836 KiB

Open AccessArticle

Quantum Probabilities as Behavioral Probabilities

by Vyacheslav I. Yukalov and Didier Sornette

Entropy 2017, 19(3), 112; https://doi.org/10.3390/e19030112 - 13 Mar 2017

Cited by 28 | Viewed by 5346

Abstract

We demonstrate that behavioral probabilities of human decision makers share many common features with quantum probabilities. This does not imply that humans are some quantum objects, but just shows that the mathematics of quantum theory is applicable to the description of human decision [...] Read more.

We demonstrate that behavioral probabilities of human decision makers share many common features with quantum probabilities. This does not imply that humans are some quantum objects, but just shows that the mathematics of quantum theory is applicable to the description of human decision making. The applicability of quantum rules for describing decision making is connected with the nontrivial process of making decisions in the case of composite prospects under uncertainty. Such a process involves deliberations of a decision maker when making a choice. In addition to the evaluation of the utilities of considered prospects, real decision makers also appreciate their respective attractiveness. Therefore, human choice is not based solely on the utility of prospects, but includes the necessity of resolving the utility-attraction duality. In order to justify that human consciousness really functions similarly to the rules of quantum theory, we develop an approach defining human behavioral probabilities as the probabilities determined by quantum rules. We show that quantum behavioral probabilities of humans do not merely explain qualitatively how human decisions are made, but they predict quantitative values of the behavioral probabilities. Analyzing a large set of empirical data, we find good quantitative agreement between theoretical predictions and observed experimental data. Full article

(This article belongs to the Special Issue Foundations of Quantum Mechanics)

Review

Jump to: Research, Other

920 KiB

Open AccessReview

Born-Kothari Condensation for Fermions

by Arnab Ghosh

Entropy 2017, 19(9), 479; https://doi.org/10.3390/e19090479 - 13 Sep 2017

Cited by 1 | Viewed by 4588

Abstract

In the spirit of Bose–Einstein condensation, we present a detailed account of the statistical description of the condensation phenomena for a Fermi–Dirac gas following the works of Born and Kothari. For bosons, while the condensed phase below a certain critical temperature, permits macroscopic [...] Read more.

In the spirit of Bose–Einstein condensation, we present a detailed account of the statistical description of the condensation phenomena for a Fermi–Dirac gas following the works of Born and Kothari. For bosons, while the condensed phase below a certain critical temperature, permits macroscopic occupation at the lowest energy single particle state, for fermions, due to Pauli exclusion principle, the condensed phase occurs only in the form of a single occupancy dense modes at the highest energy state. In spite of these rudimentary differences, our recent findings [Ghosh and Ray, 2017] identify the foregoing phenomenon as condensation-like coherence among fermions in an analogous way to Bose–Einstein condensate which is collectively described by a coherent matter wave. To reach the above conclusion, we employ the close relationship between the statistical methods of bosonic and fermionic fields pioneered by Cahill and Glauber. In addition to our previous results, we described in this mini-review that the highest momentum (energy) for individual fermions, prerequisite for the condensation process, can be specified in terms of the natural length and energy scales of the problem. The existence of such condensed phases, which are of obvious significance in the context of elementary particles, have also been scrutinized. Full article

(This article belongs to the Special Issue Foundations of Quantum Mechanics)

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3646 KiB

Open AccessReview

Program for the Special State Theory of Quantum Measurement

by Lawrence S. Schulman

Entropy 2017, 19(7), 343; https://doi.org/10.3390/e19070343 - 08 Jul 2017

Cited by 5 | Viewed by 3297

Abstract

Establishing (or falsifying) the special state theory of quantum measurement is a program with both theoretical and experimental directions. The special state theory has only pure unitary time evolution, like the many worlds interpretation, but only has one world. How this can be [...] Read more.

Establishing (or falsifying) the special state theory of quantum measurement is a program with both theoretical and experimental directions. The special state theory has only pure unitary time evolution, like the many worlds interpretation, but only has one world. How this can be accomplished requires both “special states” and significant modification of the usual assumptions about the arrow of time. All this is reviewed below. Experimentally, proposals for tests already exist and the problems are first the practical one of doing the experiment and second the suggesting of other experiments. On the theoretical level, many problems remain and among them are the impact of particle statistics on the availability of special states, finding a way to estimate their abundance and the possibility of using a computer for this purpose. Regarding the arrow of time, there is an early proposal of J. A. Wheeler that may be implementable with implications for cosmology. Full article

(This article belongs to the Special Issue Foundations of Quantum Mechanics)

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Other

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364 KiB

Open AccessConcept Paper

About the Concept of Quantum Chaos

by Ignacio S. Gomez, Marcelo Losada and Olimpia Lombardi

Entropy 2017, 19(5), 205; https://doi.org/10.3390/e19050205 - 03 May 2017

Cited by 11 | Viewed by 6058

Abstract

The research on quantum chaos finds its roots in the study of the spectrum of complex nuclei in the 1950s and the pioneering experiments in microwave billiards in the 1970s. Since then, a large number of new results was produced. Nevertheless, the work [...] Read more.

The research on quantum chaos finds its roots in the study of the spectrum of complex nuclei in the 1950s and the pioneering experiments in microwave billiards in the 1970s. Since then, a large number of new results was produced. Nevertheless, the work on the subject is, even at present, a superposition of several approaches expressed in different mathematical formalisms and weakly linked to each other. The purpose of this paper is to supply a unified framework for describing quantum chaos using the quantum ergodic hierarchy. Using the factorization property of this framework, we characterize the dynamical aspects of quantum chaos by obtaining the Ehrenfest time. We also outline a generalization of the quantum mixing level of the kicked rotator in the context of the impulsive differential equations. Full article

(This article belongs to the Special Issue Foundations of Quantum Mechanics)

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