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Open AccessArticle

Quantum–Classical Hybrid Systems and Ehrenfest’s Theorem

by

,

,

and

Entropy 2023, 25(4), 602; https://doi.org/10.3390/e25040602 - 01 Apr 2023

Viewed by 474

Abstract

The conceptual analysis of quantum mechanics brings to light that a theory inherently consistent with observations should be able to describe both quantum and classical systems, i.e., quantum–classical hybrids. For example, the orthodox interpretation of measurements requires the transient creation of quantum–classical hybrids. [...] Read more.

The conceptual analysis of quantum mechanics brings to light that a theory inherently consistent with observations should be able to describe both quantum and classical systems, i.e., quantum–classical hybrids. For example, the orthodox interpretation of measurements requires the transient creation of quantum–classical hybrids. Despite its limitations in defining the classical limit, Ehrenfest’s theorem makes the simplest contact between quantum and classical mechanics. Here, we generalized the Ehrenfest theorem to bipartite quantum systems. To study quantum–classical hybrids, we employed a formalism based on operator-valued Wigner functions and quantum–classical brackets. We used this approach to derive the form of the Ehrenfest theorem for quantum–classical hybrids. We found that the time variation of the average energy of each component of the bipartite system is equal to the average of the symmetrized quantum dissipated power in both the quantum and the quantum–classical case. We expect that these theoretical results will be useful both to analyze quantum–classical hybrids and to develop self-consistent numerical algorithms for Ehrenfest-type simulations. Full article

(This article belongs to the Special Issue Quantum Mechanics and Its Foundations III)

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Open AccessFeature PaperArticle

Adiabatic Amplification of Energy and Magnetic Moment of a Charged Particle after the Magnetic Field Inversion

by

Viktor V. Dodonov

and

Alexandre V. Dodonov

Entropy 2023, 25(4), 596; https://doi.org/10.3390/e25040596 - 31 Mar 2023

Viewed by 473

Abstract

We study the evolution of the energy and magnetic moment of a quantum charged particle placed in a homogeneous magnetic field, when this field changes its sign adiabatically. We show that after a single magnetic field passage through zero value, the famous adiabatic [...] Read more.

We study the evolution of the energy and magnetic moment of a quantum charged particle placed in a homogeneous magnetic field, when this field changes its sign adiabatically. We show that after a single magnetic field passage through zero value, the famous adiabatic invariant ratio of energy to frequency is reestablished again, but with a proportionality coefficient higher than in the initial state. The concrete value of this proportionality coefficient depends on the power index of the frequency dependence on time near zero point. In particular, the adiabatic ratio of the initial ground state (with zero radial and angular quantum numbers) triplicates if the frequency tends to zero linearly as a function of time. If the Larmor frequency attains zero more than once, the adiabatic proportionality coefficient strongly depends on the lengths of the time intervals between zero points, so that the mean energy behavior can be quasi-stochastic after many passages through zero value. The original Born–Fock adiabatic theorem does not work after the frequency passes through zero. However, its generalization is found: the initial Fock state becomes a wide superposition of many instantaneous Fock states, whose weights do not depend on time in the new adiabatic regime. Full article

(This article belongs to the Special Issue Quantum Mechanics and Its Foundations III)

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Open AccessArticle

Uncertainty Relations in the Madelung Picture Including a Dissipative Environment

by

Dieter Schuch

and

Moise Bonilla-Licea

Entropy 2023, 25(2), 312; https://doi.org/10.3390/e25020312 - 08 Feb 2023

Viewed by 439

Abstract

In a recent paper, we have shown how in Madelung’s hydrodynamic formulation of quantum mechanics, the uncertainties are related to the phase and amplitude of the complex wave function. Now we include a dissipative environment via a nonlinear modified Schrödinger equation. The effect [...] Read more.

In a recent paper, we have shown how in Madelung’s hydrodynamic formulation of quantum mechanics, the uncertainties are related to the phase and amplitude of the complex wave function. Now we include a dissipative environment via a nonlinear modified Schrödinger equation. The effect of the environment is described by a complex logarithmic nonlinearity that vanishes on average. Nevertheless, there are various changes in the dynamics of the uncertainties originating from the nonlinear term. Again, this is illustrated explicitly using generalized coherent states as examples. With particular focus on the quantum mechanical contribution to the energy and the uncertainty product, connections can be made with the thermodynamic properties of the environment. Full article

(This article belongs to the Special Issue Quantum Mechanics and Its Foundations III)

Open AccessArticle

Hermitian and Unitary Almost-Companion Matrices of Polynomials on Demand

by

Liubov A. Markovich

,

Agostino Migliore

and

Antonino Messina

Entropy 2023, 25(2), 309; https://doi.org/10.3390/e25020309 - 08 Feb 2023

Viewed by 606

Abstract

We introduce the concept of the almost-companion matrix (ACM) by relaxing the non-derogatory property of the standard companion matrix (CM). That is, we define an ACM as a matrix whose characteristic polynomial coincides with a given monic and generally complex polynomial. The greater [...] Read more.

We introduce the concept of the almost-companion matrix (ACM) by relaxing the non-derogatory property of the standard companion matrix (CM). That is, we define an ACM as a matrix whose characteristic polynomial coincides with a given monic and generally complex polynomial. The greater flexibility inherent in the ACM concept, compared to CM, allows the construction of ACMs that have convenient matrix structures satisfying desired additional conditions, compatibly with specific properties of the polynomial coefficients. We demonstrate the construction of Hermitian and unitary ACMs starting from appropriate third-degree polynomials, with implications for their use in physical-mathematical problems, such as the parameterization of the Hamiltonian, density, or evolution matrix of a qutrit. We show that the ACM provides a means of identifying the properties of a given polynomial and finding its roots. For example, we describe the ACM-based solution of cubic complex algebraic equations without resorting to the use of the Cardano-Dal Ferro formulas. We also show the necessary and sufficient conditions on the coefficients of a polynomial for it to represent the characteristic polynomial of a unitary ACM. The presented approach can be generalized to complex polynomials of higher degrees. Full article

(This article belongs to the Special Issue Quantum Mechanics and Its Foundations III)

Open AccessFeature PaperArticle

Inverted Oscillator Quantum States in the Probability Representation

by

Olga V. Man’ko

and

Vladimir I. Man’ko

Entropy 2023, 25(2), 217; https://doi.org/10.3390/e25020217 - 22 Jan 2023

Cited by 1 | Viewed by 532

Abstract

The quantizer–dequantizer formalism is used to construct the probability representation of quantum system states. Comparison with the probability representation of classical system states is discussed. Examples of probability distributions describing the system of parametric oscillators and inverted oscillators are presented. Full article

(This article belongs to the Special Issue Quantum Mechanics and Its Foundations III)

Open AccessArticle

Nonlocality in Quantum Mechanics Portrayed as a Human Twins’ Metaphor

by

Salomon S. Mizrahi

Entropy 2023, 25(2), 192; https://doi.org/10.3390/e25020192 - 18 Jan 2023

Viewed by 680

Abstract

Avoiding the use of mathematical formalism, this essay exposes the quantum mechanics phenomenon of nonlocality in terms of a metaphor involving human twins, focused on their hands’ dexterity attribute. Full article

(This article belongs to the Special Issue Quantum Mechanics and Its Foundations III)

Open AccessArticle

Locality, Realism, Ergodicity and Randomness in Bell’s Experiment

by

Alejandro Andrés Hnilo

Entropy 2023, 25(1), 160; https://doi.org/10.3390/e25010160 - 13 Jan 2023

Cited by 1 | Viewed by 648

Abstract

Assuming that there is no way of sending signals propagating faster than light and that free will exists, the loophole-free observed violation of Bell’s inequalities demonstrates that at least one of three fundamental hypotheses involved in the derivation and observation of the inequalities [...] Read more.

Assuming that there is no way of sending signals propagating faster than light and that free will exists, the loophole-free observed violation of Bell’s inequalities demonstrates that at least one of three fundamental hypotheses involved in the derivation and observation of the inequalities is false: Locality, Realism, or Ergodicity. An experiment is proposed to obtain some evidence about which one is the false one. It is based on recording the time evolution of the rate of non-random series of outcomes that are generated in a specially designed Bell’s setup. The results of such experiment would be important not only to the foundations of Quantum Mechanics, but they would also have immediate practical impact on the efficient use of quantum-based random number generators and the security of Quantum Key Distribution using entangled states. Full article

(This article belongs to the Special Issue Quantum Mechanics and Its Foundations III)

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Open AccessArticle

Entanglement and Fisher Information for Atoms–Field System in the Presence of Negative Binomial States

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,

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and

Entropy 2022, 24(12), 1817; https://doi.org/10.3390/e24121817 - 13 Dec 2022

Viewed by 680

Abstract

We developed a quantum scheme of two atoms (TAs) and field initially in a negative binomial state (NBS). We displayed and discussed the physical implications of the obtained results in terms of the physical parameters of the model. By considering that the TAs [...] Read more.

We developed a quantum scheme of two atoms (TAs) and field initially in a negative binomial state (NBS). We displayed and discussed the physical implications of the obtained results in terms of the physical parameters of the model. By considering that the TAs were initially prepared in a maximally entangled state, and that the single-mode field was in the NBS, the dynamics of quantum phenomena such TAs–field entanglement, TAs entanglement, and parameter estimation were examined. We found that the quantum quantifiers exhibited randomly quasi-periodic and periodic oscillations that depended on the success probability, photon number transition, and the intensity-dependent coupling effect. Furthermore, we analyzed the connection between the dynamical behavior of the quantifiers. This system can be compared with some other ones that are being discussed in the literature, in order to realize the quantum entanglement, and to control the precision of the parameter estimation. Full article

(This article belongs to the Special Issue Quantum Mechanics and Its Foundations III)

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Open AccessArticle

Fisher Information Perspective of Pauli’s Electron

by

Asher Yahalom

Entropy 2022, 24(12), 1721; https://doi.org/10.3390/e24121721 - 24 Nov 2022

Cited by 2 | Viewed by 618

Abstract

An electron moving at velocities much lower that the speed of light with a spin, is described by a wave function which is a solution of Pauli’s equation. It has been demonstrated that this system can be viewed as a vortical fluid which [...] Read more.

An electron moving at velocities much lower that the speed of light with a spin, is described by a wave function which is a solution of Pauli’s equation. It has been demonstrated that this system can be viewed as a vortical fluid which has remarkable similarities but also differences with classical ideal flows. In this respect, it was shown that the internal energy of the Pauli fluid can be interpreted, to some degree, as Fisher Information. In previous work on this subject, electromagnetic fields which are represented by a vector potential were ignored, here we remove this limitation and study the system under general electromagnetic interaction. Full article

(This article belongs to the Special Issue Quantum Mechanics and Its Foundations III)

Open AccessFeature PaperArticle

Better Heisenberg Limits, Coherence Bounds, and Energy-Time Tradeoffs via Quantum Rényi Information

by

Michael J. W. Hall

Entropy 2022, 24(11), 1679; https://doi.org/10.3390/e24111679 - 17 Nov 2022

Viewed by 1194

Abstract

An uncertainty relation for the Rényi entropies of conjugate quantum observables is used to obtain a strong Heisenberg limit of the form

RMSE \geq f (α) / (⟨ N ⟩ + \frac{1}{2})

, bounding the root mean square [...] Read more.

An uncertainty relation for the Rényi entropies of conjugate quantum observables is used to obtain a strong Heisenberg limit of the form

RMSE \geq f (α) / (⟨ N ⟩ + \frac{1}{2})

, bounding the root mean square error of any estimate of a random optical phase shift in terms of average photon number, where

f (α)

is maximised for non-Shannon entropies. Related simple yet strong uncertainty relations linking phase uncertainty to the photon number distribution, such as

Δ Φ \geq {max}_{n} p_{n}

, are also obtained. These results are significantly strengthened via upper and lower bounds on the Rényi mutual information of quantum communication channels, related to asymmetry and convolution, and applied to the estimation (with prior information) of unitary shift parameters such as rotation angle and time, and to obtain strong bounds on measures of coherence. Sharper Rényi entropic uncertainty relations are also obtained, including time-energy uncertainty relations for Hamiltonians with discrete spectra. In the latter case almost-periodic Rényi entropies are introduced for nonperiodic systems. Full article

(This article belongs to the Special Issue Quantum Mechanics and Its Foundations III)

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Open AccessArticle

Correspondence Rules for SU(1,1) Quasidistribution Functions and Quantum Dynamics in the Hyperbolic Phase Space

by

Miguel Baltazar

,

Iván F. Valtierra

and

Andrei B. Klimov

Entropy 2022, 24(11), 1580; https://doi.org/10.3390/e24111580 - 31 Oct 2022

Viewed by 677

Abstract

We derive the explicit differential form for the action of the generators of the

S U (1, 1)

group on the corresponding s-parametrized symbols. This allows us to obtain evolution equations for the phase-space functions on the upper sheet [...] Read more.

We derive the explicit differential form for the action of the generators of the

S U (1, 1)

group on the corresponding s-parametrized symbols. This allows us to obtain evolution equations for the phase-space functions on the upper sheet of the two-sheet hyperboloid and analyze their semiclassical limits. Dynamics of quantum systems with

S U (1, 1)

symmetry governed by compact and non-compact Hamiltonians are discussed in both quantum and semiclassical regimes. Full article

(This article belongs to the Special Issue Quantum Mechanics and Its Foundations III)

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Open AccessFeature PaperArticle

How to Secure Valid Quantizations

by

John R. Klauder

Entropy 2022, 24(10), 1374; https://doi.org/10.3390/e24101374 - 27 Sep 2022

Cited by 1 | Viewed by 560

Abstract

Canonical quantization has created many valid quantizations that require infinite-line coordinate variables. However, the half-harmonic oscillator, which is limited to the positive coordinate half, cannot receive a valid canonical quantization because of the reduced coordinate space. Instead, affine quantization, which is a new [...] Read more.

Canonical quantization has created many valid quantizations that require infinite-line coordinate variables. However, the half-harmonic oscillator, which is limited to the positive coordinate half, cannot receive a valid canonical quantization because of the reduced coordinate space. Instead, affine quantization, which is a new quantization procedure, has been deliberately designed to handle the quantization of problems with reduced coordinate spaces. Following examples of what affine quantization is, and what it can offer, a remarkably straightforward quantization of Einstein’s gravity is attained, in which a proper treatment of the positive definite metric field of gravity has been secured. Full article

(This article belongs to the Special Issue Quantum Mechanics and Its Foundations III)

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