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Quantum Mechanics and Its Foundations II
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Quantum Mechanics and Its Foundations II

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

Deadline for manuscript submissions: closed (15 January 2022) | Viewed by 30045

Special Issue Editor

Prof. Dr. Vladimir Man'ko

E-Mail Website
Guest Editor
1. P.N. Lebedev Physical Institute, Russian Academy of Sciences, Leninskii Prospect 53, 119991 Moscow, Russia
2. Moscow Institute of Physics and Technology, Institutskii Per. 9, Dolgoprudny, 141700 Moscow Region, Russia
Interests: foundations of quantum theory; quantum mechanics; quantum information; group-theoretical methods in physics; quantum tomography; entrophic-information propertiies of classical and quantum systems; associative algenras in quantum mechanics, quasiprobability and probability distributions; pseudostochasic matricies; open system evolution
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

In this Second Issue of “Quantum Mechanics and Its Foundations” we continue to consider the basic notions of quantum mechanics, which was developed during one century of intensive research into its foundations and applications in theoretical and experimental studies directed to create new quantum technologies .

In spite of the great success of quantum mechanical methods, intuition based on the classical experience of human beings still provides difficulties in obtaining a real understanding of the differences of classical mechanics notions of system states like particle position and momentum and wave functions identified with states in quantum mechanics. The wish to formulate quantum notions of states similar to those exploited in classical mechanics or classical statistical mechanics created activity to do this from very beginning of quantum mechanics development. This yielded approaches based on the Wigner function, Husimi function, and Glauber–Sudarshan function, similar to classical probability distributions in phase space determining the particle states in classical statistical mechanics, but these functions are not probability distributions in view of position momentum uncertainty relation which is valid in the quantum world and forbids the existence of such probability distributions.

Nevertheless in the last few decades, this problem of constructing probability distributions which determine the quantum state was solved both for systems with continuous variables (e.g., oscillators) and systems with discrete variables (e.g., spins). The construction was partially reviewed in the first Issue on the foundations of quantum mechanics.

In this second Issue we consider the recent development of the probability representation of quantum states and other problems related to its applications, as well as other problems of the foundations of quantum mechanics and its formalism. Additionally, examples of applications of the new formalism as well as other approaches to the foundations of quantum mechanics will be discussed in the articles of the second Issue.

We believe that the new formalism of quantum mechanics will fundamentally change the education programs in universities in courses related to the study of quantum theory and the development of quantum technologies.

Prof. Dr. Vladimir Man'ko
Guest Editor

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 (15 papers)

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Research

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13 pages, 294 KiB  
Article
Symplectic Radon Transform and the Metaplectic Representation
by Maurice A. de Gosson
Entropy 2022, 24(6), 761; https://doi.org/10.3390/e24060761 - 28 May 2022
Cited by 4 | Viewed by 1206
Abstract
We study the symplectic Radon transform from the point of view of the metaplectic representation of the symplectic group and its action on the Lagrangian Grassmannian. We give rigorous proofs in the general setting of multi-dimensional quantum systems. We interpret the Radon transform [...] Read more.
We study the symplectic Radon transform from the point of view of the metaplectic representation of the symplectic group and its action on the Lagrangian Grassmannian. We give rigorous proofs in the general setting of multi-dimensional quantum systems. We interpret the Radon transform of a quantum state as a generalized marginal distribution for its Wigner transform; the inverse Radon transform thus appears as a “demarginalization process” for the Wigner distribution. Full article
(This article belongs to the Special Issue Quantum Mechanics and Its Foundations II)
20 pages, 324 KiB  
Article
Entangled Qubit States and Linear Entropy in the Probability Representation of Quantum Mechanics
by Vladimir N. Chernega, Olga V. Man’ko and Vladimir I. Man’ko
Entropy 2022, 24(4), 527; https://doi.org/10.3390/e24040527 - 09 Apr 2022
Cited by 4 | Viewed by 1704
Abstract
The superposition states of two qubits including entangled Bell states are considered in the probability representation of quantum mechanics. The superposition principle formulated in terms of the nonlinear addition rule of the state density matrices is formulated as a nonlinear addition rule of [...] Read more.
The superposition states of two qubits including entangled Bell states are considered in the probability representation of quantum mechanics. The superposition principle formulated in terms of the nonlinear addition rule of the state density matrices is formulated as a nonlinear addition rule of the probability distributions describing the qubit states. The generalization of the entanglement properties to the case of superposition of two-mode oscillator states is discussed using the probability representation of quantum states. Full article
(This article belongs to the Special Issue Quantum Mechanics and Its Foundations II)
11 pages, 654 KiB  
Article
Helstrom Bound for Squeezed Coherent States in Binary Communication
by Evaldo M. F. Curado, Sofiane Faci, Jean-Pierre Gazeau and Diego Noguera
Entropy 2022, 24(2), 220; https://doi.org/10.3390/e24020220 - 31 Jan 2022
Cited by 2 | Viewed by 2257
Abstract
In quantum information processing, using a receiver device to differentiate between two non-orthogonal states leads to a quantum error probability. The minimum possible error is known as the Helstrom bound. In this work, we study the conditions for state discrimination using an alphabet [...] Read more.
In quantum information processing, using a receiver device to differentiate between two non-orthogonal states leads to a quantum error probability. The minimum possible error is known as the Helstrom bound. In this work, we study the conditions for state discrimination using an alphabet of squeezed coherent states and compare them with conditions using the Glauber-Sudarshan, i.e., standard, coherent states. Full article
(This article belongs to the Special Issue Quantum Mechanics and Its Foundations II)
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22 pages, 855 KiB  
Article
Quantum–Classical Entropy Analysis for Nonlinearly-Coupled Continuous-Variable Bipartite Systems
by Ángel S. Sanz
Entropy 2022, 24(2), 190; https://doi.org/10.3390/e24020190 - 27 Jan 2022
Cited by 1 | Viewed by 1834
Abstract
The correspondence principle plays a fundamental role in quantum mechanics, which naturally leads us to inquire whether it is possible to find or determine close classical analogs of quantum states in phase space—a common meeting point to both classical and quantum density statistical [...] Read more.
The correspondence principle plays a fundamental role in quantum mechanics, which naturally leads us to inquire whether it is possible to find or determine close classical analogs of quantum states in phase space—a common meeting point to both classical and quantum density statistical descriptors. Here, this issue is tackled by investigating the behavior of classical analogs arising upon the removal of all interference traits displayed by the Wigner distribution functions associated with a given pure quantum state. Accordingly, the dynamical evolution of the linear and von Neumann entropies is numerically computed for a continuous-variable bipartite system, and compared with the corresponding classical counterparts, in the case of two quartic oscillators nonlinearly coupled under regular and chaos conditions. Three quantum states for the full system are considered: a Gaussian state, a cat state, and a Bell-type state. By comparing the quantum and classical entropy values, and particularly their trends, it is shown that, instead of entanglement production, such entropies rather provide us with information on the system (either quantum or classical) delocalization. This gradual loss of information translates into an increase in both the quantum and the classical realms, directly connected to the increase in the correlations between both parties’ degrees of freedom which, in the quantum case, is commonly related to the production of entanglement. Full article
(This article belongs to the Special Issue Quantum Mechanics and Its Foundations II)
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10 pages, 1745 KiB  
Article
Classical-Quantum Transition as a Disorder-Order Process
by Andres M. Kowalski and Angelo Plastino
Entropy 2022, 24(1), 87; https://doi.org/10.3390/e24010087 - 05 Jan 2022
Cited by 1 | Viewed by 1087
Abstract
We associate here the relationship between de-coherence to the statistical notion of disequilibrium with regards to the dynamics of a system that reflects the interaction between matter and a given field. The process is described via information geometry. Some of its tools are [...] Read more.
We associate here the relationship between de-coherence to the statistical notion of disequilibrium with regards to the dynamics of a system that reflects the interaction between matter and a given field. The process is described via information geometry. Some of its tools are shown here to appropriately explain the process’ mechanism. In particular we gain some insight into what is the role of the uncertainty principle (UP) in the pertinent proceedings. Full article
(This article belongs to the Special Issue Quantum Mechanics and Its Foundations II)
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13 pages, 280 KiB  
Article
Uncertainty Relations in the Madelung Picture
by Moise Bonilla-Licea and Dieter Schuch
Entropy 2022, 24(1), 20; https://doi.org/10.3390/e24010020 - 23 Dec 2021
Cited by 2 | Viewed by 1923
Abstract
Madelung showed how the complex Schrödinger equation can be rewritten in terms of two real equations, one for the phase and one for the amplitude of the complex wave function, where both equations are not independent of each other, but coupled. Although these [...] Read more.
Madelung showed how the complex Schrödinger equation can be rewritten in terms of two real equations, one for the phase and one for the amplitude of the complex wave function, where both equations are not independent of each other, but coupled. Although these equations formally look like classical hydrodynamic equations, they contain all the information about the quantum system. Concerning the quantum mechanical uncertainties of position and momentum, however, this is not so obvious at first sight. We show how these uncertainties are related to the phase and amplitude of the wave function in position and momentum space and, particularly, that the contribution from the phase essentially depends on the position–momentum correlations. This will be illustrated explicitly using generalized coherent states as examples. Full article
(This article belongs to the Special Issue Quantum Mechanics and Its Foundations II)
17 pages, 743 KiB  
Article
No Preferred Reference Frame at the Foundation of Quantum Mechanics
by William Stuckey, Timothy McDevitt and Michael Silberstein
Entropy 2022, 24(1), 12; https://doi.org/10.3390/e24010012 - 22 Dec 2021
Cited by 5 | Viewed by 5115
Abstract
Quantum information theorists have created axiomatic reconstructions of quantum mechanics (QM) that are very successful at identifying precisely what distinguishes quantum probability theory from classical and more general probability theories in terms of information-theoretic principles. Herein, we show how one such principle, Information [...] Read more.
Quantum information theorists have created axiomatic reconstructions of quantum mechanics (QM) that are very successful at identifying precisely what distinguishes quantum probability theory from classical and more general probability theories in terms of information-theoretic principles. Herein, we show how one such principle, Information Invariance and Continuity, at the foundation of those axiomatic reconstructions, maps to “no preferred reference frame” (NPRF, aka “the relativity principle”) as it pertains to the invariant measurement of Planck’s constant h for Stern-Gerlach (SG) spin measurements. This is in exact analogy to the relativity principle as it pertains to the invariant measurement of the speed of light c at the foundation of special relativity (SR). Essentially, quantum information theorists have extended Einstein’s use of NPRF from the boost invariance of measurements of c to include the SO(3) invariance of measurements of h between different reference frames of mutually complementary spin measurements via the principle of Information Invariance and Continuity. Consequently, the “mystery” of the Bell states is understood to result from conservation per Information Invariance and Continuity between different reference frames of mutually complementary qubit measurements, and this maps to conservation per NPRF in spacetime. If one falsely conflates the relativity principle with the classical theory of SR, then it may seem impossible that the relativity principle resides at the foundation of non-relativisitic QM. In fact, there is nothing inherently classical or quantum about NPRF. Thus, the axiomatic reconstructions of QM have succeeded in producing a principle account of QM that reveals as much about Nature as the postulates of SR. Full article
(This article belongs to the Special Issue Quantum Mechanics and Its Foundations II)
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15 pages, 627 KiB  
Article
A Smooth Path between the Classical Realm and the Quantum Realm
by John R. Klauder
Entropy 2021, 23(12), 1689; https://doi.org/10.3390/e23121689 - 16 Dec 2021
Viewed by 1933
Abstract
A simple example of classical physics may be defined as classical variables, p and q, and quantum physics may be defined as quantum operators, P and Q. The classical world of p&q, as it is currently understood, is [...] Read more.
A simple example of classical physics may be defined as classical variables, p and q, and quantum physics may be defined as quantum operators, P and Q. The classical world of p&q, as it is currently understood, is truly disconnected from the quantum world, as it is currently understood. The process of quantization, for which there are several procedures, aims to promote a classical issue into a related quantum issue. In order to retain their physical connection, it becomes critical as to how to promote specific classical variables to associated specific quantum variables. This paper, which also serves as a review paper, leads the reader toward specific, but natural, procedures that promise to ensure that the classical and quantum choices are guaranteed a proper physical connection. Moreover, parallel procedures for fields, and even gravity, that connect classical and quantum physical regimes, will be introduced. Full article
(This article belongs to the Special Issue Quantum Mechanics and Its Foundations II)
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21 pages, 799 KiB  
Article
The Influence of the Symmetry of Identical Particles on Flight Times
by Salvador Miret-Artés, Randall S. Dumont, Tom Rivlin and Eli Pollak
Entropy 2021, 23(12), 1675; https://doi.org/10.3390/e23121675 - 13 Dec 2021
Cited by 6 | Viewed by 2048
Abstract
In this work, our purpose is to show how the symmetry of identical particles can influence the time evolution of free particles in the nonrelativistic and relativistic domains as well as in the scattering by a potential δ-barrier. For this goal, we [...] Read more.
In this work, our purpose is to show how the symmetry of identical particles can influence the time evolution of free particles in the nonrelativistic and relativistic domains as well as in the scattering by a potential δ-barrier. For this goal, we consider a system of either two distinguishable or indistinguishable (bosons and fermions) particles. Two sets of initial conditions have been studied: different initial locations with the same momenta, and the same locations with different momenta. The flight time distribution of particles arriving at a ‘screen’ is calculated in each case from the density and flux. Fermions display broader distributions as compared with either distinguishable particles or bosons, leading to earlier and later arrivals for all the cases analyzed here. The symmetry of the wave function seems to speed up or slow down the propagation of particles. Due to the cross terms, certain initial conditions lead to bimodality in the fermionic case. Within the nonrelativistic domain, and when the short-time survival probability is analyzed, if the cross term becomes important, one finds that the decay of the overlap of fermions is faster than for distinguishable particles which in turn is faster than for bosons. These results are of interest in the short time limit since they imply that the well-known quantum Zeno effect would be stronger for bosons than for fermions. Fermions also arrive earlier and later than bosons when they are scattered by a δ-barrier. Although the particle symmetry does affect the mean tunneling flight time, in the limit of narrow in momentum initial Gaussian wave functions, the mean times are not affected by symmetry but tend to the phase time for distinguishable particles. Full article
(This article belongs to the Special Issue Quantum Mechanics and Its Foundations II)
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13 pages, 14300 KiB  
Article
Proposal to Test a Transient Deviation from Quantum Mechanics’ Predictions for Bell’s Experiment
by Alejandro Andrés Hnilo, Monica Beatriz Agüero and Marcelo Gregorio Kovalsky
Entropy 2021, 23(12), 1589; https://doi.org/10.3390/e23121589 - 27 Nov 2021
Cited by 1 | Viewed by 1312
Abstract
Quantum mechanics predicts correlations between measurements performed in distant regions of a spatially spread entangled state to be higher than allowed by intuitive concepts of Locality and Realism. These high correlations forbid the use of nonlinear operators of evolution (which would be desirable [...] Read more.
Quantum mechanics predicts correlations between measurements performed in distant regions of a spatially spread entangled state to be higher than allowed by intuitive concepts of Locality and Realism. These high correlations forbid the use of nonlinear operators of evolution (which would be desirable for several reasons), for they may allow faster-than-light signaling. As a way out of this situation, it has been hypothesized that the high quantum correlations develop only after a time longer than L/c has elapsed (where L is the spread of the entangled state and c is the velocity of light). In shorter times, correlations compatible with Locality and Realism would be observed instead. A simple hidden variables model following this hypothesis is described. It is based on a modified Wheeler–Feynman theory of radiation. This hypothesis has not been disproved by any of the experiments performed to date. A test achievable with accessible means is proposed and described. It involves a pulsed source of entangled states and stroboscopic record of particle detection during the pulses. Data recorded in similar but incomplete optical experiments are analyzed, and found consistent with the proposed model. However, it is not claimed, in any sense, that the hypothesis has been validated. On the contrary, it is stressed that a complete, specific test is absolutely needed. Full article
(This article belongs to the Special Issue Quantum Mechanics and Its Foundations II)
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9 pages, 311 KiB  
Article
Quantum Contextual Advantage Depending on Nonzero Prior Probabilities in State Discrimination of Mixed Qubit States
by Jaehee Shin, Donghoon Ha and Younghun Kwon
Entropy 2021, 23(12), 1583; https://doi.org/10.3390/e23121583 - 26 Nov 2021
Cited by 5 | Viewed by 1716
Abstract
Recently, Schmid and Spekkens studied the quantum contextuality in terms of state discrimination. By dealing with the minimum error discrimination of two quantum states with identical prior probabilities, they reported that quantum contextual advantage exists. Meanwhile, if one notes a striking observation that [...] Read more.
Recently, Schmid and Spekkens studied the quantum contextuality in terms of state discrimination. By dealing with the minimum error discrimination of two quantum states with identical prior probabilities, they reported that quantum contextual advantage exists. Meanwhile, if one notes a striking observation that the selection of prior probability can affect the quantum properties of the system, it is necessary to verify whether the quantum contextual advantage depends on the prior probabilities of the given states. In this paper, we consider the minimum error discrimination of two states with arbitrary prior probabilities, in which both states are pure or mixed. We show that the quantum contextual advantage in state discrimination may depend on the prior probabilities of the given states. In particular, even though the quantum contextual advantage always exists in the state discrimination of two nonorthogonal pure states with nonzero prior probabilities, the quantum contextual advantage depends on prior probabilities in the state discrimination of two mixed states. Full article
(This article belongs to the Special Issue Quantum Mechanics and Its Foundations II)
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54 pages, 1990 KiB  
Article
Energy and Magnetic Moment of a Quantum Charged Particle in Time-Dependent Magnetic and Electric Fields of Circular and Plane Solenoids
by Viktor V. Dodonov and Matheus B. Horovits
Entropy 2021, 23(12), 1579; https://doi.org/10.3390/e23121579 - 26 Nov 2021
Cited by 7 | Viewed by 1540
Abstract
We consider a quantum spinless nonrelativistic charged particle moving in the xy plane under the action of a time-dependent magnetic field, described by means of the linear vector potential [...] Read more.
We consider a quantum spinless nonrelativistic charged particle moving in the xy plane under the action of a time-dependent magnetic field, described by means of the linear vector potential A=B(t)−y(1+α),x(1−α)/2, with two fixed values of the gauge parameter α: α=0 (the circular gauge) and α=1 (the Landau gauge). While the magnetic field is the same in all the cases, the systems with different values of the gauge parameter are not equivalent for nonstationary magnetic fields due to different structures of induced electric fields, whose lines of force are circles for α=0 and straight lines for α=1. We derive general formulas for the time-dependent mean values of the energy and magnetic moment, as well as for their variances, for an arbitrary function B(t). They are expressed in terms of solutions to the classical equation of motion ε¨+ωα2(t)ε=0, with ω1=2ω0. Explicit results are found in the cases of the sudden jump of magnetic field, the parametric resonance, the adiabatic evolution, and for several specific functions B(t), when solutions can be expressed in terms of elementary or hypergeometric functions. These examples show that the evolution of the mentioned mean values can be rather different for the two gauges, if the evolution is not adiabatic. It appears that the adiabatic approximation fails when the magnetic field goes to zero. Moreover, the sudden jump approximation can fail in this case as well. The case of a slowly varying field changing its sign seems especially interesting. In all the cases, fluctuations of the magnetic moment are very strong, frequently exceeding the square of the mean value. Full article
(This article belongs to the Special Issue Quantum Mechanics and Its Foundations II)
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9 pages, 472 KiB  
Article
Einstein-Podolsky-Rosen Steering for Mixed Entangled Coherent States
by Sayed Abdel-Khalek, Kamal Berrada, Mariam Algarni and Hichem Eleuch
Entropy 2021, 23(11), 1442; https://doi.org/10.3390/e23111442 - 31 Oct 2021
Cited by 1 | Viewed by 1476
Abstract
By using the Born Markovian master equation, we study the relationship among the Einstein–Podolsky–Rosen (EPR) steering, Bell nonlocality, and quantum entanglement of entangled coherent states (ECSs) under decoherence. We illustrate the dynamical behavior of the three types of correlations for various optical field [...] Read more.
By using the Born Markovian master equation, we study the relationship among the Einstein–Podolsky–Rosen (EPR) steering, Bell nonlocality, and quantum entanglement of entangled coherent states (ECSs) under decoherence. We illustrate the dynamical behavior of the three types of correlations for various optical field strength regimes. In general, we find that correlation measurements begin at their maximum and decline over time. We find that quantum steering and nonlocality behave similarly in terms of photon number during dynamics. Furthermore, we discover that ECSs with steerability can violate the Bell inequality, and that not every ECS with Bell nonlocality is steerable. In the current work, without the memory stored in the environment, some of the initial states with maximal values of quantum steering, Bell nonlocality, and entanglement can provide a delayed loss of that value during temporal evolution, which is of interest to the current study. Full article
(This article belongs to the Special Issue Quantum Mechanics and Its Foundations II)
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14 pages, 881 KiB  
Article
Wigner’s Friend Scenarios and the Internal Consistency of Standard Quantum Mechanics
by Dmitri Sokolovski and Alexandre Matzkin
Entropy 2021, 23(9), 1186; https://doi.org/10.3390/e23091186 - 09 Sep 2021
Cited by 2 | Viewed by 1895
Abstract
Wigner’s friend scenarios involve an Observer, or Observers, measuring a Friend, or Friends, who themselves make quantum measurements. In recent discussions, it has been suggested that quantum mechanics may not always be able to provide a consistent account of a situation involving two [...] Read more.
Wigner’s friend scenarios involve an Observer, or Observers, measuring a Friend, or Friends, who themselves make quantum measurements. In recent discussions, it has been suggested that quantum mechanics may not always be able to provide a consistent account of a situation involving two Observers and two Friends. We investigate this problem by invoking the basic rules of quantum mechanics as outlined by Feynman in the well-known “Feynman Lectures on Physics”. We show here that these “Feynman rules” constrain the a priori assumptions which can be made in generalised Wigner’s friend scenarios, because the existence of the probabilities of interest ultimately depends on the availability of physical evidence (material records) of the system’s past. With these constraints obeyed, a non-ambiguous and consistent account of all measurement outcomes is obtained for all agents, taking part in various Wigner’s Friend scenarios. Full article
(This article belongs to the Special Issue Quantum Mechanics and Its Foundations II)
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Review

Jump to: Research

7 pages, 314 KiB  
Review
Thermal–Statistical Odd–Even Fermions’ Staggering Effect and the Order–Disorder Disjunction
by Flavia Pennini, Angelo Plastino and Angel Ricardo Plastino
Entropy 2021, 23(11), 1428; https://doi.org/10.3390/e23111428 - 29 Oct 2021
Cited by 3 | Viewed by 1428
Abstract
We review thermal–statistical considerations on the odd–even staggering effect (OES) in fermions. There is a well known OES in nuclear binding energies at zero temperature. We discuss here a thermal OES (finite temperatures) that establishes links with the order–disorder disjunction. The present thermal [...] Read more.
We review thermal–statistical considerations on the odd–even staggering effect (OES) in fermions. There is a well known OES in nuclear binding energies at zero temperature. We discuss here a thermal OES (finite temperatures) that establishes links with the order–disorder disjunction. The present thermal considerations cannot be found in the nuclear literature. Full article
(This article belongs to the Special Issue Quantum Mechanics and Its Foundations II)
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