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Quantum Rep., Volume 4, Issue 4 (December 2022) – 19 articles

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12 pages, 296 KiB  
Article
Entropic DDoS Detection for Quantum Networks
Quantum Rep. 2022, 4(4), 604-615; https://doi.org/10.3390/quantum4040044 - 13 Dec 2022
Viewed by 1449
Abstract
Distributed Denial-of-Service (DDoS) attacks are a significant issue in classical networks. These attacks have been shown to impact the critical infrastructure of a nation, such as its major financial institutions. The possibility of DDoS attacks has also been identified for quantum networks. In [...] Read more.
Distributed Denial-of-Service (DDoS) attacks are a significant issue in classical networks. These attacks have been shown to impact the critical infrastructure of a nation, such as its major financial institutions. The possibility of DDoS attacks has also been identified for quantum networks. In this theoretical work, we introduce a quantum analogue of classical entropic DDoS detection systems and apply it in the context of detecting an attack on a quantum network. In particular, we examine DDoS attacks on a quantum repeater and harness the associated entanglement entropy for the detection system. Our results extend the applicability of quantum information from the domain of data security to the area of network security. Full article
15 pages, 6900 KiB  
Article
Quantum Work from a Pseudo-Hermitian Hamiltonian
Quantum Rep. 2022, 4(4), 589-603; https://doi.org/10.3390/quantum4040043 - 13 Dec 2022
Viewed by 1198
Abstract
In this work, we study the thermodynamics of a hybrid system based on the Da Providencia–Schütte Hamiltonian. The model consists of bosons, i.e., photons in a cavity, interacting with an ensemble of spins through a pseudo-Hermitian Hamiltonian. We compute the exact partition function [...] Read more.
In this work, we study the thermodynamics of a hybrid system based on the Da Providencia–Schütte Hamiltonian. The model consists of bosons, i.e., photons in a cavity, interacting with an ensemble of spins through a pseudo-Hermitian Hamiltonian. We compute the exact partition function of the system, and from it, we derive the statistical properties of the system. Finally, we evaluate the work that can be extracted from the system by performing an Otto cycle and discuss the advantages of the proposed pseudo-Hermitian interaction. Full article
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15 pages, 479 KiB  
Article
Detecting Quantum Critical Points of Correlated Systems by Quantum Convolutional Neural Network Using Data from Variational Quantum Eigensolver
Quantum Rep. 2022, 4(4), 574-588; https://doi.org/10.3390/quantum4040042 - 08 Dec 2022
Viewed by 1801
Abstract
Machine learning has been applied to a wide variety of models, from classical statistical mechanics to quantum strongly correlated systems, for classifying phase transitions. The recently proposed quantum convolutional neural network (QCNN) provides a new framework for using quantum circuits instead of classical [...] Read more.
Machine learning has been applied to a wide variety of models, from classical statistical mechanics to quantum strongly correlated systems, for classifying phase transitions. The recently proposed quantum convolutional neural network (QCNN) provides a new framework for using quantum circuits instead of classical neural networks as the backbone of classification methods. We present the results from training the QCNN by the wavefunctions of the variational quantum eigensolver for the one-dimensional transverse field Ising model (TFIM). We demonstrate that the QCNN identifies wavefunctions corresponding to the paramagnetic and ferromagnetic phases of the TFIM with reasonable accuracy. The QCNN can be trained to predict the corresponding ‘phase’ of wavefunctions around the putative quantum critical point even though it is trained by wavefunctions far away. The paper provides a basis for exploiting the QCNN to identify the quantum critical point. Full article
(This article belongs to the Special Issue Exclusive Feature Papers of Quantum Reports)
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8 pages, 286 KiB  
Article
Schrödinger–Newton Equation with Spontaneous Wave Function Collapse
Quantum Rep. 2022, 4(4), 566-573; https://doi.org/10.3390/quantum4040041 - 05 Dec 2022
Viewed by 1370
Abstract
Based on the assumption that the standard Schrödinger equation becomes gravitationally modified for massive macroscopic objects, two independent proposals have survived from the 1980s. The Schrödinger–Newton equation (1984) provides well-localized solitons for free macro-objects but lacks the mechanism of how extended wave functions [...] Read more.
Based on the assumption that the standard Schrödinger equation becomes gravitationally modified for massive macroscopic objects, two independent proposals have survived from the 1980s. The Schrödinger–Newton equation (1984) provides well-localized solitons for free macro-objects but lacks the mechanism of how extended wave functions collapse on solitons. The gravity-related stochastic Schrödinger equation (1989) provides the spontaneous collapse, but the resulting solitons undergo a tiny diffusion, leading to an inconvenient steady increase in the kinetic energy. We propose the stochastic Schrödinger–Newton equation, which contains the above two gravity-related modifications together. Then, the wave functions of free macroscopic bodies will gradually and stochastically collapse to solitons, which perform inertial motion without momentum diffusion: conservation of momentum and energy is restored. Full article
8 pages, 283 KiB  
Article
Energy Transition Density of Driven Chaotic Systems: A Compound Trace Formula
Quantum Rep. 2022, 4(4), 558-565; https://doi.org/10.3390/quantum4040040 - 30 Nov 2022
Viewed by 1028
Abstract
Oscillations in the probability density of quantum transitions of the eigenstates of a chaotic Hamiltonian within classically narrow energy ranges have been shown to depend on closed compound orbits. These are formed by a pair of orbit segments, one in the energy shell [...] Read more.
Oscillations in the probability density of quantum transitions of the eigenstates of a chaotic Hamiltonian within classically narrow energy ranges have been shown to depend on closed compound orbits. These are formed by a pair of orbit segments, one in the energy shell of the original Hamiltonian and the other in the energy shell of the driven Hamiltonian, with endpoints that coincide. Viewed in the time domain, the same pair of trajectory segments arises in the semiclassical evaluation of the trace of a compound propagator: the product of the complex exponentials of the original Hamiltonian and of its driven image. It is shown here that the probability density is the double Fourier transform of this trace, and that the closed compound orbits emulate the role played by the periodic orbits in Gutzwiller’s trace formula in its semiclassical evaluation. The phase of the oscillations with the energies or the evolution parameters agree with those previously obtained, whereas the amplitude of the contribution of each closed compound orbit is more compact and independent of any feature of the Weyl–Wigner representation in which the calculation was carried out. Full article
(This article belongs to the Special Issue Continuous and Discrete Phase-Space Methods and Their Applications)
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14 pages, 1436 KiB  
Article
A Study of Strong Confinement Regions Using Informational Entropy
Quantum Rep. 2022, 4(4), 544-557; https://doi.org/10.3390/quantum4040039 - 23 Nov 2022
Cited by 4 | Viewed by 1240
Abstract
We present an informational study of a spherically confined hydrogen atom, a hydrogenic ion confined in a strongly coupled plasma, a spherically confined harmonic oscillator, and a particle confined in a cage. For this, we have implemented a numerical procedure to obtain information [...] Read more.
We present an informational study of a spherically confined hydrogen atom, a hydrogenic ion confined in a strongly coupled plasma, a spherically confined harmonic oscillator, and a particle confined in a cage. For this, we have implemented a numerical procedure to obtain information entropies of these confined quantum systems. The procedure is based on the variational formalism that uses the finite element method (FEM) for the expansion of the wavefunction in terms of local base functions. Such a study is carried out in order to analyze what happens in the rigorous confinement regime. In particular, we have shown that the effects of the interaction potential is no longer important for rigorous confinements and the studied systems start to behave just like an electron confined by a impenetrable spherical cage. When possible, we compared our results with those published in the literature. Full article
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11 pages, 318 KiB  
Article
Field Form of the Dynamics of Classical Many- and Few-Body Systems: From Microscopic Dynamics to Kinetics, Thermodynamics and Synergetics
Quantum Rep. 2022, 4(4), 533-543; https://doi.org/10.3390/quantum4040038 - 20 Nov 2022
Viewed by 1264
Abstract
A method is proposed for describing the dynamics of systems of interacting particles in terms of an auxiliary field, which in the static mode is equivalent to given interatomic potentials, and in the dynamic mode is a classical relativistic composite field. It is [...] Read more.
A method is proposed for describing the dynamics of systems of interacting particles in terms of an auxiliary field, which in the static mode is equivalent to given interatomic potentials, and in the dynamic mode is a classical relativistic composite field. It is established that for interatomic potentials, the Fourier transform of which is a rational algebraic function of the wave vector, the auxiliary field is a composition of elementary fields that satisfy the Klein-Gordon equation with complex masses. The interaction between particles carried by the auxiliary field is nonlocal both in space variables and in time. The temporal non-locality is due to the dynamic nature of the auxiliary field and can be described in terms of functional-differential equations of retarded type. Due to the finiteness mass of the auxiliary field, the delay in interactions between particles can be arbitrarily large. A qualitative analysis of the dynamics of few-body and many-body systems with retarded interactions has been carried out, and a non-statistical mechanisms for both the thermodynamic behavior of systems and synergistic effects has been established. Full article
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10 pages, 501 KiB  
Article
Fricke Topological Qubits
Quantum Rep. 2022, 4(4), 523-532; https://doi.org/10.3390/quantum4040037 - 14 Nov 2022
Cited by 3 | Viewed by 2657
Abstract
We recently proposed that topological quantum computing might be based on SL(2,C) representations of the fundamental group π1(S3\K) for the complement of a link K in the three-sphere. The restriction [...] Read more.
We recently proposed that topological quantum computing might be based on SL(2,C) representations of the fundamental group π1(S3\K) for the complement of a link K in the three-sphere. The restriction to links whose associated SL(2,C) character variety V contains a Fricke surface κd=xyzx2y2z2+d is desirable due to the connection of Fricke spaces to elementary topology. Taking K as the Hopf link L2a1, one of the three arithmetic two-bridge links (the Whitehead link 512, the Berge link 622 or the double-eight link 632) or the link 732, the V for those links contains the reducible component κ4, the so-called Cayley cubic. In addition, the V for the latter two links contains the irreducible component κ3, or κ2, respectively. Taking ρ to be a representation with character κd (d<4), with |x|,|y|,|z|2, then ρ(π1) fixes a unique point in the hyperbolic space H3 and is a conjugate to a SU(2) representation (a qubit). Even though details on the physical implementation remain open, more generally, we show that topological quantum computing may be developed from the point of view of three-bridge links, the topology of the four-punctured sphere and Painlevé VI equation. The 0-surgery on the three circles of the Borromean rings L6a4 is taken as an example. Full article
(This article belongs to the Special Issue Exclusive Feature Papers of Quantum Reports)
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14 pages, 577 KiB  
Article
Coherent Phase States in the Coordinate and Wigner Representations
Quantum Rep. 2022, 4(4), 509-522; https://doi.org/10.3390/quantum4040036 - 08 Nov 2022
Cited by 1 | Viewed by 1607
Abstract
In this paper, we numerically study the coordinate wave functions and the Wigner functions of the coherent phase states (CPS), paying particular attention to their differences from the standard (Klauder–Glauber–Sudarshan) coherent states, especially in the case of the high mean values of the [...] Read more.
In this paper, we numerically study the coordinate wave functions and the Wigner functions of the coherent phase states (CPS), paying particular attention to their differences from the standard (Klauder–Glauber–Sudarshan) coherent states, especially in the case of the high mean values of the number operator. In this case, the CPS can possess a strong coordinate (or momentum) squeezing, which is roughly twice weaker than for the vacuum squeezed states. The Robertson–Schrödinger invariant uncertainty product in the CPS logarithmically increases with the mean value of the number operator (whereas it is constant for the standard coherent states). Some measures of the (non)Gaussianity of CPS are considered. Full article
(This article belongs to the Special Issue Continuous and Discrete Phase-Space Methods and Their Applications)
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23 pages, 452 KiB  
Review
Some Classical Models of Particles and Quantum Gauge Theories
Quantum Rep. 2022, 4(4), 486-508; https://doi.org/10.3390/quantum4040035 - 03 Nov 2022
Viewed by 1936
Abstract
The article contains a review and new results of some mathematical models relevant to the interpretation of quantum mechanics and emulating well-known quantum gauge theories, such as scalar electrodynamics (Klein–Gordon–Maxwell electrodynamics), spinor electrodynamics (Dirac–Maxwell electrodynamics), etc. In these models, evolution is typically described [...] Read more.
The article contains a review and new results of some mathematical models relevant to the interpretation of quantum mechanics and emulating well-known quantum gauge theories, such as scalar electrodynamics (Klein–Gordon–Maxwell electrodynamics), spinor electrodynamics (Dirac–Maxwell electrodynamics), etc. In these models, evolution is typically described by modified Maxwell equations. In the case of scalar electrodynamics, the scalar complex wave function can be made real by a gauge transformation, the wave function can be algebraically eliminated from the equations of scalar electrodynamics, and the resulting modified Maxwell equations describe the independent evolution of the electromagnetic field. Similar results were obtained for spinor electrodynamics. Three out of four components of the Dirac spinor can be algebraically eliminated from the Dirac equation, and the remaining component can be made real by a gauge transformation. A similar result was obtained for the Dirac equation in the Yang–Mills field. As quantum gauge theories play a central role in modern physics, the approach of this article may be sufficiently general. One-particle wave functions can be modeled as plasma-like collections of a large number of particles and antiparticles. This seems to enable the simulation of quantum phase-space distribution functions, such as the Wigner distribution function, which are not necessarily non-negative. Full article
(This article belongs to the Special Issue Exclusive Feature Papers of Quantum Reports)
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10 pages, 952 KiB  
Article
Topological Properties of the 2D 2-Band System with Generalized W-Shaped Band Inversion
Quantum Rep. 2022, 4(4), 476-485; https://doi.org/10.3390/quantum4040034 - 02 Nov 2022
Cited by 2 | Viewed by 1337
Abstract
We report the topological properties, in terms of the Berry phase, of the 2D noninteracting system with electron–hole band inversion, described by the two-band generalized analogue of the low-energy Bernevig–Hughes–Zhang Hamiltonian, yielding the W-shaped energy bands in the form of two intersecting cones [...] Read more.
We report the topological properties, in terms of the Berry phase, of the 2D noninteracting system with electron–hole band inversion, described by the two-band generalized analogue of the low-energy Bernevig–Hughes–Zhang Hamiltonian, yielding the W-shaped energy bands in the form of two intersecting cones with the gap along the closed continuous loop. We identify the range of parameters where the Berry phase attains qualitatively different values: (a) the integer multiplier of 2π, (b) the integer multiplier of π, and (c) the nontrivial value between the latter two, which depends on the system parameters. The system thus exhibits the anomalous quantum Hall effect associated with the nontrivial geometric phase, which is presumably tunable through the choice of parameters at hand. Full article
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14 pages, 1695 KiB  
Article
Learning Mixed Strategies in Quantum Games with Imperfect Information
Quantum Rep. 2022, 4(4), 462-475; https://doi.org/10.3390/quantum4040033 - 29 Oct 2022
Cited by 1 | Viewed by 1194
Abstract
The quantization of games expand the players strategy space, allowing the emergence of more equilibriums. However, finding these equilibriums is difficult, especially if players are allowed to use mixed strategies. The size of the exploration space expands so much for quantum games that [...] Read more.
The quantization of games expand the players strategy space, allowing the emergence of more equilibriums. However, finding these equilibriums is difficult, especially if players are allowed to use mixed strategies. The size of the exploration space expands so much for quantum games that makes far harder to find the player’s best strategy. In this work, we propose a method to learn and visualize mixed quantum strategies and compare them with their classical counterpart. In our model, players do not know in advance which game they are playing (pay-off matrix) neither the action selected nor the reward obtained by their competitors at each step, they only learn from an individual feedback reward signal. In addition, we study both the influence of entanglement and noise on the performance of various quantum games. Full article
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20 pages, 571 KiB  
Article
Excitation Spectra and Edge Singularities in the One-Dimensional Anisotropic Heisenberg Model for Δ = cos(π/n), n = 3,4,5
Quantum Rep. 2022, 4(4), 442-461; https://doi.org/10.3390/quantum4040032 - 19 Oct 2022
Viewed by 1388
Abstract
The T=0 excitation spectra of the antiferromagnetic (J>0) anisotropic Heisenberg chain of spins 1/2 are studied using the Bethe Ansatz equations for Δ=cos(π/n), n=3,4 and [...] Read more.
The T=0 excitation spectra of the antiferromagnetic (J>0) anisotropic Heisenberg chain of spins 1/2 are studied using the Bethe Ansatz equations for Δ=cos(π/n), n=3,4 and 5. The number of unknown functions is n1 for Δ=cos(π/n) and can be solved numerically for a finite external field. The low-energy excitations form a Luttinger liquid parametrized by a conformal field theory with conformal charge of c=1. For higher energy excitations, the spectral functions display deviations from the Luttinger behavior arising from the curvature in the dispersion. Adding a corrective term of the form of a mobile impurity coupled to the Luttinger liquid modes corrects this difference. The “impurity” is an irrelevant operator, which if treated non-perturbatively, yields the threshold singularities in the one-spinwave particle and hole Green’s function correctly. Full article
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8 pages, 314 KiB  
Article
Quantum-Inspired Classification Based on Voronoi Tessellation and Pretty-Good Measurements
Quantum Rep. 2022, 4(4), 434-441; https://doi.org/10.3390/quantum4040031 - 17 Oct 2022
Cited by 1 | Viewed by 1387
Abstract
In quantum machine learning, feature vectors are encoded into quantum states. Measurements for the discrimination of states are useful tools for classification problems. Classification algorithms inspired by quantum state discrimination have recently been implemented on classical computers. We present a local approach combining [...] Read more.
In quantum machine learning, feature vectors are encoded into quantum states. Measurements for the discrimination of states are useful tools for classification problems. Classification algorithms inspired by quantum state discrimination have recently been implemented on classical computers. We present a local approach combining Vonoroi-type tessellation of a training set with pretty-good measurements for quantum state discrimination. Full article
(This article belongs to the Special Issue Exclusive Feature Papers of Quantum Reports)
16 pages, 4903 KiB  
Article
Machine Learning with Quantum Matter: An Example Using Lead Zirconate Titanate
Quantum Rep. 2022, 4(4), 418-433; https://doi.org/10.3390/quantum4040030 - 03 Oct 2022
Viewed by 1617
Abstract
Stephen Wolfram (2002) proposed the concept of computational equivalence, which implies that almost any dynamical system can be considered as a computation, including programmable matter and nonlinear materials such as, so called, quantum matter. Memristors are often used in building and evaluating hardware [...] Read more.
Stephen Wolfram (2002) proposed the concept of computational equivalence, which implies that almost any dynamical system can be considered as a computation, including programmable matter and nonlinear materials such as, so called, quantum matter. Memristors are often used in building and evaluating hardware neural networks. Ukil (2011) demonstrated a theoretical relationship between piezoelectrical materials and memristors. We review that work as a necessary background prior to our work on exploring a piezoelectric material for neural network computation. Our method consisted of using a cubic block of unpoled lead zirconate titanate (PZT) ceramic, to which we have attached wires for programming the PZT as a programmable substrate. We then, by means of pulse trains, constructed on-the-fly internal patterns of regions of aligned polarization and unaligned, or disordered regions. These dynamic patterns come about through constructive and destructive interference and may be exploited as a type of reservoir network. Using MNIST data we demonstrate a learning machine. Full article
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17 pages, 348 KiB  
Article
Coarse-Graining of Observables
Quantum Rep. 2022, 4(4), 401-417; https://doi.org/10.3390/quantum4040029 - 03 Oct 2022
Cited by 1 | Viewed by 1195
Abstract
We first define the coarse-graining of probability measures in terms of stochastic kernels. We define when a probability measure is part of another probability measure and say that two probability measures coexist if they are both parts of a single probability measure. We [...] Read more.
We first define the coarse-graining of probability measures in terms of stochastic kernels. We define when a probability measure is part of another probability measure and say that two probability measures coexist if they are both parts of a single probability measure. We then show that any two probability measures coexist. We extend these concepts to observables and instruments and mention that two observables need not coexist. We define the discretization of an observable as a special case of coarse-graining and show that these have 0–1 stochastic kernels. We next consider finite observables and instruments and show that in these cases, stochastic kernels are replaced by stochastic matrices. We also show that coarse-graining is the same as post-processing in this finite case. We then consider sequential products of observables and discuss the sequential product of a post-processed observable with another observable. We briefly discuss SIC observables and the example of qubit observables. Full article
(This article belongs to the Special Issue Exclusive Feature Papers of Quantum Reports)
11 pages, 317 KiB  
Article
A Potential-Based Quantization Procedure of the Damped Oscillator
Quantum Rep. 2022, 4(4), 390-400; https://doi.org/10.3390/quantum4040028 - 21 Sep 2022
Cited by 1 | Viewed by 1482
Abstract
Today, two of the most prosperous fields of physics are quantum computing and spintronics. In both, the loss of information and dissipation play a crucial role. In the present work, we formulate the quantization of the dissipative oscillator, which aids the understanding of [...] Read more.
Today, two of the most prosperous fields of physics are quantum computing and spintronics. In both, the loss of information and dissipation play a crucial role. In the present work, we formulate the quantization of the dissipative oscillator, which aids the understanding of the abovementioned issues, and creates a theoretical frame to overcome these issues in the future. Based on the Lagrangian framework of the damped spring system, the canonically conjugated pairs and the Hamiltonian of the system are obtained; then, the quantization procedure can be started and consistently applied. As a result, the damping quantum wave equation of the dissipative oscillator is deduced, and an exact damping wave solution of this equation is obtained. Consequently, we arrive at an irreversible quantum theory by which the quantum losses can be described. Full article
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10 pages, 1838 KiB  
Article
Model-Free Deep Recurrent Q-Network Reinforcement Learning for Quantum Circuit Architectures Design
Quantum Rep. 2022, 4(4), 380-389; https://doi.org/10.3390/quantum4040027 - 21 Sep 2022
Cited by 3 | Viewed by 2467
Abstract
Artificial intelligence (AI) technology leads to new insights into the manipulation of quantum systems in the Noisy Intermediate-Scale Quantum (NISQ) era. Classical agent-based artificial intelligence algorithms provide a framework for the design or control of quantum systems. Traditional reinforcement learning methods are designed [...] Read more.
Artificial intelligence (AI) technology leads to new insights into the manipulation of quantum systems in the Noisy Intermediate-Scale Quantum (NISQ) era. Classical agent-based artificial intelligence algorithms provide a framework for the design or control of quantum systems. Traditional reinforcement learning methods are designed for the Markov Decision Process (MDP) and, hence, have difficulty in dealing with partially observable or quantum observable decision processes. Due to the difficulty of building or inferring a model of a specified quantum system, a model-free-based control approach is more practical and feasible than its counterpart of a model-based approach. In this work, we apply a model-free deep recurrent Q-network (DRQN) reinforcement learning method for qubit-based quantum circuit architecture design problems. This paper is the first attempt to solve the quantum circuit design problem from the recurrent reinforcement learning algorithm, while using discrete policy. Simulation results suggest that our long short-term memory (LSTM)-based DRQN method is able to learn quantum circuits for entangled Bell–Greenberger–Horne–Zeilinger (Bell–GHZ) states. However, since we also observe unstable learning curves in experiments, suggesting that the DRQN could be a promising method for AI-based quantum circuit design application, more investigation on the stability issue would be required. Full article
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18 pages, 391 KiB  
Article
Integral Quantization for the Discrete Cylinder
Quantum Rep. 2022, 4(4), 362-379; https://doi.org/10.3390/quantum4040026 - 21 Sep 2022
Cited by 4 | Viewed by 1676
Abstract
Covariant integral quantizations are based on the resolution of the identity by continuous or discrete families of normalized positive operator valued measures (POVM), which have appealing probabilistic content and which transform in a covariant way. One of their advantages is their ability to [...] Read more.
Covariant integral quantizations are based on the resolution of the identity by continuous or discrete families of normalized positive operator valued measures (POVM), which have appealing probabilistic content and which transform in a covariant way. One of their advantages is their ability to circumvent problems due to the presence of singularities in the classical models. In this paper, we implement covariant integral quantizations for systems whose phase space is Z×S1, i.e., for systems moving on the circle. The symmetry group of this phase space is the discrete & compact version of the Weyl–Heisenberg group, namely the central extension of the abelian group Z×SO(2). In this regard, the phase space is viewed as the right coset of the group with its center. The non-trivial unitary irreducible representation of this group, as acting on L2(S1), is square integrable on the phase space. We show how to derive corresponding covariant integral quantizations from (weight) functions on the phase space and resulting resolution of the identity. As particular cases of the latter we recover quantizations with de Bièvre-del Olmo–Gonzales and Kowalski–Rembielevski–Papaloucas coherent states on the circle. Another straightforward outcome of our approach is the Mukunda Wigner transform. We also look at the specific cases of coherent states built from shifted gaussians, Von Mises, Poisson, and Fejér kernels. Applications to stellar representations are in progress. Full article
(This article belongs to the Special Issue Continuous and Discrete Phase-Space Methods and Their Applications)
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