Quantization Map Research Articles

Quantum (matrix) geometry and quasi-coherent states

A general framework is described which associates geometrical structures to any set of D finite-dimensional Hermitian matrices X a , a = 1, …, D. This framework generalizes and systematizes the well-known examples of fuzzy spaces, and allows to extract the underlying classical space without requiring the limit of large matrices or representation theory. The approach is based on the previously introduced concept of quasi-coherent states. In particular, a concept of quantum Kähler geometry arises naturally, which includes the well-known quantized coadjoint orbits such as the fuzzy sphere and fuzzy . A quantization map for quantum Kähler geometries is established. Some examples of quantum geometries which are not Kähler are identified, including the minimal fuzzy torus.

Dependence of the affine coherent states quantization on the parametrization of the affine group

The affine coherent states quantization is a promising integral quantization of Hamiltonian systems when the phase space includes at least one conjugate pair of variables which takes values from a half plane. Such a situation is common for gravitational systems which include singularities. The construction of the quantization map includes a one-to-one mapping of the half plane onto the affine group. Particular cases of this mapping define specific parametrizations of the group. Our aim is showing that different such parametrizations lead to unitarily inequivalent quantum theories. Depending on the Hamiltonian system under consideration, this dependence could potentially be used constructively.

Blind Omnidirectional Image Quality Assessment Based on Structure and Natural Features

Omnidirectional image quality assessment (OIQA) is a hot research topic in image processing. Besides, IQA is also very important in the field of instrumentation and measurement. In this article, we propose a new no-reference OIQA model by analyzing the relationship between image features and perceived quality. A gradient weighted local phrase quantization map is built to complement the gradient map to model the structure degradation of omnidirectional images caused by various distortions. Considering the large amount of information contained in the omnidirectional image, we extract the luminance features, global entropy instead of local entropy, and color features, which are ignored in most previous works to estimate the naturalness quality of the omnidirectional image. Finally, support vector regression is utilized to train and test our model based on the image features and human subjective quality assessment scores. The experiments on two databases show that our model outperforms the state-of-the-art IQA models and correlates well with subjective scores in the two databases.

Quasi-probability distributions in loop quantum cosmology

In this paper, we introduce a complete family of parametrized quasi-probability distributions in phase space and their corresponding Weyl quantization maps with the aim to generalize the recently developed Wigner–Weyl formalism within the loop quantum cosmology (LQC) program. In particular, we intend to define those quasi-distributions for states valued on the Bohr compactification of the real line in such a way that they are labeled by a parameter that accounts for the ordering ambiguity corresponding to non-commutative quantum operators. Hence, we notice that the projections of the parametrized quasi-probability distributions result in marginal probability densities which are invariant under any ordering prescription. We also note that, in opposition to the standard Schrödinger representation, for an arbitrary character the quasi-distributions determine a positive function independently of the ordering. Further, by judiciously implementing a parametric-ordered Weyl quantization map for LQC, we are able to recover in a simple manner the relevant cases of the standard, anti-standard, and Weyl symmetric orderings, respectively. We expect that our results may serve to analyze several fundamental aspects within the LQC program, in special those related to coherence, squeezed states, and the convergence of operators, as extensively analyzed in the quantum optics and in the quantum information frameworks.

The Duflo non-commutative Fourier transform for the Lorentz group

For a quantum system whose phase space is the cotangent bundle of a Lie group, like for systems endowed with particular cases of curved geometry, one usually resorts to a description in terms of the irreducible representations of the Lie group, where the role of (non-commutative) phase space variables remains obscure. However, a non-commutative Fourier transform can be defined, intertwining the group and (non-commutative) algebra representation, depending on the specific quantization map. We discuss the construction of the non-commutative Fourier transform and the non-commutative algebra representation, via the Duflo quantization map, for a system whose phase space is the cotangent bundle of the Lorentz group.

Strict deformation quantization of the state space of Mk(ℂ) with applications to the Curie–Weiss model

Increasing tensor powers of the [Formula: see text] matrices [Formula: see text] are known to give rise to a continuous bundle of [Formula: see text]-algebras over [Formula: see text] with fibers [Formula: see text] and [Formula: see text], where [Formula: see text], the state space of [Formula: see text], which is canonically a compact Poisson manifold (with stratified boundary). Our first result is the existence of a strict deformation quantization of [Formula: see text] à la Rieffel, defined by perfectly natural quantization maps [Formula: see text] (where [Formula: see text] is an equally natural dense Poisson subalgebra of [Formula: see text]). We apply this quantization formalism to the Curie–Weiss model (an exemplary quantum spin with long-range forces) in the parameter domain where its [Formula: see text] symmetry is spontaneously broken in the thermodynamic limit [Formula: see text]. If this limit is taken with respect to the macroscopic observables of the model (as opposed to the quasi-local observables), it yields a classical theory with phase space [Formula: see text] (i.e. the unit three-ball in [Formula: see text]). Our quantization map then enables us to take the classical limit of the sequence of (unique) algebraic vector states induced by the ground state eigenvectors [Formula: see text] of this model as [Formula: see text], in which the sequence converges to a probability measure [Formula: see text] on the associated classical phase space [Formula: see text]. This measure is a symmetric convex sum of two Dirac measures related by the underlying [Formula: see text]-symmetry of the model, and as such the classical limit exhibits spontaneous symmetry breaking, too. Our proof of convergence is heavily based on Perelomov-style coherent spin states and at some stage it relies on (quite strong) numerical evidence. Hence the proof is not completely analytic, but somewhat hybrid.

Diffeomorphisms on the fuzzy sphere

Abstract Diffeomorphisms can be seen as automorphisms of the algebra of functions. In matrix regularization, functions on a smooth compact manifold are mapped to finite-size matrices. We consider how diffeomorphisms act on the configuration space of the matrices through matrix regularization. For the case of the fuzzy $$S^2$, we construct the matrix regularization in terms of the Berezin–Toeplitz quantization. By using this quantization map, we define diffeomorphisms on the space of matrices. We explicitly construct the matrix version of holomorphic diffeomorphisms on $$S^2$. We also propose three methods of constructing approximate invariants on the fuzzy $$S^2$. These invariants are exactly invariant under area-preserving diffeomorphisms and only approximately invariant (i.e. invariant in the large-$$N$ limit) under general diffeomorphisms.

Spin foam models and the Duflo map

We give a general definition of spin foam models, and then of models of 4d quantum gravity based on constraining BF theory. We highlight the construction and quantization ambiguities entering model building, among which the choice of quantization map applied to the B variables carrying metric information after imposing simplicity constraints, and the different strategies for imposing the latter constraints. We then construct a new spin foam model for 4d quantum gravity, using the flux representation of states and amplitudes, based on the Duflo quantization map and the associated non-commutative Fourier transform for Lie groups. The advantages of the new model are the geometrically transparent way in which constraints are imposed, and the underlying mathematical properties of the Duflo map itself. Finally, the presence of a closed analytical formula for the model’s amplitudes is another valuable asset for future applications.

Spatial Diversity Impact in Mobile Quantisation Mapping for Cognitive Radio Networks

Spatial Diversity Impact in Mobile Quantisation Mapping for Cognitive Radio Networks

Spatial Diversity Impact in Mobile Quantisation Mapping for Cognitive Radio Networks

Spatial Diversity Impact in Mobile Quantisation Mapping for Cognitive Radio Networks

On the von Neumann rule in quantization

We show that any linear quantization map into the space of self-adjoint operators in a Hilbert space violates the von Neumann rule on postcomposition with real functions.

Shannon entropy at avoided crossings in the quantum transition from order to chaos.

Shannon entropy is studied for the series of avoided crossings that characterize the transition from order to chaos in quantum mechanics. In order to be able to study jointly this entropy for discrete and continuous probability, calculations have been performed on a quantized map, the kicked Harper map, resulting in a different behavior, as order-chaos transition takes place, for the discrete (position representation) and continuous (coherent state representation) cases. This different behavior is analyzed in terms of the distribution of zeros of the Husimi function.

Local Ternary Cross Structure Pattern: A Color LBP Feature Extraction with Applications in CBIR

With the advent of medical endoscopes, earth observation satellites and personal phones, content-based image retrieval (CBIR) has attracted considerable attention, triggered by its wide applications, e.g., medical image analytics, remote sensing, and person re-identification. However, constructing effective feature extraction is still recognized as a challenging problem. To tackle this problem, we first propose the five-level color quantizer (FLCQ) to acquire a color quantization map (CQM). Secondly, according to the anatomical structure of the human visual system, the color quantization map (CQM) is amalgamated with a local binary pattern (LBP) map to construct a local ternary cross structure pattern (LTCSP). Third, the LTCSP is further converted into the uniform local ternary cross structure pattern (LTCSPuni) and the rotation-invariant local ternary cross structure pattern (LTCSPri) in order to cut down the computational cost and improve the robustness, respectively. Finally, through quantitative and qualitative evaluations on face, objects, landmark, textural and natural scene datasets, the experimental results illustrate that the proposed descriptors are effective, robust and practical in terms of CBIR application. In addition, the computational complexity is further evaluated to produce an in-depth analysis.

Noncommutative Fourier transform for the Lorentz group via the Duflo map

We defined a non-commutative algebra representation for quantum systems whose phase space is the cotangent bundle of the Lorentz group, and the non-commutative Fourier transform ensuring the unitary equivalence with the standard group representation. Our construction is from first principles in the sense that all structures are derived from the choice of quantization map for the classical system, the Duflo quantization map.

Quantization and the resolvent algebra

We introduce a novel commutative C*-algebra CR(X) of functions on a symplectic vector space (X,σ) admitting a complex structure, along with a strict deformation quantization that maps a dense subalgebra of CR(X) to the resolvent algebra introduced by Buchholz and Grundling [2]. The associated quantization map is a field-theoretical Weyl quantization compatible with the work of Binz, Honegger and Rieckers [1]. We also define a Berezin-type quantization map on all of CR(X), which continuously and bijectively maps it onto the resolvent algebra.The commutative resolvent algebra CR(X), generally defined on a real inner product space X, intimately depends on the finite dimensional subspaces of X. We thoroughly analyze the structure of this algebra in the finite dimensional case by giving a characterization of its elements and by computing its Gelfand spectrum.

Rotation sensing and gesture control of a robot joint via triboelectric quantization sensor

In human-machine interaction, robotic hands are expected to work like human's hands and to be even more powerful or delicate in certain situations. To operate robotic hands via human gesture instead of handle or button will make this human-robot interface more natural and precise. Here, we designed a joint motion triboelectric quantization sensor (jmTQS) for constructing a robotic hand synchronous control system. Based on the ultrahigh sensitivity of a triboelectric nanogenerator (TENG) to mechanical displacement, the jmTQS designed as grating-sliding mode realized directly quantifying a joint's flexion-extension degree/speed. Through counting the pulses induced by jmTQS and signing the positive/negative of the pulses to represent flexion/extension, the joint's angular position can be determined with absolute value on the basis of the initial human-robotic synchronizing position value. In the whole operating course, the intuitionistic human-robotic hand two-dimensional motion mapping can be preserved. The minimum resolution angle of the fabricated jmTQSs is 3.8° and can be further improved by decreasing the grating width. This direct quantization and intuitionistic mapping at the sensing stage greatly simplified the signal processing and classification algorithms, which contributes to achieving the natural, high-precision and real-time interface.

Variations à la Fourier-Weyl-Wigner on Quantizations of the Plane and the Half-Plane.

Any quantization maps linearly function on a phase space to symmetric operators in a Hilbert space. Covariant integral quantization combines operator-valued measure with the symmetry group of the phase space. Covariant means that the quantization map intertwines classical (geometric operation) and quantum (unitary transformations) symmetries. Integral means that we use all resources of integral calculus, in order to implement the method when we apply it to singular functions, or distributions, for which the integral calculus is an essential ingredient. We first review this quantization scheme before revisiting the cases where symmetry covariance is described by the Weyl-Heisenberg group and the affine group respectively, and we emphasize the fundamental role played by Fourier transform in both cases. As an original outcome of our generalisations of the Wigner-Weyl transform, we show that many properties of the Weyl integral quantization, commonly viewed as optimal, are actually shared by a large family of integral quantizations.

Information metric, Berry connection, and Berezin-Toeplitz quantization for matrix geometry

We consider the information metric and Berry connection in the context of noncommutative matrix geometry. We propose that these objects give a new method of characterizing the fuzzy geometry of matrices. We first give formal definitions of these geometric objects and then explicitly calculate them for the well-known matrix configurations of fuzzy $S^2$ and fuzzy $S^4$. We find that the information metrics are given by the usual round metrics for both examples, while the Berry connections coincide with the configurations of the Wu-Yang monopole and the Yang monopole for fuzzy $S^2$ and fuzzy $S^4$, respectively. Then, we demonstrate that the matrix configurations of fuzzy $S^n$ $(n=2,4)$ can be understood as images of the embedding functions $S^n\rightarrow \textbf{R}^{n+1}$ under the Berezin-Toeplitz quantization map. Based on this result, we also obtain a mapping rule for the Laplacian on fuzzy $S^4$.

Dirac’s magnetic monopole and the Kontsevich star product

We examine relationships between various quantization schemes for an electrically charged particle in the field of a magnetic monopole. Quantization maps are defined in invariant geometrical terms, appropriate to the case of nontrivial topology, and are constructed for two operator representations. In the first setting, the quantum operators act on the Hilbert space of sections of a nontrivial complex line bundle associated with the Hopf bundle, whereas the second approach uses instead a quaternionic Hilbert module of sections of a trivial quaternionic line bundle. We show that these two quantizations are naturally related by a bundle morphism and, as a consequence, induce the same phase-space star product. We obtain explicit expressions for the integral kernels of star-products corresponding to various operator orderings and calculate their asymptotic expansions up to the third order in the Planck constant . We also show that the differential form of the magnetic Weyl product corresponding to the symmetric ordering agrees completely with the Kontsevich formula for deformation quantization of Poisson structures and can be represented by Kontsevich’s graphs.

Quantum spaces, central extensions of Lie groups and related quantum field theories11Based on a talk presented by Poulain T at the XXVth International Conference on Integrable Systems and Quantum symmetries (ISQS-25), Prague, June 6-10 2017.

Quantum spaces with su(2) noncommutativity can be modelled by using a family of SO(3)-equivariant differential *-representations. The quantization maps are determined from the combination of the Wigner theorem for SU(2) with the polar decomposition of the quantized plane waves. A tracial star-product, equivalent to the Kontsevich product for the Poisson manifold dual to su(2) is obtained from a subfamily of differential *-representations. Noncommutative (scalar) field theories free from UV/IR mixing and whose commutative limit coincides with the usual ϕ4 theory on ℛ3 are presented. A generalization of the construction to semi-simple possibly non simply connected Lie groups based on their central extensions by suitable abelian Lie groups is discussed.