Quantum Harmonic Analysis Research Articles

Decoupling for Schatten class operators in the setting of quantum harmonic analysis

AbstractWe introduce the notion of decoupling for operators, and prove an equivalence between classical decoupling for functions and decoupling for operators on bounded sets in . We also show that the equivalence depends only on the bounded set and not on the values of nor on the partition of . The proof relies on a quantum version of Wiener's division lemma.

Quantum Harmonic Analysis for Polyanalytic Fock Spaces

We develop the quantum harmonic analysis framework in the reducible setting and apply our findings to polyanalytic Fock spaces. In particular, we explain some phenomena observed in recent work by the second author and answer a few related open questions. For instance, we show that there exists a symbol such that the corresponding Toeplitz operator is unitary on the analytic Fock space but vanishes completely on one of the true polyanalytic Fock spaces. This follows directly from an explicit characterization of the kernel of the Toeplitz quantization, which we derive using quantum harmonic analysis. Moreover, we show that the Berezin transform is injective on the set of of Toeplitz operators. Finally, we provide several characterizations of the C1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {C}_1$$\\end{document}-algebra in terms of integral kernel estimates and essential commutants.

Local structure and effective dimensionality of time series data sets

The goal of this paper is to develop novel tools for understanding the local structure of systems of functions, e.g. time-series data points. The proposed tools include a total correlation function, the Cohen class of the data set, the data operator and the average lack of concentration. The Cohen class of the data operator gives a time-frequency representation of the data set. Furthermore, we show that the von Neumann entropy of the data operator captures local features of the data set and that it is related to the notion of effective dimensionality. The accumulated Cohen class of the data operator gives us a low-dimensional representation of the data set and we quantify this in terms of the average lack of concentration and the von Neumann entropy of the data operator and an improvement of the Berezin-Lieb inequality using the projection functional of the data augmentation operator. The framework for our approach is provided by quantum harmonic analysis.

Measure-operator convolutions and applications to mixed-state Gabor multipliers

For the Weyl-Heisenberg group, convolutions between functions and operators were defined by Werner as a part of a framework called quantum harmonic analysis. We show how recent results by Feichtinger can be used to extend this definition to include convolutions between measures and operators. Many properties of function-operator convolutions carry over to this setting and allow us to prove novel results on the distribution of eigenvalues of mixed-state Gabor multipliers and derive a version of the Berezin-Lieb inequality for lattices. New results on the continuity of Gabor multipliers with respect to lattice parameters, masks and windows as well as their ability to approximate localization operators are also derived using this framework.

A Quantum Harmonic Analysis Approach to Segal Algebras

In this article, we study a commutative Banach algebra structure on the space L1(R2n)⊕T1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1(\\mathbb {R}^{2n})\\oplus {\\mathcal {T}}^1$$\\end{document}, where the T1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {T}}^1$$\\end{document} denotes the trace class operators on L2(Rn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2(\\mathbb {R}^{n})$$\\end{document}. The product of this space is given by the convolutions in quantum harmonic analysis. Towards this goal, we study the closed ideals of this space, and in particular its Gelfand theory. We additionally develop the concept of quantum Segal algebras as an analogue of Segal algebras. We prove that many of the properties of Segal algebras have transfers to quantum Segal algebras. However, it should be noted that in contrast to Segal algebras, quantum Segal algebras are not ideals of the ambient space. We also give examples of different constructions that yield quantum Segal algebras.

Symplectic and Lagrangian polar duality; applications to quantum harmonic analysis

Symplectic and Lagrangian polar duality; applications to quantum harmonic analysis

Quantum harmonic analysis on locally compact groups

On a locally compact group we introduce covariant quantization schemes and analogs of phase space representations as well as mixed-state localization operators. These generalize corresponding notions for the affine group and the Heisenberg group. The approach is based on associating to a square integrable representation of the locally compact group two types of convolutions between integrable functions and trace-class operators. In the case of non-unimodular groups these convolutions only are well-defined for admissible operators, which is an extension of the notion of admissible wavelets as has been pointed out recently in the case of the affine group.

Affine quantum harmonic analysis

We develop a quantum harmonic analysis framework for the affine group. This encapsulates several examples in the literature such as affine localization operators, covariant integral quantizations, and affine quadratic time-frequency representations. In the process, we develop a notion of admissibility for operators and extend well known results to the operator setting. A major theme of the paper is the interaction between operator convolutions, affine Weyl quantization, and admissibility.

A Wiener Tauberian theorem for operators and functions

We prove variants of Wiener's Tauberian theorem in the framework of quantum harmonic analysis, i.e. for convolutions between an absolutely integrable function and a trace class operator, or of two trace class operators. Our results include Wiener's Tauberian theorem as a special case. Applications of our Tauberian theorems are related to localization operators, Toeplitz operators, isomorphism theorems between Bargmann-Fock spaces and quantization schemes with consequences for Shubin's pseudodifferential operator calculus and Born-Jordan quantization. Based on the links between localization operators and Tauberian theorems we note that the analogue of Pitt's Tauberian theorem in our setting implies compactness results for Toeplitz operators in terms of the Berezin transform. In addition, we extend the results on Toeplitz operators to other reproducing kernel Hilbert spaces induced by the short-time Fourier transform, known as Gabor spaces. Finally, we establish the equivalence of Wiener's Tauberian theorem and the condition in the characterization of compactness of localization operators due to Fernández and Galbis.

Quantum Harmonic Analysis on Lattices and Gabor Multipliers

We develop a theory of quantum harmonic analysis on lattices in {mathbb {R}}^{2d}. Convolutions of a sequence with an operator and of two operators are defined over a lattice, and using corresponding Fourier transforms of sequences and operators we develop a version of harmonic analysis for these objects. We prove analogues of results from classical harmonic analysis and the quantum harmonic analysis of Werner, including Tauberian theorems and a Wiener division lemma. Gabor multipliers from time-frequency analysis are described as convolutions in this setting. The quantum harmonic analysis is thus a conceptual framework for the study of Gabor multipliers, and several of the results include results on Gabor multipliers as special cases.

On Accumulated Cohen’s Class Distributions and Mixed-State Localization Operators

Recently we introduced mixed-state localization operators associated with a density operator and a (compact) domain in phase space. We continue the investigations of their eigenvalues and eigenvectors. Our main focus is the definition of a time-frequency distribution that is based on the Cohen class distribution associated with the density operator and the eigenvectors of the mixed-state localization operator. This time-frequency distribution is called the accumulated Cohen class distribution. If the trace class operator is a rank-one operator, then the mixed-state localization operators and the accumulated Cohen class distribution reduce to Daubechies’ localization operators and the accumulated spectrogram. We extend all the results about the accumulated spectrogram to the accumulated Cohen class distribution. The techniques used in the case of spectrograms cannot be adapted to other distributions in Cohen’s class since they rely on the reproducing kernel property of the short-time Fourier transform. Our approach is based on quantum harmonic analysis on phase space, which also provides the tools and notions to introduce the analogues of the accumulated spectrogram for mixed-state localization operators, the accumulated Cohen’s class distributions.

Convolutions for Berezin quantization and Berezin-Lieb inequalities

Concepts and results from quantum harmonic analysis, such as the convolution between functions and operators or between two operators, are identified as the appropriate setting for Berezin quantization and Berezin-Lieb inequalities. Based on this insight, we provide a rigorous approach to the generalized phase-space representation introduced by Klauder-Skagerstam and their variants of Berezin-Lieb inequalities in this setting. Hence our presentation of the results of Klauder-Skagerstam gives a more conceptual framework, which yields as a byproduct an interesting perspective on the connection between the Berezin quantization and Weyl quantization.

Convolutions for localization operators

Quantum harmonic analysis on phase space is shown to be linked with localization operators. The convolution between operators and the convolution between a function and an operator provide a conceptual framework for the theory of localization operators which is complemented by an appropriate Fourier transform, the Fourier–Wigner transform. We link the Hausdorff–Young inequality for the Fourier–Wigner transform with Lieb's inequality for ambiguity functions. Noncommutative Tauberian theorems due to Werner allow us to extend results of Bayer and Gröchenig on localization operators. Furthermore we show that the Arveson spectrum and the theory of Banach modules provide the abstract setting of quantum harmonic analysis.

Schwartz operators

In this paper, we introduce Schwartz operators as a non-commutative analog of Schwartz functions and provide a detailed discussion of their properties. We equip them, in particular, with a number of different (but equivalent) families of seminorms which turns the space of Schwartz operators into a Fréchet space. The study of the topological dual leads to non-commutative tempered distributions which are discussed in detail as well. We show, in particular, that the latter can be identified with a certain class of quadratic forms, therefore making operations like products with bounded (and also some unbounded) operators and quantum harmonic analysis available to objects which are otherwise too singular for being a Hilbert space operator. Finally, we show how the new methods can be applied by studying operator moment problems and convergence properties of fluctuation operators.

Quantum Harmonic Analysis and Geometric Invariants

We develop two topics in parallel and show their inter-relation. The first centers on the notion of a fractional-differentiable structure on a commutative or a non-commutative space. We call this studyquantum harmonic analysis. The second concerns homotopy invariants for these spaces and is an aspect of non-commutative geometry. We study an algebra A, which will be a Banach algebra with unit, represented as an algebra of operators on a Hilbert space H. In order to obtain a geometric interpretation of A, we define a derivative on elements of A. We do this in a Hilbert space context, takingdaas a commutatorda=[Q,a]. HereQis a basic self-adjoint operator with discrete spectrum, increasing sufficiently rapidly that exp(−βQ)2has a trace wheneverβ>0. We can define fractional differentiability of orderμ, with 0<μ⩽1, by the boundedness of (Q2+I)μ/2a(Q2+I)−μ/2. Alternatively we can require the boundedness of an appropriate smoothing (Bessel potential) ofda. We find that it is convenient to assume the boundedness of (Q2+I)−β/2da(Q2+I)−α/2, where we chooseα,β⩾0 such thatα+β<1. We show that this also ensures a fractional derivative of orderμ=1−βin the first sense. We define a family of interpolation spaces Jβ,α. Each such space is a Banach algebra of operator, whose elements have a fractional derivative of orderμ=1−β>0. We concentrate on subalgebras A of Jβ,αwhich have certain additional covariance properties under a group Z2×G acting on H by a unitary representationγ×U(g). In addition, the derivativeQis assumed to be G-invariant. The geometric interpretation flows from the assumption that elements of A possess an arbitrarily small fractional derivative. We study homotopy invariants of A in terms of equivariant, entire cyclic cohomology. In fact, the existence of a fractional derivative on A allows the construction of the cochainτJLO, which plays the role of the integral of differential forms. We give a simple expression for a homotopy invariant ZQ(a;g), determined by pairingτJLO, with a G-invariant elementa∈A, such thatais a square root of the identity. This invariant is ZQ(a;g)=(1/π)∫∞−∞e−t2Tr(γU(g)ae−Q2+it da)dt. This representation of the pairing is reminiscent of the heat-kernel representation for an index. In fact this quantity is an invariant, in the following sense. We isolate a simple condition on a familyQ(λ) of differentiations that yields a continuously-differentiable familyτJLO(λ) of cochains. Since ZQ(a;g) need not be an integer, continuity ofτJLO(λ) inλis insufficient to prove the constancy of the pairing. However the existence of the derivative leads to the existence of the homotopy. As τ,a vanishes forτa coboundary, and asdτJLO(λ)/dλis a coboundary, our condition onQ(λ) ensures that ZQ(λ)(a;g) is independent ofλ. Hence it is a homotopy invariant. The theory of ZQ(a;g) reduces to the study of the Radon transform of sequences of certain functions. The fractional differentiability properties of elements of A translate into properties of the asymptotics of the sequences of Radon transforms. The condition thatτJLOfit into the framework of entire cyclic cohomology translates to the existence of some fractional derivative for functions in the algebra under study, and in particular the assumptionα+β<1. Thus the study of fractionally- differentiable structures dove-tails naturally with the theory of homotopy invariants. In our study of quantum harmonic analysis, we introduce spaces T(−β,α) of operator-valued distributions. These spaces are bounded, linear operators between Sobolev spaces. The elements of the interpolation spaces, the Banach algebras Jβ,αhave derivativesdawhich belong to the spaces T(−β,α). For a certain range ofβandα, we extend the theory of the Radon transform from products of regularized, bounded operators to products of regularized, operator-valued distributions. We sometimes wish to evaluate such an invariant at the endpoint of an interval such asλ∈(0,1], where ZQ(λ)(a;g) becomes singular asλ→0. We discuss in brief a procedure to regularize the endpoint, and a method to recover ZQ(λ)(a;g) fully from certain partial information at the endpoint. Finally, we generalize this approach to cover the case whenQcan be split into the sum of “independent” partsQ1+Q2, such that (Q1+Q2)2=(Q1)2+(Q2)2. Here we assume thatQ1and (Q2)2are G-invariant, but not necessarilyQ2. With further assumptions ona, the most important being that (Q1)2−(Q2)2commutes witha, we obtain a modified formula for an invariant, namely Z{Qj}(a;g)=(1/π)∫∞−∞e−t2Tr(γU(g)ae−(Q21+Q22)/2+itd1a)dt.

Quantum harmonic analysis on phase space

Relative to an irreducible representation of the canonical commutation relations, convolutions between quantum mechanical operators and between functions and operators are defined, for which the usual Weyl transform acts as a Fourier transform. Basic properties of these operations are developed in close analogy to harmonic analysis on R2n. Using the quantum version of Wiener’s approximation theorem, a natural one-to-one correspondence between the closed, phase-space translation invariant subspaces of classical and quantum observables is established.

Physical uniformities on the state space of nonrelativisitic quantum mechanics

Uniformities describing the distinguishability of states and of observables are discussed in the context of general statistical theories and are shown to be related to distinguished subspaces of continuous observables and states, respectively. The usual formalism of quantum mechanics contains no such physical uniformity for states. Using recently developed tools of quantum harmonic analysis, a natural one-to-one correspondence between continuous subspaces of nonrelativistic quantum and classical mechanics is established, thus exhibiting a close interrelation between physical uniformities for quantum states and compactifications of phase space. General properties of the completions of the quantum state space with respect to these uniformities are discussed.