The Zak transform on strongly proper G-spaces and its applications

Jüstel, Dominik

doi:10.1112/jlms.12097

Cited by 7 publications

(10 citation statements)

References 36 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…V C), so this formula is useful to approximate sums with integrals. A closer inspection reveals that this is a special case (H = Z and G = R) of the more general formula for evaluating the sum of a function f ∈ L 2 (G) over H [116]: This can be generalized to sums over cosets aH.…”

Section: (C20)mentioning

confidence: 93%

“…For example,Ŝ Z for a linear rotor code based on the octahedral group is a superposition of octopole ( = 4) spherical harmonics; see Eq. (116). Such harmonics are in principle present in a general interaction with a bath [72].…”

Section: A Molecular Rotorsmentioning

confidence: 98%

“…The caption of Table IV adapts these discussions to other G, and Ref. [116] rigorously formulates many of the required tools for type I unimodular second-countable groups.…”

Section: A Qubit On a Groupmentioning

confidence: 99%

“…More generally, if a group G acts transitively on the states of a quantum system, and the subgroup H of G leaves the states invariant, then the configuration space of the system is G/H. In Appendix D, we develop mathematical tools for parameterizing the position eigenstates and the dual momentum states of such a system, including orthogonality/completeness relations, and a Poisson summation formula [116].…”

Section: Other Systemsmentioning

confidence: 99%

“…We observe that this is zero unless Z (H) = 0. The participating thus form the reciprocal space [116] While the surviving are established by H ⊥ , surviving m, n depend on the choice of basis used for the irreps Z . We pick an H-admissible basis [165], for which each Z decomposes into blocks of some irreps λ( ) ∈ H when restricted to h ∈ H,…”

Section: (C20)mentioning

confidence: 99%

See 4 more Smart Citations

Encoding a qubit in a molecule

Albert

Covey

Preskill

2019

Rochester Conference on Coherence and Quantum Optics (CQO-11)

View full text Add to dashboard Cite

We construct quantum error-correcting codes that embed a finite-dimensional code space in the infinite-dimensional Hilbert state space of rotational states of a rigid body. These codes, which protect against both drift in the body's orientation and small changes in its angular momentum, may be well suited for robust storage and coherent processing of quantum information using rotational states of a polyatomic molecule. Extensions of such codes to rigid bodies with a symmetry axis are compatible with rotational states of diatomic molecules, as well as nuclear states of molecules and atoms. We also describe codes associated with general nonabelian compact Lie groups and develop orthogonality relations for coset spaces, laying the groundwork for quantum information processing with exotic configuration spaces.

show abstract

Section: (C20)mentioning

confidence: 93%

Section: A Molecular Rotorsmentioning

confidence: 98%

“…The caption of Table IV adapts these discussions to other G, and Ref. [116] rigorously formulates many of the required tools for type I unimodular second-countable groups.…”

Section: A Qubit On a Groupmentioning

confidence: 99%

Section: Other Systemsmentioning

confidence: 99%

Section: (C20)mentioning

confidence: 99%

See 3 more Smart Citations

Encoding a qubit in a molecule

Albert

Covey

Preskill

2019

Rochester Conference on Coherence and Quantum Optics (CQO-11)

View full text Add to dashboard Cite

show abstract

Aperiodic order and spherical diffraction, II: translation bounded measures on homogeneous spaces

Björklund

Hartnick²,

Pogorzelski

2021

Math. Z.

View full text Add to dashboard Cite

We study the auto-correlation measures of invariant random point processes in the hyperbolic plane which arise from various classes of aperiodic Delone sets. More generally, we study auto-correlation measures for large classes of Delone sets in (and even translation bounded measures on) arbitrary locally compact homogeneous metric spaces. We then specialize to the case of weighted model sets, in which we are able to derive more concrete formulas for the auto-correlation. In the case of Riemannian symmetric spaces we also explain how the auto-correlation of a weighted model set in a Riemannian symmetric space can be identified with a (typically non-tempered) positive-definite distribution on $$\mathbb {R}^n$$ R n . This paves the way for a diffraction theory for such model sets, which will be discussed in the sequel to the present article.

show abstract

Abstract Poisson summation formulas over homogeneous spaces of compact groups

Farashahi

2016

Anal.Math.Phys.

View full text Add to dashboard Cite

Poisson summation formulas over homogeneous spaces of compact groupsAbstract This paper presents the abstract notion of Poisson summation formulas for homogeneous spaces of compact groups. Let G be a compact group, H be a closed subgroup of G, and μ be the normalized G-invariant measure over the left coset space G/H associated to the Weil's formula. We prove that the abstract Fourier transform over G/H satisfies a generalized version of the Poisson summation formula.A. G. Farashahi groups of compact non-Abelian groups is placed as building blocks for coherent states analysis, constructive approximation, and particle physics, see [4,10,11,[15][16][17][18][19] and references therein. Over the last decades, abstract and computational aspects of Fourier transforms and Plancherel formulas over symmetric spaces have achieved significant popularity in geometric analysis, mathematical physics, and scientific computing, see [6,8,9] and references therein.Let G be a compact group and H be a closed subgroup of G. Then the left coset space G/H is a compact homogeneous space, where G acts on it via the left action. Let μ be the normalized G-invariant measure on the compact homogeneous space G/H associated to the Weil's formula with respect to the probability measures of G and H . This article consists of theoretical aspects of a unified approach to the nature of abstract Poisson summation formulas over homogeneous spaces of compact groups.This paper is organized as follows. Section 2 is devoted to fixing notation and preliminaries including a summary on non-Abelian Fourier analysis over compact groups and classical results on abstract harmonic analysis over compact homogeneous spaces. In Sect. 3, we present some abstract aspects of harmonic analysis on the Banach functions spaces over the compact homogeneous space G/H . We then introduce the abstract notion of dual space G/H for the homogeneous space G/H . Next we present the theoretical definition of the abstract operator-valued Fourier transform over the compact homogeneous space G/H . The paper is concluded by presenting an abstract generalized version of the Poisson summation formula. Preliminaries and notationLet H be a separable Hilbert space. An operator T ∈ B(H) is called a Hilbert-Schmidt operator if for one, hence for any orthonormal basis {e k } of H we have k T e k 2 < ∞. The set of all Hilbert-Schmidt operators on H is denoted by HS(H) and for T ∈ HS(H) the Hilbert-Schmidt norm of T is T 2 HS = k T e k 2 . The set HS(H) is a self adjoint two sided ideal in B(H) and if H is finite-dimensional we have HS(H) = B(H). An operator T ∈ B(H) is trace-class, whenever T tr = tr[|T |] < ∞, if tr[T ] = k T e k , e k and |T | = (T T * )

show abstract

The Zak transform on strongly proper G-spaces and its applications

Cited by 7 publications

References 36 publications

Encoding a qubit in a molecule

Encoding a qubit in a molecule

Aperiodic order and spherical diffraction, II: translation bounded measures on homogeneous spaces

Abstract Poisson summation formulas over homogeneous spaces of compact groups

Contact Info

Product

Resources

About