Applications of rigged Hilbert spaces in quantum mechanics and signal processing

A classical problem in axiomatic quantum mechanics is deducing a Hilbert space realization for a quantum logic that admits a vector space coordinatization of the Piron–McLaren type. Our aim is to show how a theorem of M. Solér [Characterization of Hilbert spaces by orthomodular spaces, Comm. Algebra23 (1995) 219–243.] can be used to get a (partial) solution of this problem. We first derive a generalization of the Wigner theorem on symmetry transformations that holds already in the Piron–McLaren frame. Then we investigate which conditions on the quantum logic allow the use of Solér's theorem in order to obtain a Hilbert space solution for the coordinatization problem.

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Linear Algebra and the Dirac Notation

An Introduction to Quantum Computing ◽

10.1093/oso/9780198570004.003.0005 ◽

2006 ◽

Author(s):

Phillip Kaye ◽

Raymond Laflamme ◽

Michele Mosca

Keyword(s):

Hilbert Space ◽

Quantum Mechanics ◽

Quantum Computing ◽

Linear Algebra ◽

Hilbert Spaces ◽

Vector Spaces ◽

Complex Vector ◽

Dual Vector ◽

Finite Dimensional ◽

Basic Facts

We assume the reader has a strong background in elementary linear algebra. In this section we familiarize the reader with the algebraic notation used in quantum mechanics, remind the reader of some basic facts about complex vector spaces, and introduce some notions that might not have been covered in an elementary linear algebra course. The linear algebra notation used in quantum computing will likely be familiar to the student of physics, but may be alien to a student of mathematics or computer science. It is the Dirac notation, which was invented by Paul Dirac and which is used often in quantum mechanics. In mathematics and physics textbooks, vectors are often distinguished from scalars by writing an arrow over the identifying symbol: e.g a⃗. Sometimes boldface is used for this purpose: e.g. a. In the Dirac notation, the symbol identifying a vector is written inside a ‘ket’, and looks like |a⟩. We denote the dual vector for a (defined later) with a ‘bra’, written as ⟨a|. Then inner products will be written as ‘bra-kets’ (e.g. ⟨a|b⟩). We now carefully review the definitions of the main algebraic objects of interest, using the Dirac notation. The vector spaces we consider will be over the complex numbers, and are finite-dimensional, which significantly simplifies the mathematics we need. Such vector spaces are members of a class of vector spaces called Hilbert spaces. Nothing substantial is gained at this point by defining rigorously what a Hilbert space is, but virtually all the quantum computing literature refers to a finite-dimensional complex vector space by the name ‘Hilbert space’, and so we will follow this convention. We will use H to denote such a space. Since H is finite-dimensional, we can choose a basis and alternatively represent vectors (kets) in this basis as finite column vectors, and represent operators with finite matrices. As you see in Section 3, the Hilbert spaces of interest for quantum computing will typically have dimension 2n, for some positive integer n. This is because, as with classical information, we will construct larger state spaces by concatenating a string of smaller systems, usually of size two.

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AN INFINITE DIMENSIONAL VERSION OF THE SCHUR CONVEXITY PROPERTY AND APPLICATIONS

Analysis and Applications ◽

10.1142/s0219530507000912 ◽

2007 ◽

Vol 05 (02) ◽

pp. 123-136 ◽

Cited By ~ 9

Author(s):

CLAUDE VALLÉE ◽

VICENŢIU RĂDULESCU

Keyword(s):

Quantum Mechanics ◽

Hilbert Spaces ◽

Symmetric Matrix ◽

Liouville Problem ◽

Convexity Property ◽

Infinite Dimensional ◽

Selfadjoint Operators ◽

Dimensional Version ◽

Sturm Liouville ◽

Schur Convexity

We extend to infinite dimensional separable Hilbert spaces the Schur convexity property of eigenvalues of a symmetric matrix with real entries. Our framework includes both the case of linear, selfadjoint, compact operators, and that of linear selfadjoint operators that can be approximated by operators of finite rank and having a countable family of eigenvalues. The abstract results of the present paper are illustrated by several examples from mechanics or quantum mechanics, including the Sturm–Liouville problem, the Schrödinger equation, and the harmonic oscillator.

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Zernike functions, rigged Hilbert spaces, and potential applications

Journal of Mathematical Physics ◽

10.1063/1.5093488 ◽

2019 ◽

Vol 60 (8) ◽

pp. 083508 ◽

Cited By ~ 2

Author(s):

E. Celeghini ◽

M. Gadella ◽

M. A. del Olmo

Keyword(s):

Hilbert Spaces ◽

Rigged Hilbert Spaces ◽

Potential Applications

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A Hybrid Forward–Backward Algorithm and Its Optimization Application

Mathematics ◽

10.3390/math8030447 ◽

2020 ◽

Vol 8 (3) ◽

pp. 447

Author(s):

Liya Liu ◽

Xiaolong Qin ◽

Jen-Chih Yao

Keyword(s):

Signal Processing ◽

Strong Convergence ◽

Hilbert Spaces ◽

Numerical Experiments ◽

Sparse Reconstruction ◽

Convergence Solution

In this paper, we study a hybrid forward–backward algorithm for sparse reconstruction. Our algorithm involves descent, splitting and inertial ideas. Under suitable conditions on the algorithm parameters, we establish a strong convergence solution theorem in the framework of Hilbert spaces. Numerical experiments are also provided to illustrate the application in the field of signal processing.

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Riesz-Like Bases in Rigged Hilbert Spaces

Zeitschrift für Analysis und ihre Anwendungen ◽

10.4171/zaa/1564 ◽

2016 ◽

Vol 35 (3) ◽

pp. 243-265 ◽

Cited By ~ 14

Author(s):

Giorgia Bellomonte ◽

Camillo Trapani

Keyword(s):

Hilbert Spaces ◽

Rigged Hilbert Spaces

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