Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2206.10602

Quantum states defined by using the finite frame quantization

Nicolae Cotfas

Published Jun 19, 2022• 0 citations
arXivHTMLPDF

Abstract

Finite frame quantization is a discrete version of the coherent state quantization. In the case of a quantum system with finite-dimensional Hilbert space, the finite frame quantization allows us to associate a linear operator to each function defined on the discrete phase space of the system. We investigate the properties of the density operators which can be defined by using this method.

DOI
10.48550/arxiv.2206.10602

Related papers

Finite-dimensional Hilbert space and frame quantization

Nicolae Cotfas, Jean‐Pierre Gazeau, Apostol Vourdas

The quantum observables used in the case of quantum systems with finite-dimensional Hilbert space are defined either algebraically in terms of an orthonormal basis and discrete Fourier transformation or by using a continuous system of coherent states. We present an alternative approach to these important quantum systems based on the finite frame quantization. Finite systems of coherent states, usually called finite tight frames, can be defined in a natural way in the case of finite quantum systems. Novel examples of such tight frames are presented. The quantum observables used in our approach are obtained by starting from certain classical observables described by functions defined on the discrete phase space corresponding to the system. They are obtained by using a finite frame and a Klauder–Berezin–Toeplitz-type quantization. Semi-classical aspects of tight frames are studied through lower symbols of basic classical observables.

Journal of Physics A Mathematical and Theoretical•Apr 4, 2011•32 citations

Phase space quantum mechanics via frame quantization on finite groups

Francis D. Delloro, Job A. Nable

In this work, the version of Wigner transforms and Wigner functions on discrete systems are formulated. Using frame quantization, the star-product of phase space functions are also computed for a particular finite group. Specifically, the Wigner transform, Wigner function and star-product are computed for the dicyclic group Dien. Frames are the discrete version of coherent states. In the literature, quasiprobability distributions are essential in the study of the phase space representations of quantum mechanics, and are naturally associated to frame quantizations. For this work, the unitary irreducible representations of a finite group play an essential role in the construction.

AIP conference proceedings•Jan 1, 2022•0 citations

Frame representation of quantum systems with finite-dimensional Hilbert space

Nicolae Cotfas

Abstract There exist many attempts to define a Wigner function for quantum systems with finite-dimensional Hilbert space, each of them coming with its advantages and limitations. The existing finite versions have simple definitions, but they are based only on the existence of a formal analogy with the continuous-variable Wigner function and do not allow an intuitive state analysis. The continuous versions have more complicated definitions, but they are closer to the original Wigner function and allow a visualization of the quantum states. The version based on the concept of tight frame we present is finite, but it has certain properties and applications similar to those of continuous versions. It allows us to present a new graphical representation of qubit states, and to define new parameters concerning them. An important advantage of frame representation follows from the use of redundant information. The values taken by the frame version of Wigner function are not independent. They have to satisfy a large number of mathematical relations, useful in error detection and correction.

arXiv:2305.19287•Journal of Physics A Mathematical and Theoretical•Aug 28, 2024•2 citations

Frame-dependent coherence of Gaussian quantum states

Nicolae Cotfas

A Hilbert space is usually described using an orthonormal basis, but a more general description can be defined using a tight frame. In the case of continuous-variable Gaussian states, the coherence is usually described using the Hermite-Gauss basis. We present an alternative description using the tight frame of all canonical coherent states. The description of Gaussian states in terms of coherent states is simpler than in terms of Hermite-Gauss basis, and this allows us to obtain an explicit formula describing the frame-dependent coherence of a mixed Gaussian state as a function of the determinant and the trace of covariance matrix. For each mixed Gaussian state, we construct a sequence of discrete-variable quantum states rapidly convergent to it (as concern the purity and coherence) when the dimension of Hilbert space becomes large enough (15 or more). In the discrete case, the coherence is measured with respect to the tight frame of all discrete coherent states.

quant-pharXiv:2610.05623•Oct 6, 2026•0 citations

Finite oscillator obtained through finite frame quantization

Nicolae Cotfas, Daniela Dragoman

The Hamiltonian of the harmonic oscillator is usually defined as a differential operator, but an integral representation can be obtained by using the coherent state quantization. The finite frame quantization is a finite counterpart of the coherent state quantization and it allows us to define a finite oscillator by starting from the integral representation of the harmonic oscillator. Our purpose is to investigate the oscillator obtained in this way, and to present a possible application to the discrete fractional Fourier transform.

arXiv:1301.0704•Journal of Physics A Mathematical and Theoretical•Aug 8, 2013•3 citations

Introduction to coherent quantization

Arnold Neumaier, Arash Ghaani Farashahi

Abstract This paper studies coherent quantization, the way operators in the quantum space of a coherent space—defined in the recent book ’Coherent Quantum Mechanics’ by the first author—can be studied in terms of objects defined directly on the coherent space. The results may be viewed as a generalization of geometric quantization, including the non-unitary case. Care has been taken to work with the weakest meaningful topology and to assume as little as possible about the spaces and groups involved. Unlike in geometric quantization, the groups are not assumed to be compact, locally compact, or finite-dimensional. This implies that the setting can be successfully applied to quantum field theory, where the groups involved satisfy none of these properties. The paper characterizes linear operators acting on the quantum space of a coherent space in terms of their coherent matrix elements. Coherent maps and associated symmetry groups for coherent spaces are introduced, and formulas are derived for the quantization of coherent maps. The importance of coherent maps for quantum mechanics is due to the fact that there is a quantization map that associates homomorphically with every coherent map a linear operator from the quantum space into itself. The quantization map generalizes the second quantization procedure for free classical fields to symmetry groups of general coherent spaces. Field quantization is obtained by specialization to Klauder spaces, whose quantum spaces are the bosonic Fock spaces. Implied by the new approach is a short, coordinate-free derivation of all basic properties of creation and annihilation operators in Fock spaces.

arXiv:1804.01400•Analysis and Mathematical Physics•Jun 17, 2022•3 citations
Finite frame quantization is a discrete version of the coherent state quantization. In the case of a quantum system with finite-dimensional Hilbert space, the finite frame quantization allows us to associate a linear operator to each function defined on the discrete phase space of the system. We investigate the properties of the density operators which can be defined by using this method.
The quantum observables used in the case of quantum systems with finite-dimensional Hilbert space are defined either algebraically in terms of an orthonormal basis and discrete Fourier transformation or by using a continuous system of coherent states. We present an alternative approach to these important quantum systems based on the finite frame quantization. Finite systems of coherent states, usually called finite tight frames, can be defined in a natural way in the case of finite quantum systems. Novel examples of such tight frames are presented. The quantum observables used in our approach are obtained by starting from certain classical observables described by functions defined on the discrete phase space corresponding to the system. They are obtained by using a finite frame and a Klauder–Berezin–Toeplitz-type quantization. Semi-classical aspects of tight frames are studied through lower symbols of basic classical observables.
In this work, the version of Wigner transforms and Wigner functions on discrete systems are formulated. Using frame quantization, the star-product of phase space functions are also computed for a particular finite group. Specifically, the Wigner transform, Wigner function and star-product are computed for the dicyclic group Dien. Frames are the discrete version of coherent states. In the literature, quasiprobability distributions are essential in the study of the phase space representations of quantum mechanics, and are naturally associated to frame quantizations. For this work, the unitary irreducible representations of a finite group play an essential role in the construction.
Abstract There exist many attempts to define a Wigner function for quantum systems with finite-dimensional Hilbert space, each of them coming with its advantages and limitations. The existing finite versions have simple definitions, but they are based only on the existence of a formal analogy with the continuous-variable Wigner function and do not allow an intuitive state analysis. The continuous versions have more complicated definitions, but they are closer to the original Wigner function and allow a visualization of the quantum states. The version based on the concept of tight frame we present is finite, but it has certain properties and applications similar to those of continuous versions. It allows us to present a new graphical representation of qubit states, and to define new parameters concerning them. An important advantage of frame representation follows from the use of redundant information. The values taken by the frame version of Wigner function are not independent. They have to satisfy a large number of mathematical relations, useful in error detection and correction.
A Hilbert space is usually described using an orthonormal basis, but a more general description can be defined using a tight frame. In the case of continuous-variable Gaussian states, the coherence is usually described using the Hermite-Gauss basis. We present an alternative description using the tight frame of all canonical coherent states. The description of Gaussian states in terms of coherent states is simpler than in terms of Hermite-Gauss basis, and this allows us to obtain an explicit formula describing the frame-dependent coherence of a mixed Gaussian state as a function of the determinant and the trace of covariance matrix. For each mixed Gaussian state, we construct a sequence of discrete-variable quantum states rapidly convergent to it (as concern the purity and coherence) when the dimension of Hilbert space becomes large enough (15 or more). In the discrete case, the coherence is measured with respect to the tight frame of all discrete coherent states.
The Hamiltonian of the harmonic oscillator is usually defined as a differential operator, but an integral representation can be obtained by using the coherent state quantization. The finite frame quantization is a finite counterpart of the coherent state quantization and it allows us to define a finite oscillator by starting from the integral representation of the harmonic oscillator. Our purpose is to investigate the oscillator obtained in this way, and to present a possible application to the discrete fractional Fourier transform.
Abstract This paper studies coherent quantization, the way operators in the quantum space of a coherent space—defined in the recent book ’Coherent Quantum Mechanics’ by the first author—can be studied in terms of objects defined directly on the coherent space. The results may be viewed as a generalization of geometric quantization, including the non-unitary case. Care has been taken to work with the weakest meaningful topology and to assume as little as possible about the spaces and groups involved. Unlike in geometric quantization, the groups are not assumed to be compact, locally compact, or finite-dimensional. This implies that the setting can be successfully applied to quantum field theory, where the groups involved satisfy none of these properties. The paper characterizes linear operators acting on the quantum space of a coherent space in terms of their coherent matrix elements. Coherent maps and associated symmetry groups for coherent spaces are introduced, and formulas are derived for the quantization of coherent maps. The importance of coherent maps for quantum mechanics is due to the fact that there is a quantization map that associates homomorphically with every coherent map a linear operator from the quantum space into itself. The quantization map generalizes the second quantization procedure for free classical fields to symmetry groups of general coherent spaces. Field quantization is obtained by specialization to Klauder spaces, whose quantum spaces are the bosonic Fock spaces. Implied by the new approach is a short, coordinate-free derivation of all basic properties of creation and annihilation operators in Fock spaces.