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

Journal of the Optical Society of America A

Quantum number theoretic transforms on multipartite finite systems

Apostol Vourdas, S. Zhang

Published May 13, 2009• 0 citations
Publisher

Abstract

A quantum system composed of p-1 subsystems, each of which is described with a p-dimensional Hilbert space (where p is a prime number), is considered. A quantum number theoretic transform on this system, which has properties similar to those of a Fourier transform, is studied. A representation of the Heisenberg-Weyl group in this context is also discussed.

DOI
10.1364/josaa.26.001360

Related papers

Quantum systems with finite Hilbert space

Apostol Vourdas

Quantum systems with finite Hilbert space are considered, and phase-space methods like the Heisenberg–Weyl group, symplectic transformations and Wigner and Weyl functions are discussed. A factorization of such systems in terms of smaller subsystems, based on the Chinese remainder theorem, is studied. The general formalism is applied to the case of angular momentum. In this context, SU (2) coherent states are used for analytic representations. Links between the theory of finite quantum systems and other fields of research are discussed.

Reports on Progress in Physics•Feb 2, 2004•321 citations

Symmetries of finite Heisenberg groups for multipartite systems

Miroslav Korbelář, J. Tolar

A composite quantum system comprising a finite number k of subsystems which are described with position and momentum variables in , i = 1, ..., k , is considered. Its Hilbert space is given by a k -fold tensor product of Hilbert spaces of dimensions n 1 , ..., n k . The symmetry group of the respective finite Heisenberg group is given by the quotient group of certain normalizer. This paper extends our previous investigation of bipartite quantum systems to arbitrary multipartite systems of the above type. It provides detailed description of the normalizers and the corresponding symmetry groups. The new class of symmetry groups represents a very specific generalization of symplectic groups over modular rings. As an application, a new proof of existence of the maximal set of mutually unbiased bases in Hilbert spaces of prime power dimensions is provided.

arXiv:1210.6167•Journal of Physics A Mathematical and Theoretical•Jun 27, 2012•6 citations

Unitarily inequivalent local and global Fourier transforms in multipartite quantum systems

Chao Lei, Apostol Vourdas

Abstract A multipartite system comprised of n subsystems, each of which is described with ‘local variables’ in $${{\mathbb {Z}}}(d)$$ Z ( d ) and with a d-dimensional Hilbert space H(d), is considered. Local Fourier transforms in each subsystem are defined and related phase space methods are discussed (displacement operators, Wigner and Weyl functions, etc). A holistic view of the same system might be more appropriate in the case of strong interactions, which uses ‘global variables’ in $${{\mathbb {Z}}}(d^n)$$ Z ( d n ) and a $$d^n$$ d n -dimensional Hilbert space $$H(d^n)$$ H ( d n ) . A global Fourier transform is then defined and related phase space methods are discussed. The local formalism is compared and contrasted with the global formalism. Depending on the values of d, n the local Fourier transform is unitarily inequivalent or unitarily equivalent to the global Fourier transform. Time evolution of the system in terms of both local and global variables, is discussed. The formalism can be useful in the general area of Fast Fourier transforms.

arXiv:2301.12137•Quantum Information Processing•Jan 19, 2023•2 citations

Duality, measurements, and factorization in finite quantum systems

Apostol Vourdas, C. Bendjaballah

Finite quantum systems are considered and dual quantities are defined with a finite Fourier transform. Ladder operators that translate the eigenstates of these quantities are shown to form a finite Weyl group. Dual measurements are introduced and shown to obey certain entropic inequalities. A factorization of these systems into subsystems with the use of number-theoretic results is also presented.

Physical Review A•May 1, 1993•49 citations

Quantum multiresolution analysis via fast Fourier transforms on Heisenbrg finite group

E. Rundblad, Valery Labunets, Péter Novák

We study classical and quantum harmonic analysis of phase space functions (classical observes) on finite Heisenberg group HW 2N+1 (Z N m n,Z N m n, Z m n) over the ring Z m n. This group is the discrete version of the real Heisenberg group HW 2N+1 (R N , R N , R), where R is the real field. These functions have one dimensional and m 1 , m 2 ,..., m n -dimensions matrix valued spectral components (for irreducible representations of HW 2N+1 . The family of all ID representations gives classical world world (CW). Various m i -dimension representations (i = 1,2,...,n) map classical world (CW) into quantum worlds QW(m i ) of ith resolution (i = 1,2,....n). Worlds QW(m 1 ) and QW(m n ) contain rough information and fine details about quantum word, respectively. In this case the Fourier transform on the Heisenberg group can be considered as Weyl quantization multiresolution procedure. We call this transform the natural quantum Fourier transform.

•Jan 1, 2005•0 citations

Quantum systems with finite Hilbert space: Galois fields in quantum mechanics

Apostol Vourdas

A 'Galois quantum system' in which the position and momentum take values in the Galois field GF( p ℓ ) is considered. It is comprised of ℓ-component systems which are coupled in a particular way and is described by a certain class of Hamiltonians. Displacements in the GF( p ℓ ) × GF( p ℓ ) phase space and the corresponding Heisenberg–Weyl group are studied. Symplectic transformations are shown to form the Sp(2, GF( p ℓ )) group. Wigner and Weyl functions are defined and their properties are studied. Frobenius symmetries, which are based on Frobenius automorphisms in the theory of Galois fields, are a unique feature of these systems (for ℓ ⩾ 2). If they commute with the Hamiltonian, there are constants of motion which are discussed. An analytic representation in the ℓ-sheeted complex plane provides an elegant formalism that embodies the properties of Frobenius transformations. The difference between a Galois quantum system and other finite quantum systems where the position and momentum take values in the ring is discussed.

Journal of Physics A Mathematical and Theoretical•Aug 1, 2007•83 citations
A quantum system composed of p-1 subsystems, each of which is described with a p-dimensional Hilbert space (where p is a prime number), is considered. A quantum number theoretic transform on this system, which has properties similar to those of a Fourier transform, is studied. A representation of the Heisenberg-Weyl group in this context is also discussed.
Quantum systems with finite Hilbert space are considered, and phase-space methods like the Heisenberg–Weyl group, symplectic transformations and Wigner and Weyl functions are discussed. A factorization of such systems in terms of smaller subsystems, based on the Chinese remainder theorem, is studied. The general formalism is applied to the case of angular momentum. In this context, SU (2) coherent states are used for analytic representations. Links between the theory of finite quantum systems and other fields of research are discussed.
A composite quantum system comprising a finite number k of subsystems which are described with position and momentum variables in , i = 1, ..., k , is considered. Its Hilbert space is given by a k -fold tensor product of Hilbert spaces of dimensions n 1 , ..., n k . The symmetry group of the respective finite Heisenberg group is given by the quotient group of certain normalizer. This paper extends our previous investigation of bipartite quantum systems to arbitrary multipartite systems of the above type. It provides detailed description of the normalizers and the corresponding symmetry groups. The new class of symmetry groups represents a very specific generalization of symplectic groups over modular rings. As an application, a new proof of existence of the maximal set of mutually unbiased bases in Hilbert spaces of prime power dimensions is provided.
Abstract A multipartite system comprised of n subsystems, each of which is described with ‘local variables’ in Z(d){{\mathbb {Z}}}(d)Z(d) Z ( d ) and with a d-dimensional Hilbert space H(d), is considered. Local Fourier transforms in each subsystem are defined and related phase space methods are discussed (displacement operators, Wigner and Weyl functions, etc). A holistic view of the same system might be more appropriate in the case of strong interactions, which uses ‘global variables’ in Z(dn){{\mathbb {Z}}}(d^n)Z(dn) Z ( d n ) and a dnd^ndn d n -dimensional Hilbert space H(dn)H(d^n)H(dn) H ( d n ) . A global Fourier transform is then defined and related phase space methods are discussed. The local formalism is compared and contrasted with the global formalism. Depending on the values of d, n the local Fourier transform is unitarily inequivalent or unitarily equivalent to the global Fourier transform. Time evolution of the system in terms of both local and global variables, is discussed. The formalism can be useful in the general area of Fast Fourier transforms.
We study classical and quantum harmonic analysis of phase space functions (classical observes) on finite Heisenberg group HW 2N+1 (Z N m n,Z N m n, Z m n) over the ring Z m n. This group is the discrete version of the real Heisenberg group HW 2N+1 (R N , R N , R), where R is the real field. These functions have one dimensional and m 1 , m 2 ,..., m n -dimensions matrix valued spectral components (for irreducible representations of HW 2N+1 . The family of all ID representations gives classical world world (CW). Various m i -dimension representations (i = 1,2,...,n) map classical world (CW) into quantum worlds QW(m i ) of ith resolution (i = 1,2,....n). Worlds QW(m 1 ) and QW(m n ) contain rough information and fine details about quantum word, respectively. In this case the Fourier transform on the Heisenberg group can be considered as Weyl quantization multiresolution procedure. We call this transform the natural quantum Fourier transform.
A 'Galois quantum system' in which the position and momentum take values in the Galois field GF( p ℓ ) is considered. It is comprised of ℓ-component systems which are coupled in a particular way and is described by a certain class of Hamiltonians. Displacements in the GF( p ℓ ) × GF( p ℓ ) phase space and the corresponding Heisenberg–Weyl group are studied. Symplectic transformations are shown to form the Sp(2, GF( p ℓ )) group. Wigner and Weyl functions are defined and their properties are studied. Frobenius symmetries, which are based on Frobenius automorphisms in the theory of Galois fields, are a unique feature of these systems (for ℓ ⩾ 2). If they commute with the Hamiltonian, there are constants of motion which are discussed. An analytic representation in the ℓ-sheeted complex plane provides an elegant formalism that embodies the properties of Frobenius transformations. The difference between a Galois quantum system and other finite quantum systems where the position and momentum take values in the ring is discussed.