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

Symplectic transformations and entanglement in finite quantum systems

Lina Wang

Published Aug 21, 2009• 2 citations
OpenAlex

Abstract

Quantum systems with finite Hilbert space are considered. Position and mo- \nmentum states and their relation through a Fourier transform, displacement \nin the position-momentum phase-space, and symplectic transformations are \nintroduced and their properties are studied. Symplectic Sp(2l;Zp) trans- \nformations in l-partite finite system are explicit constructed. The general \nmethod is applied to bi-partite and tri-partite systems. The effect of these \ntransformations on the correlations is discussed. Entanglement calculations \nbetween the subsystems in a bi-partite system and a tri-partite system are \npresented. The effect of measurements is also studied.

Related papers

Phase space methods in finite quantum systems.

Hilal Al Hadhrami

Quantum systems with finite Hilbert space where position x and momentum \np take values in Z(d) (integers modulo d) are considered. Symplectic tranformations \nS(2¿,Z(p)) in ¿-partite finite quantum systems are studied and \nconstructed explicitly. Examples of applying such simple method is given \nfor the case of bi-partite and tri-partite systems. The quantum correlations \nbetween the sub-systems after applying these transformations are discussed \nand quantified using various methods. An extended phase-space x¿p¿X¿P \nwhere X, P ¿ Z(d) are position increment and momentum increment, is introduced. \nIn this phase space the extended Wigner and Weyl functions are \ndefined and their marginal properties are studied. The fourth order interference \nin the extended phase space is studied and verified using the extended \nWigner function. It is seen that for both pure and mixed states the fourth \norder interference can be obtained.

•Jan 1, 2009•0 citations

Symplectic transformations and quantum tomography in finite quantum systems

Apostol Vourdas, Cyril Banderier

Quantum systems where the position and momentum are in the ring Zd (d is an odd integer) are considered. Symplectic transformations are studied, and the order of Sp(2, Zd ) is calculated. Quantum tomography is also discussed. It is shown that measurements (used in the inverse Radon transform) need to be made on J2(d) lines (where J2(d) is the Jordan totient function).

Journal of Physics A Mathematical and Theoretical•Mar 14, 2011•12 citations

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

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

Galois quantum systems

Apostol Vourdas

A finite quantum system in which the position and momentum take values in the Galois field GF( p ℓ ) is constructed from a smaller quantum system in which the position and momentum take values in , using field extension. The Galois trace is used in the definition of the Fourier transform. The Heisenberg–Weyl group of displacements and the Sp(2, GF( p ℓ )) group of symplectic transformations are studied. A class of transformations inspired by the Frobenius maps in Galois fields is introduced. The relationship of this 'Galois quantum system' with its subsystems in which the position and momentum take values in subfields of GF( p ℓ ) is discussed.

Journal of Physics A Mathematical and General•Sep 14, 2005•40 citations

Entangling transformations in composite finite quantum systems

Apostol Vourdas

Phase space methods are applied in the context of finite quantum systems. 'Galois quantum systems' (with a dimension which is a power of a prime number) are considered, and symplectic transformations are studied. Composite systems comprising two finite quantum systems are also considered. Symplectic transformations are classified into local and entangling ones and the necessary matrices which perform such transformations are calculated numerically.

Journal of Optics B Quantum and Semiclassical Optics•Oct 16, 2003•5 citations
Quantum systems with finite Hilbert space are considered. Position and mo- \nmentum states and their relation through a Fourier transform, displacement \nin the position-momentum phase-space, and symplectic transformations are \nintroduced and their properties are studied. Symplectic Sp(2l;Zp) trans- \nformations in l-partite finite system are explicit constructed. The general \nmethod is applied to bi-partite and tri-partite systems. The effect of these \ntransformations on the correlations is discussed. Entanglement calculations \nbetween the subsystems in a bi-partite system and a tri-partite system are \npresented. The effect of measurements is also studied.
Quantum systems with finite Hilbert space where position x and momentum \np take values in Z(d) (integers modulo d) are considered. Symplectic tranformations \nS(2¿,Z(p)) in ¿-partite finite quantum systems are studied and \nconstructed explicitly. Examples of applying such simple method is given \nfor the case of bi-partite and tri-partite systems. The quantum correlations \nbetween the sub-systems after applying these transformations are discussed \nand quantified using various methods. An extended phase-space x¿p¿X¿P \nwhere X, P ¿ Z(d) are position increment and momentum increment, is introduced. \nIn this phase space the extended Wigner and Weyl functions are \ndefined and their marginal properties are studied. The fourth order interference \nin the extended phase space is studied and verified using the extended \nWigner function. It is seen that for both pure and mixed states the fourth \norder interference can be obtained.
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 '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.
A finite quantum system in which the position and momentum take values in the Galois field GF( p ℓ ) is constructed from a smaller quantum system in which the position and momentum take values in , using field extension. The Galois trace is used in the definition of the Fourier transform. The Heisenberg–Weyl group of displacements and the Sp(2, GF( p ℓ )) group of symplectic transformations are studied. A class of transformations inspired by the Frobenius maps in Galois fields is introduced. The relationship of this 'Galois quantum system' with its subsystems in which the position and momentum take values in subfields of GF( p ℓ ) is discussed.