Usage de l'Algèbre de Lie su (<i>n</i>) dans l'Etude des Systèmes Quantiques à <i>n</i> Etats. II. Transformation de l'Espace des Observables, Problèmes d'Evolution

We examine and compare ways of dividing into subspaces the space whose elements are density matrices or other operators for the class of model problems defined by a Finite one-particle basis set. One method of decomposition makes the significance of the subspaces apparent. We show that this decomposition is also complete, in the group-theoretic sense, for the group of unitary transformations of the set of one-electron basis functions. The irreducible subspaces are labeled by particle number and by an additional integer we call the reduction index. For spaces of particle-number-conserving operators, all subspaces with the same reduction index are isomorphic, and an analogous isomorphism exists for non-particle-numberconserving cases. The general linear group also plays a key role, and we introduce the term "canonical superoperators" to characterize those superoperators which commute with this group. When an appropriate basis set is chosen for the matrix spaces, the supermatrices corresponding to these superoperators have a particularly simple form: a block structure with the only nonzero blocks being multiples of unit matrices.The superoperators of interest can be constructed in terms of two operators, A+, and these two have been expressed simply in terms of creation and annihilation operators. When only real orthogonal transformations of the basis are considered, a further decomposition is possible. We have introduced superoperators associated with this decomposition.

Section: The (S) Vectors {S(tb)} and The (3 Vectors {X(j)} Span Thmentioning

confidence: 99%

Geometry of density matrices. VI. Superoperators and unitary invariance

Casida

Harriman

1986

“…I1 convient de remarquerAque ces conditions ne mettent pas en jeu la partie antisymktrique de la matrice M. Cette erreur ne modifie pas la suite de l'article [5].…”

Section: P * and C ( N -1 ) P N (172)unclassified

Usage de l'algebre de lie su(n) dans l'etude des systems à n etats. VI. Equations linéaires d'evolution positive

Groenendael

1986

The n-level quantum systems density-operators evolution is not usually given by the Liouville equation, but by a more general one. This equation must keep density-operators trace, hermiticity, and positiveness. These threx conditions restrict the available kinds of evolution equations. In this paper we investigate linear equations for systems without memory effects. By using the first two conditions and the formalism we introduced in earlier papers, the evolution equation takes the form of a first order differential equation concerning the n2 -1 dimension "coherence-vector." The third one is the essential object of this paper. Moreover, we obtain a canonical splitting of this equation into four parts that may be separately studied.

“…En introduisant dans 8, une F base {$} ou base orthonormke au sens de Fano [8], on peut Ccrire tout opkateur-densit6 sous la forme n oh le vecteur de coherence P [6,9], de composantes Pf , peut Ctre considCrC comme un vecteur dans un espace euclidien %, (isomorphe B 8,) que nous appellerons dksormais l'espace de cohkrence. Chaque opCrateur-densit6 dCfinit un vecteur P unique.…”

Section: (3)unclassified

“…(2) Lorsque l'espace % du systkme 9, est de dimension finie n , l'ensemble des opkrateurs hermitiques de trace nulle~, muni de l'opkration i [. , .I, forme une alg&bre de Lie 8, isomorphe a su(n) [5,6,7]. Cet espace Ce,, de dimension r = n2 -1, a ktk appelC espace des observables [5,8].…”

Section: Introductionunclassified

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Usage de l'algégbre de lie su(n) dans l'étude des systémes quantiques á n états. IV. Geometrie du domaine des vecteurs de cohérence

Groenendael

1985

The density operator of an n level quantum system is known to be a positive semidefinite, hermitian operator of trace one. In a previous article we have established, through su(n) algebras, a formalism where density operators are built from coherence vectors in a nz -1 dimension, real, Euclidean space. The last two conditions are then automatically satisfied. Being positive semidefinite means a restriction to the domain of coherence vectors. In this article we clarify this domain and obtain several equivalent tests to know if a given vector is part of it.Par dtfinition, l'operateur-densit6 d'un systkme quantique a n Ctats est soumis trois conditions: il doit