Ensembles and Density Operators

Giuseppe Pileio

In this lecture, I lay down the basics for the study of the dynamics of a large collection of identical spin systems by introducing the concept of a density operator and an equation for its evolution. Ensembles of spin systems represent macroscopic samples; in quantum mechanics, these can only be described in statistical terms through a density operator. This latter can be expressed in terms of spin angular momentum operators. The density operator allows all measurable properties of an ensemble of spin systems to be determined.

A. I. Panin

ABSTRACT: Interrelation of the Coleman’s representabilty theory for 1-density operators and abstract algebraic form of the Hohenberg-Kohn theorem is studied in detail. Convenient realization of the Hohenberg-Kohn set of classes of 1-electron operators and the Coleman’s set of ensemble representable 1-density operators is presented. Dependence of the Hohenberg-Kohn class structure on the boundary properties of the ground state 1-density operator is established and is illustrated on concrete simple examples. Algorithm of restoration of many electron determinant ensembles from a given 1-density diagonal is described. Complete description of the combinatorial structure of Coleman’s polyhedrons is obtained. Key words: density operators; representability problem; density functional theory I.

Romàn R. Zapatrin

A density operator of a bipartite quantum system is called robustly separable if it has a neighborhood of separable operators. Given a bipartite density matrix, its property to be robustly separable is reduced, using the continuous ensemble method, to a finite number of numerical equations. The solution of this system exists for any robustly separable density operator and provides its representation by a continuous mixture of pure product states.