Classification and time evolution of density matrices for a N-state system

Abstract

A possible method of classifying all density matrices for a N-state quantum system into different types, using the sameness of the characteristic polynomial of density matrices belonging to a given type, is discussed. The method employs the SU(N) generator expansion of density matrices. It is demonstrated that density matrices belonging to a given type, although different, share many common properties including the same eigenvalues, the same shape of the region in the parametric space in which lie the allowed parameter values, and the same global averages of the pertinent observables. Furthermore, we discuss the time evolution of a given type of density matrices and show that, in the appropriately defined time-dependent comoving matrix basis, a density matrix of a given type belongs, during entire unitary evolution, to the same type. In this way one is able to group and organize the multitude of different density matrices with certain analogous properties, into a smaller number of types, thus systematizing and cataloguing different mixed and pure states of a N-state system. The general discussion is illustrated throughout with the special case of a N=4-state quantum system. Two- and three-dimensional cross sections of the space of generalized Bloch vectors are determined by the Monte Carlo sampling method for several types of density matrices, providing some insight into the intricate and complex geometric structure of the space of density matrices.