The density matrix ρ for an n-level system is decomposed into the minimum number of pure states necessary to calculate physical observables. The corresponding physical system is first represented by a set B of n pure states |βi〉, together with their statistical weights. The time evolution of the system is therefore calculated as B(t)=UB(t0), with the propagator U operating on each member of the set, in contrast to the more laborious ρ(t)=Uρ(t0)U†. At least one of the states can be eliminated from the set by reducing its weight to zero via a simple transformation of the density matrix. When there are degenerate weights, the transformation is applied using the weight with the largest degeneracy. Thus, even if none of the original statistical weights are equal to zero, so that rank(ρ)=n, the system can be described by a set of m states with m
Skinner, T. E.,
& Glaser, S. J.
(2002). Representation of a Quantum Ensemble as a Minimal Set of Pure States. Physical Review A, 66 (3), 32112.