Structure and Parametrization of Stochastic Maps of Density Matrices

The generic linear evolution of the density matrix of a system with a finite-dimensional state space is by stochastic maps which take a density matrix linearly into the set of density matrices. These dynamical stochastic maps form a linear convex set that may be viewed as supermatrices. The property of hermiticity of density matrices renders an associated supermatrix hermitian and hence diagonalizable; but the positivity of the density matrix does not make this associated supermatrix positive. If it is positive, the map is called completely positive and they have a simple parametrization. This is extended to all positive (not completely positive) maps. A contraction of a norm-preserving map of the combined system can be contracted to obtain all dynamical maps. The reconstruction of the extended dynamics is given.