Conditions for strictly purity-decreasing quantum Markovian dynamics
  The purity, Tr(rho^2), measures how pure or mixed a quantum state rho is. It is well known that quantum dynamical semigroups that preserve the identity operator (which we refer to as unital) are strictly purity-decreasing transformations. Here we provide an almost complete characterization of the class of strictly purity-decreasing quantum dynamical semigroups. We show that in the case of finite-dimensional Hilbert spaces a dynamical semigroup is strictly purity-decreasing if and only if it is unital, while in the infinite dimensional case, unitality is only sufficient. 
D.A. Lidar (1), A. Shabani (1), R. Alicki (2) ((1) Center for Quantum   Information and Quantum Control, University of Toron...