Stochastic evolution of finite level systems: classical vs. quantum

Quantum dynamics of the density operator in the framework of a single probability vector is analyzed. In this framework quantum states define a proper convex quantum subset in an appropriate simplex. It is showed that the corresponding dynamical map preserving a quantum subset needs not be stochastic contrary to the classical evolution which preserves the entire simplex. Therefore, violation of stochasticity witnesses quantumness of evolution.