Nonpositive evolutions in open system dynamics

The long-time evolution of a system in interaction with an external environment is usually described by a family of linear maps ${\gamma }_t,$ generated by master equations of Block-Redfield type. These maps are, in general, nonpositive; a widely adopted cure for this physical inconsistency is to restrict the domain of definition of the dynamical maps to those states for which ${\gamma }_t$ remains positive. We show that this prescription has to be modified when two systems are immersed in the same environment and evolve with the factorized dynamics ${\gamma }_t\otimes {\gamma }_t$ starting from an entangled initial state.