where the F i are operators on the Hilbert space of the system alone, and no reference is made to the environment. In particular, if the system and environment are initially in a product state ρ ρ S E ( 0 ) = ρ ρ S ( 0 ) ⊗ ⊗ ρ ρ E ( 0 ) , it can be shown that the reduced dynamics are completely positive. However, the most general possible reduced dynamics are not completely positive.