The system time evolution can also be regarded dually as time evolution of the statistical state space. The evolution of the statistical state is given by a family of operators {β t } t such that Tr ⁡ ⁡ ( β β t ( S ) E ) = Tr ⁡ ⁡ ( S α α − − t ( E ) ) = Tr ⁡ ⁡ ( S U t E U t ∗ ∗ ) = Tr ⁡ ⁡ ( U t ∗ ∗ S U t E ) .