Two Spin-1/2 Particles Interacting with a Common Environment

We know that the quantum system evolves to different physical phenomena according to initial states. However, the situation is very different in the open system. We consider the system and the environment governed by the Markovian process. The density matrix (the reduced density matrix) of the system satisfies the Kossakowski-Lindblad equation. We calculate the expectation value of the z-direction-magnetization in the system for various types of 1) initial states and 2) interactions between the system and the environment. In all the cases, the values of the magnetization approached asymptotically to the same point even though the different initial states and different interaction types between the system and the environment are applied. The paths to asymptotically approach the final values are different according to the initial states and the interactions. These facts show that the physical results of expectation value are independent of the initial states.