We consider two spin 1/2 systems that are described by the following Hamiltonian: $$ H^{12} = jZ^{1} \otimes Z^{2} \tag{1} $$
The composite system is initialised in the state: $|\psi^{12}(0)\rangle=|x_+,x_+\rangle,$ which written in the $z$ basis ($z_+$ being spin up state along the z dimension), is: $$|\psi^{12}(0)\rangle = 1/2(|z_+,z_+\rangle+|z_+,z_-\rangle+|z_-,z_+\rangle+|z_-,z_-\rangle).\tag{2} $$
Given the Hamiltonian in $(1)$, the unitary time evolution is $U(t)=e^{-iH^{12}t/\hbar},$ and after a time $t_0=\pi \hbar/4j,$ $U(t)$ becomes
$$ U(t_0)=e^{-i\frac{\pi}{4}Z^1\otimes Z^2} \tag{3} $$
Knowing $U$ at $t_0,$ the composite state evolved by this Hamiltonian upto $t_0$ becomes:
\begin{align*} |\psi^{12}(t_0)\rangle =& U(t_0)|\psi^{12}(0)\rangle \\ =& 1/2(e^{-i\pi/4}|z_+,z_+\rangle+e^{i\pi/4}|z_+,z_-\rangle+e^{i\pi/4}|z_-,z_+e^{-i\pi/4}\rangle+e^{-i\pi/4}|z_-,z_-\rangle) \tag{4} \end{align*}
So it seems that just by evolving the separable initial state $(2)$ that was not an eigenstate, for a certain amount of time, the composite system suddenly becomes entangled $(4).$
- Once the system has become entangled, will further evolving the system according to $U(t)$ dis-entangle the system again at some point in time? I.e. do we expect regular transitions between separable to entangled states of the composite system? Or, once the system has become entangled, it maintains its entangled state?
- If the change between separable and entangled is expected to occur during the time evolution of this system, then it means the involved subsystems are constantly undergoing transitions from pure (when in separable state) to mixed states (when the composite state is entangled), is such behaviour physically allowed?