quantum mechanics - Unitarily evolving a separable state into an entangled one

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?

Answer

In general, there's nothing we "expect" about how often transitions between separable and entangled states happen, except that if the Hamiltonian is non-interacting, i.e. of the form $H = H_1\otimes\mathbf{1}_2 + \mathbf{1}_1\otimes H_2$, then separable states will never become entangled. However, the set of separable states has zero measure in the total space of states regardless of what measure you pick, so we probably shouldn't expect entangled states to become separable again unless the time evolution is cyclic (which it is in your case).

As for whether transitions between pure and mixed states are "allowed", this is asking the wrong question: Since your systems are interacting, you cannot meaningfully talk about the evolution of one of the systems without talking about the other, that is, it doesn't really make sense to claim that the subsystems are undergoing transitions. Sure, with ordinary unitary time evolution such transitions are forbidden, but when you try to restrict the evolution operator to one of the subsystems, you don't get something unitary - your subsystem is open, and none of the usual assumptions hold. Just keep looking at the full system, it's much easier.