Time Evolution of a Maximally Entangled State [duplicate]

My question is basically this:

Does a maximally entangled state stay maximally entangled under the time evolution?

Assume our Hilbert space is $\mathcal{H}_A \otimes \mathcal{H}_B$. At the time $t=0$, the states is $\rho_{AB}$ such that $\text{Tr}_A \rho_{AB}=\frac{1}{n_A}id_B$ and $\text{Tr}_B \rho_{AB}=\frac{1}{n_B}id_A$. Now assume at the time $t$, the density matrix is $\rho_{AB}(t)$. Is it right that $\rho_{AB}(t)$ is a maximally entangled state? (If not, under what assumptions one can guarantee this?)

TEMP: Is it right?

$\text{Tr}_A \rho_{AB}(t)=\sum_A \left<e^A_i(t),\rho_{AB}(t)e^A_i(t)\right>=\left<U(t)e^A_i(0),U(t)\rho_{AB}(0)U^{-1}(t)U(t)e^A_i(0)\right>=\left<e^A_i(0),U^{-1}(t)U(t)\rho_{AB}(0)e^A_i(0)\right>=\left<e^A_i(0),\rho_{AB}(0)e^A_i(0)\right>=\text{Tr}_A \rho_{AB}(0)=\frac{1}{n_A}id_B$

So Maximally entangled states remains maximally entangled.

• Surely there are evolutions that map entangled states in non-entangled states (the easy example is the following: take a non-entangled state, and evolve it with an interacting hamiltonian; the resulting state is entangled for almost all times $t$, this one evolved back with $-t$ gives you a non-entangled state). Therefore my guess is that also maximal entanglement is not in general preserved by time evolution Commented May 10, 2018 at 16:37
• @yuggib I got your intuition and it makes sense but what about "maximally" entangled state? Commented May 10, 2018 at 17:13

The situation is different if the Hamiltonian acts on the two parts separately. Then it creates a unitary evolution $U_A\otimes U_B$, which cannot change the entanglement.