Does the entanglement entropy work for mixed states?

Question

I know for a pure state $\rho$, if its subsystems are mixed then $\rho$ is entangled.

Is the result true if $\rho$ is mixed state?

Is the above law generalized for a $n$-qubit state?

Answer 1

No, entanglement entropy is only reliable as an entanglement quantifier for pure states. You can see it easily considering what it gives for $\frac{1}{\sqrt2}(|00\rangle+|11\rangle)$ and $\frac12(|00\rangle\!\langle00|+|11\rangle\!\langle 11|)$. Both give the same reduced states, but clearly have entirely different entanglement properties. See also this related answer on physics.SE.