# Maximal entanglement and overall pure state

Alice's quantum system $\rho$ lives in Hilbert space $H^A$. Bob's quantum system $\sigma$ lives in Hilbert space $H^B$. Say, I can represent the their overall state as a pure state $\psi$ that lives in $H^A\bigotimes H^B$. Does this imply that Alice and Bob's quantum states are maximally entangled with each other? My line of reasoning was that Alice and Bob's states cannot be correlated/entangled with any other quantum state. Otherwise, you could not write the overall state as a pure state. This is strikingly similar to the principle of monogamy of (maximal) entanglement. Any sense here in saying their states must be maximally entangled?