# 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?

## 1 Answer

No. Alice's and Bob's state could, for instance, be equally well in a product state.

• Is the statement that if the overall state is a pure state, then neither Alice's nor Bob's state can have any correlations/etanglement with any other state? – IanDsouza Dec 18 '16 at 2:58
• Yes. (Nor their joint state.) – Norbert Schuch Dec 18 '16 at 3:02