I have two ebits $e1$ and $e2$ and my 4-qubit system is in the state:
$$(|00\rangle + |11\rangle) (|00\rangle + |11\rangle)$$
Ignoring the normalizing constants for now. As mentioned, I need to obtain $|0000\rangle + |1111\rangle$.
I have the additional restriction that I can only use single gates on all the qubits, and controlled gates (CNOT, etc) only on the second and third qubit (otherwise I could just use inverse operations to disentangle all qubits, and proceed to get what I want in same way we create the GHZ state).
My idea to do this was to entangle second and third qubits, using CNOT and H gates, but it doesn't work because the 2nd and third qubits are in the state $|00\rangle + |01\rangle + |10\rangle + |11\rangle$, and I only know how to entangle if we were starting in $|00\rangle$ (or even if we're in $|0\rangle(|0\rangle + |1\rangle)$, which is what we get after H gate).
Is this possible to do?