Let $U$ be an $n$ qubit Haar random unitary and $\mathbb{I}_n$ is the $n$ qubit identity operator. I want to find the density matrix corresponding to the following:
$$ \rho = \underset{U}{\mathbb{E}}\left[ (\mathbb{I}_{n-1} \otimes U \otimes \mathbb{I})(U \otimes \mathbb{I}_{n})|0^{2n}\rangle\langle 0^{2n}| (U^{*} \otimes \mathbb{I}_{n}) (\mathbb{I}_{n-1} \otimes U^{*} \otimes \mathbb{I})\right] $$
That is, the same unitary is applied twice; firstly, to the first $n$ qubits, and then to another configuration of qubits (applied between $n^{\mathsf{th}}$ qubit from the top and second last qubit).
I think the state should still be very close to being maximally mixed. How to prove this?
