Calculating an average over the Haar measure

Question

Suppose $\rho$ is an $n$-qubit density matrix and $V$ is a unitary. Is there a closed form expression for $$ \int dU\,(U\otimes U)(\rho \otimes (V\rho V^\dagger))(U^\dagger\otimes U^\dagger), $$ where $dU$ is the Haar measure over the unitary group $U\left(2^n\right)$ on $n$ qubits?

Answer 1

Using Weingarten calculus (1, 2, 3): $$ \int_{U(d)} (U\otimes U) A (U^\dagger \otimes U^\dagger)\,dU = \frac{\mathrm{tr}(A) - \frac1d \mathrm{tr}(\mathbb{F}A)}{d^2-1} \mathbb{1} + \frac{\mathrm{tr}(\mathbb{F}A) - \frac1d \mathrm{tr}(A)}{d^2-1} \mathbb{F} \,, $$ where $\mathbb{1}$ and $\mathbb{F}$ are the identity and flip/swap operator, respectively.

In particular, if $A=\rho\otimes V\rho V^\dagger$, then we have $\mathrm{tr}(A)=1$ and we can use the swap trick, $$ \mathrm{tr}(\mathbb{F}\rho\otimes V\rho V^\dagger) = \mathrm{tr}(\rho V\rho V^\dagger)\,, $$ to obtain $$ \int_{U(d)} (U\otimes U) (\rho\otimes V\rho V^\dagger) (U^\dagger \otimes U^\dagger)\,dU = \frac{1 - \frac1d \mathrm{tr}(\rho V\rho V^\dagger)}{d^2-1} \mathbb{1} + \frac{\mathrm{tr}(\rho V\rho V^\dagger) - \frac1d }{d^2-1} \mathbb{F} \,. $$ Note that this form is especially useful if you want to contract the average again with a product operator.