Let $H$ be the Haar measure on $\mathcal{U}(d)$, let $\left|\psi_1\right>$ and $\left|\psi_2\right>$ be two orthogonal states (unit vectors) and $\left|m\right>$ an arbitrary state. Let $t$ be a natural number.
I am trying to find a useful expression for the following integral
$$\intop_U \left<\psi_1 \mid U\mid m \right>\left<m \mid U^\dagger\mid \psi_2 \right> (U\left|m\right>\left<m\right|U^\dagger)^{\otimes t} dH$$
Would appreciate a push in the right direction.
Edit: the broader context is that I am trying to minimize $Tr(Y)$ over all matrices $Y$ for which $Y\otimes I_V\ge \sum_{i,j} M^{i,j} \otimes \left|j\right>\left<i\right|$ where $\{\left|i\right>\}_{i=1,\ldots,d}$ is an orthonormal basis for $V$ and $M^{i,j}$ is the matrix above with $\left|\psi_1\right> = \left|i\right>$ and $\left|\psi_2\right> = \left|j\right>$
Edit 2: It seems that one can use Weingarten functions to get that, in the computational basis, all elements of this matrix are $O(2^{-d(t+1)})$ and that the support of each row/column is polynomial in $t$. It follows from Gershgorin's theorem that $\lambda_\max(A)\le O(2^{-d(t+1)})$, so by choosing $Y=\lambda_\max(A) I$ I get that $Y\times I \ge A$ and that $Tr(Y) = O(2^{-d})$, which is exactly what I needed.