Consider arbitrary unit vectors $w,x,y,z \in \mathbb{C}^d$. Is there an explicit formula for what this average is? $$ \int \mathrm{Tr}( \psi \psi^* \, \, w x^* \,\, \psi \psi^* \,\, y z^*) d\psi $$ where the average is over a Haar-random unit vector $\psi \in \mathbb{C}^d$. Here, $\psi \psi^*$ is the rank-one matrix formed by the outer product of $\psi$ with itself. I'm looking for a formula that relates this average to inner products between $w, x, y, ,z$, for example.

A related question I'm wondering about is what the expected value of the following inner product is: suppose you pick a Haar-random vector $\psi$, followed by a Haar-random vector $\psi^\perp$ that's orthogonal to $\psi$, and then look at $\langle \psi, x \rangle \langle y, \psi^\perp \rangle$.

Thanks!