Do you have a more precise estimate of the error (as a function of $d$) when you try to treat the elements of $\psi$ as independent Gaussians?

Great, thanks! Where did you get the first formula, though? Also, as a followup: would you know what kind of measure of concentration is satisfied by these inner products? That is, what is the probability that $\mathrm{Tr}( \psi \psi^* \, \, w x^* \,\, \psi \psi^* \,\, y z^*)$ deviates from the average by more than some amount $\epsilon$?

Thanks, quid, for your answer, and the links to Hamidoune's work.