Hilbert Schmidt inner product of random isotropic unitary matrices

Question

Does anybody know if there exists results on the probability distribution of the Hilbert Schmidt inner product of random unitary matrices?

To be more specific, given two random isotropically distributed unitary matrices $U_1 \in \mathbb{C}^{n \times n}$ and $U_2 \in \mathbb{C}^{n \times n}$, is something known about the distribution of $\mathrm{trace}\left(U_1^{H} U_2 \right)$? The superscript $H$ denotes conjugate transposition.

Actually, I am only interested in the real-part of this quantity, since it comes up when calculating the Frobenius norm $\left\|U_1 - U_2 \right\|_F$. But I guess the distribution of real-part and imaginary-part would be equal in that case.

Many thanks for any hint!

Answer 1

I assume the matrices $U_1$ and $U_2$ are independently uniformly distributed in the unitary group. The product $V=U_1^HU_2$ is then itself uniformly distributed in the unitary group. The distribution of the trace ${\rm Tr}\,V=\sum_{p=1}^n e^{i\phi_p}$ follows from the known joint distribution of the eigenvalues $e^{i\phi_p}$ of $V$, given by $$P(\phi_1,\phi_2,\ldots\phi_n)\propto\prod_{p<q}\left|e^{i\phi_p}-e^{i\phi_q}\right|^2.$$ For $n\gg 1$ the real and imaginary parts of ${\rm Tr}\,V$ have independent Gaussian distributions (zero mean, variance $1/2$), basically because the diagonal elements of $V$ have independent Gaussian distributions in the large-$n$ limit.