Distribution of inner product of random unit vectors

Question

Suppose I have two $2\times2$ Haar random unitary matrices $u_1$ and $u_2$, then I can define a diagonal matrix $$\begin{pmatrix}(u_1\cdot u_2)_{11}&0\\ 0&(u_1\cdot X\cdot u_2)_{11}\end{pmatrix}.$$ Here $X$ is the Pauli matrix $\begin{pmatrix} 0 & 1 \\ 1& 0 \end{pmatrix}$. Now I want to ask the probability distribution of this diagonal matrix. How to solve it?

It seems that I can view $(u_1\cdot u_2)_{11}$ as the inner product of two random unit vectors. Using the fact that $u_1$ and $u_2$ are chosen Haar randomly, will this problem be solvable?

Answer 1

Parameterize $u_n\in\text{U}(2)$ by $$u_n=e^{i\phi_n}\left( \begin{array}{cc} e^{i (\alpha_n+\alpha'_n)}\cos \theta_n & - e^{i (\alpha_n-\alpha'_n)}\sin \theta_n \\ e^{-i (\alpha_n-\alpha'_n)}\sin \theta_n & e^{-i (\alpha_n+\alpha'_n)}\cos \theta_n \\ \end{array} \right),$$ with $\alpha_n,\alpha'_n,\phi_n\in(0,2\pi)$ and $\theta_n\in(0,\pi/2)$. The Haar measure is $$P(\alpha_n,\alpha'_n,\phi_n,\theta_n)=(2\pi)^{-3}\sin(2\theta_n).$$ The matrix $\Omega$ constructed by the OP has elements $$\Omega=\begin{pmatrix} (u_1 u_2)_{11}&0\\ 0&(u_1X u_2)_{11} \end{pmatrix},\;\;\text{with}\;\;X=\begin{pmatrix} 0&1\\ 1&0\end{pmatrix}. $$ Its distribution assuming independent $u_1,u_2$ is $$P(\alpha_1,\alpha'_1,\phi_1,\theta_1,\alpha_2,\alpha'_2,\phi_2,\theta_2)=(2\pi)^{-6}\sin(2\theta_1)\sin(2\theta_2).$$

A few phases may be eliminated, $$\Omega=\begin{pmatrix}a&0\\ 0&b\end{pmatrix},$$ $$a=e^{i\phi}\cos\theta\cos\theta'-e^{i\phi'-i\phi''}\sin\theta\sin\theta',$$ $$b=e^{i\phi'}\cos\theta\sin\theta'-e^{i\phi-i\phi''}\sin\theta\cos\theta',$$ $$P(\phi,\phi',\phi'',\theta,\theta')=(2\pi)^{-3}\sin(2\theta)\sin(2\theta').$$