Parametrization of a matrix drawn randomly from $SU(n)$ (using Haar measure)

Question

I have been trying to find a (simple) parametrization of a random Unitary matrix, drawn from $SU(n)$, in terms of random variables.

A trivial example would be a matrix drawn from $U(1)$,

$$M = [e^{i\theta}]$$

where $\theta$ is a random variable uniformly drawn from $[0, 2\pi)$.

Any reference would be appreciated.

Answer 1

Not an easy question. See for instance http://home.lu.lv/~sd20008/papers/essays/Random%20unitary%20[paper].pdf

Answer 2

$\mathrm{SU}(2)$ is simple: $$U=\begin{bmatrix} \cos\theta&e^{i\psi}\sin\theta\\ -e^{-i\psi}\sin\theta&\cos\theta \end{bmatrix},\quad \theta\in(0,\pi/2), \psi\in(0,2\pi),$$ with Haar measure $$\mu(d U)\propto\sin\theta\cos\theta d\theta d\psi.$$ For general $\mathrm{SU}(n)$ see this paper. The idea is to generalise the Euler parametrisation using the previous $2\times 2$ matrix as building blocks.