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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.

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2 Answers 2

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

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  • $\begingroup$ If I understand correctly, the referenced report (which I liked to read, thanks) gives a way to uniformly sample from $U(n)$, but no way to parameterize it. $\endgroup$
    – G Frazao
    Mar 15, 2023 at 16:10
  • $\begingroup$ Parametrizing is a different question which is not the one addressed by the article I linked to. One can of course push the idea further and get a parametrization by analogues of Euler angles but it is quite messy. You can find such explicit parametrizations in the book "Log-gases and random matrices" by Peter Forrester press.princeton.edu/books/hardcover/9780691128290/… This MO answer also cites some references for the closely related SU(N) mathoverflow.net/questions/430835/… $\endgroup$ Mar 16, 2023 at 14:50
  • $\begingroup$ Thanks. Meanwhile I found the article Generalized Euler Angle Paramterization for SU(N) which actually goes through the mess! $\endgroup$
    – G Frazao
    Mar 16, 2023 at 14:54
  • $\begingroup$ Okay, I just noticed the same article is referred to in the MO answer you point. Still, I'll leave it here in case someone finds this answer and not the MO one (as happened with me). $\endgroup$
    – G Frazao
    Mar 16, 2023 at 14:56
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$\mathrm{SU}(2)$ 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.

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