# The square modulus of coordinates of a uniformly chosen point in complex projective space is uniform in the simplex

I can't recall where I learned this (beautiful) fact, and I would like a reference (if possible, in a textbook):

Let $$(z_0:\cdots:z_n) \in \mathbb{P}^n(\mathbb{C})$$ be chosen uniformly at random w.r.t. the Fubini-Study metric, and normalized so that $$|z_0|^2 + \cdots |z_n|^2 = 1$$. Then the point $$(|z_0|^2, \ldots, |z_n|^2)$$ is uniformly distributed on the $$n$$-simplex (the set of $$(p_0,\ldots,p_n)\in\mathbb{R}^{n+1}$$ such that all $$p_i\geq 0$$ and $$p_0+\cdots+p_n = 1$$) w.r.t. its Euclidean metric.

This affords the quantum-mechanical interpretation that if we draw a random quantum entanglement of $$n+1$$ pure states, and we observe the pure state it is in, we get a uniform probability measure on the pure states. However, I'm not asking for a reference in relation to quantum mechanics.

This is, for example, proposition 1 in this paper but the only reference the authors give after calling the fact “known” is a 600-page book without any specific page number (shame!).

It is also mentioned in this blog post (and attributed to Bill Wootters); and the particular case $$n=1$$ is mentioned in this other blog post (in relation to the Box-Muller transformation); but I would like something more tangible than a blog post.

As a bonus question, if we take a uniformly random $$(n+1)\times(n+1)$$ unitary matrix (uniformly w.r.t. the Haar measure) and we look at the square norms of its columns, we get $$n+1$$ points on the $$n$$-simplex, each uniformly distributed by the above fact: does this distribution of $$n+1$$ points on the $$n$$-simplex have a standard name, and where might I learn more about it?

Long ago, I learned this from a MO answer by Greg Kuperberg. A more general fact is that the moment map of a projective toric variety is measure-preserving. As explained in the linked email by Yael Karshon, this is a special case of the Duistermaat-Heckman formula (see Corollary 3.3 in Duistermaat and Heckman's 1982 paper and note that $$n=l$$, where $$l$$ is the dimension of the torus acting on the symplectic manifold of dimension $$2n$$.