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?

1 Answer

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 $$2n$$.

Aside: I recommend the paper of Atiyah and Bott on this topic for a nice discussion of Duistermaat-Heckman in the context of equivariant cohomology and Witten's work on supersymmetry and Morse theory. There ought to be some nice textbook treatments on this aspect (maybe Guillemin and Sternberg's "Supersymmetry and Equivariant de Rham Theory"?), but I haven't looked to see whether your formula is mentioned explicitly.