# How to derive the probability distribution of reduced density matrix eigenvalues for randomly chosen pure states in Page's theorem?

Motivation
I am trying to reproduce the proof in Page's theorem as conjectured in the seminal paper Average Entropy of a Subsystem by Don N. Page. It is crucial in various resolutions of black hole information paradox involving the Page curve.

The Problem
In the original paper, Don Page is trying to derive an expression for coarse grained entropy $$S_{mn}=\langle S_A \rangle$$ where $$S_A$$ is the Von-Neumann entropy associated with a system $$A$$ with Hilbert space dimension $$m$$, where the average is taken respect to the unitary invariant Haar measure on the space of unitary vectors in the $$mn$$ dimensional Hilbert space of the total system. For the calculation he was was led to consider the probability distribution of the eigenvalues of the reduced density matrix $$\rho_A$$ for the random pure states $$\rho$$ of the entire system. He used $$P(p_1,p_2,...,p_m)=N~\delta(1-\sum_{i=1}^m p_i) \prod_{1\le i\le j \le m} (p_i-p_j)^2 \prod_{k=1}^m p_k^{n-m} dp_k \tag{1}$$ in an intermediate step. This was partially derived in an earlier paper by Lloyd and Pagels and completed in a "private communication" between Page and Lloyds. I am unable to derive the above expression having no access to the private communication (obviously!) and also unable to salvage some help from the seemingly cryptic Lloyd-Pagels paper.

However, each of this assumed the step I am having difficulty in because the authors focussed in the calculation of an integral which Page could not perform for generic cases in his paper. So, How to derive the probability distribution of reduced density matrix eigenvalues for randomly chosen pure states which is given in eq. ($$1$$) above?