I'm trying to compute the expectation value of entanglement entropy of composite system in a random pure state, but I'm running into some problems.
The system we are considering is composed of two subsystems $\mathcal{H} = \mathcal{H}_A \otimes \mathcal{H}_B$ with dimensions $N_A$ and $_B$. Let's say that system A is the smaller of the two: $N_A \leq N_B$. We are considering random pure states $|\psi\rangle \in \mathcal{H}$ these are generated as follows:
For basis $\{e_i\}$ of $\mathcal{H}_A$ and basis $\{f_j\}$ of $\mathcal{H}_B$ we van write $$ |\psi\rangle = > \sum_{i=1}^{N_A}\sum_{j=1}^{N_a} \Psi_{ij} |e_i\rangle \otimes > |f_j\rangle. $$ $\Psi_{ij}$ can be seen as the coordinates of a point on the unit sphere $S^{N_AN_B-1}$ in $\mathbb{C}^{N_ANB}$. So for each $\psi$ there is a corresponding point on the unit sphere. It is this point that is chosen uniformly at random.
Equivalently we can construct the random states as $U|\psi_\rangle$ where U is a random unitary matrix chosen with the Haar measure.
The reduced density matrix of A is $\rho_A = \text{Tr}_B |\psi\rangle\langle\psi|$ with corresponding entanglement entropy $S_A (\psi) = -\text{Tr} \rho_A \log \rho_A$.
I want to compute the expectation value of $S_A$, given by $$ \mathbb{E}(S_A) = \int_{S^{(N_AN_B-1)}} d\sigma(\psi) S_A(\psi), $$ where $d\sigma(\psi)$ is the uniform measure on the unit sphere $S^{(N_AN_B-1)}$.
I tried two different things:
Using the Schmidt decomposition
Every state $\psi$ can be Schmidt decomposed: there exist orthonormal families $\{e_1, e_2, ..., e_{N_A}\}\in \mathcal{H}_A$ and $\{f_1, f_2, ..., f_{N_A}\} \in \mathcal{H}_B$ and real numbers $c_1, c_2, ..., c_{N_A} \geq 0$ with $\sum_i c_i^2 = 1$ such that $$ |\psi \rangle = \sum_{i=1}^{N_A} c_i |e_i\rangle \otimes |f_i\rangle. $$ The entanglement entropy in this case is given by $ S_A (\psi) = \sum_i c_i ^2 \log c_i ^2 $.
I thought I could generate a random state by taking a random Schmidt decomposition, by which I mean, take all $c_i$ uniformly with $\sum_i c_i^2 = 1$, take a random orthonormal basis of $\mathcal{H}_A$ (using a random unitary matrix with the Haar measure to generate one from some fixed basis) and a random orthogonal family in $\mathcal{H}_B$ (again using a random unitary matrix U with the Haar measure to generate one, but since we would only care about the $N_A$ first collums I guess I should adapt the measure in some way to compensate for this).
I fear however that this is not correct: I have no dependence on the choice of orthonormal families so when computing the expectation value the integrals over the unitary matrices would just be trivial. So my first question is: Do my "random Schmidt decomposed states" coincide with (normal) random states? And if not, why?
Usnig a uniform measure on the unit sphere
My second try (which I didn't complete yet) was just to use the uniform measure on the unit sphere as described above.
Using this I could give a probability density of $\rho_A = \Psi\Psi^\dagger$ and then I could write $\rho_A = U\Lambda U^\dagger$ with U some unitary matrix and $\Lambda = \text{diag}(p_1, p_2, ..., p_{N_A})$. I could then give a probability density for $ \Lambda$ as $$P(p_1, p_2, ..., p_{N_A}) = \int d\sigma (U) P(U\Lambda U^\dagger) $$ where $d\sigma (U)$ is the Haar measure. But I'm stuck a bit with this.
Once I find this I could conmute the expectation value as $$ \mathbb{E}(S_A) = -\int dp_1dp_2, ... dp_{N_A} P(p_1, p_2, ..., p_{N_A}) \sum_i p_i \log p_i $$
My second question is Is this a correct way to do it? Can anyone help me with the parametrisation of $\Psi$ in terms of angles on the unit sphere, or with another method to obtain $P(p_1,...,p_{N_A})$ and maybe some of the subsequent integrals?
I found something in this article, but most of the steps are a bit vague to me.
Should this kind of question rather be posted in the math stack exchange? I reposted it over there since its actually a technical question on the math and there isn't so much physics involved. Should I remove it here?