Random pure states: quantifying bipartite entanglement beyond the linear statistics

We analyze the properties of entangled random pure states of a quantum system partitioned into two smaller subsystems of dimensions $N$ and $M$. Framing the problem in terms of random matrices with a fixed-trace constraint, we establish, for arbitrary $N \leq M$, a general relation between the $n$-point densities and the cross-moments of the eigenvalues of the reduced density matrix, i.e. the so-called Schmidt eigenvalues, and the analogous functionals of the eigenvalues of the Wishart-Laguerre ensemble of the random matrix theory. This allows us to derive explicit expressions for two-level densities, and also an exact expression for the variance of von Neumann entropy at finite $N,M$. Then we focus on the moments $\mathbb{E}\{K^a\}$ of the Schmidt number $K$, the reciprocal of the purity. This is a random variable supported on $[1,N]$, which quantifies the number of degrees of freedom effectively contributing to the entanglement. We derive a wealth of analytical results for $\mathbb{E}\{K^a\}$ for $N = 2$ and $N=3$ and arbitrary $M$, and also for square $N = M$ systems by spotting for the latter a connection with the probability $P(x_{min}^{GUE} \geq \sqrt{2N}\xi)$ that the smallest eigenvalue $x_{min}^{GUE}$ of a $N\times N$ matrix belonging to the Gaussian Unitary Ensemble is larger than $\sqrt{2N}\xi$. As a byproduct, we present an exact asymptotic expansion for $P(x_{min}^{GUE} \geq \sqrt{2N}\xi)$ for finite $N$ as $\xi \to \infty$. Our results are corroborated by numerical simulations whenever possible, with excellent agreement.