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We analyze the properties of entangled random pure states of a quantum systempartitioned into two smaller subsystems of dimensions $N$ and $M$. Framing theproblem in terms of random matrices with a fixed-trace constraint, weestablish, for arbitrary $N \leq M$, a general relation between the $n$-pointdensities and the cross-moments of the eigenvalues of the reduced densitymatrix, i.e. the so-called Schmidt eigenvalues, and the analogous functionalsof the eigenvalues of the Wishart-Laguerre ensemble of the random matrixtheory. 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 effectivelycontributing 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 forsquare $N = M$ systems by spotting for the latter a connection with theprobability $P(x_{min}^{GUE} \geq \sqrt{2N}\xi)$ that the smallest eigenvalue$x_{min}^{GUE}$ of a $N\times N$ matrix belonging to the Gaussian UnitaryEnsemble is larger than $\sqrt{2N}\xi$. As a byproduct, we present an exactasymptotic expansion for $P(x_{min}^{GUE} \geq \sqrt{2N}\xi)$ for finite $N$ as$\xi \to \infty$. Our results are corroborated by numerical simulationswhenever possible, with excellent agreement.

Random pure states: quantifying bipartite entanglement beyond the linear statistics