Exact distribution of largest Schmidt eigenvalue for quantum entanglement

We derive exact analytical formulae for the distribution of the largest Schmidt eigenvalue as an explicit piecewise polynomial with rational coefficients and for its moments as rational numbers by using random matrix theory based on the fixed-trace ensemble in order to study the quantum entanglement. The derivation utilizes a new connection between the multivariate hypergeometric functions and the Painlevé systems. The formulae are compared to numerical experiments performed in the coupled-kicked top system to reveal their sensitivity to the type of underlying dynamics, regular or chaos.