Largest Schmidt eigenvalue of entangled random pure states and conductance distribution in chaotic cavities

A strategy to evaluate the distribution of the largest Schmidt eigenvalue for entangled random pure states of bipartite systems is proposed. We point out that the multiple integral defining the sought quantity for a bipartition of sizes N, M is formally identical (upon simple algebraic manipulations) to the one providing the probability density of Landauer conductance in open chaotic cavities supporting N and M electronic channels in the two leads. Known results about the latter can then be straightforwardly employed in the former problem for both systems with broken ({\beta} = 2) and preserved ({\beta} = 1) time reversal symmetry. The analytical results, yielding a continuous but not everywhere analytic distribution, are in excellent agreement with numerical simulations.