  The degree of entanglement of random pure states in bipartite quantum systems
can be estimated from the distribution of the extreme Schmidt eigenvalues. For
a bipartition of size M\geq N, these are distributed according to a
Wishart-Laguerre ensemble (WL) of random matrices of size N x M, with a
fixed-trace constraint. We first compute the distribution and moments of the
smallest eigenvalue in the fixed trace orthogonal WL ensemble for arbitrary
M\geq N. Our method is based on a Laplace inversion of the recursive results
for the corresponding orthogonal WL ensemble by Edelman. Explicit examples are
given for fixed N and M, generalizing and simplifying earlier results. In the
microscopic large-N limit with M-N fixed, the orthogonal and unitary WL
distributions exhibit universality after a suitable rescaling and are therefore
independent of the constraint. We prove that very recent results given in terms
of hypergeometric functions of matrix argument are equivalent to more explicit
expressions in terms of a Pfaffian or determinant of Bessel functions. While
the latter were mostly known from the random matrix literature on the QCD Dirac
operator spectrum, we also derive some new results in the orthogonal symmetry
class.
