  The class of quantum operations known as Local Operations and Classical
Communication (LOCC) induces a partial ordering on quantum states. We present
the results of systematic numerical computations related to the volume (with
respect to the unitarily invariant measure) of the set of LOCC-convertible
bipartite pure states, where the ordering is characterised by an algebraic
relation known as majorization. The numerical results, which exploit a
tridiagonal model of random matrices, provide quantitative evidence that the
proportion of LOCC-convertible pairs vanishes in the limit of large dimension,
and therefore support a previous conjecture by Nielsen. In particular, we show
that the problem is equivalent to the persistence of a non-Markovian stochastic
process and the proportion of LOCC-convertible pairs decays algebraically with
a nontrivial persistence exponent. We extend this analysis by investigating the
distribution of the maximal success probability of LOCC-conversions. We show a
dichotomy in behaviour between balanced and unbalanced bipartitions. In the
latter case the asymptotics is somehow surprising: in the limit of large
dimensions, for the overwhelming majority of pairs of states a perfect
LOCC-conversion is not possible; nevertheless, for most states there exist
local strategies that succeed in achieving the conversion with a probability
arbitrarily close to one. We present strong evidences of a universal scaling
limit for the maximal probability of successful LOCC-conversions and we suggest
a connection with the typical fluctuations of the smallest eigenvalue of
Wishart random matrices.
