Volume of the set of LOCC-convertible quantum states

The class of quantum operations known as local operations and classical com- munication (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 rela- tion known as majorization. The numerical results, which exploit a tridiag- onal model of random matrices, provide quantitative evidence that the pro- portion of LOCC-convertible pairs vanishes in the limit of large dimensions, and therefore support a previous conjecture by Nielsen. In particular, we show that the problem is equivalent to the persistence of a non-Markovian stochas- tic process and the proportion of LOCC-convertible pairs decays algebraically with a nontrivial persistence exponent. We extend this analysis by investigat- ing the distribution of the maximal success probability of LOCC-conversions. We show a dichotomy in behaviour between balanced and unbalanced bipar- titions. 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 arbitrar- ily close to one. We present strong evidence of a universal scaling limit for the maximal probability of successful LOCC-conversions and we suggest a connec- tion with the typical fluctuations of the smallest eigenvalue of Wishart random matrices.