Volume of the set of LOCC-convertible quantum states

TL;DR

This paper investigates the volume of LOCC-convertible quantum states, revealing that the proportion diminishes in high dimensions and exploring the distribution of maximal success probabilities, with implications for quantum state transformation theory.

Contribution

It provides numerical evidence supporting the conjecture that LOCC-convertible states become rare in large systems and introduces a stochastic process perspective to understand this phenomenon.

Findings

Proportion of LOCC-convertible pairs vanishes as dimension increases

Distribution of maximal success probability exhibits universal scaling behavior

Balanced and unbalanced bipartitions show different asymptotic behaviors

Abstract

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.