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 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 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 evidence 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.
Highlights
Introduction and summary of resultsThe theoretical and experimental development of quantum information protocols has motivated the study of ‘manipulations’ of quantum states using classes of operations restricted by some physical constraints
The subset of quantum operations describing this scenario is the so-called class of Local Operations assisted by Classical Communication (LOCC)
In this paper we have studied the probability of LOCC-convertibility between pairs of random pure states
Summary
The theoretical and experimental development of quantum information protocols has motivated the study of ‘manipulations’ of quantum states using classes of operations restricted by some physical constraints. If the probability measure is unitarily invariant, the question can be rephrased in terms of volumes of the set of convertible states. Questions about the volume of LOCC-convertible states can be relaxed to include local convertibility that succeed with some probability p according to Definition 1. The investigation of local convertibility of quantum states has led to precise criteria based on the majorization [43, 44,52,53], a relation in the technical mathematical sense that formalizes the idea that a probability distribution can be more ‘disordered’ than another [4, 39]. We remark at this stage that the Schmidt decomposition is symmetric under the interchange of A and B; without loss of generality we can assume n ≤ m
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