Volume of the set of LOCC-convertible quantum states. (7th April 2020)

• Record Type: Journal Article
• Authors: Cunden, Fabio Deelan; Facchi, Paolo; Florio, Giuseppe; Gramegna, Giovanni
• Is Part Of: Journal of physics. Volume 53:Number 17(2020)
• Publication Date: 2020-04-07
• Subjects: entanglement -- local operations and classical communication -- random matrices -- majorization; Mathematical physics -- Periodicals; Statistical physics -- Periodicals; Quantum theory -- Periodicals; Matter -- Properties -- Periodicals; 530.105
• DOI: 10.1088/1751-8121/ab7b21
• Languages: English
• ISSNs: 1751-8113

Abstract

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.