Majorization lattice and entanglement transformations

BOSYK, GUSTAVO MARTÍN; SERGIOLI, GIUSEPPE

Bruselas

Simposio; International symposium "Worlds of Entanglement"; 2017

Free University of Brussels (VUB)

Majorization is nowadays a well-established and powerful mathematical tool with many and different applications in several disciplines as in economy, biology, physics among others. The spread applicability of majorization in the quantum realm emerges as a consequence of deep connections among majorization, partially ordered probability vectors, unitary matrices and the probabilistic structure of quantum mechanics. In this talk, we will review basics aspects of majorization focusing on the problem of interconversion of bipartite pure states applying local operation and classical communications (LOCC). More precisely, this problem consists in two parties, Alice and Bob, that share an (initial) entangled pure-state and their goal is to transform it in another entangled pure-state (target state), by applying LOCC. A celebrated result of Nielsen gives the necessary and sufficient condition that does this entanglement transformation process be possible [1]. Indeed, this process can be achieved if and only if there exists majorization relation between the initial and target states. In general, this condition is not fulfilled, but one can look for an approximate target state. Vidal et. al have proposed a deterministic transformation using LOCC in order to obtain a state most approximate to target in terms of maximal fidelity between them [2]. We present an alternative proposal by exploiting the fact that majorization is indeed a lattice for the set of probability vectors [3].[1] M.A. Nielsen, Phys. Rev. Lett. 83, 436 (1999)[2] G. Vidal, D. Jonathan, M.A. Nielsen, Phys. Rev. A 62, 012304 (2000)[3] G.M. Bosyk, G. Sergioli, H. Freytes, F. Holik, G. Bellomo, Physica A 473 , 403 (2017)