State transformations on the majorization lattice

BOSYK, G. M.

Cagliari

Workshop; Quantum Cagliari 2018: Quantum structures & quantum information theory; 2018

Universidad de Cagliari

We present some results and discussions regarding the implications of the theory of majorization in the field of quantum information. Majorization provides a way to order a set of probability vectors according to their degree of uncertainty. However, this is not a total order, since there are vectors that are not comparable under this criterion. In this case, the infimum and the supremum are well defined so that a lattice structure with a natural distance is obtained. On the other hand, given the probabilistic structure of quantum mechanics, it is not surprising that the theory of majorization plays a fundamental role in certain quantum processing tasks. For instance, there are several quantum resources theories where the free transformations between resources are fully characterized by a majorization relation. The paradigmatic case is the result of Nielsen [2] about interconversion of bipartite pure entangled quantum states by means of local operations and classical communication (LOCC). We discuss here this kind of problems by exploiting the lattice properties of the majorization [3].[1] F. Cicalese U. Vaccaro, IEEE Trans. Inf. Theory 48, 933 (2002).[2] M.A. Nielsen, Phys. Rev. Lett. 83, 436 (1999).[3] G.M. Bosyk, G. Sergioli, H. Freytes, F. Holik G. Bellomo, Physica A 473, 403(2017).