INSTITUTO DE INVESTIGACION EN CIENCIAS DE LA COMPUTACION
Unidad Ejecutora - UE
Extremal elements of a sublattice of the majorization lattice and approximate majorization
HOLIK, F.; MASSRI, C.; BOSYK, G.M.; BELLOMO, G.
JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL
IOP PUBLISHING LTD
Año: 2020 vol. 53
Given a probability vector x with its components sorted in non-increasing order, we consider the closed ball Bp (x) with p 1 formed by the probability vectors whose p-norm distance to the center x is less than or equal to a radius . Here, we provide an order-Theoretic characterization of these balls by using the majorization partial order. Unlike the case p = 1 previously discussed in the literature, we nd that the extremal probability vectors, in general, do not exist for the closed balls Bp (x) with 1 p. On the other hand, we show that B (x) is a complete sublattice of the majorization lattice. As a consequence, this ball also has extremal elements. In addition, we give an explicit characterization of those extremal elements in terms of the radius and the center of the ball. This allows us to introduce some notions of approximatemajorization and discuss its relation with previous results of approximate majorization given in terms of the 1-norm. Finally, we apply our results to the problem of approximate conversion of resources within the framework of quantumresource theory of nonuniformity.