Extremal elements of a sublattice of the majorization lattice and approximate majorization

Abstract

Given a probability vector x with its components sorted in non-increasing order, we consider the closed ball B p (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 find that the extremal probability vectors, in general, do not exist for the closed balls B p (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 has also 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 approximate majorization 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 quantum resource theory of nonuniformity.

Generalized coherence vector applied to coherence transformations and quantifiers

“…Our proposal is based on the coherence vector of an arbitrary state introduced in Def. 15. The fact that the coherence vector satisfies the properties given in Prop.…”

“…The majorization relation is preorder relation on the set ∆ d and a partial order on the set ∆ ↓ d . Moreover, the set ∆ ↓ d together with the majorization relation is a complete lattice 1 [12,13], and the algorithms to obtain the supremum and infimum can be found in [13][14][15]. In particular, the supremum of a set U ⊆ ∆ ↓ d , denoted as U, can be computed as follows.…”

“…This characterization is given in terms of the majorization relation [11] between the coherence vectors of the pure states. Motivated by this fact, we propose a generalization of the coherence vector to general quantum states, and we advance on the characterization of the state conversion by means of incoherent operations by appealing to the majorization lattice theory [12][13][14][15]. More precisely, given a pure state decomposition of a quantum state, we define the coherence vector of the decomposition in terms of the coherence vectors of the pure states.…”

Generalized coherence vector applied to coherence transformations and quantifiers

“…The algorithms to obtain the supremum and infimum of any subset of the majorization lattice can be found in [16][17][18]. In particular, the supremum of a set U ⊆ ↓ d , denoted as U , can be computed as follows.…”

“…This characterization is given in terms of the majorization relation [14] between the coherence vectors of the pure states. Motivated by this fact, we propose a generalization of the coherence vector applicable to arbitrary quantum states, and we advance on the characterization of the state conversion by means of incoherent operations by appealing to the majorization lattice theory [15][16][17][18]. More precisely, given a pure state decomposition of a quantum state, we define the coherence vector of the decomposition in terms of the coherence vectors of the pure states.…”