标题： 主序格的子格的极值元素与近似主序 显示英文标题

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

摘要： Given a probability vector $x$ with its components sorted in non-increasing order, we consider the closed ball ${\mathcal{B}}^p_\epsilon(x)$ with $p \geq 1$ formed by the probability vectors whose $\ell^p$-norm distance to the center $x$ is less than or equal to a radius $\epsilon$. Here, we provide an order-theoretic characterization of these balls by using the majorization partial order. 与文献中先前讨论的情况 $p=1$ 不同，我们发现对于一般的闭球 ${\mathcal{B}}^p_\epsilon(x)$ （其中 $1<p<\infty$），极值概率向量通常不存在。 另一方面，我们证明了 ${\mathcal{B}}^\infty_\epsilon(x)$ 是主要化格的完全子格。 因此，这个球也具有极值元素。 此外，我们通过球的半径和中心显式地刻画了这些极值元素。 这使我们能够引入一些近似主要化的概念，并讨论它与之前用 $\ell^1$-范数表示的近似主要化结果之间的关系。 最后，我们将我们的结果应用于非均匀性量子资源理论中的近似资源转换问题。 显示英文摘要

摘要： Given a probability vector $x$ with its components sorted in non-increasing order, we consider the closed ball ${\mathcal{B}}^p_\epsilon(x)$ with $p \geq 1$ formed by the probability vectors whose $\ell^p$-norm distance to the center $x$ is less than or equal to a radius $\epsilon$. 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 ${\mathcal{B}}^p_\epsilon(x)$ with $1<p<\infty$. On the other hand, we show that ${\mathcal{B}}^\infty_\epsilon(x)$ 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 $\ell^1$-norm. Finally, we apply our results to the problem of approximate conversion of resources within the framework of quantum resource theory of nonuniformity.