论文标题
大型晶格的sublattice的极端元素和近似多数化
Extremal elements of a sublattice of the majorization lattice and approximate majorization
论文作者
论文摘要
给定概率向量$ x $，其组件按非侵入顺序排序，我们考虑了封闭的球$ {\ Mathcal {b}}}^p_ε（x）$，$ p \ geq 1 $由$ \ ell^p $ norm dos to to the Centry $ x $ x $ by best and best y hadius $ a的概率向量形成。在这里，我们通过使用主要化部分顺序提供了这些球的顺序理论表征。与以前在文献中讨论过的情况不同的情况不同，我们发现封闭的球$ {\ MATHCAL {\ MATHCAL {b}}^p_is（x）$，带有$ 1 <p <\ p <\ infty $的极概率向量不存在。另一方面，我们表明$ {\ Mathcal {b}}^\infty_ε（x）$是多数化晶格的完整sublattice。结果，这个球也具有极端元素。此外，我们从半径和球中心方面给出了这些极端元素的明确表征。这使我们能够介绍一些近似大多数的概念，并讨论其与先前根据$ \ ell^1 $ norm给出的近似大多数结果的关系。最后，我们将结果应用于量子不一度资源理论框架内资源转化的问题。
Given a probability vector $x$ with its components sorted in non-increasing order, we consider the closed ball ${\mathcal{B}}^p_ε(x)$ 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 $ε$. 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 ${\mathcal{B}}^p_ε(x)$ with $1<p<\infty$. On the other hand, we show that ${\mathcal{B}}^\infty_ε(x)$ 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.
