基于屋顶构造的凸壳簇与纠缠测度
Convex Hulls of Varieties and Entanglement Measures Based on the Roof Construction
摘要：
本文研究一类仿射代数簇的凸壳计算问题。正如我们所解释的，考虑这个问题的动机是，量子纠缠的某些纯态测度，我们称之为多项式纠缠测度，可以表示为仿射代数簇。我们认为这些多项式纠缠措施，即凸和凹屋顶的某些混合状态扩展的评价。我们表明，基于屋顶的混合状态扩展的评价相当于计算一个超平面，该超平面是乘切线的各种在一些地方等于在最佳分解的措施的条款的数量。通过这种方式，我们提供了一个隐式表示的最佳分解的混合态纠缠措施的基础上的屋顶建设。
英文摘要：
In this paper we study the problem of calculating the convex hull of certain affine algebraic varieties. As we explain, the motivation for considering this problem is that certain pure-state measures of quantum entanglement, which we call polynomial entanglement measures, can be represented as affine algebraic varieties. We consider the evaluation of certain mixed-state extensions of these polynomial entanglement measures, namely convex and concave roofs. We show that the evaluation of a roof-based mixed-state extension is equivalent to calculating a hyperplane which is multiply tangent to the variety in a number of places equal to the number of terms in an optimal decomposition for the measure. In this way we provide an implicit representation of optimal decompositions for mixed-state entanglement measures based on the roof construction.
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