We study extreme points of the set of finite-outcome positive-operator-valued measures (POVMs) on finite-dimensional Hilbert spaces and particularly the possible ranks of the effects of an extreme POVM. We give results discussing ways of deducing new rank combinations of extreme POVMs from rank combinations of known extreme POVMs and, using these results, show ways to characterize rank combinations of extreme POVMs in low dimensions. We show that, when a rank combination together with a given dimension of the Hilbert space solve a particular packing problem, there is an extreme POVM on the Hilbert space with the given ranks. This geometric method is particularly effective for constructing extreme POVMs with desired rank combinations.

中文翻译：

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构造极端量子可观物的有效方法

我们研究了有限维希尔伯特空间上的一组有限结果正算子值测度（POVM）的极点，特别是极端POVM的影响的可能等级。我们给出结果，讨论从已知的极端POVM的等级组合推导极端POVM的新等级组合的方法，并使用这些结果，展示表征低维极端POVM的等级组合的方法。我们证明，当等级组合与希尔伯特空间的给定维数解决一个特定的包装问题时，希尔伯特空间上具有给定等级的极端POVM。这种几何方法对于构建具有所需等级组合的极端POVM特别有效。