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arXiv quant-ph/0612103quant-ph
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关于Gleason定理和Kochen-Specker定理的几点注记

Some remarks on the theorems of Gleason and Kochen-Specker

Helena Granstrom

摘要:

在量子比特情形下,证明了两类限制性信息完备POVM的Gleason型定理。一个特殊的(不完全)Kochen-Specker着色,建议Appleby在三维,推广到任意尺寸。我们调查其有效性作为一个功能的尺寸,使用两种不同的措施。特别是,我们将得出一个限制的分数的领域,可以令人满意地着色使用广义Appleby建设的维数接近无穷大。第二个,也是物理上更相关的有效性测量,是查看适当着色的可能ON碱基的分数。利用这个测度,我们将推导出三维和四维实空间中的“上界的下界”。

英文摘要:

A Gleason-type theorem is proved for two restricted classes of informationally complete POVMs in the qubit case. A particular (incomplete) Kochen-Specker colouring, suggested by Appleby in dimension three, is generalized to arbitrary dimension. We investigate its effectivity as a function of dimension, using two different measures of this. In particular, we will derive a limit for the fraction of the sphere that can be satisfactorily coloured using the generalized Appleby construction as the number of dimensions approaches infinity. The second, and physically more relevant measure of effectivity, is to look at the fraction of possible ON-bases properly coloured. Using this measure, we will derive a 'lower bound for the upper bound' in three and four real dimensions.

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