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arXiv 0904.3472quant-ph
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研究非正常混合物的量子逻辑和几何方法

A quantum logical and geometrical approach to the study of improper mixtures

Graciela Domenech, Federico Holik, Cesar Massri

备注：
submitted to the Journal of Mathematical Physics
发表信息：
Journal of Mathematical Physics 51 (2010) 521081
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摘要：

我们从量子逻辑和几何的角度来研究不适当的混合物。考虑到不适当的混合不承认无知的解释，必须被视为国家在自己的权利，我们不遵循标准的方法，认为不适当的混合作为措施的代数投影。取而代之的是，我们使用凸集的状态，以构建一个新的晶格，其原子都是物理状态：纯态和不适当的混合物。这样做是为了克服标准量子逻辑形式主义中出现的一个问题，即对于处于纠缠态的较大系统的子系统，子系统的所有实际属性的结合并不产生其实际状态。事实上，它的状态是一个不适当的混合物，不能在冯诺依曼晶格中表示为一个最小的属性，它决定了所有其他属性，就像纯态或经典系统一样。新格也包含了冯诺依曼格的所有命题。我们认为，这种扩展表示在一个代数形式的事实，即-类似的经典情况下-量子相互作用系统之间产生非平凡的相关性。最后，我们研究了复合系统的扩张格与其子系统格之间的映射。

英文摘要：

We study improper mixtures from a quantum logical and geometrical point of view. Taking into account the fact that improper mixtures do not admit an ignorance interpretation and must be considered as states in their own right, we do not follow the standard approach which considers improper mixtures as measures over the algebra of projections. Instead of it, we use the convex set of states in order to construct a new lattice whose atoms are all physical states: pure states and improper mixtures. This is done in order to overcome one of the problems which appear in the standard quantum logical formalism, namely, that for a subsystem of a larger system in an entangled state, the conjunction of all actual properties of the subsystem does not yield its actual state. In fact, its state is an improper mixture and cannot be represented in the von Neumann lattice as a minimal property which determines all other properties as is the case for pure states or classical systems. The new lattice also contains all propositions of the von Neumann lattice. We argue that this extension expresses in an algebraic form the fact that -alike the classical case- quantum interactions produce non trivial correlations between the systems. Finally, we study the maps which can be defined between the extended lattice of a compound system and the lattices of its subsystems.

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