论文标题
熵理论概念
A Topos Theoretic Notion of Entropy
论文作者
论文摘要
在Topos的量子理论方法中，光谱前eaf起着量子系统状态空间的作用。我们展示了如何使用状态与光谱前eak eak efor的等效性在Topos形式主义中定义熵的概念。我们展示了这种结构如何统一香农和冯·诺伊曼的熵以及古典和量子雷尼熵。主要结果是，从有限维系统的量子状态的上下文熵的知识中，可以（数学上）重建量子状态，即，即密度矩阵，如果希尔伯特空间为尺寸$ 3 $或更高。我们为该州重建提供了一种明确的算法，并将我们的结果与格里森定理联系起来。
In the topos approach to quantum theory, the spectral presheaf plays the role of the state space of a quantum system. We show how a notion of entropy can be defined within the topos formalism using the equivalence between states and measures on the spectral presheaf. We show how this construction unifies Shannon and von Neumann entropy as well as classical and quantum Renyi entropies. The main result is that from the knowledge of the contextual entropy of a quantum state of a finite-dimensional system, one can (mathematically) reconstruct the quantum state, i.e., the density matrix, if the Hilbert space is of dimension $3$ or greater. We present an explicit algorithm for this state reconstruction and relate our result to Gleason's theorem.
