AI 中文总结
本文针对量子信息局域化问题，证明源自非交换几何的调整后冯·诺依曼熵是量化局域化信息的最小量子熵，且是唯一能通过冗余信息表征贝尔态的量子熵，推广了经典情况。
AI 中文摘要
信息可局域化的观点在物理学中普遍存在：热力学中熵从一个热库流向另一个热库，量子引力中某些空间区域的熵含量存在边界；量子信息理论中信息在不同地点间传输，当传输受约束时纠缠可提供优势。香农熵在边际化定义经典多体系统各部分集合上的外测度的意义下，可量化局域化信息；而冯·诺依曼熵在该意义下无法量化局域化信息，但源自非交换几何的冯·诺依曼熵变体可以。我们证明该调整后的冯·诺依曼熵是量化局域化信息的最小量子熵，还证明其是唯一能通过冗余信息表征贝尔态的量子熵，直接推广了经典情况。我们全程使用有限维$C^*$-代数，对可能具有超选择 sector 的有限量子系统进行建模。
英文摘要
The idea that information can be localized is pervasive in physics. In thermodynamics, entropy flows from one reservoir to another, and in quantum gravity, the entropy content of some spatial regions is bounded. In quantum information theory, information is transmitted from one place to another, and entanglement can provide an advantage when there are constraints on such transmission. Shannon entropy quantifies localizable information in the sense that marginalization defines an outer measure on the set of parts of a classical multipartite system. In contrast, von Neumann entropy does not quantify localizable information in this sense. However, a variant of von Neumann entropy, which originates in noncommutative geometry, does. We prove that this adjusted von Neumann entropy is the minimum quantum entropy that quantifies localizable information. We also prove that this is the unique quantum entropy that characterizes Bell states in terms of redundant information, directly generalizing the classical case. We work with finite-dimensional $C^*$-algebras throughout, modeling finite quantum systems that may have superselection sectors.
Comments27 pages
