AI 中文总结
本文将回溯推断的范畴定义扩展到冯·诺依曼代数的无限维情形，探讨Petz恢复映射的相关问题，为量子贝叶斯推断的结构必然性提供理论依据。
AI 中文摘要
回溯推断是从结果反推原因的行为，最常见的例子是贝叶斯推断，其结构过程理论性质可通过范畴论进行数学刻画。近期研究表明，回溯推断的这种范畴定义可能将Petz恢复映射分离为量子贝叶斯推断的唯一通用候选。本文将这些结果扩展到冯·诺依曼代数的无限维情形，过程中对无限维下的Petz恢复映射及其与有限维常用表达式的关系进行了教学性综述，将“回溯推断的范畴公理是否确实唯一刻画Petz恢复映射”这一开放问题形式化。若该刻画成立，将表明贝叶斯反演与Petz恢复映射是结构必然性，而非仅适用于经典和量子推断的有用算法。
英文摘要
Retrodiction is the act of inferring a cause from its effects, the most common example of which is Bayesian inference. Retrodiction can be defined by its structural process-theoretic properties, which are mathematically captured by category theory. This categorical definition of retrodiction has recently been shown to potentially isolate the Petz recovery map as a unique universal candidate for quantum Bayesian inference. This paper extends these results to the infinite-dimensional setting on von Neumann algebras. In the process, we provide a pedagogical review of the Petz recovery map in infinite dimensions and its relation to the more commonly used expression in the finite-dimensional setting. We formalize the open question as to whether these categorical axioms for retrodiction do in fact uniquely characterize the Petz recovery map. If such a characterization holds, this would show that Bayesian inversion and the Petz recovery map are structural necessities and not simply useful algorithms for classical and quantum inference.
CommentsComments are welcome! 30 pages + appendix + bibliography
