论文标题
非交流性贝叶斯定理
A non-commutative Bayes' theorem
论文作者
论文摘要
使用贝叶斯定理的图表重新制定，在有限维$ C^*$ - 代数的情况下，我们为存在贝叶斯推断提供了必要和充分的条件。换句话说，我们在联合古典和量子环境中证明了贝叶斯定理的类似物。分类概率理论的最新进展证明了我们的类似物，该理论提供了对经典贝叶斯定理的抽象表述。在此过程中，我们进一步发展了几乎无与伦比的非交流性，并说明了其在非交流性贝叶斯反演中的重要作用。当这种贝叶斯倒置存在时，构建涉及解决Choi矩阵的正分半基质完成问题的构建。这解决了解决贝叶斯倒置的开放问题，以实现没有完全支持的密度矩阵上的完全积极的Unital图。我们说明了该过程如何适用于与量子信息理论相关的几个示例。
Using a diagrammatic reformulation of Bayes' theorem, we provide a necessary and sufficient condition for the existence of Bayesian inference in the setting of finite-dimensional $C^*$-algebras. In other words, we prove an analogue of Bayes' theorem in the joint classical and quantum context. Our analogue is justified by recent advances in categorical probability theory, which have provided an abstract formulation of the classical Bayes' theorem. In the process, we further develop non-commutative almost everywhere equivalence and illustrate its important role in non-commutative Bayesian inversion. The construction of such Bayesian inverses, when they exist, involves solving a positive semidefinite matrix completion problem for the Choi matrix. This gives a solution to the open problem of constructing Bayesian inversion for completely positive unital maps acting on density matrices that do not have full support. We illustrate how the procedure works for several examples relevant to quantum information theory.
