一个非交换贝叶斯定理
A non-commutative Bayes' theorem
摘要：
利用贝叶斯定理的一个图解形式，给出了有限维$C^*$ -代数上贝叶斯推断存在的一个充分必要条件.换句话说，我们证明了一个类似的贝叶斯定理在联合经典和量子的背景下。我们的类比是合理的最近进展，在分类概率论，它提供了一个抽象的制定经典贝叶斯定理。在这个过程中，我们进一步发展了非交换几乎处处等价，并说明了它在非交换贝叶斯反演中的重要作用。这种贝叶斯逆的构造，当它们存在时，涉及解决Choi矩阵的半正定矩阵完成问题。这给出了一个解决的开放问题的构造贝叶斯反演的完全正的单位映射的密度矩阵，没有充分的支持。我们说明了如何工作的过程中的几个例子相关的量子信息理论。
英文摘要：
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.
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