Bayes' rule $\mathbb{P}(B|A)\mathbb{P}(A)=\mathbb{P}(A|B)\mathbb{P}(B)$ is one of the simplest yet most profound, ubiquitous, and far-reaching results of classical probability theory, with applications in any field utilizing statistical inference. Many attempts have been made to extend this rule to quantum systems, the significance of which we are only beginning to understand. In this work, we develop a systematic framework for defining Bayes' rule in the quantum setting, and we show that a vast majority of the proposed quantum Bayes' rules appearing in the literature are all instances of our definition. Moreover, our Bayes' rule is based upon a simple relationship between the notions of state over time and a time-reversal symmetry map, both of which are introduced here.

翻译：贝叶斯规则 $\mathbb{P}(B|A)\mathbb{P}(A)=\mathbb{P}(A|B)\mathbb{P}(B)$ 是经典概率论中最简单却最深刻、普遍且影响深远的结果之一，广泛应用于任何涉及统计推断的领域。人们多次尝试将该规则推广至量子系统，其重要意义我们才刚刚开始理解。在本工作中，我们发展了一个系统化的框架来定义量子情景下的贝叶斯规则，并表明文献中提出的大多数量子贝叶斯规则均属于我们定义的实例。此外，我们的贝叶斯规则基于“时间上的状态”与“时间反演对称性映射”这两个概念之间的简单关系，这两个概念均在此首次提出。