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 decision making, artificial intelligence, weather forecasts, betting strategies, and more generally 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 \emph{state over time} and a time-reversal symmetry map, both of which are introduced here.

翻译：巴伊斯规则$mathbb{P}(B ⁇ A)\mathbb{P}(A) ⁇ mathbb{P}(A ⁇ B)\mathbb{P}(B)$(B)$是古典概率理论最简单、最深刻、无处不在和意义深远的结果之一,在决策应用、人工智能、天气预报、赌注策略以及更广义的统计推论方面,我们多次试图将这一规则扩大到量子系统,而我们刚刚开始理解量子系统的意义。在这项工作中,我们为界定量子环境中的Bayes规则制定了一个系统框架,我们表明文献中出现的绝大多数拟议量子湾规则都是我们定义的事例。此外,我们的贝伊斯规则基于一个简单的关系,即时间-反向对称图,这两个概念都是在这里介绍的。