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:巴伊斯规则$mathbb{P}(B<unk> A)\mathbb{P}(A)<unk> mathbb{P}(A<unk> B)\mathbb{P}(B)$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,在任何领域的应用都利用统计推论。许多尝试都试图将这一规则扩大到量子系统,而我们只是开始理解它的重要性。在这项工作中,我们制定了一个系统框架,在量子设置中界定贝伊斯规则,我们表明在文献中出现的绝大多数拟议的量子贝伊规则都是我们定义的例子。此外,我们海湾规则的基础是时间概念和时间-反对称图之间的简单关系,两者都是在这里介绍的。</s>