[论文解读] Quantum Causal Models

Jonathan Barrett|arXiv (Cornell University)|Jul 30, 2018

Quantum Mechanics and Applications参考文献 74被引用 14

一句话总结

本文提出了一种基于酉量子线路的量子因果模型框架，将因果关系内在地定义于量子理论之中，而非通过经典变量。它证明了任意酉线路均会诱导出一个有向无环图（DAG）结构，且由此产生的量子过程相对于该图满足马尔可夫条件；此外，它证明了d-分离在量子条件独立性上是可靠且完备的，将Pearl的do-演算推广至量子领域。

ABSTRACT

It is known that the classical framework of causal models is not general enough to allow for causal reasoning about quantum systems. While the framework has been generalized in a variety of different ways to the quantum case, much of this work leaves open whether causal concepts are fundamental to quantum theory, or only find application at an emergent level of classical devices and measurement outcomes. Here, we present a framework of quantum causal models, with causal relations defined in terms intrinsic to quantum theory, and the central object of study being the quantum process itself. Following Allen et al., Phys. Rev. X 7, 031021 (2017), the approach defines quantum causal relations in terms of unitary evolution, in a way analogous to an approach to classical causal models that assumes underlying determinism and situates causal relations in functional dependences between variables. We show that any unitary quantum circuit has a causal structure corresponding to a directed acyclic graph, and that when marginalising over local noise sources, the resulting quantum process satisfies a Markov condition with respect to the graph. We also prove a converse to this statement. We introduce an intrinsically quantum notion that plays a role analogous to the conditional independence of classical variables, and (generalizing a central theorem of the classical framework) show that d-separation is sound and complete for it in the quantum case. We present generalizations of the three rules of the classical do-calculus, in each case relating a property of the causal structure to a formal property of the quantum process, and to an operational statement concerning the outcomes of interventions. In addition, we introduce and derive similar results for classical split-node causal models, which are more closely analogous to quantum causal models than the classical causal models that are usually studied.

研究动机与目标

• 开发一种内在基于量子力学的因果框架，而非依赖于经典因果概念。
• 解决经典因果模型在解释量子关联方面的局限性，特别是在贝尔型场景和多源量子网络中。
• 利用酉动力学和过程算符，形式化量子系统中的因果影响、干预和条件独立性。
• 将经典do-演算推广至量子领域，将因果结构与操作性干预结果联系起来。
• 探索时间对称性与量子过程中的因果结构，包括未来工作中可能存在的不确定因果序。

提出的方法

• 将量子因果模型定义为以酉线路作为基本动力学演化，输入和输出作用于量子系统。
• 通过编码动力学并允许对子系统进行边缘化的过程算符来表示量子过程。
• 基于从酉线路导出的因果DAG结构，为量子过程引入马尔可夫条件。
• 通过类似于经典d-分离的三元关系定义量子条件独立性，推广条件独立性的概念。
• 证明d-分离在量子条件独立性上是可靠且完备的，将经典d-分离定理扩展至量子情形。
• 将Pearl的do-演算三条规则推广至量子干预，将因果结构、形式过程属性与操作性干预结果联系起来。

实验结果

研究问题

• RQ1 量子理论中的因果关系能否被内在定义，而不还原为经典变量或测量结果？
• RQ2 d-分离如何推广至量子过程？它在量子条件独立性上是否可靠且完备？
• RQ3 经典do-演算的规则在多大程度上可被适应于量子因果模型？它们保有何种操作意义？
• RQ4 酉演化在定义因果结构中起什么作用？它与量子过程中马尔可夫条件有何关系？
• RQ5 时间对称性在量子因果模型中如何体现？导致观测到的时间不对称性的假设是什么？

主要发现

• 任意酉量子线路均会诱导出一个有向无环图（DAG），表示所涉及量子系统之间的因果结构。
• 当对局部噪声源进行边缘化后，所得的量子过程相对于因果DAG满足马尔可夫条件。
• 反之亦然：任何相对于DAG满足马尔可夫条件的过程均可由酉线路实现。
• 通过三元关系定义了量子条件独立性的概念，且证明d-分离对该概念是可靠且完备的。
• 经典do-演算的三条规则被推广至量子情形，每条规则均将因果结构、形式过程属性与操作性干预结果联系起来。
• 该框架允许对酉过程中因果结构进行时间对称处理，但时间不对称性源于对仅作用于后代的噪声源和干预的假设。