Quantum Bayesian networks provide a mathematical formalism to describe causal relations, to analyse correlations, and to predict the probabilities of measurement outcomes, in systems involving both classical and quantum data. They generalize Pearl's Bayesian networks -- prominent graphical models for classical probabilistic reasoning and inference. The goal of this paper is to bring compositional principles and a typing discipline into this setting. A key feature of our compositional semantics is that when all causes are classical, it coincides with the standard factor-based semantics of Bayesian networks, while in the purely quantum case it reduces to tensor networks. We then propose a typed formalism based on linear logic proof-nets, where types ensure well-behaved composition of systems, and which we prove sound and complete with respect to quantum Bayesian networks.

翻译：量子贝叶斯网络提供了一种数学形式体系，用于描述涉及经典与量子数据的系统中的因果关系、分析相关性并预测测量结果的概率。该体系推广了Pearl提出的贝叶斯网络——这一经典概率推理与推断领域的重要图模型。本文旨在将组合性原则与类型化规则引入该框架。我们组合语义的一个关键特征在于：当所有因果因素均为经典时，该语义与贝叶斯网络的标准因子化语义一致；而在纯量子情形下，它退化为张量网络。在此基础上，我们提出了一种基于线性逻辑证明网的类型化形式体系，其中类型系统确保了系统组合的良好行为，并且我们证明该体系相对于量子贝叶斯网络具有可靠性与完备性。