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. Our paper brings 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.

翻译：量子贝叶斯网络提供了一种数学形式，用于描述因果关联、分析相关性以及预测包含经典与量子数据的系统中的测量结果概率。该框架推广了珀尔贝叶斯网络——经典概率推理与推断中重要的图模型。本文将此领域引入了组合原理和类型化规范。我们组合语义的一个关键特性在于：当所有原因均为经典时，它回归至贝叶斯网络的标准因子语义；而在纯量子情形下，则简化为张量网络。在此基础上，我们提出了一种基于线性逻辑证明网的带类型形式化方法，其中类型确保了系统组合的良好行为。