Quantum Bayesian Networks: Compositionality and Typing via Linear Logic
Provides a foundational formalism for causal reasoning in quantum-classical hybrid systems, addressing a key bottleneck in unifying probabilistic graphical models with quantum mechanics.
The paper introduces compositional principles and a typing discipline to quantum Bayesian networks, unifying classical Bayesian networks and quantum tensor networks under a single framework based on linear logic.
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.