Quantum Bayesian Networks: Compositionality and Typing via Linear Logic

TL;DR

This paper introduces a formalism for quantum Bayesian networks that combines classical and quantum data analysis with compositional and typing principles based on linear logic.

Contribution

It develops a typed, compositional framework for quantum Bayesian networks that unifies classical and quantum probabilistic reasoning.

Findings

When causes are classical, the semantics match standard Bayesian networks.

In the quantum case, the semantics reduce to tensor networks.

The typed formalism ensures well-behaved composition of quantum systems.

Abstract

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.