Bounding Classical and Quantum Correlations in Bayesian Networks with Quasiprobabilities

Abstract: Bell's theorem reveals that quantum theory is in tension with classical causal reasoning and, in particular, the notion of local causality. This is now understood as a particular example of non-classicality in the study of correlations in (Bayesian) networks with both unobserved and observed nodes: the correlations are probability distributions over the observed nodes. There is a great deal of work aiming to understand the bounds on quantum and classical correlations in such networks and one approach is to consider outer approximations to the former. Along these lines, we consider quasiprobabilistic models for Bayesian networks, which can be seen as classical models but the probability distributions involving unobserved nodes are "replaced" with quasiprobabilities that respect normalisation but not positivity. We denote the set of correlations resulting from these models as the quasi set. Such models have a history in the study of Bell-type non-classicality where it has been shown that they can produce all non-signalling correlations. We show a generalisation of this result for a broad class of networks, which motivates a conjecture that the quasi set recovers the so called nested Markov model. Our work utilises a connection to tensor network decompositions, which may be of independent interest.

• The paper presents a novel framework using quasiprobabilities to generalize Bell scenarios and characterize bounds on classical and quantum correlations in Bayesian networks.
• It rigorously establishes the equivalence between quasilatents, quasiresponses, and quasinoise models, linking observed marginals with tensor network decompositions.
• The study shows that in tree-structured scenarios the quasi set exactly coincides with the nested Markov model, offering practical insights for causal inference.

Bounding Classical and Quantum Correlations in Bayesian Networks with Quasiprobabilities

Overview and Motivation

This paper investigates the structure and bounds of classical and quantum correlations in Bayesian networks via the introduction and analysis of quasiprobabilistic models. Bell's theorem, which exposes the incompatibility between quantum theory and classical notions of local causality, serves as the foundational context for the work. The study of non-classical correlations in directed graphical models with observed and unobserved nodes has had significant scrutiny, but exact characterization of quantum sets remains elusive, due to their undecidability and non-polytope structure. The paper seeks to understand and generalize bounds on correlations by considering "quasi sets," wherein distributions over latent variables are replaced by quasiprobabilities—real-valued functions that respect normalization but may be negative.

Classical and Quantum Marginal Models in Causal Graphs

Directed acyclic graphs (DAGs) are central to Bayesian networks, providing a formalism for representing causal relationships among random variables. When all variables are observed, distributions consistent with a DAG satisfy equivalent Markov properties (factorization, local, global). In the presence of latent variables, the classical marginal model—the set of observed distributions extendable to a fully Markov distribution—is strictly contained within the constraints imposed by polynomial equalities (Verma constraints) and inequalities, with the latter being complex and difficult to characterize. Relaxations of these constraints yield the ordinary Markov and nested Markov models, the latter characterized by equality constraints and used effectively as approximations to the classical set due to their tractability.

Quantum causality introduces further complexity. Quantum sets in scenarios such as bipartite Bell networks are not finite polytopes and exhibit pathological features, challenging their exact description. The classical set is always simulatable via quantum separable states, but the quantum set contains distributions violating Bell-type inequalities, signifying nonlocality.

Quasiprobabilistic Models and the Quasi Set

The paper generalizes prior results showing that in the Bell scenario, all non-signaling conditional distributions (including many not quantum-realizable) can be expressed via factorization with negative probabilities. The authors develop a comprehensive framework for quasiprobabilistic models in arbitrary networks, formalizing the equivalence between three distinct approaches:

• Quasilatents: Allowing negative distributions for latent nodes only.
• Quasiresponses: Allowing negative distributions for response functions only.
• Quasinoise: Introducing independent noise variables, some with negative distributions.

The main theorem establishes that all three models are equivalent in terms of their observed marginals, confirming a duality between latent and response negativity in quasiprobabilistic networks. The quasi set, denoted $\mathcal{W}(\mathcal{G})$, comprises all observed distributions which are marginals of such models.

Relationship to the Nested Markov Model

The authors prove that $$\mathcal{W}(\mathcal{G}) \subseteq \mathcal{N}(\mathcal{G})$$, i.e., the quasi set is always contained in the nested Markov model for any DAG. The paper conjectures, and proves for tree-structured correlation scenarios (TCS), that equality holds: $$\mathcal{W}(\mathcal{G}) = \mathcal{N}(\mathcal{G})$$. This is significant, as the nested Markov model is a subset of distributions defined by polynomial (equality) constraints, and typically comprises a convenient approximation for classical marginal models, but typically, not all distributions in this set are quantum-realisable.

Tensor Networks as a Technical Bridge

A critical component of the proof is the connection to tensor network decompositions, a tool from quantum many-body physics and graphical models. Every distribution in the quasi set for TCS admits a tensor network decomposition whose bond dimensions correspond to latent variable domain sizes. The proof leverages this correspondence to show that any distribution in the nested Markov set for TCS can be realized as a marginal of a quasiprobabilistic model.

Numerical and Structural Highlights

• Equivalence of Quasimarginal Definitions: Demonstrated rigorously, allowing generalization beyond Bell-type scenarios.
• Tree-Structured Correlation Scenarios: Proof that for these cases, the quasi set coincides exactly with both the nested and ordinary Markov models.
• Perfect Correlation in Triangle Scenario: Direct constructions show that non-classical (non-quantum) distributions can be achieved in the quasi set, reinforcing its strict inclusion over quantum sets.

Implications and Theoretical Consequences

The conjectured equivalence between the quasi set and nested Markov model, if proven for general DAGs, would indicate that relaxing positivity constraints essentially makes all polynomial equalities obtainable, without limitations from inequality constraints. This demonstrates the extraordinary expressive power of quasiprobabilistic models, surpassing classical and quantum sets, and general GPTs (generalized probabilistic theories) lacking no-restriction. Practically, this formalism could provide algorithmic simplifications for causal inference and marginal problems in networks with latent variables, since inequality constraints—commonly the computational bottleneck—are eliminated in the quasi set.

Moreover, this work delineates the boundary of physically permissible correlations, suggesting that quantum theory imposes structure not merely by equality constraints, but fundamentally via positivity and properties of quantum states. The mathematical formalism presented offers a unified treatment for outer approximation of quantum sets, situating the nested Markov model as an expressive, algebraically constrained upper bound.

Speculation on Future Directions

• Proof of Conjecture for General DAGs: Extending the equivalence result to all networks remains an open challenge, potentially requiring deeper exploration of algebraic geometry and de Finetti-type theorems for quasiprobabilities.
• Connection to Inflation Technique: The quasi set may facilitate simpler or more complete inflation techniques in marginal problems, offering tractable solutions in quasiprobability.
• Foundational Implications for Probabilistic Theories: Investigating why the quasi set is so expressive compared to physically motivated theories could yield insight into the structure of physical laws, perhaps leading to alternative models beyond quantum mechanics.

Conclusion

This paper establishes a comprehensive framework for bounding classical and quantum correlations in Bayesian networks using quasiprobabilities, generalizing known results and introducing new equivalences via tensor network decompositions. The quasi set is shown to be equivalent to the nested Markov model in tree-structured correlation scenarios, suggesting, via conjecture, a broader equivalence for arbitrary graphs. The technical depth and rigor clarify the role of positivity and algebraic constraints in classical and quantum causal inference, laying the foundation for further theoretical development in causal networks, marginal problems, and the structure of non-signaling correlations.