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Bounding Classical and Quantum Correlations in Bayesian Networks with Quasiprobabilities

This paper investigates quasiprobabilistic models for Bayesian networks, demonstrating that they can generate all non-signalling correlations for a broad class of networks and conjecturing that this "quasi set" recovers the nested Markov model, utilizing connections to tensor network decompositions.

Original authors: Paul Becsi, Matty J. Hoban

Published 2026-06-23

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Paul Becsi, Matty J. Hoban

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: The Mystery of Hidden Connections

Imagine you are a detective trying to figure out how a group of people are connected. You can see what they say and do (the observed nodes), but you cannot see the private notes they pass to each other or the secret rules they follow (the latent/unobserved nodes).

In the world of science, we use "Bayesian Networks" (which are just fancy flowcharts) to map out these cause-and-effect relationships. Usually, we assume that everything follows the rules of standard probability: things have a 0% to 100% chance of happening. Nothing can be "minus 50% likely."

However, Quantum Physics (the rules that govern atoms and light) sometimes breaks these rules. It allows for "spooky" connections that classical logic can't explain. Scientists have been trying to draw a line between what is possible in our everyday world (Classical) and what is possible in the quantum world.

The Problem: The Quantum Puzzle is Too Hard

The authors point out that figuring out exactly what a quantum network can do is incredibly difficult. It's like trying to solve a maze where the walls keep moving. The set of all possible quantum behaviors is messy, complex, and mathematically "undecidable" (meaning there's no simple formula to check if a specific behavior is possible or not).

Because the quantum puzzle is so hard, scientists often look for "outer approximations." Think of this like drawing a large, simple circle around a complex, jagged island. The circle isn't the island itself, but it tells you the island is definitely inside it. If you can prove something is outside the circle, you know it's not on the island.

The Solution: Introducing "Quasiprobabilities" (The Magic Ink)

The authors propose a new way to draw that "circle." They introduce a concept called Quasiprobabilities.

Imagine you have a standard probability map where every number must be positive (0 to 1). Now, imagine you have a magic ink that allows you to write negative numbers on the map.

Normal Probability: "There is a 50% chance of rain."
Quasiprobability: "There is a -20% chance of rain."

In the real world, negative rain doesn't make sense. But in this mathematical model, allowing these "negative probabilities" (which still add up to 100% total) creates a much larger, more flexible set of possibilities. The authors call this the "Quasi Set."

The Main Discovery: Two Ways to Use the Magic Ink

The paper proves something surprising about this magic ink. You can use it in two different ways, and they result in the exact same set of outcomes:

Magic Hidden Variables: You can let the secret notes (the hidden variables) be written in magic ink (negative probabilities).
Magic Rules: You can let the rules that turn those notes into actions (response functions) be written in magic ink.

The authors show that it doesn't matter which one you choose to be "magic." If you allow either the hidden notes or the rules to use negative numbers, you get the exact same range of possible outcomes. This is a "duality," like looking at a coin from the front or the back—it's the same coin.

The Big Conjecture: The Perfect Map

The authors have a bold guess (a Conjecture):
They believe that if you allow this magic ink (quasiprobabilities), your "Quasi Set" becomes exactly equal to something called the Nested Markov Model.

The Nested Markov Model is a very strict, orderly mathematical structure that describes all the logical rules a network must follow, ignoring the messy inequality constraints that usually make things hard.
The Claim: By allowing negative probabilities, you strip away all the messy "inequality" rules. What's left is a clean, orderly structure that perfectly matches the Nested Markov Model.

If this is true, it means the "Quasi Set" is the perfect, simple outer boundary for quantum correlations. It suggests that the only reason classical and quantum correlations have "inequality" constraints (like the famous Bell inequalities) is simply because we are forced to use positive numbers. If we allow negative numbers, those constraints disappear.

The Proof: Trees and Tangled Strings

The authors couldn't prove this for every possible network, but they did prove it for a specific, very common type of network called a Tree-Structured Correlation Scenario.

The Analogy: Imagine a family tree where the parents are hidden, and the children are the people you can see. If the family tree has no loops (no one is their own ancestor), it's a "tree."
The Tool: To prove their point for these tree networks, the authors used a tool from computer science and physics called Tensor Networks.
- Think of a Tensor Network as a way to untangle a knot of strings.
- They showed that any pattern of behavior in these tree networks can be "untangled" and represented using their magic ink (quasiprobabilities).
- This proved that for tree-like networks, the "Quasi Set" and the "Nested Markov Model" are indeed identical.

Summary

The Goal: Find a simple way to describe the limits of quantum correlations in networks.
The Trick: Allow probabilities to be negative (Quasiprobabilities).
The Result: This trick creates a "Quasi Set" that is mathematically identical to a clean, orderly structure called the Nested Markov Model (at least for tree-shaped networks).
The Implication: The complex "inequalities" that separate classical and quantum worlds might just be an artifact of forcing probabilities to be positive. If you allow negative numbers, the boundary becomes much simpler and more predictable.

The paper essentially says: "If you let your math use negative numbers, the messy puzzle of quantum networks becomes a neat, solvable pattern."

Technical Summary: Bounding Classical and Quantum Correlations in Bayesian Networks with Quasiprobabilities

Problem Statement
The study of correlations in Bayesian networks (directed acyclic graphs or DAGs) with latent (unobserved) variables is central to causal inference and quantum foundations. A primary challenge is characterizing the "classical set" $\mathcal{C}(G)$ , defined as the set of observed probability distributions consistent with a given causal structure $G$ . While the classical set is semialgebraic (defined by polynomial equalities and inequalities), it is notoriously difficult to characterize due to complex inequality constraints.

Parallel to this, the "quantum set" $\mathcal{Q}(G)$ , representing correlations achievable via quantum resources, is even more complex; it is not a finite polytope, not closed, and undecidable in general. Researchers often seek "outer approximations" to bound these sets. A known result in the Bipartite Bell Scenario (a specific DAG) is that allowing latent variables to follow quasiprobability distributions (distributions that sum to 1 but may take negative values) recovers the set of all non-signaling correlations. This paper investigates whether this result generalizes to arbitrary Bayesian networks and explores the relationship between the "quasi set" and the "nested Markov model" $\mathcal{N}(G)$ , a tractable algebraic approximation of the classical set.

Methodology
The authors employ a combination of causal modeling, quasiprobabilistic analysis, and tensor network theory.

Quasiprobabilistic Models: The paper defines three equivalent formulations of a "quasimarginal set" $\mathcal{W}(G)$ , where the observed distribution is the marginal of a full distribution factorizing over the graph, but with relaxed positivity constraints:
- Quasilatents ( $\mathcal{W}_l$ ): Latent variables follow quasidistributions, while response functions (conditional probabilities of observed nodes) remain non-negative.
- Quasiresponses ( $\mathcal{W}_r$ ): Response functions follow quasidistributions, while latents remain non-negative.
- Quasinoise ( $\mathcal{W}_n$ ): Observed nodes are modeled with deterministic response functions dependent on classical latents and independent "noise" variables, where the noise variables follow quasidistributions.
Graph Reduction: The authors utilize results from classical causality (e.g., exogenization of latents, elimination of redundant variables) to show that these reduction steps hold for quasiprobabilistic models, allowing them to assume "reduced" graphs without loss of generality.
Tensor Network Decompositions: To prove the main conjecture for specific graph structures, the authors map the problem to tensor networks. They treat the observed probability distribution as a high-order tensor and utilize the correspondence between the causal structure and an underlying undirected graph (where edges represent shared latent parents). They leverage theorems regarding the existence of minimal tensor decompositions for trees.

Key Contributions and Results

Equivalence of Quasiprobabilistic Definitions: The paper proves Theorem 4, establishing that for any DAG $G$ , the sets $\mathcal{W}_l(G)$ , $\mathcal{W}_r(G)$ , and $\mathcal{W}_n(G)$ are identical. This demonstrates a duality: allowing negativity in latent variables is mathematically equivalent to allowing negativity in response functions (or noise) regarding the set of achievable non-negative observed distributions.
Inclusion in Nested Markov Model: The authors prove Proposition 6, showing that $\mathcal{W}(G) \subseteq \mathcal{N}(G)$ . Since the nested Markov model is defined purely by factorization constraints (polynomial equalities) without positivity requirements, any distribution factorizing with quasiprobabilities (which satisfy normalization but not positivity) must satisfy the nested Markov constraints.
Conjecture on Equality: The paper proposes Conjecture 1: For any DAG $G$ , $\mathcal{W}(G) = \mathcal{N}(G)$ . This would imply that the quasi set is exactly the set of distributions satisfying all polynomial equality constraints (the nested Markov model), and that inequality constraints in classical and quantum sets arise solely from the non-negativity requirement of probabilities.
Proof for Tree-Structured Correlation Scenarios (TCS): The authors provide a rigorous proof of Conjecture 1 for the subclass of correlation scenarios where the underlying undirected graph (formed by observed nodes connected if they share a latent parent) is a tree.
- They demonstrate that any distribution in the ordinary Markov model (which equals the nested Markov model for TCSs) can be decomposed into a tensor network.
- By analyzing the minimal bond dimensions of this decomposition, they construct a quasiprobabilistic latent variable model that reproduces the distribution, thereby proving $\mathcal{M}(G) \subseteq \mathcal{W}(G)$ for TCSs. Combined with the general inclusion $\mathcal{W}(G) \subseteq \mathcal{N}(G)$ , this establishes $\mathcal{W}(G) = \mathcal{N}(G)$ for TCSs.

Significance and Claims
The paper claims that its work generalizes the known result that quasiprobabilities recover non-signaling correlations in the Bell Scenario to a broader class of networks.

Characterization of Inequality Constraints: If Conjecture 1 holds, it provides a structural interpretation of inequality constraints in classical and quantum causal models. The authors argue that these inequalities are not inherent to the causal structure or the factorization criterion itself, but are artifacts of the non-negativity constraint on probabilities. In the quasi set, which relaxes positivity, these inequalities vanish, leaving only the algebraic (equality) constraints of the nested Markov model.
Expressivity of Generalized Probabilistic Theories (GPTs): The results suggest that a theory allowing latent variables (or response functions) to take arbitrary real values (a non-free GPT) is expressive enough to generate all distributions in the nested Markov model. This contrasts with standard GPTs (like quantum theory), which are known to be insufficient to generate all nested Markov correlations (e.g., in the Triangle Scenario).
Connection to Tensor Networks: The work highlights a novel link between causal inference and tensor network decompositions, suggesting that the rank of a tensor (in the context of the underlying undirected graph) corresponds to the minimal size of latent variables required in a quasiprobabilistic model.

The authors remain modest regarding the general case, noting that while they have proven the conjecture for tree-structured scenarios and provided a proof for the Triangle Scenario (in the Appendix), a full proof for arbitrary DAGs remains an open direction for future work. They suggest that the "Inflation Technique" might be a viable path for a general proof, though the completeness of inflation for quasiprobabilities currently lacks a supporting de Finetti theorem.