Bounding Classical and Quantum Correlations in Bayesian Networks with Quasiprobabilities

The gist The authors formalize the set of quasidistributions consistent with a graph, generalizing results from Bell scenarios to show that this quasi set coincides with the nested Markov model for tree-structured correlation scenarios.

Directed Graphical Models and Classical Sets

Directed graphical models are a tool used frequently in all fields related to statistical and causal modeling, such as medicine [1], psychology [2], economics [3], and machine learning [4] (Page 1). In directed acyclic graph (DAG) modeling, each vertex v of a graph G is associated with a random variable Xv from a sample space Xv (Page 2). A distribution consistent with a latent variable DAG must satisfy some independence constraints, but these are not sufficient. In the simplest setting where all variables are observed, consistency is given by the three equivalent Markov properties: the factorisation criterion, the local Markov property, and the global Markov property (Page 2). For a given graph G, characterising this marginal model, which we will call the classical set and denote C(G), turns out to be much more difficult and has been the focus of extensive study (Page 2).

Quantum Correlations and Quasiprobabilities

Quantum theory has developed tremendously in the last century, scientists now generally believing it to be the correct theory to describe Nature, as recognised by the Nobel Prize in Physics in 2022 (Page 3). Bell inequalities [21] play a crucial role in understanding nonlocality, asserting that quantum correlations cannot be explained via hidden variables (Page 3). Researchers noticed graphical models are useful in the study of nonlocality (Page 3). The idea to use graphical reasoning dates back to papers such as [30] and [31] (Page 2). It is known in any Bell Scenario (i.e. one latent variable and any number of observed variables with separate inputs) that if the latent variable is allowed to follow a quasiprobability distribution (a distribution with possibly negative probabilities), all nonsignaling correlations can be achieved [50, 51] (Page 3).

The Quasimarginal Set Definition

The paper defines the quasimarginal model or the quasi set as the set of observed distributions which are marginals of a full quasidistribution that factorises with respect to the graph (Page 7). This is defined by three equivalent definitions:

1. Definition 1 (Distributions with Quasilatents and Quasiresponses): A distribution p has quasilatents if p can be written as p(xV ∪L) = Y v∈V ∪L p(xvxpaG(v)) for some nonnegative distributions p(xvxpaG(v)) for all v ∈ V and quasidistributions p(xlxpaG(l)) for all l ∈ L (Page 8).

2. Definition 2 (Distributions with Quasinoise): A distribution p has quasinoise if there exist random variables Nv for each v ∈ V (independent of everything in the graph except Xv), called noise variables, such that p can be written as p(xV ∪L) = X nv Y v∈V p(xvxpaG(v), nv) Y l∈L p(xlxpaG(l)) Y v∈V p(nv) for some deterministic distributions p(xvxpaG(v), nv), nonnegative distributions p(xlxpaG(l)) and quasidistributions p(nv) (Page 8).

3. Definition 3 (The Three Quasimarginal Sets): The sets Wl(G), Wr(G), and Wn(G) are defined based on these types of full quasidistributions, where the observed distributions have entries in [0, 1] (Page 9).

Equivalence and Nested Markov Model

Theorem 4 states that for any DAG G, Wl(G) = Wr(G) = Wn(G) (Page 9). This means the quasimarginal set is defined by any of these three types of full quasidistributions (Page 9). The authors conjecture that this quasi set coincides with the nested Markov model for any DAG, stating Conjecture 1: For any DAG G, W(G) = N (G) (Page 9).

Tree-Structured Correlation Scenarios

The paper focuses on correlation scenarios where observed variables have no children and latent variables have exactly two children (Page 10). For a tree-structured correlation scenario G, the Markov model and the nested Markov model have particularly simple forms, with N (G) = M(G) (Page 10). The main result in this context is Theorem 14: For any Tree-structured Correlation Scenario G, a distribution is in the quasimarginal model if and only if it satisfies the observed independence constraints enforced by the graph, meaning W(G) = N (G) = M(G) (Page 9).

Tensor Network Decompositions and Proof for TCS

The proof for Theorem 14 relies on tensor network decompositions. The paper shows that a distribution in the ordinary Markov model p ∈ M(G) can be written as a tensor decomposition with respect to the undirected graph Gt (Page 13). By using Lemma 34, which states that if PxV T(xV)!= 0, then there exists a decomposition where Xxv T(v) is non-zero for any adjacent indices (Page 14). The final step involves showing that the columns of M(B) must be linearly independent, leading to the conclusion that W(G) = M(G) = N (G) (Page 14).

Conclusion and Future Work

The work formalizes the set of quasidistributions consistent with a graph, generalizing results about the Bipartite Bell Scenario (Page 5). The paper proves that the quasi set is the same as the nested Markov model for any correlation scenario with a tree structure (Page 9). A proof of Conjecture 1 would provide an affirmative answer to Proposition 3.4 of [33] regarding GPTs and nested Markov model correlations (Page 10). Future work suggests studying Conjecture 1 in general DAGs, perhaps using the Inflation Technique (Page 5). The paper concludes by showing that for tree-structured correlation scenarios, the quasi set coincides with the nested Markov model (Page 9).

References

The paper cites numerous foundational works on causality [6], Bell inequalities [21], and graphical models [7] (Page 1). Key references include those by Pearl [6], Richardson et al. [58], and others concerning quantum correlations like those by Greenberger, Horne, Shimony, and Zeilinger [27]. The paper also references tensor network decompositions from Robeva and Seigal [61] (Page 40).

1. Generalization of correlation bounds for Bayesian networks: The system can now handle correlations from both classical and quantum theories by using quasiprobabilities that respect normalisation but not positivity.

2. Characterization of non-signaling distributions: The AI system can determine if a given distribution is in the quasi set, which is equivalent to the nested Markov model for any graph, by checking if it satisfies the factorisation criterion with quasiprobabilistic factors.

3. Simulation of all nonsignaling correlations: The system can simulate all nonsignaling correlations in a Bell scenario by allowing latent variables or response functions to be negative, as shown in any non-signaling distribution in the Bell Scenario (including non-classical and non-quantum ones) can be written in the form described in (1).

4. Discovery of causal structures from distributions: The system can answer Given a distribution, what are the causal structures it is consistent with? by identifying which directed acyclic graphs are consistent with the distribution, even when latent variables are present.

5. Analysis of graph structure using tensor network decompositions: The AI can use tensor network decompositions to analyze directed graphical models and understand their correlation scenarios by relating them to undirected graphs and bond dimensions.

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.