Beyond Bell's Theorem II: Scenarios with arbitrary causal structure

1. Introduction

• The framework defines classical and quantum correlations for scenarios including Bell, many-party, bilocality, and other causal structures.It also shows that its definitions recover the usual classical and quantum notions for Bell scenarios.Source: §1 · p. 2
• Sources, choices of measurement setting, and measurements are all instances of the same concept: an event.Events are modeled as spacetime-located measurement apparatuses that probe and potentially modify the outside world and output classical outcomes.Source: §1 · pp. 2–3
• The framework also contributes to causal inference by treating unobserved information as potentially propagating through spacetime events without being revealed by probing.The paper argues that hidden variables should be considered behind every event relevant to the scenario, not only behind particle sources.Source: §1 · pp. 3–4

• Some arguments are presented only as sketches because the authors did not achieve perfect rigor in all proofs.Source: Summary and statement of results · p. 4
• The triangle scenario [10] can be realized with lightcone-induced causal structure by placing its six events at regular-hexagon vertices and scheduling x, y, and z earlier than a, b, and c.Source: Figure 4 · p. 5
• The proposed correlation conditions require variables with disjoint causal pasts to be independent, reducing to no-signalling in Bell scenarios.They express Reichenbach’s common-cause principle as a minimal requirement for distributions arising from a causal structure.Source: Summary and statement of results · pp. 4–5
• Classical correlations are modeled by hidden variables propagated along causal links and processed at nodes, which may take values in arbitrary measurable spaces.The paper conjectures, with a partial proof, that processing at nodes with incoming hidden variables can be assumed deterministic.Source: Summary and statement of results · pp. 5–6
• The authors describe the two-party sequential correlation scenario as involving sequences of two measurements.Source: Figure 5 · p. 6
• The set of correlations depends only on the causal structure’s induced partial order, not the full directed graph.This applies to the general C-correlation construction.Source: Summary and statement of results · pp. 4, 6
• The quantum-correlation definition agrees with the conventional one in Bell scenarios; the claim that every classical correlation is quantum has only a partial proof.The inclusion is stated as Conjecture 6.4.Source: Summary and statement of results · p. 6
• Classical correlations are exactly the distributions modelable by hidden Bayesian networks.The paper also characterizes Bell-scenario classical correlations as those whose associated conditional distributions are local, and shows arbitrary classical correlations can be approximated by ones with finite-valued hidden variables.Source: Summary and statement of results · pp. 5–6

• The graphs considered are directed acyclic graphs, with nodes and events treated as essentially synonymous.Source: Terminology and Notation · p. 6
• Edges denote causal links along which physical systems propagate, except for edges in hidden Bayesian networks, which are not interpreted this way.Source: Terminology and Notation · p. 7
• pst(v) denotes the causal past of v: the nodes that can reach v along a directed path.Source: Terminology and Notation · p. 7
• For a node v, o[v] denotes its outcome variable, distinguishing the node from the variable it carries.The paper uses P(o[v]) for that variable’s distribution.Source: Terminology and Notation · p. 7

• The definitions are provisional: other approaches may be useful, and the paper does not provide a way to handle indefinite causal structure.Source: Disclaimer · p. 7
• The approach assumes independent repeated trials, which may be difficult to justify in applications such as neurophysiology where neurons could retain hidden information between trials.The authors do not know when this assumption can be relaxed.Source: Disclaimer · p. 7

2. What is causal structure?

• A causal structure is a finite directed graph whose nodes are events and whose edges represent possible influence; relaxing finiteness may add technical challenges.Source: §2 · p. 8
• Indirect influence can pass along directed paths, and an intermediate node’s observed outcome need not screen off correlations because it only probes the environment.Source: §2 · pp. 8–9
• The causal-past relation is transitive, and each node is included in its own causal past.Source: §2 · p. 9
• A directed acyclic graph excludes causal loops, so no event can influence itself through a nontrivial path.Source: §2 · p. 9
• In Bell’s graph, source, setting, and measurement nodes can all be treated as observations whose outcomes process incoming hidden variables and emit them along outgoing links.The hidden variables may be classical, quantum, or other kinds of systems.Source: §2 · p. 9
• The framework can test trial causal structures beyond spacetime, but the paper does not solve the general compatibility problem or prescribe how to find candidate structures.Source: §2 · pp. 8–9

3. Correlations

• A correlation is the joint probability distribution over the finite-valued outcomes observed at all nodes.Source: §3 · p. 10
• For any collection of outcome subsets with pairwise disjoint causal pasts, their joint distribution factors into the product of their marginal distributions.Source: §3 · p. 10
• To derive concrete constraints, it is enough to impose factorization for pairs of disjoint-past sets; maximal pairs suffice to characterize a correlation.Source: §3 · pp. 10–11
• In an n-party Bell scenario, P is a correlation if and only if it satisfies the free-will equation and the no-signalling equation for every party.Source: §3 · p. 11
• The formalism includes distributions for settings and source outcomes, which the paper notes can matter in practical information-processing protocols.Source: §3 · p. 11
• The set of correlations on a graph depends only on the partial order of causal influence that the graph induces.Source: §3 · p. 12

4. Classical correlations

• Each node can transform incoming edge hidden variables into an outcome and outgoing variables by an arbitrary conditional distribution, allowing hidden information and invasive observations.The authors note that this edge-based formalism does not assume noninvasiveness and therefore does not comprise Leggett-Garg scenarios [26].Source: §4 · p. 13
• A classical correlation is generated by edge hidden variables and node-level conditional distributions, and every such distribution is a correlation.The model allows hidden variables on general probability spaces, including infinite ones.Source: §4 · pp. 13–14
• In Bell scenarios, classical correlations correspond to local conditional distributions satisfying the free will equation and a factorization with one shared hidden variable.The section states that these conditional distributions form the local polytope.Source: §4 · pp. 15–16
• For general causal structures, deciding whether a correlation is classical is difficult, and classical correlations are nonconvex even in Bell scenarios when represented as unconditional distributions.For the triangle scenario with deterministic first-layer outcomes, the cited discussion describes entropic inequalities as the only known generally applicable conditions.Source: §4 · p. 16
• For every ε > 0, any classical correlation on a finite causal structure can be approximated pointwise within ε using finite hidden-variable spaces.Whether every classical correlation has an exact finite-space representation remains open.Source: §4 · p. 17
• The authors conjecture that information processing can be deterministic at every node with parents, but prove this only for models with finite hidden-variable spaces.The conjecture would leave randomness only at nodes without parents; the proposed mechanism is to push randomness back to parent nodes.Source: §4 · p. 19
• In the induction step, incoming hidden variables are treated as outcomes at predecessor nodes, yielding a classical correlation on the graph with w removed.The induction hypothesis gives deterministic processing at non-root nodes, which is combined with deterministic processing P′ at w.Source: §4 · p. 21
• Every classical correlation can be arbitrarily well approximated by one whose conditional distributions P(o[v]λout(v)|λin(v)) are deterministic.The passage attributes this result to Theorem 4.4.Source: §4 · p. 21
• The authors suspect Conjecture 4.5 holds without approximation, but have no proof; a later theorem might help complete one.They note that extending their partial proof may require the category of measurable spaces to be cartesian closed, which appears to be an open problem.Source: §4 · p. 21

5. Correlations within any theoretical framework

• A framework for C-correlations is a symmetric monoidal category with a distinguished normalized subcategory, enabling edge systems, node operations, and scalar-valued graph evaluations.The construction also requires morphism sets to support nonnegative scaling and addition, and a unique normalized discard operation for each system.Source: §5 · pp. 22–23
• Evaluating a graph whose edges carry objects and whose nodes carry C-instruments assigns joint-outcome probabilities that form a correlation on the graph.Each instrument’s outcome-indexed operations sum to a normalized operation; the graph evaluation is a nonnegative real number.Source: §5 · pp. 24–25
• Marginalizing outcomes yields normalized distributions, and disjoint subgraphs satisfy the binary correlation equation.The proof establishes normalization by evaluating the empty graph and obtains the binary equation from disjoint-union composition.Source: §5 · p. 27
• The set of C-correlations on a graph is independent of the chosen orderings of incoming and outgoing edges at each node.The proof handles adjacent edge swaps by composing the node instrument with the category’s symmetry morphism.Source: §5 · pp. 27–28
• The set of C-correlations depends only on the partial order induced by the graph, not on whether causal links are direct or indirect.An added transitive edge can be represented by relaying its information through the intermediate node.Source: §5 · pp. 28–29

• StOp-correlations coincide exactly with classical correlations from Definition 4.1.The integrals in (4.2) arise from composing stochastic operations.Source: §5 · p. 29
• Certain scenarios have correlations that are not general probabilistic correlations, although a suitable category for general probabilistic operations remains undeveloped.This contrasts with Bell scenarios, where correlations are no-signaling boxes and every such box arises from box world.Source: §5 · p. 29
• A linear symmetric monoidal functor F : C →D with F(C1) ⊆D1 guarantees that every C-correlation is also a D-correlation.Source: §5 · p. 29

6. Quantum correlations

• Quantum correlations are modeled using Hilbert spaces on edges and quantum operations at nodes, with systems allowed to have finite or infinite dimension.The authors identify these with distributions obtainable by interpreting the graph in an operator-tensor formulation, while noting that formulation has only been worked out for finite-dimensional systems.Source: §6 · p. 31
• A quantum correlation is a correlation in the paper’s general framework.Source: §6 · pp. 31–32
• A square-root assignment of stochastic operations is monoidal but nonlinear, while the linear decoherence assignment fails to preserve identities.These are the two obstacles described for constructing a suitable functor from classical to quantum operations.Source: §6 · p. 32
• The paper states that every classical correlation is quantum, but acknowledges that it lacks a general proof; it gives a proof sketch for finite hidden-variable spaces.The finite-space construction uses decoherence in the canonical basis to recover the original probabilities.Source: §6 · pp. 32–33
• In Bell scenarios, the general framework translates back to the usual definitions of quantum correlations.The paper compares its definition with the conventional Bell-scenario formulation and explains how to translate between them.Source: §6 · pp. 33, 35
• Further study of quantum correlations on arbitrary causal structures, including finding nonclassical examples, is left to future work.The authors also call for checking that the general definition covers other previously studied scenarios with precise definitions.Source: §6 · p. 35

7. Classical correlations as hidden Bayesian networks

• In the edge-emitting hidden Markov model, hidden variables form a Markov chain, while observed outputs need not have that Markov property.Every stochastic process admits a classical-correlation representation, though not necessarily one with finite hidden-variable spaces.Source: §7 · pp. 36–37
• The paper uses “hidden Markov model” without requiring finite hidden spaces or time-independent stochastic maps.Source: §7 · p. 36
• On a finite directed acyclic graph, a correlation is classical if and only if it has a hidden Bayesian network representation.The representation uses a Bayesian network over hidden variables and node-wise stochastic maps producing observed outcomes.Source: §7 · pp. 37–38
• The hidden Bayesian network generalizes the state-emitting hidden Markov model to arbitrary finite directed acyclic graphs.The section presents the hidden Bayesian network for Popescu’s hidden-nonlocality scenario as an example.Source: §7 · p. 37, Figure 10 · p. 37
• Whether a corresponding equivalence holds for quantum correlations using quantum Bayesian networks is left as an open question.Source: §7 · p. 39

Appendix A. Measure-theoretic technicalities

• The appendix recalls probability structures that allow classical hidden-variable value spaces to be finite or infinite.It frames this material as the classical analogue of concepts familiar from quantum theory.Source: Measurable spaces and measures · p. 40

A.1. Definition.

• A measurable space pairs a set with a σ-algebra of subsets, and the paper assumes that this σ-algebra separates points.Source: §A.1 · p. 40

A.2. Definition.

• A measure assigns nonnegative values to measurable sets and is countably additive; a probability measure is normalized to one.The paper also allows measures whose total mass differs from one.Source: §A.2 · p. 40

A.3. Definition.

• Measurable functions provide the functions that can be integrated against a measure to form expectation values.The appendix defines measurability through inverse images of real intervals.Source: §A.3 · pp. 40–41

Stochastic operations.

• A stochastic operation maps each input to a measure over outputs, with measurable dependence on the input.Its normalization may depend on the input; normalized stochastic operations are called conditional distributions.Source: §A.4 · p. 41

A.4. Definition.

• Stochastic operations compose by integrating the later operation against the measure produced by the earlier one.For finite spaces with normalized operations, the resulting category is FinStoch.Source: §A.4 · pp. 41–42

Products of measure spaces and stochastic operations.

• Product stochastic operations describe parallel execution by assigning product measures to pairs of inputs.This product makes StOp a symmetric monoidal category.Source: Products of measure spaces and… · p. 42

Appendix B. Finite approximations of probability spaces

• The appendix’s technical results are described as crucial to the proof of Theorem 4.4.It refers to for additional measure-theory terminology and background.Source: Appendix B · p. 43
• The appendix includes a passage concerning the symmetric difference.Source: Appendix B · p. 43

B.1. Lemma.

• For every δ > 0, given a probability space (X, Σ, P), an algebra Γ ⊆ Σ, and a measurable function f: X → S for finite S, there exist sets B_s ∈ Γ that partition X.Source: §B.1 · p. 43
• The key requirement is that the approximating sets B_s belong to Γ, rather than merely to the larger σ-algebra Σ.Measurability already places each f^-1(s) in Σ; the lemma approximates these fibers using Γ.Source: §B.1 · p. 43
• The proof proceeds by induction on |S|, selecting one fiber for approximation and applying induction on its complement.It adjusts the restricted function on a set of measure less than δ/2 before applying the induction hypothesis with δ/2.Source: §B.1 · p. 43

B.2. Proposition.

• The finite coarse-grainings and a function g reproduce f with probability greater than 1 − ε.Each factor X_k is mapped measurably to a finite set S_k, and g maps the product of those finite sets to S.Source: §B.2 · p. 43
• The construction approximates each f^-1(s) by a partition whose members are finite unions of products of measurable sets.Those sets generate finite Boolean algebras whose atoms define the finite sets S_k.Source: §B.2 · p. 44
• The map g assigns each tuple of atoms to the unique s whose partition member contains their product.The proof chooses δ ≤ ε/|S| to obtain the stated agreement bound.Source: §B.2 · p. 44