Classical Explanations in (and of) General Probabilistic Theories

This paper introduces a categorical notion of "explanation" between probabilistic models using spans, demonstrating that every locally-finite probabilistic theory admits a canonical, sharp classical representation through a functorial construction.

The Gist

Imagine you are trying to explain a complex, mysterious game to a friend who only understands simple, classic board games like Chess or Checkers.

This paper, written by John Harding and Alex Wilce, is essentially a guide on how to translate the weird, quantum world into the language of classical probability, and then figuring out exactly where that translation breaks down.

Here is the breakdown using simple analogies:

1. The Problem: The "Incompatible Experiments"

In our everyday world (Classical Probability), if you flip a coin and roll a die, you can do both at the same time. You can combine them into one big experiment. Everything fits together neatly.

But in the Quantum World (QM), things are weird. You have experiments that cannot be done together. It's like trying to measure the "color" and the "taste" of a particle at the exact same time; the act of measuring one changes the other. The authors call these "incompatible experiments."

Scientists have tried two main ways to fix this:

• The Logic Approach: Changing the rules of math (like swapping a square for a hexagon).
• The Shape Approach: Looking at all possible states as a weird, multi-sided shape (a convex set) rather than a simple line.

The authors say: "Let's stop fighting the shape. Let's just see if we can explain these weird shapes using our old, familiar classical tools."

2. The Solution: The "Universal Translator" (Borelification)

The authors introduce a magical machine they call Borelification.

Think of a Probabilistic Model (like a quantum system) as a black box with a bunch of buttons (experiments) and lights (outcomes). Sometimes, pressing two buttons at once is impossible.

The authors prove that every such black box can be "explained" by a Classical Model.

• The Analogy: Imagine you have a secret code (the quantum model). The authors show you can build a massive, classical library (a "Borel model") that contains every possible version of that code.
• The Mechanism: They create a "span" (a bridge). On one side is your weird quantum box. On the other is a giant, boring, classical library. In the middle, they build a bridge that connects them.
• The Result: They prove that for any local, finite quantum system, you can build a canonical (standard) classical explanation. It's like saying, "Yes, this quantum thing can be described as a classical thing, provided you are willing to look at a very specific, huge library of possibilities."

3. The Catch: The "Hidden Variables" and Locality

Here is where it gets spicy. Just because you can translate the quantum world into classical terms doesn't mean the translation is "nice."

In the classical library, the "explanation" relies on hidden variables (let's call them "secret ingredients").

• The Analogy: Imagine a magician (Quantum) makes a rabbit appear. The classical explanation says, "Well, the magician is actually using a secret trapdoor (hidden variable) that we can't see."
• The Twist: The authors show that for this translation to work, the "secret ingredients" in the classical library often have to be signaling.
  • Non-Signaling: In a real quantum experiment, if Alice measures her particle, she can't instantly send a message to Bob on the other side of the universe.
  • Signaling: In the classical explanation, the "hidden ingredients" often do send messages instantly.

The Big Discovery:
If you try to explain a composite system (two entangled particles, like Alice and Bob's particles) using this classical translation:

1. The classical explanation works mathematically.
2. BUT, the "hidden ingredients" required to make it work are signaling. They break the rule that "nothing travels faster than light."
3. Therefore, if you demand that your classical explanation is local (no faster-than-light signaling), it fails. You cannot explain entangled particles with a local classical model.

4. The "Entangled" Metaphor

Imagine two dice, Alice's and Bob's.

• Classical World: If they are entangled, it means they were glued together at the factory. If Alice rolls a 6, Bob must roll a 6 because they are physically connected.
• Quantum World: They aren't glued. They are far apart. Yet, when Alice rolls a 6, Bob instantly rolls a 6, even though no signal passed between them.

The authors say: "We can explain this quantum magic using a classical story, but the story requires that the dice are secretly telepathic (signaling). If you forbid telepathy (locality), the story falls apart."

5. The Conclusion: What Does This Mean?

The paper ends with a philosophical "so what?"

They show that Bell's Theorem (the famous proof that quantum mechanics is weird) isn't just about "spooky action at a distance." It's about a conflict between two things:

1. Classicality: The idea that the world is made of definite, pre-existing facts (like the hidden ingredients).
2. Locality: The idea that things can't influence each other instantly across space.

The authors suggest three ways to look at this:

1. The "It's Weird" View: The world is non-classical. The "hidden ingredients" in our classical explanation are fake (unphysical) because they require impossible signaling. The universe just doesn't work like a classical machine.
2. The "It's Local" View: The world is classical (it has hidden ingredients), but it is non-local. The universe allows for that secret telepathy.
3. The "It's Both" View: "Classicality" includes the idea of locality. Since the world isn't local, it isn't classical.

Summary in One Sentence

The authors built a universal translator that can turn any quantum system into a classical story, but they proved that for entangled systems, that story only works if you allow for "magic" (instant signaling), confirming that the quantum world is fundamentally different from our everyday classical intuition.

Technical Summary

Here is a detailed technical summary of the paper "Classical Explanations in (and of) General Probabilistic Theories" by John Harding and Alex Wilce.

1. Problem Statement

The paper addresses the foundational tension between classical probability theory and generalized probabilistic theories (such as Quantum Mechanics).

• The Core Issue: Classical probability assumes that all experiments can be jointly performed (common refinements exist). Quantum Mechanics (QM) and other generalized theories reject this, allowing for incompatible experiments.
• The Challenge: While various theorems exist showing that non-classical models can be represented by classical ones (often via "hidden variables" or "ontological models"), these results are scattered across different frameworks (quantum logic, convex sets, ontological models) and often rely on specific assumptions like contextuality failure or loss of locality.
• The Goal: The authors aim to provide a unified, general framework to define what it means for one probabilistic model to "explain" another. They seek to determine if every generalized probabilistic theory admits a canonical classical representation and to analyze the implications of this representation for locality and entanglement.

2. Methodology and Framework

The authors construct their analysis within the category Prob of concrete probabilistic models.

A. Probabilistic Models (PMs)

• Test Spaces: A model is defined by a "test space" $\mathcal{M}$ (a collection of non-empty sets representing outcomes of possible experiments) and a state space $\Omega$ (a set of probability weights on the outcomes).
• Morphisms: They define morphisms between models that preserve the test structure (events and orthogonality) and map states to sub-normalized states.
• Embeddings and Quotients:
  • An embedding restricts a larger model to a smaller one (identifying states that cannot be distinguished).
  • A quotient maps a model to a coarser one (neglecting distinctions between outcomes).

B. Definition of "Explanation"

The central innovation is defining an explanation of model $A$ by model $B$ as a specific span in the category Prob:
$$C \xrightarrow{q} A \quad \text{and} \quad C \xrightarrow{e} B$$
Where:

• $q: C \to A$ is a quotient morphism.
• $e: C \to B$ is an embedding.
• Interpretation: Model $B$ "explains" $A$ if $A$ can be obtained from $B$ by restricting access to certain tests and identifying indistinguishable states, mediated by an intermediate model $C$.

C. Composition and Pullbacks

• The authors show that while the category Prob does not support arbitrary pullbacks, it does support pullbacks for sub-quotients (diagrams involving an embedding and a quotient).
• This allows explanations to compose associatively (up to isomorphism), forming a bicategory structure.

3. Key Contributions and Results

A. The Borelification Functor (Canonical Classical Explanation)

The paper's primary positive result is the construction of a canonical classical explanation for any locally finite probabilistic model.

• Semiclassical Cover: Every probabilistic model $A$ has a "semiclassical cover" $\tilde{A}$, which is a refinement where outcomes are tagged with the specific test they belong to. This cover is always a quotient of $A$.
• Sharp Classical Embedding: The authors prove that any locally finite, semiclassical model can be embedded into a Borel model (a classical model based on a measurable space).
  • They utilize the set $S$ of dispersion-free (d.f.) states (extreme points of the state space) of the semiclassical cover.
  • They construct a Borel test space $B(S)$ and an embedding $\phi: \tilde{A} \to B(S)$.
• The Functor $\text{Bor}$: This construction defines an endofunctor $\text{Bor}: \text{Prob} \to \text{Prob}$. For any locally finite model $A$, $\text{Bor}(A)$ is a sharp, classical Borel model.
• Result: Every locally finite probabilistic model $A$ admits a canonical, sharp classical explanation via the span:
  $$\tilde{A} \xrightarrow{q} A \quad \text{and} \quad \tilde{A} \xrightarrow{e} \text{Bor}(A)$$
  This generalizes previous results by Beltrametti, Bugajski, and the framework of ontological models.

B. Ontological Representations

The paper clarifies the relationship between their "explanation" framework and ontological models (hidden variable theories).

• A sharp ontological representation corresponds exactly to an embedding of the semiclassical cover into a sharp Borel model.
• The authors show that every probabilistic model arises as the quotient of a model that admits a sharp ontological model.

C. Locality and Entanglement (The Negative Result)

The authors investigate how this classical explanation interacts with composite systems (multipartite systems).

• Bell-Locality Definition: They define a classical explanation (or embedding) as Bell-local if the dispersion-free states of the classical model restrict to product states (non-signalling) on the component systems.
• The Conflict:
  • If a composite system $AB$ admits a Bell-local classical explanation, then all its states must be separable (convex combinations of product states).
  • Conversely, if $AB$ contains entangled states (states that are not separable), it cannot admit a Bell-local classical explanation.
• Signaling in the Classical Model: The canonical classical explanation (Borelification) of a composite system represents non-signalling entangled states as weighted averages of dispersion-free, signaling states. In the classical "explanation," the hidden variables (d.f. states) are themselves non-local (signaling), even though the observed states are not.

4. Significance and Implications

1. Unification: The paper unifies disparate approaches (quantum logic, convex geometry, ontological models) under a single categorical framework. It demonstrates that "classicalization" is a universal property of locally finite probabilistic theories, provided one accepts a specific definition of explanation.
2. Clarification of "Classicality": The results show that the failure of classical explanations in QM is not due to a lack of mathematical representation (since a representation always exists), but due to the failure of locality in that representation.
3. Reframing Bell's Theorem: The authors reframe Bell's Theorem not as a proof that the world is non-classical per se, but as a proof that the world is non-local.
   • If one insists on locality, the world cannot be classical (in the sense of having a local hidden variable model).
   • If one accepts non-locality, the world can be explained classically (via the Borelification functor), but the "hidden variables" in that explanation are necessarily signaling (unphysical in a relativistic sense).
4. Philosophical Stance: The paper concludes that the tension lies between locality and classicality. One must choose:
   • The world is non-classical (observables are incompatible) and potentially local.
   • The world is classical (has an ontological model) but non-local (hidden variables are signaling).
   • Locality is a defining feature of classicality, making the world non-classical because it is non-local.

Summary

Harding and Wilce demonstrate that every locally finite probabilistic theory has a canonical sharp classical representation. However, for composite systems, this representation is Bell-local if and only if the system contains no entangled states. Consequently, the existence of entanglement in quantum mechanics implies that any classical explanation of the theory must be non-local, resolving the "no-go" theorems by showing they are constraints on locality, not on the existence of a classical representation itself.