Decoupling local classicality from classical explainability: A noncontextual model for bilocal classical theory and a locally-classical but contextual theory

This paper constructs an ontological model for bilocal classical theory to demonstrate that local classicality does not guarantee classical explainability, thereby refuting a prior conjecture, exposing the limitations of local tomography assumptions in structural theorems, and proving via counterexample that the two concepts are fundamentally distinct.

The Gist

Imagine you are trying to understand the rules of a game. In physics, we usually have two main ways of looking at the world: the "Classical" way (like a standard deck of cards where every card has a definite face) and the "Quantum" way (where cards can be in a superposition of faces until you look).

This paper is about a specific, weird game called Bilocal Classical Theory (BCT). The authors are asking a deep question: Can we explain this weird game using simple, everyday logic (classicality), even though the game's rules for combining players seem to break the usual laws of physics?

Here is the breakdown of their discovery, using simple analogies.

1. The Two Ways to Be "Classical"

The paper distinguishes between two types of "classicalness":

• Locally Classical: Imagine every single player in the game is a normal, predictable person. If you look at them one by one, they make perfect sense.
• Classically Explainable: Imagine you can explain the entire game, including how players interact and combine, using a simple, logical story (an "ontological model") that fits our everyday understanding of cause and effect.

Usually, physicists thought that if a game looked "weird" when players combined (specifically, if it violated a rule called Local Tomography), it would be impossible to explain it with a simple story. Local Tomography is like saying: "To know the state of a team, you just need to check every individual player." If a game violates this, it means the team has a "secret handshake" or a hidden connection that you can't see just by looking at the individuals.

2. The First Discovery: The "Bilocal" Game Is Explainable

The authors looked at Bilocal Classical Theory (BCT).

• The Setup: In BCT, every individual player is perfectly normal (locally classical). However, when two players join forces, the rules for their combination are strange. It's like if two normal people holding hands suddenly created a third, invisible person that only exists because they are holding hands. You can't figure out what the pair is doing just by looking at Person A and Person B separately; you need to look at the pair together.
• The Old Belief: A previous theory suggested that because BCT has this "invisible third person" (violating Local Tomography), it was impossible to create a simple, logical story to explain it. It was thought to be too weird for a classical explanation.
• The New Result: The authors built a map (an ontological model) that translates the weird BCT game into a standard, boring classical game.
  • The Analogy: Imagine BCT is a complex magic trick. The authors found a way to show that the trick is actually just a standard card shuffle, but performed on a table that is twice as big as you thought. They showed that even though the "team" behavior looks mysterious, you can explain it perfectly by adding a few extra "hidden slots" (ontic states) to your mental model of the players.
  • The Takeaway: You can have a game that looks weird when players combine, but it can still be explained by simple, local logic. The "weirdness" isn't a fundamental mystery; it's just a matter of looking at the right level of detail.

3. The Second Discovery: Not All Weird Games Are Explainable

If the first result was "Yes, this weird game is explainable," the authors asked: "Is every weird game explainable?"

• The Counter-Example: They constructed a different type of game called Latent Classical Theories (LCT).
• The Setup: Like BCT, every individual player is normal. But the rules for how they combine are even more twisted. In this game, when two players combine, they sometimes create a "ghost" connection that is so strong it can cancel out any normal interaction between them.
• The Result: The authors proved that for these specific games, no simple story exists. You cannot build a map that translates this game into a standard classical one.
  • The Analogy: Imagine a game where two normal people shake hands, and suddenly they become a single entity that can "erase" any other person in the room. No matter how you try to explain this using standard logic (like "they are just holding hands"), the math breaks down. The "ghost" connection is too fundamental to be explained away by adding hidden slots.
  • The Takeaway: Just because a game looks classical when you look at the players individually doesn't mean the whole game can be explained classically. Sometimes, the way things combine creates a "weirdness" that is truly unexplainable by simple logic.

4. The Big Picture

The paper draws a line in the sand:

• Old View: If a theory fails the "Local Tomography" test (meaning you can't understand the whole by just looking at the parts), it must be fundamentally non-classical and unexplainable.
• New View: That is false.
  • Some theories (like BCT) fail the test but are explainable.
  • Some theories (like LCT) fail the test and are not explainable.

The Conclusion:
The authors show that there is no simple, straight-line relationship between "looking classical on the inside" and "being explainable by a simple story." The way systems combine (composition) is a crucial, independent factor. You can't just assume that because the parts are simple, the whole must be simple—or that if the whole is weird, it's unexplainable. You have to look at the specific rules of the combination to know for sure.

In short: Being "locally classical" is not enough to guarantee that a theory is "classically explainable." The devil is in the details of how things come together.

Technical Summary

Technical Summary: Decoupling Local Classicality from Classical Explainability

Problem Statement
The paper addresses the relationship between two distinct notions of "classicality" within the framework of Operational Probabilistic Theories (OPTs): local classicality and classical explainability.

• Local Classicality: A theory is locally classical if every individual system within it is classical (i.e., its state space is a simplex and it satisfies the no-restriction hypothesis), even if the rule for composing these systems differs from standard classical theory.
• Classical Explainability: A theory is classically explainable if it admits an ontological model. This is defined as a linear, diagram-preserving, and determinacy-preserving map from the theory to standard Classical Theory (CT). This concept is equivalent to the theory being generalized noncontextual.

A central question arises: Does every locally classical theory admit an ontological model? Previous work on Bilocal Classical Theory (BCT) [1] suggested that because BCT violates local tomography (the principle that composite states are fully determined by local measurements), it might not admit a local-realist ontological model. Conversely, other constructions of locally classical theories might inherently fail to be classically explainable. The paper aims to decouple these concepts by constructing explicit models and counterexamples.

Methodology
The authors employ the formalism of OPTs and the framework of ontological models as defined in Refs. [2, 6]. The methodology involves two primary constructions:

1. Construction of an Ontological Model for BCT:

   • The authors define a linear map $\tilde{\xi}$ from BCT to standard Classical Theory ($\Delta$).
   • System Mapping: A non-trivial system $n$ in BCT is mapped to a classical system of dimension $2n$ (specifically, a tensor product of an $n$-dimensional space and a binary space).
   • State/Effect Mapping: Pure states and effects in BCT, which involve binary variables ($s \in \{0,1\}$) alongside standard indices, are mapped to classical states/effects using "copy" and "controlled-NOT" (CNOT) gates on the binary subspaces.
   • Transformation Mapping: Atomic transformations in BCT are mapped to substochastic matrices in classical theory, preserving the specific structure of the BCT composition rule (which involves a "twisting" of indices via the binary variable).
   • Consistency Checks: The authors rigorously verify that this map satisfies the required properties: linearity, diagram preservation (preserving sequential and parallel composition, identity, and swap), and determinacy preservation (mapping deterministic transformations to deterministic ones).

2. Construction of a Counterexample (Latent Classical Theories - LCTs):

   • The authors utilize the "latent theory" construction from Ref. [76], adapted to classical systems.
   • Definition: LCTs are defined by a composition rule where the composite of two systems $C_1$ and $C_2$ includes an extra "latent" factor space $L_{12}$ (i.e., $C_1 \otimes C_2 = L_{12} \otimes C_1 \otimes C_2$).
   • Condition: The latent state $|\kappa\rangle_{L_{12}}$ is chosen to be non-full-rank (not spanning the entire latent space).
   • Proof of Non-Existence: The authors prove that if the latent state is not full-rank, there exists a non-null bipartite effect that annihilates every product state. They demonstrate that no ontological model can exist for such a theory because the diagram-preserving property of the map would require this effect to map to a null effect in classical theory, contradicting the probability preservation of the non-null effect on a specific composite state.

Key Contributions and Results

1. Existence of an Ontological Model for BCT (Theorem 2.3):
   The paper constructs the first ontological model for an entire theory that fails to be locally tomographic. This result refutes the conjecture in Ref. [1] that BCT might lack a local-realist ontological model.

   • Implication: The failure of local tomography does not necessarily preclude classical explainability. The non-locally tomographic degrees of freedom in BCT can be accounted for by adding extra ontic states locally.

2. Non-Existence of Ontological Models for LCTs (Theorem 3.2):
   The paper proves that Latent Classical Theories with non-full-rank latent states do not admit any ontological model.

   • Implication: Not every locally classical theory is classically explainable. There is a fundamental tension between certain compositional rules (specifically those allowing strong violations of local tomography) and the existence of a noncontextual hidden-variable model.

3. Decoupling of Concepts:
   The results demonstrate that local classicality and classical explainability are independent properties. A theory can be locally classical but not classically explainable (LCTs), or locally classical and classically explainable despite failing local tomography (BCT).

Significance and Claims
The paper claims that these findings have significant implications for the foundational understanding of classicality:

• Limitation of Structure Theorems: The result shows that the assumption of local tomography, which underpins the structure theorem for generalized noncontextual models in Ref. [2], is a genuine limitation of that theorem. The theorem does not hold for theories that violate local tomography, even if they are locally classical.
• Compositional Properties are Central: The authors argue that the way systems compose (the composition rule) is not a technical side-issue but a central, unavoidable question for understanding classicality. The distinction between theories that are locally classical but globally non-classical (in the sense of explainability) highlights the importance of compositional structure.
• Clarification of Classicality: The paper provides a concrete example (BCT) where a theory is more simply understood via its underlying ontological model than via its original operational formulation, similar to how Spekkens' toy theory clarifies odd-dimensional stabilizer subtheories.
• No Straightforward Relationship: The findings establish that there is no straightforward relationship between a theory being locally classical and it being classically explainable. Future work is suggested to find physical principles that distinguish which locally classical theories admit ontological models.

The paper concludes that the status of compositional properties is fundamental to a coherent understanding of classicality itself, moving beyond the traditional view that local tomography is a necessary condition for classical explainability.