⚛️ quantum physics

Analysing causal structures in generalised probabilistic theories

This paper proposes a method using measurement entropy to analyze and derive new constraints that distinguish classical, quantum, and more general theories within various causal structures, while proving that the set of achievable entropy vectors forms a convex cone for any causal structure and generalized probabilistic theory.

Original authors: Mirjam Weilenmann, Roger Colbeck

Published 2026-07-01

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Mirjam Weilenmann, Roger Colbeck

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

Imagine you are a detective trying to solve a mystery: Why do two things happen at the same time?

In the world of physics, this is called finding the "causal structure." Did Person A cause Person B? Did they both happen because of a secret third thing (a hidden cause) that neither of them knew about?

For a long time, detectives only had one set of rules to solve these mysteries: Classical Physics. Think of this like a standard deck of cards. If you shuffle and deal, the outcomes are random, but they follow strict, predictable rules.

Then, Quantum Physics arrived. This is like a deck of cards that can be in two places at once, or where looking at one card instantly changes the value of another card miles away. The old rules didn't work anymore. Scientists needed a new way to explain how these "spooky" connections happen.

But here is the big question: Is Quantum Physics the only weird way the universe can work? Or are there even stranger possibilities?

This paper by Mirjam Weilenmann and Roger Colbeck introduces a new detective tool to answer that. They look at a whole family of possible worlds, ranging from our everyday classical world, to our quantum world, to "super-quantum" worlds that are even weirder than anything we've seen yet.

Here is how they do it, using simple analogies:

1. The "Entropy" Ruler

To measure these different worlds, the authors use a tool called Measurement Entropy.

The Analogy: Imagine you are trying to guess the contents of a sealed box.
- If the box is empty or full of identical items, it's very easy to guess. It has low entropy (low uncertainty).
- If the box is full of random, chaotic items, it's very hard to guess. It has high entropy (high uncertainty).
The Twist: In the quantum world, the "box" behaves differently than in the classical world. The authors developed a special ruler (based on "measurement entropy") that can measure the uncertainty of these boxes, no matter what kind of physics governs them.

2. The "Causal Map"

The authors draw a map of the mystery.

The Nodes: These are the things we can see (like the results of an experiment).
The Arrows: These show who influences whom.
The Hidden Nodes: These are the secret causes we can't see (like the hidden cards in the deck).

In some maps, the hidden causes are classical (standard cards). In others, they are quantum (magic cards). In the most general maps, they could be "Box-World" cards—hypothetical cards that break even more rules than quantum cards do.

3. The "Shape" of Possibility

The paper's biggest discovery is about the shape of the answers.

The Analogy: Imagine you are trying to draw the outline of all the possible shapes a piece of clay can take.
- The authors prove that for any causal structure (any mystery map), the set of all possible answers forms a perfect, smooth shape called a convex cone.
- Think of this like a traffic cone. No matter how you slice it, the shape is consistent. This is a huge deal because it means we can compare different theories (Classical vs. Quantum vs. Box-World) by simply looking at the edges of these cones.

4. What They Found

The authors applied their ruler to several different "mystery maps" to see if they could tell the theories apart.

Case A: The "Same" Map. In some scenarios (like the "Instrumental" scenario), the rules are so strict that Classical, Quantum, and even the weirdest "Box-World" theories all produce the exact same results. You can't tell them apart just by looking at the data.
Case B: The "Different" Map. In other scenarios (like the "Bilocality" scenario, which is used in quantum internet technology), the theories do diverge.
- Classical Physics allows a certain range of outcomes.
- Quantum Physics allows a slightly wider range.
- Box-World (the super-weird theory) allows an even wider range.
- The authors found specific mathematical "inequalities" (rules) that Box-World breaks, which Quantum and Classical physics obey. This proves that if we see a violation of these rules, we know we are dealing with something even stranger than quantum mechanics.

5. The "Information Causality" Test

They also tested a famous principle called Information Causality.

The Rule: If I send you a message with 1 bit of information, you shouldn't be able to learn more than 1 bit of my secret data, no matter what tricks we use.
The Result: This rule holds true for Classical and Quantum physics. However, in "Box-World," this rule breaks. The authors showed that even in this super-weird world, there is still a minimal version of causality that holds true. It's not total chaos; there are still some rules, just different ones.

Summary

Think of this paper as building a universal stress test for the laws of physics.

Before, we could only test if a mystery was "Classical" or "Quantum."
Now, the authors have a method to test if a mystery fits into any possible theory of cause-and-effect.
They proved that the "space of possibilities" for these theories has a very specific, predictable shape (a convex cone).
They showed that while some mysteries look the same to all theories, others reveal that our universe might be more constrained than the most extreme "what-if" scenarios we can imagine.

In short, they gave us a better magnifying glass to see exactly how much "weirdness" is allowed in the universe's causal structure, and they proved that even the weirdest possible worlds still have to follow some basic rules of logic.

Technical Summary: Analysing Causal Structures in Generalised Probabilistic Theories

Problem Statement
Causal structures provide a framework for understanding the origin of observed correlations. While well-developed for classical scenarios, the violation of Bell inequalities and other quantum phenomena necessitate a generalisation of these structures to accommodate non-classical resources. Existing analyses have extensively compared classical and quantum correlations within specific causal structures, often deriving classical constraints and demonstrating their quantum violations. However, there is a limited understanding of the constraints imposed by quantum systems themselves and, more broadly, the behaviour of theories beyond quantum mechanics, known as Generalised Probabilistic Theories (GPTs). The challenge lies in systematically analysing the differences between these theories (classical, quantum, and arbitrary GPTs) to identify minimal requirements for causal explanations and potentially single out quantum mechanics via physical principles.

Methodology
The authors propose a method for deriving constraints on observable correlations in various causal structures based on measurement entropy ( $H^+$ ). This approach extends previous entropic techniques used for classical and quantum causal structures to the broad class of GPTs.

Key methodological components include:

Measurement Entropy: The authors utilize the minimal Shannon entropy of outcome distributions after fine-grained measurements. They define conditional measurement entropy ( $H^+(A|B)$ ) and establish a set of properties (Positivity, Reduction to Shannon entropy, Data processing, Independence, Classical subsystem inequalities, Subadditivity, and relaxations of the chain rule) that hold within GPTs.
Entropy Vector Method: For a given causal structure, the authors construct an entropy vector comprising the entropies and conditional entropies of all coexisting sets of systems. They impose a system of linear inequalities derived from the properties of measurement entropy and the causal independencies (e.g., nodes without common ancestors are independent).
Convexity and Projection: The set of achievable entropy vectors is projected onto the space of observed variables using Fourier-Motzkin elimination to derive marginal inequalities. A central technical insight is the proof that for any causal structure and any GPT, the closure of the set of achievable entropy vectors forms a convex cone.
Post-selection: The method is applied to "post-selected" causal structures, where parentless observed nodes are split into alternatives (e.g., $Y|X=1, \dots, Y|X=n$ ), allowing for the derivation of stronger inequalities that distinguish between theories.

Key Contributions

Generalisation to GPTs: The paper successfully generalises entropic causal analysis to arbitrary GPTs, overcoming the issue that no single entropy measure in GPTs retains all properties of the von Neumann entropy (specifically additivity under composition). The authors resolve this by explicitly including conditional entropy in the analysis.
Convex Cone Proof: The authors prove that the set of achievable entropy vectors for observed variables in any causal structure involving GPT resources is a convex cone. This was previously established only for classical resources. This structural insight facilitates the comparison of entropic sets across different theories and allows for completeness proofs by verifying extremal points.
Refinement of Quantum Analysis: The paper demonstrates that the existing entropic method for quantum causal structures can be refined. By imposing the positivity of conditional entropies (rather than relying solely on weak monotonicity constraints), the authors show that the resulting inequalities for observed variables are valid, simplifying the quantum method.
Proposition 1: The authors prove that for causal structures where no node has two or more unobserved parents, any correlations achievable in an arbitrary GPT are also achievable in "box-world" (the GPT with the largest state space consistent with no-signalling). This implies that for such structures, box-world serves as a universal upper bound for correlations across all GPTs.

Results
The authors apply their method to several causal structures, yielding the following results:

Instrumental Scenario: The set of achievable entropy vectors is identical for classical, quantum, and arbitrary GPT resources. The derived inequalities (Shannon inequalities plus $I(X:YZ) \leq H(Z)$ ) completely characterise the scenario for all these theories.
Structures with Coinciding Approximations: For certain structures (e.g., Figure 2 in the paper), the outer approximations of the entropy cones are identical for classical, quantum, and box-world resources.
Structures with Divergent Approximations: For other structures (e.g., Figures 3a and 3b), the outer approximations differ between classical/quantum and box-world resources. However, the authors note that no specific GPT correlations violating the classical/quantum inequalities have been identified for these cases yet; the gaps may reflect limitations of the outer approximation method rather than true differences in the entropy cones.
Bilocality (with Post-selection): The method successfully distinguishes between classical, quantum, and locally tomographic GPTs.
- Classical sources yield a cone constrained by Shannon inequalities and 53 other classes of linear inequalities.
- Quantum sources satisfy tighter constraints (e.g., $I(X_0Y_0:Z_0) \leq H(Y_0|X_1)$ ).
- Box-world resources (specifically PR-boxes) violate the quantum inequalities, providing a certified separation between quantum and GPT correlations in this scenario.
Information Causality: The authors derive inequalities for the Information Causality scenario in arbitrary GPTs. While the standard Information Causality principle (which holds for classical and quantum theories) is violated in box-world, the authors identify minimal notions of causation (inequalities (23) and (24)) that hold for all GPTs, improving upon previous analyses.

Significance
The paper claims significance in providing a systematic technique for comparing causal structures across different physical theories. By deriving constraints that are minimal requirements for any causal explanation within a GPT, the work aids in the search for physical principles that single out quantum mechanics. The authors emphasize that their method improves upon previous entropic approaches (specifically regarding the Information Causality scenario) and provides a rigorous framework for identifying when correlations achievable in one theory (e.g., box-world) are impossible in another (e.g., classical or quantum). The proof of the convex cone property for GPTs is highlighted as a fundamental structural insight that enables more robust comparisons of entropic sets. The authors remain modest regarding the "gaps" found in outer approximations, acknowledging that these may be artifacts of the method rather than features of the true entropy cones, and note that current techniques cannot distinguish theories in structures like the triangle causal structure without post-selection.