⚛️ quantum physics

Picturing general quantum subsystems

This paper extends the process-theoretic framework for quantum subsystems from factor systems to general finite-dimensional von Neumann algebras by introducing splitting maps, which establish a comprehension preorder and trace that align with standard algebraic notions and prove that the equivalence between semi-causality and semi-localisability holds for all subsystems.

Original authors: Octave Mestoudjian, Matt Wilson, Augustin Vanrietvelde, Pablo Arrighi

Published 2026-02-03

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Octave Mestoudjian, Matt Wilson, Augustin Vanrietvelde, Pablo Arrighi

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 trying to understand a complex machine, like a car. In the old, standard way of doing physics (specifically quantum physics), we usually assume the car is made of distinct, separate parts: an engine, wheels, and a chassis. We can easily describe how the engine works on its own and how it connects to the wheels. This is like looking at a system as a simple tensor product—a neat multiplication of independent parts.

However, real quantum systems are often messier. Sometimes, parts of a system are entangled in ways that don't fit into neat, separate boxes. Maybe the "engine" and the "wheels" are so deeply intertwined that you can't even point to where one ends and the other begins. Or perhaps the system has rules (like conservation laws) that prevent it from being split into simple independent pieces. The standard "box" model breaks down here.

This paper, titled "Picturing general quantum subsystems," proposes a new, more flexible way to draw and understand these messy, non-separable quantum systems. Here is the breakdown of their ideas using everyday analogies:

1. The Problem: The "Box" Doesn't Fit

Traditionally, physicists draw quantum systems as boxes that can be split into smaller boxes (A and B). This works great for simple cases. But in complex scenarios—like when dealing with superpositions of different geometries in quantum gravity or particles with strict conservation rules—this "box" model fails. The system cannot be cleanly divided into independent parts.

2. The Solution: "Splitting Maps" (The Magic Slide)

Instead of forcing the system into a box, the authors introduce a tool called a Splitting Map.

The Analogy: Imagine you have a lump of clay (the whole system). Usually, you might try to cut it perfectly in half to get two separate pieces. But what if the clay is sticky and won't separate cleanly?
The New Tool: A "Splitting Map" is like a special slide or a projector. You don't cut the clay; you slide it through a machine that projects it onto a screen where it looks like it has a left side and a right side.
How it works: The map is an "isometry" (a mathematical way of saying it preserves the shape and information of the original system) that embeds your messy system into a larger, cleaner space where it can be seen as having a left and right part. It's a way of saying, "If we look at this system through this specific lens, we can see a 'left' part and a 'right' part, even if they are deeply entangled."

3. "Comprehension": The Hierarchy of Views

Once you have these splitting maps, the authors ask: "How do these different views relate to each other?"

The Analogy: Imagine looking at a sculpture. You can look at it from a distance (a broad view) or zoom in with a magnifying glass (a detailed view).
The Concept: They define a relationship called Comprehension. If View A can be transformed into View B by adding a little bit of extra "noise" or detail, then View A is "comprehended" by View B.
The Result: This creates a hierarchy. It turns out that this "Comprehension" relationship perfectly matches the mathematical rules for how one quantum system can be "inside" another (inclusion of algebras). It's a way of organizing all possible ways to split a system, from the most general to the most specific.

4. "Balanced" and "Lean" Maps: Finding the Perfect Fit

Not all splitting maps are created equal. Some are messy; some are perfect.

Balanced Maps: These are maps where the "left" side and the "right" side are perfectly symmetrical in their relationship. If you look at the left side, it tells you everything about the right side, and vice versa. The authors prove that these "Balanced" maps are the exact mathematical equivalent of the standard algebraic way physicists describe quantum systems.
Lean Maps: These are the "minimal" or "efficient" versions. They don't have any extra fluff or redundancy.
- Why it matters: The authors show that "Lean" maps are special because they allow you to perform a Trace (a mathematical operation that essentially "sums up" or "ignores" one part of the system to see what's left).
- The Analogy: If you have a photo of a crowded room and you want to know what the person in the center is doing, you might "trace out" (blur out) everyone else. A "Lean" map is like a photo where blurring out the background leaves you with a perfectly clear, undistorted image of the foreground.

5. The Big Discovery: Causality and "No-Signaling"

The ultimate test of a good theory is whether it can explain how information flows (causality).

The Old Rule: In simple, "factor" systems (the neat boxes), physicists knew a rule: If information cannot be sent from System A to System B (non-signaling), then the process can be broken down into local actions (semi-localisability).
The New Proof: The authors used their new "Splitting Map" tools to prove that this rule holds true even for the messy, non-factor systems.
The Analogy: Imagine a secret handshake between two people in a crowded, chaotic room. In the old model, you could only prove they weren't cheating if the room was empty. The authors proved that even in the chaotic, crowded room (the general quantum system), if you can't send a secret message from one person to another, it implies that their actions are still locally independent. They didn't need to change the rules; they just needed a better way to draw the room.

Summary

This paper doesn't just offer a new formula; it offers a new language (a diagrammatic language) to talk about quantum systems that don't fit into simple boxes.

Splitting Maps let us draw messy systems as if they have parts.
Comprehension lets us organize these drawings into a logical hierarchy.
Balanced/Lean Maps ensure our drawings match the rigorous math of quantum mechanics and allow us to perform calculations (like tracing) correctly.
The Result: They proved that the fundamental rules of causality (how cause and effect work) apply to these complex, messy systems just as they do to simple ones.

In short, they built a new set of "diagrammatic tools" that let physicists draw and reason about the most complex, entangled quantum systems without losing their minds.

Technical Summary: Picturing General Quantum Subsystems

Problem Statement
Standard quantum theory typically models subsystems via the tensor product of Hilbert spaces, assuming that the algebra of operators on a subsystem is a factor (a von Neumann algebra with a trivial center). While this framework supports powerful diagrammatic tools in categorical quantum mechanics (e.g., for locality and causality), it fails to capture many physical scenarios. These include systems with superselection rules (e.g., fermion parity conservation), quantum evolutions requiring a mix of tensor products and direct sums, and quantum gravity models involving superpositions of geometries where node neighbors exist in superposition. In these cases, the relevant operator algebras are non-factor von Neumann algebras. Existing approaches to general subsystems (e.g., routed circuits, CP* constructions) lack a unified, intuitive pictorial representation for the fundamental notions of splitting a system, locality, and causality in the non-factor setting.

Methodology
The authors introduce a purely diagrammatic framework based on splitting maps within dagger symmetric monoidal categories to generalize the treatment of subsystems beyond the factor case.

Splitting Maps: Instead of requiring a unitary isomorphism to a tensor product (as in the factor case), a splitting map $\chi$ is defined as an isometry $\chi: H \to H_L^\chi \otimes H_R^\chi$ . This allows for the embedding of a system into a larger space with a standard factor structure, from which non-factor decompositions can be derived.
Locality and Consistency: The authors define $\chi$ -local operators as those acting on the global system that can be represented as acting only on the left branch of the splitting. They introduce $\chi$ -consistency to ensure that the image of the splitting map is invariant under the action of these operators, leading to the definition of strictly $\chi$ -local operators.
Comprehension Preorder: A preorder relation called comprehension ( $\zeta \sqsubseteq \chi$ ) is defined between splitting maps. Intuitively, $\zeta$ is comprehended in $\chi$ if the subsystem extracted by $\zeta$ can be embedded into the subsystem extracted by $\chi$ via auxiliary isometries. This preorder is shown to correspond to the inclusion order of von Neumann algebras.
Trace Adaptation: A $\chi$ -trace is defined as the partial trace over the right branch of the splitting map. The authors investigate conditions under which this diagrammatic trace aligns with the standard partial trace over a von Neumann algebra.
Special Classes of Splitting Maps:
- Balanced splitting maps: Those where the algebra of strictly left-local operators is the commutant of the algebra of strictly right-local operators ( $stloc_L(\chi) = stloc_R(\chi)'$ ).
- Lean splitting maps: A subclass of balanced maps where the shared states in the decomposition are separable, ensuring the $\chi$ -trace exactly recovers the algebraic trace.

Key Contributions and Results

Equivalence of Diagrammatic and Algebraic Subsystems: The paper establishes a rigorous equivalence between the algebraic approach (von Neumann subalgebras) and the diagrammatic approach (splitting maps).
- There exists a preorder-preserving map from splitting maps to von Neumann algebras (via strictly local operators) and a reverse map (via canonical splitting maps derived from the Artin-Wedderburn theorem).
- The authors prove that the class of balanced splitting maps forms the largest class of splitting maps that is equivalent (as a preordered set) to the theory of general algebraic quantum subsystems.
Trace Recovery: It is shown that for lean splitting maps, the diagrammatic $\chi$ -trace is equivalent (up to isometry) to the standard partial trace over the corresponding von Neumann algebra. This validates the diagrammatic tools for information-theoretic operations.
Causal Decomposition Theorem: The authors generalize a known result regarding the equivalence between semi-causality (non-signalling) and semi-localisability.
- In the factor case, it is known that a channel is non-signalling from one subsystem to another if and only if it can be decomposed into local operations.
- Using the splitting map formalism, the authors prove this equivalence holds for arbitrary finite-dimensional quantum subsystems (including non-factor algebras). They demonstrate that a channel is semi-causal if and only if it is semi-localisable between the corresponding lean splitting maps.

Significance and Claims
The paper claims to provide the first "pictorial representation" of the basic notion of splitting a general quantum system into parts, along with the associated notions of locality and causality. By leveraging splitting maps, the authors offer a route to extend reasoning tools from categorical quantum mechanics (which traditionally rely on tensor products) to the non-factor setting.

The significance lies in:

Unification: Providing a diagrammatic language that recovers the algebraic approach to general subsystems, bridging the gap between abstract algebraic definitions and compositional diagrammatic reasoning.
Generalization: Extending fundamental structural theorems about causality (specifically the equivalence of semi-causality and semi-localisability) from factor systems to general von Neumann algebras, which are necessary for describing complex physical scenarios like superpositions of geometries.
Foundation for Future Work: The authors suggest that this framework could serve as a basis for generalizing non-factor subsystems to theories beyond quantum mechanics (e.g., operational or categorical probabilistic theories) and for exploring new ways to decompose dagger monoidal categories into 2-categories.

The authors remain modest regarding the current state of the diagrammatic representation, acknowledging that while the definitions of splitting, locality, and trace are diagrammatically simple, the representation of "balanced" and "lean" maps currently relies on algebraic characterizations. They intend to develop fully equational diagrammatic representations for these classes in future work.