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Partitions in quantum theory

This paper argues for extending quantum decompositional theories to include non-factor subsystems by defining general partitions of C*-algebras, demonstrating their representation via routed quantum circuits, and highlighting how certain partitions, such as those of fermionic systems, inherently retain a form of pseudo-nonlocality.

Original authors: Augustin Vanrietvelde, Octave Mestoudjian, Pablo Arrighi

Published 2026-10-30

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CC BY 4.0

Original authors: Augustin Vanrietvelde, Octave Mestoudjian, 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

In the standard way physicists think about the universe, the whole is built from parts. We imagine a system as a collection of smaller, independent pieces that we can put together, like bricks in a wall or notes in a chord. This bottom-up view has served science well for over a century, allowing researchers to construct complex models from simple, fundamental units. However, there is another way to look at the same reality: starting with the whole and asking how it can be broken apart. This top-down perspective is not just a matter of semantics; it is essential for understanding how different ways of dividing a system relate to one another. It allows scientists to ask whether a specific division into two parts is compatible with a division into three, or how information flows when the boundaries between parts shift. For decades, this approach relied on a strict assumption: that every piece of a system must be a "factor," a mathematical term meaning the piece is completely independent and free of hidden internal constraints. But this assumption, while convenient, turns out to be a mistake when dealing with real-world quantum systems that are governed by conservation laws or symmetries.

A new study by Augustin Vanrietvelde, Octave Mestoudjian, and Pablo Arrighi challenges this long-held view by arguing that the most natural way to describe quantum systems often involves parts that are not independent factors. These "non-factor" subsystems arise naturally when a system is subject to rules that prevent certain changes, such as the conservation of particle number. In these cases, the system cannot be cleanly split into completely separate pieces; instead, the pieces share a subtle, classical-like connection that restricts what can be done locally. The researchers have developed a rigorous mathematical framework to describe how to divide a quantum system into any number of these constrained parts, moving beyond the simple two-part splits that previous theories could handle. Their work reveals that while these complex divisions can be described algebraically with perfect clarity, they often resist being translated into the standard language of quantum mechanics, which relies on splitting a single space into independent dimensions.

The core of the researchers' discovery lies in redefining what it means to partition a system. In the past, if a scientist wanted to divide a quantum system into two parts, they would look for two sets of operations that do not interfere with each other. If the system was a "factor," this was straightforward. But when the system has constraints, the two parts are not completely independent; they share a common "center," a set of operations that both parts can perform. The authors show that for a division to make sense, the classical information held by the whole system must be accessible through the combination of the local information held by the parts. They prove that this condition, combined with a careful definition of how the parts relate to each other within their shared constraints, creates a consistent theory for dividing systems into three, four, or even more parts. This is a significant step forward because previous attempts to generalize the concept of a partition to more than two parts often failed or produced contradictions when non-factor systems were involved.

One of the most striking findings of the study is that this new theory exposes a fundamental limitation in how we visualize quantum systems. The researchers demonstrate that while they can define these complex partitions perfectly using algebra, they cannot always represent them using the standard tool of Hilbert spaces, which are the geometric spaces where quantum states usually live. In many cases, they can find a representation where each individual part acts locally on its own piece of space. However, when they try to describe the combined action of two or more parts together, the mathematics forces them to introduce a strange, residual "pseudo-nonlocality." This means that even though the parts are defined as separate, their combined behavior cannot be fully captured by simply putting their individual spaces side by side; there is a hidden connection that refuses to disappear.

To illustrate this, the authors examine a specific physical scenario involving a system of fermions, such as electrons, divided into local modes. In this setup, the rules of physics dictate that the number of particles must remain even or odd in specific ways, creating a non-factor system. They prove that for a system with three or more modes, it is impossible to find a representation where the physical operations on any group of modes act strictly locally on a corresponding set of dimensions. This is not a flaw in their theory, but rather a feature of nature itself. It shows that the familiar picture of quantum mechanics, where everything can be neatly sliced into independent spatial dimensions, is sometimes an illusion. The algebraic structure of the system is perfectly well-behaved and logical, but the attempt to force it into a geometric mold of independent parts fails.

The implications of this work extend far beyond abstract mathematics. The ability to handle non-factor partitions is crucial for understanding several active areas of research, including quantum causal modeling, where the order of events can be indefinite, and the study of quantum reference frames, which deals with how different observers define the boundaries of their systems. The authors note that their framework has already been used to prove that certain one-dimensional quantum systems can be broken down into causal components, a result that was previously out of reach. By accepting that subsystems can be non-factor and by providing the tools to work with them, the researchers have opened the door to a more accurate description of quantum reality. They have shown that the universe does not always oblige us to see it as a collection of independent bricks; sometimes, the bricks are fused together in ways that only become clear when we stop trying to separate them and start looking at the whole structure as it truly is.

The study concludes by acknowledging that while their algebraic framework is complete, the question of when a partition can be fully represented in a geometric space remains open. They suspect that there may be no simple algebraic rule that predicts when this geometric representation will fail, suggesting that the "good behavior" of a system at the algebraic level does not always guarantee a "good behavior" at the geometric level. This disconnect implies that the standard Hilbert space picture, while powerful, is not the final word on the structure of quantum systems. The researchers suggest that future work should explore how these ideas connect to operational definitions of subsystems and whether the concept of time-thin subsystems, or events, can be integrated into this framework. Ultimately, this paper provides a necessary correction to the way physicists think about the parts of a whole, replacing a rigid, factor-based view with a flexible, algebraic one that can accommodate the complex, interconnected nature of the quantum world.

Technical Summary: Partitions in Quantum Theory

Problem Statement
In quantum theory, the standard approach to decomposing a global system is "bottom-up" (compositional), where subsystems are defined as tensor factors of a global Hilbert space. However, many physical scenarios—such as those involving symmetries, superselection rules, or quantum reference frames—require a "top-down" (decompositional) perspective where the partition of a system is not fixed but depends on the observer or the physical constraints.

Existing decompositional frameworks typically restrict subsystems to factor C-algebras* (algebras isomorphic to $Lin(\mathcal{H})$ ). This paper argues that this restriction is a mistake. Physical systems often naturally correspond to non-factor C-algebras* (direct sums of factors), arising from conservation laws or partial classicality. The primary challenge addressed is that defining partitions into three or more parts (multipartitions) involving non-factor algebras is non-trivial. Unlike factor bipartitions, multipartitions of non-factors suffer from a "Failure Of Local Tomography" (FOLT), where global information exists that cannot be reconstructed even from correlations between local measurements. Furthermore, it is unclear how to rigorously define such partitions or represent them within the standard Hilbert space formalism without introducing "pseudo-nonlocality."

Methodology
The authors adopt an algebraic framework where quantum systems are modeled as finite-dimensional C*-algebras ( $\omega$ ) and subsystems as sub-C*-algebras.

Mathematical Foundations: They utilize the Wedderburn-Artin theorem to characterize finite-dimensional C*-algebras as direct sums of factors. They introduce routes (partial equivalence relations on sector indices) to diagrammatically represent sectorial constraints and commutants within these algebras.
Defining Bipartitions: For a general C*-algebra $\omega$ $ω$ , a bipartition $(A_1, A_2)$ $(A_{1}, A_{2})$ is defined not merely by commutants ( $A_2 = A_1'$ $A_{2} = A_{1}^{'}$ ), which fails for non-factors, but by two conditions:
- Quantum Splitting: Within each block (sector) of the global center $Z(\omega)$ , the subsystems must be mutual commutants.
- Classical Splitting: The global center $Z(\omega)$ must be contained within the algebraic closure of the local centers ( $Z(\omega) \subseteq Z(A_1) \vee Z(A_2)$ ), ensuring classical information is accessible via the conjunction of local parts.
Defining Multipartitions: A partition into an arbitrary number of parts is defined as a mapping from the power set of labels to sub-algebras, satisfying the bipartition condition for any disjoint union of parts.
Representation Theory: The authors investigate whether these algebraic partitions can be represented as operators on a split Hilbert space using the framework of routed quantum circuits. They distinguish between "well-localized" representations (where individual parts act on specific tensor factors) and "fully representable" partitions (where joint parts also act exactly on the corresponding tensor factors without residual non-locality).

Key Contributions

Formalization of Non-Factor Subsystems: The paper provides a self-contained presentation of modeling subsystems as sub-C*-algebras, explicitly validating the inclusion of non-factor algebras arising from symmetry and conservation rules.
Definition of Multipartitions: The authors propose the first rigorous definition of partitions into three or more parts involving non-factor algebras. This definition resolves the ambiguity of "conjunctions of systems" (e.g., $A_{\{1,2\}}$ ) in the presence of FOLT.
Structure of Centres: They prove that the centers of joint subsystems are rigidly determined by the centers of individual subsystems and the global center. Specifically, the center of a conjunction $A_S$ is derived from the intersection of the centers of its components and the global structure, revealing how classical information "disappears" (becomes quantum) when viewed from a global perspective.
Representation Theorem: The paper establishes a representation theorem showing that any partition can be mapped to operators on a Hilbert space with sectorial constraints (routes). However, it proves that not all partitions are fully representable.
Counterexample (Fermionic Systems): The authors demonstrate that the partition of a fermionic system into local modes is not fully representable for $N \ge 3$ . While individual modes can be localized, the algebra of physical operators for joint modes necessarily retains a "residual pseudo-nonlocality" in any Hilbert space representation. This is not a failure of the algebraic framework but a limitation of the Hilbert space formalism in capturing the locality of fermionic systems.

Results

FOLT is Inevitable: The paper confirms that FOLT is a necessary consequence of admitting non-factor subsystems. Global information (like relative phases between superselection sectors) may be inaccessible to local agents and their correlations, yet accessible globally.
Centres as Redundancy: In a partition of a factor, the center of a subsystem corresponds to redundant information shared with its complement. In non-factor partitions, the center also encodes the global classical information.
Partial Representability: While individual subsystems in a partition can always be represented as acting locally on a Hilbert space factor, joint subsystems may require "dephasings" (unitary conjugations by elements of the global center) to be represented. If these dephasings cannot be eliminated, the partition is not fully representable.
Fermionic Non-Representability: The partition of fermionic modes into local sets is a concrete example of a physically valid partition that is not fully representable. This aligns with previous results regarding the impossibility of locality-preserving encodings for fermions on non-tree graphs.

Significance and Claims
The paper claims that its primary contribution is the realization that defining partitions in the presence of non-factor subsystems is a non-trivial task that requires a new theoretical framework. The authors argue that:

Non-factor algebras are fundamental: They are not pathological cases but arise naturally in physics (e.g., quantum causal models, reference frames, superselection rules).
Algebraic vs. Hilbert Space: The algebraic structure of partitions is well-behaved and rich, but the Hilbert space formalism sometimes fails to capture this "good behavior" without introducing artificial non-locality (pseudo-nonlocality).
Practical Utility: The framework is already being used to prove the causal decomposability of 1D quantum cellular automata.
Future Directions: The authors modestly suggest that their framework could elucidate physical facts in fields involving symmetries and causal structures. They explicitly note that they have not yet found an algebraic characterization that is equivalent to "full representability," leaving open the question of whether "good behavior" at the Hilbert space level has a purely morphological (algebraic) counterpart. They also identify the extension to "time-thin subsystems" (events) as a significant future challenge.

The paper does not propose new experimental setups but rather provides the mathematical infrastructure to analyze existing physical scenarios (like the Quantum Switch controversy or fermionic systems) where the standard tensor-product decomposition is insufficient.

Technical Summary: Partitions in Quantum Theory

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