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Partitions in quantum theory

Augustin Vanrietvelde, Octave Mestoudjian, Pablo J. Arrighi

Published Sep 2, 2025• 0 citations
OpenAlex

Abstract

Decompositional theories describe the ways in which a global physical system can be split into subsystems, facilitating the study of how different possible partitions of a same system interplay, e.g. in terms of inclusions or signalling. In quantum theory, subsystems are usually framed as sub-C* algebras of the algebra of operators on the global system. However, most decompositional approaches have so far restricted their scope to the case of systems corresponding to factor algebras. We argue that this is a mistake: one should cater for the possibility for non-factor subsystems, arising for instance from symmetry considerations. Building on simple examples, we motivate and present a definition of partitions into an arbitrary number of parts, each of which is a possibly non-factor sub-C* algebra. We discuss its physical interpretation and study its properties, in particular with regards to the structure of algebras' centres. We prove that partitions, defined at the C*-algebraic level, can be represented in terms of a splitting of Hilbert spaces, using the framework of routed quantum circuits. For some partitions, however, such a representation necessarily retains a residual pseudo-nonlocality. We provide an example of this behaviour, given by the partition of a fermionic system into local modes.

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Decompositional theories describe the ways in which a global physical system can be split into subsystems, facilitating the study of how different possible partitions of a same system interplay, e.g. in terms of inclusions or signalling. In quantum theory, subsystems are usually framed as sub-C* algebras of the algebra of operators on the global system. However, most decompositional approaches have so far restricted their scope to the case of systems corresponding to factor algebras. We argue that this is a mistake: one should cater for the possibility for non-factor subsystems, arising for instance from symmetry considerations. Building on simple examples, we motivate and present a definition of partitions into an arbitrary number of parts, each of which is a possibly non-factor sub-C* algebra. We discuss its physical interpretation and study its properties, in particular with regards to the structure of algebras' centres. We prove that partitions, defined at the C*-algebraic level, can be represented in terms of a splitting of Hilbert spaces, using the framework of routed quantum circuits. For some partitions, however, such a representation necessarily retains a residual pseudo-nonlocality. We provide an example of this behaviour, given by the partition of a fermionic system into local modes.
One of the most basic notions in physics is the partitioning of a system into subsystems and the study of correlations among its parts. In this Letter, we explore this notion in the context of quantum reference frame (QRF) covariance, in which this partitioning is subject to a symmetry constraint. We demonstrate that different reference frame perspectives induce different sets of subsystem observable algebras, which leads to a gauge-invariant, frame-dependent notion of subsystems and entanglement. We further demonstrate that subalgebras which commute before imposing the symmetry constraint can translate into noncommuting algebras in a given QRF perspective after symmetry imposition. Such a QRF perspective does not inherit the distinction between subsystems in terms of the corresponding tensor factorizability of the kinematical Hilbert space and observable algebra. Since the condition for this to occur is contingent on the choice of QRF, the notion of subsystem locality is frame dependent.
The characterization of entanglement is a fundamental issue for Quantum Information Theory. But the definition of entanglement depends on the notion of locality, and thus on the tensor product structure of the state space of the composite system. This notion is affected by the presence of superselection rules that restrict the accessible Hilbert space to a direct sum of subspaces. Indistinguishability of particles imposes one such restriction, namely to totally symmetric or totally antisymmetric states. The entanglement can in this case be defined with respect to partitions of modes in the second quantization formalism. For fermionic systems the Fock space of m modes is isomorphic to the space of m qubits, but the action of creation and annihilation operators is not local, due to their anticommutation. Conservation of the parity of fermion number imposes another relevant superselection rule. It requires that local physical observables commute with the local parity operator. Taking into account the considerations above, it is possible to define the set of separable states or equivalently the concept of entanglement for fermionic systems in a number of ways. Here we analyze systematically these possibilities and the relation among the various sets of separable states. We also discuss the behavior of the different classes when taking several copies of the state, as well as the characterization of the sets in terms of the usual criteria regarding the tensor product.
What theoretical structures characterise our notions of what it means for a whole physical system to be the sum of its parts? We explore various aspects of this question, with emphasise on quantum theory, from the following perspective: A physical theory constitutes the definition of a set of propositions with logical relationships between them such that probabilities can be used to judge the merits of the theory, given empirical evidence. The logical relationships between the propositions characterise our semantic notions such as individuation. We distinguish between the ontological and epistemological notions of individuation. We introduce the notion of ontological individuation within the context of ontological models. A subsystem is defined as a proposition that provides necessary and sufficient information for making useful predictions. This definition to some extent generalises the notions of separability and local causality to define systems that may not be local. It highlights the unified motivations for many assumptions in the literature, such as measurement independence, and points to hidden assumptions of causality and mutual exclusivity that occur when defining systems. An epistemic no-signalling criterion gives rise to a tensor product structure for epistemic systems in quantum theory. We utilise a symmetry emergent from this structure to derive a bound on the observability of a large class of operations, dependent on only the purity of the quantum state. In particular, we derive an upper bound on the trace distance between an untransformed state and the state transformed by any trace preserving, unital operation. Our notions of epistemic systems also relate to distinguishability. It is shown that for measurements of any two (_nite outcome) particle quantum numbers on multi-fermion states, the total uncertainty, minimised over all states, is zero. This is because despite indistinguishability characterised as a constraint on the observable algebra, the appropriate correspondence between states of distinguishable and indistinguishable particles is one where the Hilbert space of indistinguishable particles relates to particles with an extra quantum number such that effective distinguishability can emerge. This Hilbert space is larger than the one for distinguishable particles and hence can (does) contain states for which the uncertainty relations for distinguishable particles are no longer valid. We conclude by arguing that in general, epistemic systems are not necessarily distinguishable; They may be only effectively distinguishable. This would imply a theory of indistinguishable modes. Several motivations are presented and the form of the theory suggested. It is hypothesised that such a theory may provide a relativistic quantum theory, alternative to quantum field theory, that satisfies cluster decomposition.
Abstract The operator algebra of fermionic modes is isomorphic to that of qubits, the difference between them is twofold: the embedding of subalgebras corresponding to mode subsets and multiqubit subsystems on the one hand, and the parity superselection in the fermionic case on the other. We discuss these two fundamental differences extensively, and illustrate these through the Jordan–Wigner representation in a coherent, self-contained, pedagogical way, from the point of view of quantum information theory. Our perspective leads us to develop useful new tools for the treatment of fermionic systems, such as the fermionic (quasi-)tensor product, fermionic canonical embedding, fermionic partial trace, fermionic products of maps and fermionic embeddings of maps. We formulate these by direct, easily applicable formulas, without mode permutations, for arbitrary partitionings of the modes. It is also shown that fermionic reduced states can be calculated by the fermionic partial trace, containing the proper phase factors. We also consider variants of the notions of fermionic mode correlation and entanglement, which can be endowed with the usual, local operation based motivation, if the parity superselection rule is imposed. We also elucidate some other fundamental points, related to joint map extensions, which make the parity superselection inevitable in the description of fermionic systems.
We extend the usual process-theoretic view on locality and causality in subsystems (based on the tensor product case) to general quantum systems (i.e.\ possibly non-factor, finite-dimensional von Neumann algebras). To do so, we introduce a primitive notion of splitting maps within dagger symmetric monoidal categories. Splitting maps give rise to subsystems that admit comparison via a preorder called comprehension, and support an adaptation of the usual categorical trace. We show that the comprehension preorder precisely captures the inclusion partial order between von Neumann algebras, and that the splitting map trace captures the natural notion of von Neumann algebra trace. As a consequence of the development of these diagrammatic tools, we prove that the known equivalence between semi-causality and semi-localisability for factor subsystems extends to all (including non-factor) subsystems.
Symmetries impose structure on the Hilbert space of a quantum mechanical model. The mathematical units of this structure are the irreducible representations of symmetry groups and I consider how they function as conceptual units of interpretation. For models with symmetry, the properties of irreducible representations constrain the possibilities of Hilbert space arithmetic, i.e.\ how a Hilbert space can be decomposed into sums of subspaces and factored into products of subspaces. Partitioning the Hilbert space is equivalent to parsing the system into subsystems, and these emergent subsystems provide insight into the kinematics, dynamics, and informatics of a quantum model. This article provides examples of how complex models can be built up from irreducible representations that correspond to `natural' ontological units like spins and particles. It also gives examples of the reverse process in which complex models are partitioned into subsystems that are selected by the representations of the symmetries and require no underlying ontological commitments. These techniques are applied to a few-body model in one-dimension with a Hamiltonian depending on an interaction strength parameter. As this parameter is tuned, the Hamiltonian runs dynamical spectrum from integrable to chaotic, and the subsystems relevant for analyzing and interpreting the dynamics shift accordingly.