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arXiv:2512.07468

On the emergence of preferred structures in quantum theory

Antoine Soulas, Guilherme Franzmann, Andrea Di Biagio

quant-phhep-thmath-phSubmitted Dec 8, 2025• Announced Dec 9, 2025• 0 citations• Revised Dec 22, 2025 (v2)
v1 · Dec 8, 2025v2 · Dec 22, 2025
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Abstract

We assess the possibilities offered by Hilbert space fundamentalism, an attitude towards quantum physics according to which all physical structures (e.g. subsystems, locality, spacetime, preferred observables) should emerge from minimal quantum ingredients (typically a Hilbert space, Hamiltonian, and state). As a case study, we first mainly focus on the specific question of whether the Hamiltonian can uniquely determine a tensor product structure, a crucial challenge in the growing field of quantum mereology. The present paper reviews, clarifies, and critically examines two apparently conflicting theorems by Cotler et al. and Stoica. We resolve the tension, show how the former has been widely misinterpreted and why the latter is correct only in some weaker version. We then propose a correct mathematical way to address the general problem of preferred structures in quantum theory, relative to the characterization of emergent objects by unitary-invariant properties. Finally, we apply this formalism in the particular case we started with, and show that a Hamiltonian and a state are enough structure to uniquely select a preferred tensor product structure.

Comments
14 pages, minor attribution fixes

Related papers

Comment on "On the emergence of preferred structures in quantum theory" by Soulas, Franzmann, and Di Biagio

Ovidiu Cristinel Stoica

This reply is also a friendly introduction to the impossibility of emergence of preferred structures from the Hamiltonian $\mathsf{H}$ and the unit vector $|ψ\rangle$ only. The obstructions to emergence are illustrated on the concrete construction of a tensor product structure (TPS) from Soulas et al., 2025 (arXiv:2512.07468v2). Soulas et al. offer their TPS as a counterexample to the proof from Stoica, 2022a (arXiv:2102.08620) that structures constructed only from $\mathsf{H}$ and $|ψ\rangle$ either contradict physical observations or can't describe them unambiguously. Soulas et al.'s construction of a unique TPS can't be both invariant and compatible with physical observations, so it can't be a counterexample. Its incompatibility becomes visible by examining how the relation between $|ψ(t)\rangle$ and the TPS, encoding the entanglement, changes in time. Therefore their TPS doesn't refute, but confirms (Stoica, 2022a). Besides this, since Soulas et al.'s method to construct preferred structures consists of choosing their invariants, by the same logic one could claim as well that the masses of elementary particles emerge uniquely just by fixing their values by hand. Soulas et al.'s construction is concrete and can illustrate the major obstructions for emergent structures, confirming them despite doing the best possible to avoid them. This makes it an excellent pedagogical tool to illustrate the trilemma, but also the relational and structural aspects of quantum theory and its symmetries.

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We assess the possibilities offered by Hilbert space fundamentalism, an attitude towards quantum physics according to which all physical structures (e.g. subsystems, locality, spacetime, preferred observables) should emerge from minimal quantum ingredients (typically a Hilbert space, Hamiltonian, and state). As a case study, we first mainly focus on the specific question of whether the Hamiltonian can uniquely determine a tensor product structure, a crucial challenge in the growing field of quantum mereology. The present paper reviews, clarifies, and critically examines two apparently conflicting theorems by Cotler et al. and Stoica. We resolve the tension, show how the former has been widely misinterpreted and why the latter is correct only in some weaker version. We then propose a correct mathematical way to address the general problem of preferred structures in quantum theory, relative to the characterization of emergent objects by unitary-invariant properties. Finally, we apply this formalism in the particular case we started with, and show that a Hamiltonian and a state are enough structure to uniquely select a preferred tensor product structure.
This reply is also a friendly introduction to the impossibility of emergence of preferred structures from the Hamiltonian H\mathsf{H}H and the unit vector ∣ψ⟩|ψ\rangle∣ψ⟩ only. The obstructions to emergence are illustrated on the concrete construction of a tensor product structure (TPS) from Soulas et al., 2025 (arXiv:2512.07468v2). Soulas et al. offer their TPS as a counterexample to the proof from Stoica, 2022a (arXiv:2102.08620) that structures constructed only from H\mathsf{H}H and ∣ψ⟩|ψ\rangle∣ψ⟩ either contradict physical observations or can't describe them unambiguously. Soulas et al.'s construction of a unique TPS can't be both invariant and compatible with physical observations, so it can't be a counterexample. Its incompatibility becomes visible by examining how the relation between ∣ψ(t)⟩|ψ(t)\rangle∣ψ(t)⟩ and the TPS, encoding the entanglement, changes in time. Therefore their TPS doesn't refute, but confirms (Stoica, 2022a). Besides this, since Soulas et al.'s method to construct preferred structures consists of choosing their invariants, by the same logic one could claim as well that the masses of elementary particles emerge uniquely just by fixing their values by hand. Soulas et al.'s construction is concrete and can illustrate the major obstructions for emergent structures, confirming them despite doing the best possible to avoid them. This makes it an excellent pedagogical tool to illustrate the trilemma, but also the relational and structural aspects of quantum theory and its symmetries.
We study the question of how to decompose Hilbert space into a preferred tensor-product factorization without any preexisting structure other than a Hamiltonian operator, in particular the case of a bipartite decomposition into ``system'' and ``environment.'' Such a decomposition can be defined by looking for subsystems that exhibit quasiclassical behavior. The correct decomposition is one in which pointer states of the system are relatively robust against environmental monitoring (their entanglement with the environment does not continually and dramatically increase) and remain localized around approximately classical trajectories. We present an in-principle algorithm for finding such a decomposition by minimizing a combination of entanglement growth and internal spreading of the system. Both of these properties are related to locality in different ways. This formalism is relevant to questions in the foundations of quantum mechanics and the emergence of spacetime from quantum entanglement.
Hilbert-Space Fundamentalism (HSF) states that the only fundamental structures are the quantum state vector and the Hamiltonian, and from them everything else emerge uniquely, including the 3D-space, a preferred basis, and a preferred factorization of the Hilbert space. In this article it is shown that whenever such a structure emerges from the Hamiltonian and the state vector alone, if it is physically relevant, it is not unique. Moreover, HSF leads to strange effects like "passive" travel in time and in alternative realities, realized simply by passive transformations of the Hilbert space. The results from this article affect all theories that adhere to HSF, whether they assume branching or state vector reduction (in particular the version of Everett's Interpretation coined by Carroll and Singh "Mad-dog Everettianism"), various proposals based on decoherence, proposals that aim to describe everything by the quantum structure alone, and proposals that spacetime emerges from a purely quantum theory of gravity.
In Hamiltonian systems, the only basis-independent quantity is the spectrum. Given a spectrum, quantum mereology seeks preferred tensor-product decompositions into local subsystems, leading to the emergence of locality. Causal order, however, remains fixed by the Hamiltonian time evolution. In contrast, higher-order quantum theory permits more general processes that need not possess a definite global causal order. In the process-matrix framework, specifying a process requires a choice of subsystems corresponding to the input and output Hilbert spaces of each agent. A change of basis redefines both the agents and the corresponding decompositions into subsystems, while leaving the spectrum of the process matrix invariant. Here, we study how causality arises from this spectrum. First, we derive spectral constraints on processes compatible with definite causal order. Second, we show that, in the thermodynamic limit, generic quantum processes admit a preferred decomposition with a definite causal order. This suggests a mechanism for the emergence of classical causality only from algebraic ingredients.
Abstract We study a mechanism by which the most basic structures of quantum physics can emerge from the most meager of starting points, a Hilbert space, lacking any preassigned structure such as a tensor decomposition, and a loss function. In a simple toy model of the universe, we hypothesize a fundamental loss functional for the combined Hamiltonian and quantum state, and then minimize this loss functional by gradient descent. We find that this minimization gives rise to a co-emergence of locality, i.e. a tensor product structure simultaneously respected by both the Hamiltonian and the state, suggesting that locality can emerge by a process analogous to spontaneous symmetry breaking. We discuss the relevance of this program to the arrow of time problem. In our toy model, we interpret the emergence of a tensor factorization as the appearance of individual degrees of freedom within a previously undifferentiated (raw) Hilbert space. Earlier work [5, 6] looked at the emergence of locality in Hamiltonians only, and in that context found strong numerical confirmation of the hypothesis that raw Hilbert spaces of dim = n are unstable and prefer to settle on tensor factorization when n is not prime, expressing, for example, n = pq, and in [6] even primes were seen to “factor” after first shedding a small summand, e.g. 7 = 1 + 2 · 3. This was found in the context of a rather general potential functional F on the space of metrics {gij} on su \mathfrak{su} su su (n), the Lie algebra of symmetries. This emergence of qunits through operator-level spontaneous symmetry breaking (SSB) may help us understand why the world seems to consist of myriad interacting degrees of freedom. But understanding why the universe has an initial Hamiltonian H0 with a many-body structure is of limited conceptual value unless the initial state, ∣ψ0〉, is also structured by this tensor decomposition. Here we adapt F to become a functional on {g, | ψ0〉} = (metrics) × (initial states), and find SSB now produces a conspiracy between g and ∣ψ0〉, where they simultaneously attain low entropy by jointly settling on the same qubit decomposition. Extreme scaling of the computational problem has confined us to studying ℂ4 breaking to ℂ2 ⊗ ℂ2 and ℂ8 breaking to ℂ2 ⊗ ℂ4 or ℂ2 ⊗ ℂ2 ⊗ ℂ2.