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arXiv:2610.20790

Infinite magic and the emergence of the continuum

Leo Shaposhnik, Lorenzo Leone, Lennart Bittel, Dimitris Saraidaris, Alexander Jahn, Jens Eisert

quant-phhep-thmath-phSubmitted Oct 14, 2026• Announced Oct 16, 2026• 0 citations

Abstract

Which features of a quantum state are needed for a relativistic quantum field theory to emerge from a lattice system? We address this question by establishing a quantitative relationship between the type classification of hyperfinite von Neumann algebras and the quantum-information-theoretic resource of nonstabilizerness, also called magic. Our results establish two complementary parts of this relationship. First, we prove that any stabilizer system on a lattice defines a canonical GNS Hilbert space in which local von Neumann algebras do not contain type III factors in their central decomposition. As an explicit example, we prove that for stabilizer systems with short-range order, the local algebras are isomorphic to tensor products of hyperfinite factors of type I or type II. This demonstrates that the absence of magic is accompanied by the absence of type III factors in the local algebras, therefore prohibiting the appearance of a quantum field theory. Second, for the hyperfinite factors of Araki, Woods and Powers, we prove a quantitative equivalence: The limiting factor is of type III if and only if the locally optimized min-relative entropy of magic diverges. This divergence is necessary to generate the continuous spectrum associated to a type III factor. Thus, in this setting, the emergence of a type III factor requires not merely the presence of magic, but an infinite amount of it. These results provide a rigorous setting supporting the proposal that infinite magic is a necessary structural ingredient for the emergence of a relativistic continuum from lattice systems.

Comments: 9+33 pages, 5 figures

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Which features of a quantum state are needed for a relativistic quantum field theory to emerge from a lattice system? We address this question by establishing a quantitative relationship between the type classification of hyperfinite von Neumann algebras and the quantum-information-theoretic resource of nonstabilizerness, also called magic. Our results establish two complementary parts of this relationship. First, we prove that any stabilizer system on a lattice defines a canonical GNS Hilbert space in which local von Neumann algebras do not contain type III factors in their central decomposition. As an explicit example, we prove that for stabilizer systems with short-range order, the local algebras are isomorphic to tensor products of hyperfinite factors of type I or type II. This demonstrates that the absence of magic is accompanied by the absence of type III factors in the local algebras, therefore prohibiting the appearance of a quantum field theory. Second, for the hyperfinite factors of Araki, Woods and Powers, we prove a quantitative equivalence: The limiting factor is of type III if and only if the locally optimized min-relative entropy of magic diverges. This divergence is necessary to generate the continuous spectrum associated to a type III factor. Thus, in this setting, the emergence of a type III factor requires not merely the presence of magic, but an infinite amount of it. These results provide a rigorous setting supporting the proposal that infinite magic is a necessary structural ingredient for the emergence of a relativistic continuum from lattice systems.

The number of non-Clifford gates needed to perform a task, or simply \textit{magic}, is a resource for fault-tolerant quantum computation. Von Neumann algebras provide a formal mathematical structure to describe infinite-dimensional quantum systems, such as those in quantum field theory or quantum statistical mechanics. A particularly important class of von Neumann algebras is called Type III algebras, for which the standard notions of finite-dimensional systems such as density matrices and traces break down. In this work, we argue that Type III von Neumann algebras fundamentally require an infinite amount of magic. Specifically, we consider the thermodynamic limit of finite-dimensional quantum systems, such as lattice systems, and show that if the states in the thermodynamic limit possess only a bounded amount of magic, the resulting local von Neumann algebra cannot be of Type III. Our result has direct implications for the quantum simulations of quantum field theories, for which the algebra of a local subregion is known to be of Type III.

We show that no stabilizer state in a discrete realization of a local quantum field theory can flow in the continuum to the vacuum or to any state that resembles the vacuum at short distances. The argument rests on the fact that the entanglement spectrum is flat for stabilizer states but non-flat for cyclic and separating states in a local QFT as a consequence of the type III

_1

nature of the von Neumann algebras associated with arbitrary subregions. Our result implies that simulating physically relevant QFT states necessarily requires quantum resources beyond stabilizer states and Clifford operations. We comment on the implications for holography.

One of von Neumann’s motivations for developing the theory of operator algebras and his and Murray’s 1936 classification of factors was the question of possible decompositions of quantum systems into independent parts. For quantum systems with a finite number of degrees of freedom the simplest possibility, i.e., factors of type I in the terminology of Murray and von Neumann, are perfectly adequate. In relativistic quantum field theory (RQFT), on the other hand, factors of type III occur naturally. The same holds true in quantum statistical mechanics of infinite systems. In this brief review some physical consequences of the type III property of the von Neumann algebras corresponding to localized observables in RQFT and their difference from the type I case will be discussed. The cumulative effort of many people over more than 30 years has established a remarkable uniqueness result: The local algebras in RQFT are generically isomorphic to the unique, hyperfinite type III1 factor in Connes ’ classification of 1973. Specific theories are characterized by the net structure of the collection of these isomorphic algebras for different space-time regions, i.e., the way they are embedded into each other. John von Neumann was the father of the Hilbert space formulation of quantum mechanics [1] that has been the basis of almost all mathematically rigorous investigations of the theory to this day. We start by recalling the main concepts and explaining some notations.

Typicality arguments for emergent spacetime rely on the Lubkin-Page bounds, which show that generic quantum states have vanishing correlations between subsystems. These bounds assume a tensor-product Hilbert space (a Type~I von~Neumann algebra), but the observable algebras in quantum field theory and quantum gravity are generically Type~II or Type~III, raising the question of whether the bounds survive. We prove that they do for all Type~II von~Neumann factors. For the hyperfinite Type~II

_1

factor with a tripartite decomposition

R \cong A \otimes B \otimes E

, the mutual information between subsystems

A

and

B

vanishes as

O((d_A d_B / d_E)^2)

in finite-dimensional approximations, provided

d_A d_B \leq d_E

(Theorem~1). For Type~II

_\infty

factors, including the gravitational algebras constructed via the crossed-product method by Witten and by Chandrasekaran, Longo, Penington, and Witten, the bound acquires an additional exponential suppression controlled by the Bekenstein-Hawking entropy (Theorem~2). We identify the obstructions to extending the result to Type~III factors and discuss the open question of whether the commutant of the observable algebra can serve as a natural thermal bath that tightens the bound further.

In quantum systems with infinitely many degrees of freedom, states can be infinitely entangled across a pair of subsystems. But are there different forms of infinite entanglement? To answer this, we provide a rigorous framework for studying information-theoretic properties in infinite systems, formulated in terms of von Neumann algebras. In this framework, we show that there are operational tasks that can distinguish between different forms of infinite entanglement. We find that the type classification of von Neumann algebras (types I, II, III, and their respective subtypes) is in one-to-one correspondence with operational entanglement properties. For instance, the Connes classification of type III factors corresponds to the system’s capability at the operational task of “embezzling” entanglement. Our findings promote the type classification from algebraic bookkeeping to a classification of infinite quantum systems based on their operational entanglement properties.

Entanglement has long been the subject of discussion by philosophers of quantum theory, and has recently come to play an essential role for physicists in their development of quantum information theory. In this paper we show how the formalism of algebraic quantum field theory (AQFT) provides a rigorous framework within which to analyze entanglement in the context of a fully relativistic formulation of quantum theory. What emerges from the analysis are new practical and theoretical limitations on an experimenter’s ability to perform operations on a field in one spacetime region that can disentangle its state from the state of the field in other spacelike-separated regions. These limitations show just how deeply entrenched entanglement is in relativistic quantum field theory, and yield a fresh perspective on the ways in which the theory differs conceptually from both standard nonrelativistic quantum theory and classical relativistic field theory. “...despite its conservative way of dealing with physical principles, algebraic QFT leads to a radical change of paradigm. Instead of the Newtonian view of a space-time filled with a material content one enters the reality of Leibniz created by relation (in particular inclusions) between ‘monads ’ (∼ the hyperfinite type III1 local von Neumann factors A(O) which as single algebras are nearly void of physical meaning) ” Schroer (1998, p. 302).