Infinite magic and the emergence of the continuum

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.

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BibTeX RIS

BibTeX

@misc{shaposhnik2026infinite,
  title = {{Infinite magic and the emergence of the continuum}},
  author = {Leo Shaposhnik and Lorenzo Leone and Lennart Bittel and Dimitris Saraidaris and Alexander Jahn and Jens Eisert},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.20790},
  eprint = {2610.20790},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.20790},
}

RIS

TY  - UNPB
TI  - Infinite magic and the emergence of the continuum
AU  - Leo Shaposhnik
AU  - Lorenzo Leone
AU  - Lennart Bittel
AU  - Dimitris Saraidaris
AU  - Alexander Jahn
AU  - Jens Eisert
PY  - 2026
DA  - 2026/10/14
PB  - arXiv
DO  - 10.48550/arxiv.2610.20790
UR  - https://arxiv.org/abs/2610.20790
AB  - 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.
ER  -

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