Lubkin-Page typicality bounds for Type~II von~Neumann factors

• URL: http://arxiv.org/abs/2607.01873v1
• Date: Thu, 02 Jul 2026 08:27:45 GMT
• Title: Lubkin-Page typicality bounds for Type~II von~Neumann factors
• Abstract summary: Lubkin-Page bounds show generic quantum states have vanishing correlations between subsystems.<n> observable algebras in quantum field theory and quantum gravity are generically TypeII or TypeIII.<n>We prove that they do for all TypeII vonNeumann factors.
• Score: 5.559015632900393
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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