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

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 $d_A{d}_B\le d_E$ (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.

6 pages, 0 figure

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