Information loss, mixing and emergent type III$_1$ factors

TL;DR

The work links the black hole information loss puzzle to the emergence of type $\mathrm{III}_{1}$ von Neumann algebras via mixing operators whose time correlations cluster. It establishes that discrete-energy (type I) algebras forbid mixing, while continuous modular spectra in the thermodynamic or large-$N$ limit permit mixing and force a type $\mathrm{III}_{1}$ structure, provided the mixing set closes under multiplication. The authors prove a general theorem: if mixing operators form an algebra, the resulting von Neumann algebra is type $\mathrm{III}_{1}$ with trivial centralizer, and they explicitly realize these algebras in Generalized Free Field theories and holographic contexts (via HKLL) including time-band algebras and bulk causally complete regions. These results connect clustering-in-time observables to the emergence of bulk locality and hydrodynamics, suggesting a deep link between information flow, modular dynamics, and the algebraic structure of observables in quantum gravity. The work also clarifies the Leutheusser–Liu conjecture in the GFF setting and maps the boundary operator algebras to bulk geometries, with implications for holographic reconstruction and the structure of spacetime at large scales.

Abstract

A manifestation of the black hole information loss problem is that the two-point function of probe operators in a large Anti-de Sitter black hole decays in time, whereas, on the boundary CFT, it is expected to be an almost periodic function of time. We point out that the decay of the two-point function (clustering in time) holds important clues to the nature of observable algebras, states, and dynamics in quantum gravity. We call operators that cluster in time "mixing" and explore the necessary and sufficient conditions for mixing. The information loss problem is a special case of the statement that in type I algebras, there exists no mixing operators. We prove that, in a thermofield double (KMS state), if mixing operators form an algebra (close under multiplication) the resulting algebra must be a von Neumann type III$_1$ factor. In other words, the physically intuitive requirement that all non-conserved operators should diffuse is so strong that it fixes the observable algebra to be an exotic algebra called a type III$_1$ factor. More generally, for an arbitrary out-of-equilibrium state of a general quantum system (von Neumann algebra), we show that if the set of operators that mix under modular flow forms an algebra it is a type III$_1$ von Neumann factor. In a theory of Generalized Free Fields (GFF), we show that if the two-point function of GFF clusters in time all operators are mixing, and the algebra is a type III$_1$ factor. For instance, in $\mathscr{N=4}$ SYM, above the Hawking-Page phase transition, clustering of the single trace operators implies that the algebra is a type III$_1$ factor, settling a recent conjecture of Leutheusser and Liu. We explicitly construct the C$^*$-algebra and von Neumann subalgebras of GFF associated with time bands and more generally, open sets of the bulk spacetime using the HKLL reconstruction map.

Figures (2)

• Figure 1: (a) In the Poincaré-patch of AdS, we can associate C$^*$-algebras $\mathcal{B}_I(0)$ to a time-band $I$ defined at any constant $z$ in the bulk. In general, operator insertions on the red dots do not commute. (b) Similarly, using the HKLL reconstruction for an eternal black hole, we can associate various algebras to time intervals on time-like surfaces in the bulk.
• Figure 2: (a) In the Poincaré patch of AdS, we can associate C$^*$-algebras with any open set in the bulk, e.g. blue regions. By the timelike tube theorem, the double-commutant of these algebras are the von Neumann algebras of causally-complete regions in the bulk, i.e., $\mathcal{B}"_{\mathcal{D}}=\mathcal{A}_{\mathcal{E}(\mathcal{D})}$ and $\mathcal{B}"_{I}=\mathcal{A}_{I}$. (b) Similarly, we can define such C$^*$ and von Neumann algebras in other geometries such as an eternal black hole.

Theorems & Definitions (62)

• Lemma 1
• proof
• Definition 3.1
• Definition 3.2
• Lemma 2: Mixing Operator Systems
• proof
• Lemma 3: Maximal Mixing Operator System
• proof
• Lemma 4: Measurability criterion
• proof