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Recent works by Chandrasekaran, Penington, and Witten have shown in variousspecial contexts that the quantum-corrected Ryu-Takayanagi (RT) entropy (or itscovariant Hubeny-Rangamani-Takayanagi (HRT) generalization) can be understoodas computing an entropy on an algebra of bulk observables. These arguments donot rely on the existence of a holographic dual field theory. We show thatanalogous-but-stronger results hold in any UV-completion of asymptoticallyanti-de Sitter quantum gravity with a Euclidean path integral satisfying asimple and familiar set of axioms. We consider a quantum context in which astandard Lorentz-signature classical bulk limit would have Cauchy slices withasymptotic boundaries $B_L \sqcup B_R$ where both $B_L$ and $B_R$ are compactmanifolds without boundary. Our main result is then that (the UV-completion of)the quantum gravity path integral defines type I von Neumann algebras ${\calA}^{B_L}_L$, ${\cal A}^{B_R}_{R}$ of observables acting respectively at $B_L$,$B_R$ such that ${\cal A}^{B_L}_L$, ${\cal A}^{B_R}_{R}$ are commutants. Thepath integral also defines entropies on ${\cal A}^{B_L}_L, {\cal A}^{B_R}_R$.Positivity of the Hilbert space inner product then turns out to require theentropy of any projection operator to be quantized in the form $\ln N$ for some$N \in {\mathbb Z}^+$ (unless it is infinite). As a result, our entropies canbe written in terms of standard density matrices and standard Hilbert spacetraces. Furthermore, in appropriate semiclassical limits our entropies arecomputed by the RT-formula with quantum corrections. Our work thus provides aHilbert space interpretation of the RT entropy. Since our axioms do notseverely constrain UV bulk structures, they may be expected to hold equallywell for successful formulations of string field theory, spin-foam models, orany other approach to constructing a UV-complete theory of gravity.

Algebras and Hilbert spaces from gravitational path integrals: Understanding Ryu-Takayanagi/HRT as entropy without invoking holography