  Recent works by Chandrasekaran, Penington, and Witten have shown in various
special contexts that the quantum-corrected Ryu-Takayanagi (RT) entropy (or its
covariant Hubeny-Rangamani-Takayanagi (HRT) generalization) can be understood
as computing an entropy on an algebra of bulk observables. These arguments do
not rely on the existence of a holographic dual field theory. We show that
analogous-but-stronger results hold in any UV-completion of asymptotically
anti-de Sitter quantum gravity with a Euclidean path integral satisfying a
simple and familiar set of axioms. We consider a quantum context in which a
standard Lorentz-signature classical bulk limit would have Cauchy slices with
asymptotic boundaries $B_L \sqcup B_R$ where both $B_L$ and $B_R$ are compact
manifolds without boundary. Our main result is then that (the UV-completion of)
the quantum gravity path integral defines type I von Neumann algebras ${\cal
A}^{B_L}_L$, ${\cal A}^{B_R}_{R}$ of observables acting respectively at $B_L$,
$B_R$ such that ${\cal A}^{B_L}_L$, ${\cal A}^{B_R}_{R}$ are commutants. The
path 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 the
entropy 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 can
be written in terms of standard density matrices and standard Hilbert space
traces. Furthermore, in appropriate semiclassical limits our entropies are
computed by the RT-formula with quantum corrections. Our work thus provides a
Hilbert space interpretation of the RT entropy. Since our axioms do not
severely constrain UV bulk structures, they may be expected to hold equally
well for successful formulations of string field theory, spin-foam models, or
any other approach to constructing a UV-complete theory of gravity.
