  We study measures of quantum information when the space spanned by the set of
accessible observables is not closed under products, i.e., we consider systems
where an observer may be able to measure the expectation values of two
operators, $\langle O_1 \rangle$ and $\langle O_2 \rangle$, but may not have
access to $\langle O_1 O_2 \rangle$. This problem is relevant for the study of
localized quantum information in gravity since the set of approximately-local
operators in a region may not be closed under arbitrary products. While we
cannot naturally associate a density matrix with a state in this setting, it is
still possible to define a modular operator for a state, and distinguish
between two states using a relative modular operator. These operators are
defined on a little Hilbert space, which parameterizes small deformations of
the system away from its original state, and they do not depend on the
structure of the full Hilbert space of the theory. We extract a class of
relative-entropy-like quantities from the spectrum of these operators that
measure the distance between states, are monotonic under contractions of the
set of available observables, and vanish only when the states are equal.
Consequently, these distance-measures can be used to define measures of
bipartite and multipartite entanglement. We describe applications of our
measures to coarse-grained and fine-grained subregion dualities in AdS/CFT and
provide a few sample calculations to illustrate our formalism.
