Quantum information measures for restricted sets of observables

Abstract

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, ?O1© and ?O2©, but may not have access to ?O1O2©. 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 parametrizes 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.