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, $\langle O_1 \rangle$ 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.
Highlights
In studying entanglement and other quantum information measures associated with a system, we often assume that the Hilbert space factorizes as H ⊗ H with a factor H associated with the system of interest and another factor H associated with the rest of the world
We have described measures of quantum information that are applicable when we can only probe a system with a limited number of observables, and when the space spanned by these observables, A, does not close to form a von Neumann algebra
A corollary to this fact is that the Hilbert space of gravity does not factorize into a Hilbert space associated with a region, and its complement
Summary
In studying entanglement and other quantum information measures associated with a system, we often assume that the Hilbert space factorizes as H ⊗ H with a factor H associated with the system of interest and another factor H associated with the rest of the world. In theories of quantum gravity, except for some special regions, the set of approximately local operators associated with a region does not form an algebra [1,2]. The presence of this center creates ambiguities in quantum information measures, since the center can be considered either to belong to the region or to belong to its complement This ambiguity does not alter the fact that the set of local operators is closed under multiplication. Consider an ordinary local quantum field theory without gauge fields, where the Hilbert space factorizes as H ⊗ H and where, with respect to the full set of operators (including operators in H), the system is in a pure state. We use the modular and the relative modular operators to define measures of distance between states. In the Appendix, we provide some sample calculations to illustrate our formalism
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
