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, $⟨O_1⟩$ and $⟨O_2⟩$, but may not have access to $⟨O_1{O}_2⟩$. 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.

44 pages (v2) refs added

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• Entanglement entropy for a Maxwell field: Numerical calculation on a two dimensional lattice
• Reconstructing quantum states from local data
• Scanning Tunneling Macroscopy, Black Holes, and AdS/CFT Bulk Locality
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Cited by in corpus (11)

• Notes on Some Entanglement Properties of Quantum Field Theory
• Information Transfer with a Gravitating Bath
• Effective entropy of quantum fields coupled with gravity
• Reanalyzing an Evaporating Black Hole
• Entanglement versus entwinement in symmetric product orbifolds
• Holographic Relative Entropy in Infinite-dimensional Hilbert Spaces
• Quantum State Reduction: Generalized Bipartitions from Algebras of Observables
• Black Hole and de Sitter Microstructures from a Semiclassical Perspective
• A Toy Model of the Information Paradox in Empty Space
• Reductions in finite-dimensional quantum mechanics: from symmetries to operator algebras and beyond
• Extremality as a Consistency Condition on Subregion Duality