Abstract
We present a mathematical framework which underlies the connection be- tween information theory and the bulk spacetime in the AdS3/CFT2 correspondence. A key concept is kinematic space: an auxiliary Lorentzian geometry whose metric is defined in terms of conditional mutual informations and which organizes the entanglement pattern of a CFT state. When the field theory has a holographic dual obeying the Ryu-Takayanagi proposal, kinematic space has a direct geometric meaning: it is the space of bulk geodesics studied in integral geometry. Lengths of bulk curves are computed by kinematic volumes, giving a precise entropic interpretation of the length of any bulk curve. We explain how basic geometric concepts - points, distances and angles - are reflected in kinematic space, allowing one to reconstruct a large class of spatial bulk geometries from boundary entan- glement entropies. In this way, kinematic space translates between information theoretic and geometric descriptions of a CFT state. As an example, we discuss in detail the static slice of AdS3 whose kinematic space is two-dimensional de Sitter space.
Highlights
The last decade has taught us to be mindful of entanglement
The strikingly simple encoding of CFT entanglement entropies as bulk geodesics hints at the existence of a novel, promising framework for describing geometry — and maybe even gravitational dynamics — from the perspective of information theory
There exists, a “language barrier,” which hinders progress in further extending the translation: traditional general relativity is formulated in the local language of differential geometry whereas the fundamental information theoretic quantities are represented holographically by non-local geometric data
Summary
The last decade has taught us to be mindful of entanglement. One way to see the importance of entanglement in local quantum field theories is to contemplate the area law [1, 2]: the physically relevant states obeying the area law are a small island in the morass of the full Hilbert space, most of which comprises states with too much or too little entanglement. A promising place to start looking for such a formulation of quantum field theory is holographic duality [3] This is because the bulk gravitational dual encodes the boundary field theory entanglement in an explicit and convenient way in terms of areas of minimal surfaces [4,5,6]. It translates lengths of bulk curves into boundary quantities, which have an entropic interpretation in the field theory This property of kinematic space was first discovered in [17] under the guise of the differential entropy formula. The central theme of the introductory expositions is describing objects on the (Euclidean and hyperbolic) plane in terms of straight lines (geodesics) that intersect them This is relevant for AdS3/CFT2, because in that case the Ryu-Takayanagi proposal equates boundary entanglement entropies with lengths of geodesics
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