Abstract
The Ryu-Takayanagi formula relates entanglement entropy in a field theory to the area of extremal surfaces anchored to the boundary of a dual AdS space. It is interesting to ask if there is also an information theoretic interpretation of the areas of non-extremal surfaces that are not necessarily boundary-anchored. In general, the physics outside such surfaces is associated to observers restricted to a time-strip in the dual boundary field theory. When the latter is two-dimensional, it is known that the differential entropy associated to the strip computes the length of the dual bulk curve, and has an interpretation in terms of the information cost in Bell pairs of restoring correlations inaccessible to observers in the strip. A general realization of this formalism in higher dimensions is unknown. We first prove a no-go theorem eliminating candidate expressions for higher dimensional differential entropy based on entropic c-theorems. Then we propose a new formula in terms of an integral of shape derivatives of the entanglement entropy of ball shaped regions. Our proposal stems from the physical requirement that differential entropy must be locally finite and conformally invariant. Demanding cancelation of the well-known UV divergences of entanglement entropy in field theory guides us to our conjecture, which we test for surfaces in AdS4. Our results suggest a candidate c-function for field theories in arbitrary dimensions.
Highlights
Restricting the boundary observer to making finite-time measurements
It is interesting to ask if there is an information theoretic interpretation of the areas of non-extremal surfaces that are not necessarily boundary-anchored. The physics outside such surfaces is associated to observers restricted to a time-strip in the dual boundary field theory. When the latter is two-dimensional, it is known that the differential entropy associated to the strip computes the length of the dual bulk curve, and has an interpretation in terms of the information cost in Bell pairs of restoring correlations inaccessible to observers in the strip
We propose that the differential entropy is given by the integral of certain shape derivatives of the entanglement entropy associated to the family of balls, B(σ)
Summary
In the body of this work, we will consider states of a CF TD=d+1 with time-reflection symmetry across a d-dimensional spatial slice on which the Ryu-Takayanagi (RT) formula for holographic entanglement entropy applies [31].1 Since we are working in a time-refelection symmetric setting the extremal RT surfaces in AdSD+1 that compute entanglement entropy will be restricted to a spatial slice of the geometry. This implicit dependence on the tangents to the surface can be written in terms of the normal vector n contracted into the volume form as. The embedding of a surface N can be described locally by specifying it as the level set z = z(xi) The area of this surface is given by computing the induced metric ds2ind δij. Integrating this form on Ncorresponds to setting z = z(xi) and zi = ∂iz, which reproduces (2.7) In this language, identifying the boundary dual of the area of a surface in the bulk is equivalent to the question of finding the dual of the area form. In the following subsection we will reformulate this question in terms of information theoretic boundary CFT quantities
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