Abstract
In the context of the surface-state correspondence we propose the geodesic curvature of a convex curve as a local measure of factorization of the dual CFT state. Its integral will be interpreted as computing the bipartite entanglement among degrees of freedom with support on the chosen domain. We will derive results through application of the Gauss-Bonnet theorem and show quantitative agreement with computations using the MERA tensor network and the formalism of entanglement density.
Highlights
AND MOTIVATIONSSince the proposal from Ryu and Takayanagi to compute entanglement entropy holographically [1,2,3], an impressive effort has been done to further develop the relation between entanglement and gravity
A step forward in this direction has been done with the conjectural surface-state correspondence [9], where CFT states jψiΣ are associated to generic convex surfaces Σ embedded in the holographic space
The entanglement density can be defined for finite cutoff, generalizing the original computation in [21], 9It is perhaps interesting to note that the form (18) for entanglement entropy can be reduced to the usual logarithmic scaling if we introduce an effective cutoff zcðl; zcÞ
Summary
Since the proposal from Ryu and Takayanagi to compute entanglement entropy holographically [1,2,3], an impressive effort has been done to further develop the relation between entanglement and gravity. The aim will be to propose a holographic measure of bipartite entanglement, for a generic 2d CFT state ργ dual to a convex curve γ, as an integral of a local geometrical quantity on γ. We claim that JKðγÞ can be represented holographically, at least for some choices of measure K, as local geometrical quantities on γ, and we will present an example of this. Following the quantum information argument the integral of the geodesic curvature along some interval on γ should represent a holographic measure of bipartite entanglement of ργ there. A final note is that throughout the paper we will consider a holographic description of entanglement at the classical level in the bulk coupling constant GN This will have an impact on the physical interpretation and results of our analysis. Some discussion on quantum corrections can be found in the last section
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