Deriving covariant holographic entanglement (1607.07506v1)

Abstract: We provide a gravitational argument in favour of the covariant holographic entanglement entropy proposal. In general time-dependent states, the proposal asserts that the entanglement entropy of a region in the boundary field theory is given by a quarter of the area of a bulk extremal surface in Planck units. The main element of our discussion is an implementation of an appropriate Schwinger-Keldysh contour to obtain the reduced density matrix (and its powers) of a given region, as is relevant for the replica construction. We map this contour into the bulk gravitational theory, and argue that the saddle point solutions of these replica geometries lead to a consistent prescription for computing the field theory Renyi entropies. In the limiting case where the replica index is taken to unity, a local analysis suffices to show that these saddles lead to the extremal surfaces of interest. We also comment on various properties of holographic entanglement that follow from this construction.

• The paper derives a gravitational argument for extending the static RT prescription to dynamic, time-dependent scenarios.
• It employs the Schwinger-Keldysh contour and replica method to compute entanglement entropy through extremal bulk surfaces.
• The findings pave the way for using holographic techniques in complex dynamic systems within quantum gravity and field theory.

Deriving Covariant Holographic Entanglement

The paper "Deriving Covariant Holographic Entanglement" addresses the challenging problem of calculating entanglement entropy in time-dependent settings using the AdS/CFT correspondence. The authors, Dong, Lewkowycz, and Rangamani, provide a gravitational argument supporting the Hubeny-Rangamani-Takayanagi (HRT) proposal, which posits that in general dynamic states, the entanglement entropy is given by a quarter of the area of an extremal surface in the bulk, expressed in Planck units.

Key Arguments and Methodology

1. Schwinger-Keldysh Formalism: The paper begins with the implementation of the Schwinger-Keldysh contour to construct the density matrix and its powers. This approach is crucial for setting up the replica method necessary for calculating R\'enyi entropies, which are instrumental in deriving the von Neumann entanglement entropy in the limit where the replica index approaches unity.
2. Covariant Holographic Proposal: The authors revisit the Ryu-Takayanagi (RT) prescription originally applicable to static cases and extend it using the insights from the HRT proposal. The need for a covariant generalization is addressed by considering the principle of general covariance, which guides the proposal's extension to time-dependent circumstances involving extremal surfaces.
3. Replica Construction in the Bulk: The authors map the Schwinger-Keldysh contour into the gravitational bulk, arguing that the saddle point solutions of these replica geometries provide a consistent framework for computing entanglement. A pivotal part of this mapping involves analyzing the local region around a codimension-2 surface, showing that it leads to the extremal surfaces anticipated by HRT.
4. Boundary Conditions and Analysis: In the gravitational context, the paper emphasizes the importance of correct boundary conditions derived from the Schwinger-Keldysh formalism. The authors argue that these conditions ensure consistent application of the gravitational path integral, naturally giving rise to extremal surfaces in the entanglement entropy prescription.

Implications and Speculative Developments

• Theoretical Implications: This research provides a deeper understanding of the geometric nature of quantum entanglement in the holographic framework. It solidifies the role of extremal surfaces in the AdS/CFT correspondence, reinforcing the notion that entanglement entropy is strongly tied to geometric quantities in gravity.
• Practical Implications: From a practical standpoint, the results hint at new methods for calculating entanglement entropy in more general settings, potentially increasing the utility of holographic techniques in quantum field theory and quantum gravity applications.
• Future Directions: The paper opens avenues for exploring higher-dimensional and more complex bulk geometries, particularly those including higher derivative corrections. Given the complex nature of quantum corrections, these results encourage the investigation of their interplay with the geometrical extremal surfaces, which could yield novel insights into quantum gravity.

In summary, the paper contributes a rigorous extension of holographic entanglement entropy calculations into time-dependent regimes, offering validated prescriptions that align with the principles of the AdS/CFT correspondence. Its findings and methodology suggest potential advancements in both theoretical and practical frameworks within high-energy physics.