nLab entanglement wedge

Idea

In the context of holography/AdS-CFT duality and holographic entanglement entropy, by an entanglement wedge $W(A)$ one refers (since HHLR 2014) to a subspace of the bulk spacetime which is the holographic dual of a given subspace $A$ of the asymptotic boundary.

Concretely, the entanglement wedge of $A$ is the causal hull of the homology surface of $A$, hence the maximal globally hyperbolic spacetime whose Cauchy surface is the homology surface of $A$. (Here, the homology surface of $A$ is the minimal area bulk region whose boundary is the union of $A$ with its Ryu-Takayanagi surface.)

Closely related to but crucially different from the entanglement wedge is the causal wedge of $A$, which is the intersection of the causal future and past of $A$ inside the bulk. In contrast to the entanglement wedge, the causal wedge is generally not globally hyperbolic (cf. Rangamani & Takayanagi 2017 §9.2.2).

Related entries

• holographic entanglement entropy

References

The notion was introduced in:

• Bartlomiej Czech, Joanna L. Karczmarek, Fernando Nogueira, Mark Van Raamsdonk: The Gravity Dual of a Density Matrix, Classical and Quantum Gravity 29 (2012) 155009 [arXiv:1204.1330, doi:10.1088/0264-9381/29/15/155009]

and the term “entanglement wedge” was coined in:

• Matthew Headrick, Veronika E. Hubeny, Albion Lawrence, Mukund Rangamani: Causality & holographic entanglement entropy, J. High Energ. Phys. 2014 162 (2014) [doi:10.1007/JHEP12(2014)162, arXiv:1408.6300 hep-th]

The entanglement wedge reconstruction property was more comprehensively argued in:

• Xi Dong, Daniel Harlow, Aron C. Wall: Reconstruction of Bulk Operators within the Entanglement Wedge in Gauge-Gravity Duality, Phys. Rev. Lett. 117 021601 (2016) [doi:10.1103/PhysRevLett.117.021601, arXiv:1601.05416]

• Daniel L. Jafferis, Aitor Lewkowycz, Juan Maldacena, S. Josephine Suh: Relative entropy equals bulk relative entropy, J. High Energ. Phys. 2016 4 (2016) [doi:10.1007/JHEP06(2016)004, arXiv:1512.06431]

Review:

• Mukund Rangamani, Tadashi Takayanagi; §9.2.2 in: Holographic Entanglement Entropy, Lecture Notes in Physics 931, Springer (2017) [doi:10.1007/978-3-319-52573-0, arXiv:1609.01287]