Spacetime topology from holographic entanglement

An asymptotically AdS geometry connecting two or more boundaries is given by a entangled state, that can be expanded in the product basis of the Hilbert spaces of each CFT living on the boundaries. We derive a prescription to compute this expansion for states describing spacetimes with general spatial topology in arbitrary dimension. To large N, the expansion coincides with the Schmidt decomposition and the coefficients are given by n-point correlation functions on a particular Euclidean geometry.We show that this applies to all spacetime that admits a Hartle-Hawking type of wave functional, which via a standard hypothesis on the spatial topology, can be (one to one) mapped to CFT states defined on the asymptotic boundary. It is also observed that these states are endowed with quantum coherence properties.Applying this as holographic engineering, one can to construct an emergent space geometry with certain predetermined topology by preparing an entangled state of the dual quantum system. As an example, we apply the method to calculate the expansion and characterize a spacetime whose initial spatial topology is a (genus one) handlebody.

Automated summary

We derive a prescription to compute the expansion of states describing spacetimes with general spatial topology in arbitrary dimension, which coincides with the Schmidt decomposition for large N. The coefficients of the expansion are given by n-point correlation functions on a specific Euclidean geometry. We show that this applies to all spacetimes that admit a Hartle-Hawking type of wave functional and can be mapped to CFT states defined on the asymptotic boundary through a standard hypothesis on the spatial topology. It is also observed that these states exhibit quantum coherence properties. By applying this as holographic engineering, one can construct an emergent space geometry with a predetermined topology by preparing an entangled state of the dual quantum system. As an example, we calculate the expansion and characterize a spacetime with an initial spatial topology of a genus one handlebody.