Entanglement renormalization and holography

We show how recent progress in real space renormalization group methods can be used to define a generalized notion of holography inspired by holographic dualities in quantum gravity. The generalization is based upon organizing information in a quantum state in terms of scale and defining a higher-dimensional geometry from this structure. While states with a finite correlation length typically give simple geometries, the state at a quantum critical point gives a discrete version of anti-de Sitter space. Some finite temperature quantum states include black hole-like objects. The gross features of equal time correlation functions are also reproduced.

Figure 1

The tensor network structure of entanglement renormalization. Circles are lattice sites at various coarse-grained scales. Squares with four lines are unitary disentanglers and triangles with three lines are isometric coarse graining transformations. The network shown here represents a $2\to 1$ coarse graining scheme and has a characteristic fractal structure. In principle, each tensor can be different, but requiring translation and scale invariance provides strong constraints.

Figure 2

(Upper left) A piece of the causal cone of a small block. (Upper right) Reversing the flow to proceed from three sites to six sites. (Lower left) Shaded sites are outside the causal cone of the two-site block and can be traced out. (Lower right) Four sites remain and we can now apply the next layer of disentanglers to reach the two-site block of interest.