Consistency conditions for an AdS multiscale entanglement renormalization ansatz correspondence

Abstract

The multiscale entanglement renormalization ansatz (MERA) is a tensor network that provides an efficient way of variationally estimating the ground state of a critical quantum system. The network geometry resembles a discretization of spatial slices of an anti-de Sitter (AdS) spacetime and "geodesics" in the MERA reproduce the Ryu-Takayanagi formula for the entanglement entropy of a boundary region in terms of bulk properties. It has therefore been suggested that there could be an AdS/MERA correspondence, relating states in the Hilbert space of the boundary quantum system to ones defined on the bulk lattice. Here we investigate this proposal and derive necessary conditions for it to apply, using geometric features and entropy inequalities that we expect to hold in the bulk. We show that, perhaps unsurprisingly, the MERA lattice can only describe physics on length scales larger than the AdS radius. Further, using the covariant entropy bound in the bulk, we show that there are no conventional MERA parameters that completely reproduce bulk physics even on super-AdS scales. We suggest modifications or generalizations of this kind of tensor network that may be able to provide a more robust correspondence.

Exploring the tensor networks/AdS correspondence

“…The Renyi entropy of some boundary region A is then given by 19) JHEP08 (2016)086 r=r h r=r h Figure 10. The left panel illustrates the spin-configuration corresponding to the Renyi entropy of a large region, such that the domain wall covers the "horizon" as well as its complement.…”

“…To remedy the problem, we explore the 1 Interested readers are referred to [13,14] for a comprehensive review on tensor networks and its diverse applications in various branches of theoretical physics. Also we list out some references [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30] where the connection between reconstruction of bulk spacetime and entanglement in the context of holography have been explored. This list is by no means complete and the readers are encouraged to check the citations of these papers and the references within.…”

Continuous Multiscale Entanglement Renormalization Ansatz as Holographic Surface-State Correspondence

“…Refer also to e.g. [8][9][10][11] for various refinements and limitations of the connection between AdS/CFT and tensor networks. In general, a tensor network describes a wave function of a quantum state as a network diagram which fills a discretized space.…”

Entanglement Renormalization and Integral Geometry

“…1, and u(p A ∩ p B ) is its left end point on the boundary. In the last step, we define S A,B (u) as the length (divided by 4G) of the geodesic from the bulk point A (or B) to u and use the following relation 18) which is continuous in either branch. We note that in contrast to (A.5) the derivative with respect to one of the end points becomes a total derivative when the other end point is held fixed.…”