We show that the multiscale entanglement renormalization ansatz (MERA) can be reformulated in terms of a causality constraint on discrete quantum dynamics. This causal structure is that of de Sitter space with a flat space-like boundary, where the volume of a spacetime region corresponds to the number of variational parameters it contains. This result clarifies the nature of the ansatz, and suggests a generalization to quantum field theory. It also constitutes an independent justification of the connection between MERA and hyperbolic geometry which was proposed as a concrete implementation of the AdS-CFT correspondence.
We show that the multiscale entanglement renormalization ansatz (MERA) can be reformulated in terms of a causality constraint on discrete quantum dynamics. This causal structure is that of de Sitter space with a flat space-like boundary, where the volume of a spacetime region corresponds to the number of variational parameters it contains. This result clarifies the nature of the ansatz, and suggests a generalization to quantum field theory. It also constitutes an independent justification of the connection between MERA and hyperbolic geometry which was proposed as a concrete implementation of the AdS-CFT correspondence.
We show that the multiscale entanglement renormalization ansatz (MERA) can be reformulated in terms of a causality constraint on discrete quantum dynamics. This causal structure is that of de Sitter space with a flat space-like boundary, where the volume of a spacetime region corresponds to the number of variational parameters it contains. This result clarifies the nature of the ansatz, and suggests a generalization to quantum field theory. It also constitutes an independent justification of the connection between MERA and hyperbolic geometry which was proposed as a concrete implementation of the AdS-CFT correspondence.
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.
The multi-scale entanglement renormalization ansatz (MERA) is a tensor network representation for ground states of critical quantum spin chains, with a network that extends in an additional dimension corresponding to scale. Over the years several authors have conjectured, both in the context of holography and cosmology, that MERA realizes a discrete version of some geometry. However, while one proposal argued that the tensor network should be interpreted as representing the hyperbolic plane, another proposal instead equated MERA to de Sitter spacetime. In this \letter we show, using the framework of path integral geometry [A. Milsted, G. Vidal, arXiv:1807.02501], that MERA on the real line (and finite circle) can be given a rigorous interpretation as a two-dimensional geometry, namely a light sheet (respectively, a light cone). Accordingly, MERA describes neither the hyperbolic plane nor de Sitter spacetime. However, we also propose euclidean and lorentzian generalizations of MERA that correspond to a path integral on these two geometries.
This thesis reviews the multiscale entanglement renormalisation ansatz or MERA, a numerical tool for the study of quantum many-body systems and a discrete realisation of the AdS/CFT duality. The thesis covers an introduction to the necessary background concepts of entanglement, entanglement entropy and tensor network states, the structure and main features of MERA and its applications in condensed matter theory and holography. Also covered are details on the algorithmic implementation of MERA and some of its generalisations and extensions. MERA belongs to a class of variational ansatze for quantum many-body states known as tensor network states. It is especially well-suited for the study of scale invariant critical points. MERA is based on a real-space renormalisation group procedure called entanglement renormalisation, designed to systematically handle entanglement at different length scales along the coarse-graining flow. Entanglement renormalisation has be used for example to efficiently describe Kitaev states of the toric code, the prime example of topological order, and numerically study the ground state of the highly frustrated spin-1/2 Heisenberg model on a kagome lattice and various other one- and two-dimensional lattice models. The geometric and causal structure of MERA, which underlies its effectiveness as a numerical tool, also makes it a discrete version of the AdS/CFT duality. This duality describes a conformal field theory by a gravity theory in a higher dimensional space, and vice versa. The duality is manifest in the scaling of entanglement entropy in MERA, which is governed by a law highly analogous to the Ryu-Takayanagi formula for holographic entanglement entropy, in the connection between thermal states and a black-hole-like MERA and in the connection between correlation functions and holographic geodesics in a scale invariant MERA. The aim of this thesis is to lead the reader to an understanding of what MERA is, how it works and how it can be used. MERA's core features and uses are presented in a comprehensive and explicit way, and a broad view of possible applications and further directions is given. Plenty of references are also offered to direct the reader to further research on how MERA may relate to his/her interests.
We investigate the proposed connection between de Sitter spacetime and the multiscale entanglement renormalization ansatz (MERA) tensor network, and ask what can be learned via such a construction. We show that the quantum state obeys a cosmic no-hair theorem: the reduced density operator describing a causal patch of the MERA asymptotes to a fixed point of a quantum channel, just as spacetimes with a positive cosmological constant asymptote to de Sitter space. The MERA is potentially compatible with a weak form of complementarity (local physics only describes single patches at a time, but the overall Hilbert space is infinite dimensional) or, with certain specific modifications to the tensor structure, a strong form (the entire theory describes only a single patch plus its horizon, in a finite-dimensional Hilbert space). We also suggest that de Sitter evolution has an interpretation in terms of circuit complexity, as has been conjectured for anti--de Sitter space.
We propose a new class of tensor network state as a model for the AdS/CFT correspondence and holography. This class is demonstrated to retain key features of the multiscale entanglement renormalization ansatz (MERA), in that they describe quantum states with algebraic correlation functions, have free variational parameters, and are efficiently contractible. Yet, unlike the MERA, they are built according to a uniform tiling of hyperbolic space, without inherent directionality or preferred locations in the holographic bulk, and thus circumvent key arguments made against the MERA as a model for AdS/CFT. Novel holographic features of this tensor network class are examined, such as an equivalence between the causal cones C(R) and the entanglement wedges E(R) of connected boundary regions R.