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arXiv:1611.08581 · International Journal of Modern Physics D

Towards a dS/MERA correspondence

Raj Sinai Kunkolienkar, Kinjal Banerjee

Published Jun 20, 2017• 14 citations
arXivHTMLPDF

Abstract

Recent advances have suggested that spacetime itself emerges from the entanglement of the quantum degrees of freedom living on the boundary. In the case of the anti-de Sitter (AdS) spacetimes, a particular class of tensor networks has been shown to realize the same via Multi-Scale Entanglement Renormalization Ansatz (MERA). In this paper, we suggest a prescription for the dS/MERA correspondence and recover a discrete version of dS Penrose diagram by using the MERA on conformal theories identified with the future/past boundaries ([Formula: see text]) of the dS spacetime. In this case, as anticipated, time appears as the emergent direction. We comment on the possible interpretation that the dS cosmological horizon entropy involves entanglement with degrees of freedom across the cosmological horizon as well as the implications of our construction for cosmology.

DOI
10.1142/s0218271817501437

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Recent advances have suggested that spacetime itself emerges from the entanglement of the quantum degrees of freedom living on the boundary. In the case of the anti-de Sitter (AdS) spacetimes, a particular class of tensor networks has been shown to realize the same via Multi-Scale Entanglement Renormalization Ansatz (MERA). In this paper, we suggest a prescription for the dS/MERA correspondence and recover a discrete version of dS Penrose diagram by using the MERA on conformal theories identified with the future/past boundaries ([Formula: see text]) of the dS spacetime. In this case, as anticipated, time appears as the emergent direction. We comment on the possible interpretation that the dS cosmological horizon entropy involves entanglement with degrees of freedom across the cosmological horizon as well as the implications of our construction for cosmology.
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.
We show that the quantum entanglement between the conformal field theories (CFTs) living on the future and past boundaries in the de Sitter/conformal field theory (dS/CFT) correspondence can be described by Wick rotating to a geometry with two timelike directions. We propose a new realization of the dS/CFT correspondence in which quantum gravity on this two-time geometry is holographically dual to a CFT defined on a Lorentzian torus. We show that this duality reproduces key features of dS holography, including the dS entropy, correlation functions, and pseudoentropy. Finally, by extending the framework of path-integral optimization, we explain how dS spacetime emerges from the CFT on the Lorentzian torus.
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.
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.
Extending the holographic principle to de Sitter (dS) spacetimes remains one of the most vital open frontiers in quantum gravity, where a microscopic, bottom-up tensor-network framework that relates boundary quantum data to emergent de Sitter spacetime is still lacking. In this work, we first show the emergence of de Sitter spacetime from boundary entanglement by formulating a non-unitary continuous multi-scale entanglement renormalization ansatz (cMERA) for a concrete non-Hermitian critical fermion chain. Within this emergent spacetime, we analyze the associated geodesics and show that they act as extremal Ryu-Takayanagi (RT) surfaces undergoing a smooth timelike-to-null transition. Remarkably, we demonstrate that this continuum trajectory suggests a distinct tensor-network architecture in which the bond-counting contribution naturally truncates at the discrete timelike-to-null transition toward the deep infrared. In the resulting architecture, the null ray along the horizon is represented by zero-cost links, since the associated cut severs no tensor legs. This network structure successfully reproduces the logarithmic scaling of non-unitary critical entanglement entropy, offering a bond-counting picture for the de Sitter RT formula. Our results provide the long-sought dS/(c)MERA correspondence at the level of both emergent spacetime and discrete holographic entanglement.
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.