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Causal structure of the entanglement renormalization ansatz

Cédric Bény

TL;DR

This work reframes the MERA tensor-network ansatz as a causality-constrained discrete quantum dynamics, showing that the causal structure alone suffices to define MERA and that the variational parameter count corresponds to a spacetime volume. By mapping this causal order to discretized de Sitter space, it clarifies the MERA–hyperbolic geometry connection and enables a continuum generalization to quantum field theory on de Sitter space via a cMERA-like construction. The paper also provides a rigorous local implementation principle for MERA steps through pure causality and Stinespring dilation, discusses connections and distinctions with AdS-CFT, and outlines avenues for higher-dimensional and more general lattice realizations. Together, these results unify MERA with a geometric, spacetime-respecting framework and suggest robust paths to continuum, quantum-field-theoretic applications.

Abstract

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 spacelike 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.

Causal structure of the entanglement renormalization ansatz

TL;DR

This work reframes the MERA tensor-network ansatz as a causality-constrained discrete quantum dynamics, showing that the causal structure alone suffices to define MERA and that the variational parameter count corresponds to a spacetime volume. By mapping this causal order to discretized de Sitter space, it clarifies the MERA–hyperbolic geometry connection and enables a continuum generalization to quantum field theory on de Sitter space via a cMERA-like construction. The paper also provides a rigorous local implementation principle for MERA steps through pure causality and Stinespring dilation, discusses connections and distinctions with AdS-CFT, and outlines avenues for higher-dimensional and more general lattice realizations. Together, these results unify MERA with a geometric, spacetime-respecting framework and suggest robust paths to continuum, quantum-field-theoretic applications.

Abstract

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 spacelike 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.

Paper Structure

This paper contains 7 sections, 24 equations, 2 figures.

Table of Contents

  1. A quick introduction on quantum channels
  2. MERA from causal order
  3. Connection with de Sitter space
  4. Continuous MERA
  5. Continuum limit of MERA
  6. AdS-CFT correspondence
  7. Causality and locality

Figures (2)

  • Figure 1: Partial ordered set corresponding to the binary MERA in one dimension, embedded in $\mathbb R^2$ such that the speed of light is equal to $1$ everywhere. The output lattice is the top row of dots and time flows upward. The black circles are events and the lines segments are causal links.
  • Figure 2: The shaded area is the causal past of two disconnected regions of the $\tau=0$ spacelike surface in the static coordinates $(\tau,\zeta)$. The dashed lines indicate the horizon at $|\zeta| = \alpha = 1$.