Space from Hilbert Space: Recovering Geometry from Bulk Entanglement

TL;DR

This paper explores how to reconstruct spatial geometry from the entanglement structure of quantum states in Hilbert space, revealing emergent geometries and their relation to Einstein's equations.

Contribution

It introduces a method to derive emergent spatial geometries from entanglement patterns in Hilbert space and links entanglement perturbations to curvature changes obeying Einstein-like equations.

Findings

Mutual information defines a distance measure for emergent geometry.

Entanglement perturbations induce local curvature modifications.

Emergent geometries exhibit properties analogous to classical spacetime, including a quantum wormhole interpretation.

Abstract

We examine how to construct a spatial manifold and its geometry from the entanglement structure of an abstract quantum state in Hilbert space. Given a decomposition of Hilbert space $\mathcal{H}$ into a tensor product of factors, we consider a class of "redundancy-constrained states" in $\mathcal{H}$ that generalize the area-law behavior for entanglement entropy usually found in condensed-matter systems with gapped local Hamiltonians. Using mutual information to define a distance measure on the graph, we employ classical multidimensional scaling to extract the best-fit spatial dimensionality of the emergent geometry. We then show that entanglement perturbations on such emergent geometries naturally give rise to local modifications of spatial curvature which obey a (spatial) analog of Einstein's equation. The Hilbert space corresponding to a region of flat space is finite-dimensional and scales as the volume, though the entropy (and the maximum change thereof) scales like the area of the boundary. A version of the ER=EPR conjecture is recovered, in that perturbations that entangle distant parts of the emergent geometry generate a configuration that may be considered as a highly quantum wormhole.