Space from Hilbert space: Recovering geometry from bulk entanglement

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 $\mathrm{ER}=\mathrm{EPR}$ 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.

Figure 1

An “information graph” in which vertices represent factors in a decomposition of Hilbert space, and edges are weighted by the mutual information between the factors. In redundancy-constrained states, the entropy of a group of factors (such as the shaded region $\mathbf{B}$ containing ${\mathcal{H}}_1\otimes {\mathcal{H}}_2\otimes {\mathcal{H}}_3\otimes {\mathcal{H}}_4\otimes {\mathcal{H}}_5$) can be calculated by summing over the mutual information of the cut edges, as in (9). In the following section we put a metric on graphs of this form by relating the distance between vertices to the mutual information, in (13) and (14).

Figure 2

Multidimensional scaling results for the $1\text{−}d$ antiferromagnetic Heisenberg chain. On the left we plot the eigenvalues of the matrix (24). The fact that the first eigenvalue is much greater than the others indicates that we have a $1\text{−}d$ embedding. On the right we show the reconstructed geometry by plotting the first three coordinates of the graph vertices.

Figure 3

Multidimensional scaling results for a coarse-grained $2\text{−}d$ toric code ground state. Again, the left shows the eigenvalues of (24) and the right shows the reconstructed geometry. The two dominant eigenvalues show that the geometry is two-dimensional, though the fit is not as close as it was in the $1\text{−}d$ example. Similarly, the reconstructed geometry shows a bit more distortion.

Figure 4

For a graph $\tilde{G}$ embedded in some manifold, we assign subregion $A_p$ (dark grey region) to the vertex $p$, which is connected to adjacent vertices $q$ (red solid line). An entanglement perturbation that decreases the mutual information between $A_p$ with its neighbors elongates the connected edges (dashed black lines), creating an angular deficit which is related to the curvature perturbation at $p$.

Figure 5

For perturbations that slightly entangle two regions of the emergent space, as represented by the vertices, a positive curvature perturbation is induced locally near each perturbed site. We may interpret this as a highly “quantum wormhole.” The dotted red line joining $p$ and $p^{\prime }$ denotes some trace amount of entanglement between the two subsystems.