Bulk entanglement gravity without a boundary: Towards finding Einstein’s equation in Hilbert space

We consider the emergence from quantum entanglement of spacetime geometry in a bulk region. For certain classes of quantum states in an appropriately factorized Hilbert space, a spatial geometry can be defined by associating areas along codimension-one surfaces with the entanglement entropy between either side. We show how radon transforms can be used to convert these data into a spatial metric. Under a particular set of assumptions, the time evolution of such a state traces out a four-dimensional spacetime geometry, and we argue using a modified version of Jacobson’s “entanglement equilibrium” that the geometry should obey Einstein’s equation in the weak-field limit. We also discuss how entanglement equilibrium is related to a generalization of the Ryu-Takayanagi formula in more general settings, and how quantum error correction can help specify the emergence map between the full quantum-gravity Hilbert space and the semiclassical limit of quantum fields propagating on a classical spacetime.

Figure 1

An example of the 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 (8).

Figure 2

The radon transform of a scalar function $f(x,y)$ defined on some compact domain (shaded area) is done by integrating the function over the submanifold $\mathcal{S}$, which is a line in two-dimensional flat space. The transformed data $\mathcal{R}[f](p,\alpha )$ corresponds to the value at a point in the space of $\mathcal{S}$ (the space of lines in this case). $p$ is the perpendicular distance between the plane and the origin and $\alpha$ parametrizes the direction of the unit normal $\widehat{n}$.