ER for typical EPR

Abstract

What do the typical entangled states of two black holes look like? Do they contain semiclassical interiors? We approach these questions constructively, providing ensembles of states which densely explore the black hole Hilbert space. The states contain very long Einstein-Rosen (ER) caterpillars: semiclassical wormholes with large numbers of matter inhomogeneities. Distinguishing these ensembles from the typical entangled states of the black holes is hard. We quantify this by deriving the correspondence between a microscopic notion of quantum randomness and the geometric length of the wormhole. This formalizes a ``complexity = geometry'' relation.

Figures (5)

• Figure 1: Complex time contour $\Gamma_t$ defining the operator $U(\Gamma_t)$. The Hamiltonian $H_c(s)$ is taken to be $H_0$ on the black parts of the contour and $H_0 + H(s)$ on the blue parts. The timefolds effectively implement the interaction Hamiltonian $H(s)$.
• Figure 2: The ER caterpillar is a long bumpy wormhole supported by an inhomogeneous matter distribution, with correlation scale set by $\ell_{\Delta}$ and average length set by $\ell(t)$.
• Figure 3: Complex time contour $\mathcal{C}_t$ composed of forward ($1$) and backward ($\bar{1}$) contours.
• Figure 4: On top, the Euclidean saddle point $M_t$ of topology $D \times \mathbf{S}^{d-1}$, where $D$ is a disk, has an approximate translation isometry in the $\tau$ direction. At each constant-$\tau$ (blue) slice the path integral prepares the semiclassical dual to $\ket{\text{GS}}$. On the bottom, the Lorentzian continuation across the red slice preparing a two-sided black hole with an approximate translation symmetry in the interior. The length of the wormhole $\ell(t)$ is proportional to the circuit time $t-t_\star$.
• Figure 5: Euclidean two-boundary wormhole providing the late-time "plateau" value of $G_1(t,t')$. The saddle-point topology is $\mathbf{S}^1\times \mathbf{R}\times \mathbf{S}^{d-1}$. The wormhole geometry includes two long regions with a Euclidean time $\tau$ translation isometry geometrically equivalent to the one for the disk in Fig. \ref{['fig:caterpi']}.