Random matrix product state models of gravitationally prepared states

TL;DR

This work introduces random matrix product state (RMPS) models as a tractable framework to study gravitationally prepared states in two dimensions, enabling exact treatment of higher topology and replica effects via a transfer-matrix formalism. A central outcome is the spectral gapping condition, which guarantees a bra-ket wormhole phase transition and an entropy bound consistent with semiclassical intuition; the authors instantiate this in O(k) RMPS models and verify with numerical simulations. The RMPS framework further exposes off-shell wormhole contributions that yield nonzero long-distance correlators and regulate divergences, and it extends to continuous RMPS and de Sitter analogues, offering a versatile toolbox for nonperturbative quantum gravity and holographic reasoning. The results illuminate how nonperturbative wormhole physics can emerge from random-state ensembles and provide a concrete bridge between gravity, quantum chaos, and information-theoretic bounds. These models open avenues for exploring gravity-like dynamics in higher dimensions and for connecting with cosmological scenarios such as inflation and de Sitter holography.

Abstract

Gravitationally prepared states are quantum field theoretic states prepared by gravitational path integrals with spatial boundaries that have fixed boundary conditions for gravity but not for matter fields. They can be interpreted as quantum field theoretic states of closed universes encoding quantum gravitational effects of the past. We propose a method of modelling gravitationally prepared states in two dimensions with random matrix product states (RMPS). Such RMPS models allow us to exactly define and compute contributions of higher topologies and replica geometries in the gravitationally prepared state to all orders. We show that the bra-ket wormhole phase transition, a crucial physical property of gravitationally prepared states, is ensured if the transfer matrix of the RMPS satisfies the spectral gapping property, which we define, and define a class of models called $\mathrm{O}(k)$ models satisfying this property. A novel advantage of RMPS models is that they allow us to compute the effects of off-shell wormholes, i.e., wormhole topologies without semiclassical solutions. In particular, using RMPS models, we find that off-shell wormholes lead to nonzero long-distance correlators in gravitationally prepared states. We also define RMPS models in continuous space, and discuss implications for studying de Sitter gravitationally prepared states.

Figures (46)

• Figure 1: Picture of the disc and BKWH geometries in $\braket{\psi}{\psi}$. The meeting boundaries of the bra and the ket are denoted by a double line.
• Figure 2: Picture of a spatial subregion $R$ given by the union of two intervals $R_1$, $R_2$ and the dominant choice of island $I$ given by the union of two line segments $I_1$, $I_2$ atop the bra-ket wormhole geometry of the AdS JT+CFT prepared state. We assume the lengths of $R_1$, $R_2$ and the distances between them are larger than $d_{\text{crit}}$.
• Figure 3: Picture of the maximally disjoint $2$-Renyi replica wormhole geometry related to the spatial subregion $R = R_1 \cup R_2$ in the AdS JT+CFT prepared state. When the lengths of $R_1$, $R_2$ and the distances between them are larger than $d_{\text{crit},n}$, this geometry becomes the dominant contribution to $\Tr (\rho_R^n)$.
• Figure 4: Top: A contribution to the RMPS inner product $\braket{\psi}{\psi}$ in the double disk topology. Bottom: A contribution to the inner product $\braket{\psi}{\psi}$ in the bra-ket wormhole topology. Here $L=6$, $k=2$, and $V(A_1,A_2) = A_1^2 + A_2^2 + A_1^2 A_2^2$. We use 't Hooft's double line notation. The matrix label index $i=1,2$ is indicated by blue and turquoise double lines, respectively.
• Figure 5: Tensor diagram for a spatial subregion $R$ containing the points $x= 0,1,2,3$ in the inner product $\braket{\psi}{\psi}$. The matrix $P_R$ is represented in orange and the matrix $Q_R$ is represented in blue. Cutting along the horizontal blue line gives $\rho_R = P_R Q_R$, and cutting along the vertical red lines give $\rho_{R,\text{compressed}} = Q_R P_R$, which is an $N^2 \times N^2$ matrix, implying the entropy bound $S[R] \leq 2 \ln N$.