Building Holographic Entanglement by Measurement

TL;DR

This work introduces a Gaussian quench-and-measure protocol that can engineer boundary states with holographic entanglement patterns reflecting a chosen bulk geometry. By discretizing bulk spaces (e.g., a hyperbolic disk or a wormhole) into graphs and performing a fixed-time quench followed by bulk measurements, the resulting boundary entanglement entropies closely follow the Ryu-Takayanagi predictions, with central charges and Rényi entropies tunable via initial squeezing and graph decoration. The decorated construction yields cleaner power-law correlations and, in the strong-squeezing regime, central charge values near that of a free 1+1D boson (c ≈ 1), while undeco-rated graphs show α-dependent Rényi behavior and larger effective c scaling with μ. The protocol is experimentally accessible in photonics, atomic ensembles, or superconducting circuits and provides a scalable platform for studying holographic entanglement and potential quantum simulations of AdS/CFT phenomena, including bulk reconstruction via mutual information.

Abstract

We propose a framework for preparing quantum states with a holographic entanglement structure, in the sense that the entanglement entropies are governed by minimal surfaces in a chosen bulk geometry. We refer to such entropies as holographic because they obey a relation between entropies and bulk minimal surfaces, known as the Ryu-Takayanagi formula, that is a key feature of holographic models of quantum gravity. Typically in such models, the bulk geometry is determined by solving Einstein's equations. Here, we simply choose a bulk geometry, then discretize the geometry into a coupling graph comprising bulk and boundary nodes. Evolving under this graph of interactions and measuring the bulk nodes leaves behind the desired pure state on the boundary. We numerically demonstrate that the resulting entanglement properties approximately reproduce the predictions of the Ryu-Takayanagi formula in the chosen bulk geometry. We consider graphs associated with hyperbolic disk and wormhole geometries, but the approach is general. The minimal ingredients in our proposal involve only Gaussian operations and measurements and are readily implementable in photonic and cold-atom platforms.

Figures (8)

• Figure 1: (a) The quench-and-measure protocol. The input state is a set of independent squeezed Gaussian modes, with some designated as the bulk (blue circles) and others as the boundary (green circles). After quenching on interactions with a specified coupling graph (black lines), chosen here to discretize a hyperbolic disk, measurement of the bulk nodes prepares the boundary in a pure state. (b) Entanglement entropies $S$ of connected boundary regions of length $\ell$ for initial squeezing parameter $\mu=0.2$, compared with the fit to the prediction for a (1+1)D CFT on a circle (dashed black line), where the central charge $c=6.5(3)$ and vertical offset $\epsilon = 5.36(8)$ are free parameters. Through the Ryu-Takayanagi formula, the entanglement entropy $S$ has a geometrical interpretation as proportional to the minimal length in a holographic bulk (inset schematic). (c) Rényi entropy (normalized by entanglement entropy) vs index $\alpha$ for a region size $\ell = 4$ on the boundary and various squeezing parameters $\mu$ (colored lines), compared to the CFT prediction (dashed black line).
• Figure 2: (a) A wormhole graph, formed from gluing together portions of two hyperbolic disk graphs. In this case, each boundary has $L=64$ sites. (b) One-sided entanglement entropy of the wormhole for $\mu = 0.2$, with the fit yielding inverse temperature $\beta=2.9(1)\times10^1$, central charge $c=5.8(2)$, and offset $\epsilon=2.1(1)$. The fit is shown in red and blue, which correspond to two different candidate curves for the minimal surface (see inset). (c) Two-sided entanglement entropy of the wormhole for $\mu = 0.2$. The approximate condition used to find the crossover between the red and blue candidate two-sided entanglement entropies for the model curve is given in supp.
• Figure 3: (a) Entanglement entropies for the boundary of a depth-5 decorated graph for $\mu = 0.05$, along with the CFT fit (dashed black line) yielding $c=1.004(9)$ and vertical offset $\epsilon=1.954(2)$. The inset shows the decorated graph, where green are boundary nodes, blue are the original bulk nodes, and yellow are additional bulk nodes. The color of the edge indicates the sign of the interaction (red for positive, blue for negative). (b) Position (green) and momentum (blue) covariances of a pair of boundary sites vs distance $d$, where the boundary circumference is $L=16$. Lines show power-law fits at large $\sin(\pi d / L)$, yielding scaling dimensions $\Delta_x = 0.87(4)$ and $\Delta_p = 1.03(3)\times10^{-2}$.
• Figure 4: Approximate reconstruction of the entanglement wedge through mutual information. The color of each bulk node shows the normalized mutual information $I(A:b) / S(b)$ between a subset of boundary nodes (light pink nodes on the ring) and an unmeasured probe $b$ connected to the bulk node. Note that $2S(b)$ bounds $I(A:b)$. This undecorated construction uses $\mu=1$, which leads to a sharper surface compared to stronger initial squeezing. The expected minimal surface is drawn in dark pink. The boundary regions $R$ are (a) a small connected region (b) a large connected region (c) two disconnected regions with an expected disconnected entanglement wedge, and (d) two disconnected regions with an expected connected entanglement wedge.
• Figure S1: An illustration of the decoration construction. Left: The initial bulk graph (blue nodes) used to form the decorated graph. In this example, the outermost four sites are chosen to be adjacent to the boundary. Right: The added boundary nodes (green) are connected to the pre-selected adjacent bulk nodes. All bulk-to-bulk edges are also divided with split nodes (yellow), with the adjacent pair of links assigned $+$ or $-$ in the coupling matrix.