Building Holographic Entanglement by Measurement

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.

💡 Research Summary

The paper introduces a concrete, experimentally feasible protocol for engineering quantum many‑body states whose entanglement structure obeys the holographic Ryu‑Takayanagi (RT) formula. The authors begin by selecting a desired two‑dimensional bulk geometry—such as a hyperbolic disk or an eternal‑black‑hole wormhole—and discretizing it into a graph whose vertices represent bosonic modes (oscillators) and whose edges encode quadratic x‑x couplings. The graph is partitioned into bulk and boundary vertices.

An initially unentangled Gaussian product state is prepared, with each mode squeezed by a common parameter μ (μ=1 corresponds to the vacuum, μ<1 to a squeezed state). The system then undergoes a “quench”: it evolves for a fixed time t=1 under the quadratic Hamiltonian H_q = ½∑{ij} J{ij} x_i x_j, where J is the adjacency matrix of the graph (or a signed version thereof). This evolution is a symplectic transformation S(t) that updates the covariance matrix from V₀ to V_quenched = S V₀ S^T. Importantly, the resulting correlations depend only on the ratio t/μ, so the squeezing strength effectively controls the entangling power of the quench.

After the quench, all bulk modes are measured in the momentum basis (homodyne detection). Because the state is Gaussian, the conditional covariance matrix of the unmeasured boundary modes can be computed analytically: V_{bdy|bulk}=V_{bdy}−C(Π_p V_{bulk} Π_p)^{+}C^T, where C encodes bulk‑boundary correlations and Π_p projects onto momentum quadratures. The conditional mean depends linearly on the measurement outcomes, but a simple feed‑forward displacement can zero it, yielding a pure boundary state that is independent of the specific measurement results.

The authors then evaluate the entanglement entropy S(ℓ) of contiguous boundary subregions of length ℓ. For the hyperbolic‑disk graph, the numerically obtained S(ℓ) fits the 1+1‑dimensional CFT ground‑state formula S(ℓ)= (c/3) ln