Holographic Entanglement Achieves Pure States Via Measurement and Minimal Surfaces

Scientists are increasingly exploring the connections between quantum entanglement and gravity, and a new study details a method for building entanglement with properties predicted by holographic models. Jonathan Jeffrey, Lucien Gandarias, and Monika Schleier-Smith from Stanford University, alongside Brian Swingle from Brandeis University, demonstrate how to create quantum states exhibiting ‘holographic entanglement’ through a process of measurement and interaction. Their research is significant because it moves beyond simply observing these connections, offering a pathway to engineer entanglement structured by the geometry of a chosen ‘bulk’ space, validating key predictions like the Ryu-Takayanagi formula. By using only Gaussian operations and measurements on a discretised geometry, the team shows this approach is potentially realisable using existing quantum technologies, opening exciting possibilities for simulating gravity with quantum systems.

Holographic entanglement from discretised bulk geometries

At the fundamental level, gravity dominates at large scales, while quantum theory governs at the smallest scales, and this work seeks to bridge these regimes with a more general framework for Quantum gravity. The persistent challenge of directly accessing quantum effects in gravity has driven theoretical exploration of conceptual puzzles arising from combining quantum field theory and general relativity, such as the realisation that black holes possess entropy proportional to their area. This led to the concept of holographic duality, where spacetime geometry emerges from microscopic degrees of freedom residing on a lower-dimensional boundary. The study builds upon the anti-de Sitter space / conformal field theory correspondence, a precise duality between a quantum system without gravity and a quantum gravitational system.

Entanglement is central to this correspondence, encoding the emergence of spacetime, and the Ryu-Takayanagi formula connects boundary states to bulk states via the area of minimal surfaces. Researchers have long sought to extend this understanding beyond systems governed by Einstein gravity, aiming for a boundary side amenable to quantum simulation, as evidenced by several theoretical proposals and experiments. This work addresses the open challenge of preparing holographic boundary states for arbitrary bulk geometries, achieving this through a constant-time quench-and-measure protocol illustrated in Figure 1. Crucially, the resulting boundary state exhibits a holographic entanglement structure, with boundary subregion entropies approximately matching the Ryu-Takayanagi formula predictions for the chosen geometry. This framework not only offers a theoretical laboratory for investigating holographic entanglement but also provides an experimentally accessible method for engineering designer bulk geometries using photonic systems, atomic ensembles, or superconducting circuits. Furthermore, the generated states exhibit R enyi entropies that vary with the R enyi index and demonstrate power-law correlations, surpassing the capabilities of many existing holographic entanglement models.

Holographic state generation via quench dynamics offers insights

Initialising a collection of N quadrature operators, each with associated canonical momentum, into an unentangled Gaussian state with squeezing parameter μ, formed the basis of their experimental setup. The system’s evolution was then governed by a quench Hamiltonian, defined as Hq = 1/2 Σi,j Jijxixj, where J represents a real, symmetric coupling matrix corresponding to the adjacency matrix of the graph. Experiments employed a quench time of t = 1, after which measurements were performed on all bulk nodes in the momentum basis. Researchers meticulously tracked the transformation of the covariance matrix, V = ⟨ξξT⟩ − ⟨ξ⟩⟨ξ⟩T, throughout the quench and measurement stages, beginning with an initial covariance matrix of V0 = diag(1/(2μ)I, μ2I), where I denotes the identity matrix.

Solving the Heisenberg equation of motion, dξ/dt = i[Hq, ξ], revealed how the operators evolved under the quench, providing a detailed understanding of the resulting entanglement. Analysis focused on the entanglement entropies of connected boundary regions, comparing results with predictions from a (1+1)D conformal field theory with a fitted central charge of c = 6.5(3) and vertical offset ε = 5.36(8). Furthermore, the research extended beyond entanglement entropy to explore Rényi entropies, revealing their dependence on the Rényi index and identifying power-law correlations in certain observables, features often absent in previous holographic entanglement models. This innovative quench-and-measure protocol enables the engineering of designer bulk geometries and provides a theoretical laboratory for investigating holographic entanglement, surpassing the limitations of earlier tensor network models which typically lacked these crucial features. The resulting boundary states exhibit Rényi entropies that vary with the Rényi index and demonstrate clean power laws in correlation functions, confirming the success of this novel approach.

Holographic entanglement via Ryu-Takayanagi formula validation

For the disk graph, the fit value of the scaling parameter ‘c’ varies with ln 1 μ when μ is small, and the shape of the entropy curve is influenced by the magnitude of μ. These findings extend beyond simple tensor-network models, showcasing a richer entanglement structure. To further validate the framework, the team investigated a wormhole geometry constructed by joining two hyperbolic disk graphs. Measurements of entanglement entropy for regions on one side and both sides of the wormhole consistently aligned with minimal curves derived from the (2+1)D eternal AdS black hole geometry. The calculated entanglement entropy exhibited a rapid crossover between candidate minimal surfaces, precisely matching predictions from the RT formula, with a sharp change observed when these surfaces swapped dominance.

The study also introduced a ‘decoration’ procedure to generate power-law correlations, constructing graphs with additional bulk nodes and assigning alternating signs to edges. Applying this to a decorated hyperbolic disk graph with strong initial squeezing (μ = 0.05), the team found that both CFT ground-state entanglement entropy and position-position two-point function were approximately satisfied, yielding a scaling dimension of ∆x ≈1. Momentum-momentum correlations remained approximately constant with distance, indicating a scaling dimension ∆p ≈0, and the central charge fit saturated at c = 1, consistent with a (1+1)D free boson CFT. These results suggest a potential lattice regularization of the free boson ground state and open avenues for directly detecting the putative RT surface through bulk reconstruction.

Holographic entanglement from graph-based quantum states emerges

This research establishes a practical method for simulating aspects of holographic entanglement, potentially offering insights into the connection between quantum information and spacetime geometry. Future work may focus on refining the analytical understanding of the generated entanglement structure, exploring entropy inequalities, investigating metric reconstruction, and probing connections between wormholes and random Gaussian states, as suggested by the authors.