Space from Hilbert Space: Recovering Geometry from Bulk Entanglement (1606.08444v3)

Abstract: We examine how to construct a spatial manifold and its geometry from the entanglement structure of an abstract quantum state in Hilbert space. Given a decomposition of Hilbert space $\mathcal{H}$ into a tensor product of factors, we consider a class of "redundancy-constrained states" in $\mathcal{H}$ that generalize the area-law behavior for entanglement entropy usually found in condensed-matter systems with gapped local Hamiltonians. Using mutual information to define a distance measure on the graph, we employ classical multidimensional scaling to extract the best-fit spatial dimensionality of the emergent geometry. We then show that entanglement perturbations on such emergent geometries naturally give rise to local modifications of spatial curvature which obey a (spatial) analog of Einstein's equation. The Hilbert space corresponding to a region of flat space is finite-dimensional and scales as the volume, though the entropy (and the maximum change thereof) scales like the area of the boundary. A version of the ER=EPR conjecture is recovered, in that perturbations that entangle distant parts of the emergent geometry generate a configuration that may be considered as a highly quantum wormhole.

• The paper introduces redundancy-constrained states that allow spatial geometry to emerge directly from mutual information between quantum subsystems.
• It employs a methodology that uses graph representations of Hilbert space and classical multidimensional scaling to reconstruct emergent spatial manifolds.
• The findings imply that perturbations in entanglement reflect curvature changes analogous to Einstein's equations, offering a fresh perspective on quantum gravity.

Insights into "Space from Hilbert Space: Recovering Geometry from Bulk Entanglement"

The paper "Space from Hilbert Space: Recovering Geometry from Bulk Entanglement" by Cao, Carroll, and Michalakis develops a framework for deriving spatial geometry directly from the entanglement structure of quantum states in Hilbert space. This investigation is situated within the broader context of foundational questions about the nature of geometry and gravitation in a quantum universe. By examining quantum states where geometry emerges from entanglement properties, the authors attempt to address key aspects of quantum gravity without the conventional need for a preceding spacetime manifold.

Research Findings

The authors propose the concept of "redundancy-constrained" (RC) states. A quantum state is identified as redundancy-constrained if the spatial geometry can be derived purely from the mutual information of its subsystems. Their approach involves the following key elements:

1. Hilbert Space Decomposition and Graph Representation: The paper begins with the assumption of a given decomposition of Hilbert space into a tensor product of finite-dimensional factors. These factors yield a graph where vertices correspond to subregions, and edge weights are determined by mutual information. This establishes a conceptual move from quantum states to geometric information graphs.
2. Distance Measure via Mutual Information: The authors propose using mutual information as a proxy for physical distance, assuming higher mutual information signifies closer interaction in the emergent spatial sense. They conceptualize an ansatz for the distance function based on mutual information, forming a graph metric to reconstruct a semblance of spatial geometry.
3. Numerical Reconstruction Using Classical Multidimensional Scaling: For redundant states exhibiting area-law behavior, multidimensional scaling is employed to approximate the reconstructed spatial manifold. This is exemplified through known states associated with one and two-dimensional systems, demonstrating a practical approach to identifying emergent dimensions.
4. Implications of Entanglement Perturbations: The research explores how small perturbations in entanglement alter the reconstructed spatial curvature, suggesting analogs to Einstein's equations in the spatial domain. This work closely examines the implications of both local and nonlocal perturbations on an emergent spatial geometry, connecting changes in modular Hamiltonian to localized curvature perturbations.
5. Speculative Bridge to Reaction Forces: By considering the entanglement first law in quantum systems, the authors propose a relationship between modular energy perturbations and the geometric curvature seen in classical gravity. This establishes a formal step towards associating entanglement structure with gravitational dynamics.

Theoretical Implications

The paper provides a theoretical pathway for understanding how spatial geometry can arise within a purely quantum framework, specifically without the prior existence of a defined spacetime manifold. This approach goes beyond the traditional bounds of considering geometry, specifically in quantum gravity research, and contributes to ongoing discourse about the gravitational manifestation of quantum phenomena. The implications suggest that the high-level structures dominant in quantum states could inherently contain the seeds for classical geometry.

Practical Outlook and Future Directions

While addressing the nontrivial task of emergent geometry from quantum states, the research opens gateways for exploring the geometry-entropy-energy relationship from first principles. The authors identify areas for future work, including developing a deeper understanding of Lorentz invariance emergence and dynamical field considerations. A thorough understanding of time evolution with possible extensions of this framework to encompass spacetime dynamics could yield insights into reconciling general relativity with quantum mechanics. By doing so, this research holds potential for reshaping fundamental theories that govern our universe’s structure at high energy scales.