2016/06/27 by ChunJun Cao, Sean M. Carroll, Spyridon Michalakis · 2 voices · 215 citations
Mathematics · Physics and Astronomy · #Black Holes and Theoretical Physics #Cosmology and Gravitation Theories #Curvature #Geometry #Hilbert space #Mathematics #Physics #Projective Hilbert space #Quantum #Quantum entanglement #Quantum many-body systems #Quantum mechanics #Reproducing kernel Hilbert space #gr-qc #hep-th #quant-ph
paper · pdf · doi:10.1103/physrevd.95.024031
published in Physical review. D/Physical review. D. 95(2) (American Physical Society) · 37 pages, 5 figures. Updated notation, references, and acknowledgement
arxiv created 2016/12/15 · openalex publication_date 2017/01/27 · arxiv updated 2017/02/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
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 H into a tensor product of factors, we consider a class of ``redundancy-constrained states'' in 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.