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arXiv:1609.01295 · Fortschritte der Physik

SpaceTime from Hilbert Space: Decompositions of Hilbert Space as Instances of Time

Mahdiyar Noorbala

Published Feb 1, 2018• 2 citations
arXivHTMLPDF

Abstract

Abstract There has been recent interest in identifying entanglement as the fundamental concept from which space may emerge. We note that the particular way that a Hilbert space is decomposed into tensor factors is important in what the resulting geometry looks like. We then propose that time may be regarded as a variable that parameterizes a family of such decompositions, thus giving rise to a family of spatial geometries. As a proof of concept, this idea is demonstrated in two toy models based on Kitaev's toric code, which feature a dynamical change of dimension and topology.

DOI
10.1002/prop.201800002

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Abstract There has been recent interest in identifying entanglement as the fundamental concept from which space may emerge. We note that the particular way that a Hilbert space is decomposed into tensor factors is important in what the resulting geometry looks like. We then propose that time may be regarded as a variable that parameterizes a family of such decompositions, thus giving rise to a family of spatial geometries. As a proof of concept, this idea is demonstrated in two toy models based on Kitaev's toric code, which feature a dynamical change of dimension and topology.
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\mathcal{H}H into a tensor product of factors, we consider a class of ``redundancy-constrained states'' in H\mathcal{H}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\mathrm{ER}=\mathrm{EPR}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.
Parameterized Families of Toric Code Phase: ememem-duality family and higher-order anyon pumping
Within the toric-code phase, we study parameterized families of topologically ordered states. We construct 111- and 222-parameter families of local Hamiltonians and confirm their non-triviality via topological pumping. For the 111-parameter family, we show that the ememem-exchange defect is pumped into the bond Hilbert space of a tensor-network representation. For the 222-parameter case, we construct a ``pump of a pump'' that transports an S1S^1S1-family of a system in one lower spatial dimension. Using similar methods, we also present a 111-parameter family with a higher-order anyon pump that produces corner-localized anyon modes. These constructions provide explicit lattice realizations and concrete diagnostics of family-level topology. We use recently developed boundary algebra methods to study the non-triviality of these families.
Causal set theory starts out from a central result in general relativity according to which the causal structure determines, up to a so-called ‘conformal factor’, the geometry of a relativistic spacetime. Space and time, it seems, must be part and parcel of the ontology of any physical theory, of any theory with a credible claim to being a physical theory, that is. After all, physics is the science of the fundamental constitution of the material bodies, their motion in space and time, and indeed of space and time themselves. There are two areas in physics in which space or spacetime may emerge from something qualitatively rather different: wave function monism in quantum mechanics and quantum theories of gravity. The former case concerns a particular interpretative issue in non-relativistic quantum mechanics. The severity and the characteristics of the disappearance of spacetime differ from one approach to quantum gravity to another.
Quantum entanglement, a phenomenon where two or more particles become linked and share the same fate, has captivated physicists and philosophers alike. This research explores the potential of quantum entanglement to form the foundation for a novel framework termed "Time Geometry," proposing that entangled states can encode and manipulate time itself. We propose a conceptual model leveraging the inherent non-locality of entanglement to construct a new analytical tool for understanding time sequences. This work explores the implications of this concept, specifically focusing on how entangled states might be used to represent and analyze temporal patterns, offering a departure from traditional methods and potentially revealing unique quantum effects in time.
A definition for the entanglement entropy in a gauge theory was given recently in arXiv:1501.02593. Working on a spatial lattice, it involves embedding the physical state in an extended Hilbert space obtained by taking the tensor product of the Hilbert space of states on each link of the lattice. This extended Hilbert space admits a tensor product decomposition by definition and allows a density matrix and entanglement entropy for the set of links of interest to be defined. Here, we continue the study of this extended Hilbert space definition with particular emphasis on the case of Non-Abelian gauge theories. We extend the electric centre definition of Casini, Huerta and Rosabal to the Non-Abelian case and find that it differs in an important term. We also find that the entanglement entropy does not agree with the maximum number of Bell pairs that can be extracted by the processes of entanglement distillation or dilution, and give protocols which achieve the maximum bound. Finally, we compute the topological entanglement entropy which follows from the extended Hilbert space definition and show that it correctly reproduces the total quantum dimension in a class of Toric code models based on Non-Abelian discrete groups.