Colafrancheschi, Eugenia
(2022)
Emergent spacetime properties from entanglement.
PhD thesis, University of Nottingham.
Abstract
The last century saw the development of two revolutionary theories: Einstein’s general relativity, which taught us that spacetime is a physical system, i.e. the gravitational field; and quantum mechanics, which revealed properties of microscopic systems foreign to the way we perceive reality in our everyday life, such as nonlocality. We are now trying to reach a new turning point: the formulation of a theory of quantum gravity, capable of reconciling these revolutionary discoveries from microscopic to macroscopic scales, i.e. describing physical scenarios in which both gravity and quantum mechanics do enter, and matching validated theories for those where only one (or none) of the two does.
Various backgroundindependent approaches to the problem of quantum gravity postulate that, at the Plack scale, spacetime dissolves into a microstructure of discrete, pre geometric quantum entities, leading to the picture of continuum spacetime and geometry emerging from their collective behaviour. Entanglement, and more generally all quantum correlations generated by the interactions of the fundamental entities, are thus expected to play a key role in this phenomenon; moreover, in such a background independent hence purely relational setting, continuum spacetime and geometry have to be reconstructed from them. In recent years, several results supported this view by uncovering a close relationship between entanglement and spacetime geometry and topology. Furthermore, entanglement turned out to be deeply tied to a likely constitutive aspect of gravity: holography, which has been a guiding theme of research in quantum gravity since the formulation of the BekensteinHawking area law for the black hole entropy. Understanding the origin of the threefold connection among gravity, holography and entanglement would therefore be a major step towards the formulation of a theory of quantum gravity.
The work we are going to present tackles this issue from an informationtheoretic perspective, inspired by the cited view of spacetime properties emerging from the quantum information generated and transferred by the interactions among the fundamental entities. We specifically investigate the entanglement origin of the holographic behaviour of finite regions of quantum space modelled by spin networks. These are graphs coloured by quantum data encoding the geometric properties of elementary portions of space dual to their vertices, primarily known for providing a basis of the kinematic Hilbert space of loop quantum gravity. Our work starts with a characterization of spin networks as the entanglement skeleton of manybody states describing collections of “space quanta” in group field theory, a quantum gravity approach conceived to be a field theory of spacetime and interpreted as a secondquantization of loop quantum gravity. On the basis of the characterization of spin networks as graphs built up from the entanglement of space quanta, we establish a solid correspondence between this quantum gravity formalism and tensor networks, a quantum information language that realizes an efficient encoding of the entanglement structure of manybody states in the geometry of a network.
We rely on the correspondence we established between spin networks regarded as “entanglement graphs” and tensor networks to investigate the entanglement origin of holography in finite regions of space. On the basis of a bipartition of the quantum degrees of freedom associated to spin networks into bulk and boundary ones, we point out that every spin network state can be regarded as a map between these two sets, in the spirit of the ChoiJamiołkowski isomorphisms of quantum information theory. This enables the translation of the “static” properties of a spin network state into the “dynamic” properties of the corresponding flow of information from the bulk to the boundary. In particular, we show that requiring such a flow to be an isometry  which is a necessary condition for holography  translates into the reduced bulk state maximising the entanglement entropy. The latter is computed by leveraging the tensor network reading of spin networks: assuming a random distribution of weights associated to the individual vertices, we exploit random tensor network techniques to map the computation of the entropy to the evaluation of the free energy of a classical Ising model defined on the spin network graph. The analysis of such a statistical model allows us to highlight the relation between the combinatorial structure and colouring of a spin network and the holographic character of the bulktoboundary flow of information it defines.
We deepen the investigation of holography on spin networks composed of random vertices by studying the impact of the bulk entanglement on the boundary entropy, with the latter computed through the random tensor network technique mentioned above. We highlight a regime (in terms of colouring and combinatorics of the spin network) in which the boundary entropy is determined by the area of a bulk surface (corresponding to the domain wall of the dual Ising model), with corrections given by the entropy of the bulk region enclosed by that surface. We also show that increasing the entanglement content of the bulk triggers a phase transition from such a “holographic regime” to a regime in which the boundary entropy satisfies a “area+volume” law, up to the emergence of a black hole like region in the bulk (in the dual statistical picture, a region the Ising domain wall cannot access).
The work described so far focuses on spacetime properties emergent from the quantum correlations of the space atoms at the level of quantum discrete geometry. The continuum, classical limit of the emergent spacetime scenario, a major issue in any quantum gravity approach, is not considered. The last part of this thesis is however dedicated to a comparable issue addressed in quantum information theory: the quantumtoclassical transition problem, i.e. how classical reality (i.e. what we perceive in our everyday life) emerges from the quantum microstructure of our world. We specifically study features of the emergence of objectivity of observables within the framework of quantum Darwinism, where multiple observers acquire information on a quantum system by probing fragments of its environment.
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