Abstract

In recent years, the import of quantum information techniques in quantum gravity opened new perspectives in the study of the microscopic structure of spacetime. We contribute to such a program by establishing a precise correspondence between the quantum information formalism of tensor networks (TN), in the case of projected entangled-pair states (PEPS) generalised to a second-quantized framework, and group field theory (GFT) states, and by showing how, in this quantum gravity approach, discrete spatial manifolds arise as entanglement patterns among quanta of space, having a dual representation in terms of graphs and simplicial complexes. We devote special attention to the implementation and consequences of the label independence of the graphs/networks, corresponding to the indistinguishability of the space quanta and representing a discrete counterpart of the diffeomorphism invariance of a consistent quantum gravity formalism. We also outline a relational setting to recover distinguishability of graph/network vertices at an effective and physical level, in a partial semi-classical limit of the theory.

Highlights

  • Seems to be played by the quantum phenomenon of entanglement

  • In recent years, the import of quantum information techniques in quantum gravity opened new perspectives in the study of the microscopic structure of spacetime. We contribute to such a program by establishing a precise correspondence between the quantum information formalism of tensor networks (TN), in the case of projected entangled-pair states (PEPS) generalised to a second-quantized framework, and group field theory (GFT) states, and by showing how, in this quantum gravity approach, discrete spatial manifolds arise as entanglement patterns among quanta of space, having a dual representation in terms of graphs and simplicial complexes

  • In this paper we detail a correspondence between the quantum states of the quantum gravity formalism of group field theory (GFT) [10] and the quantum information language of tensor networks (TN) [11,12,13,14], which defines a promising framework to perform both tasks, thanks to the importing of quantum information techniques in a proper quantum gravity setting

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Summary

The GFT formalism

A GFT is a field theory whose domain is given by (d copies of) a group manifold and characterized by combinatorially non-local interactions. Additional conditions are imposed, normally at the level of the GFT dynamics, in 4d gravitational models, where the group is taken to be SU(2) or SL(2, C) or Spin(4), to ensure the proper geometric interpretation of the GFT quanta and the discrete structures they form. These geometric aspects, while crucial for the interpretation of the formalism in a quantum gravity context, are not directly relevant for our present purposes. The GFT Fock space is constructed starting from a vacuum state |0 annihilated by φ(gx), with the fundamental simplices created by the action of φ†(gx) on |0

From the single-vertex Hilbert space to the Fock space
Spin-representation of the GFT wavefunctions
Graphs and their adjacency matrix description
GFT labelled-graph states
Quantum geometry states associated to graphs
Embedding Hγ into HV
Constructing graph states with arbitrary combinatorial pattern in HV
Comparing graph states of equal size
Labelled-graph states from individually weighted vertices
GFT unlabelled-graph states
Unlabelled-graph states from individually weighted vertices
Combinatorial scalar product
Effective distinguishability of vertices
The quantum information tool of Tensor Networks
Matrix product states
Projected entangled-pair states
GFT graph states as PEPS
10 Discussion
11 Conclusions and outlook
A Scalar product between graph basis states
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