# Holographic coherent states from random tensor networks

## TL;DR

This paper introduces holographic coherent states as superpositions of various bulk geometries in random tensor networks, revealing their error correction properties and exponential suppression of overlaps, advancing holographic duality models.

## Contribution

It generalizes random tensor networks to include superpositions of geometries, establishing a basis of holographic coherent states with error correction features.

## Key findings

- Superpositions of geometries form an overcomplete basis for boundary states.
- Small fluctuations define code subspaces with error correction properties.
- Overlap between different geometries is exponentially suppressed based on geometric difference.

## Abstract

Random tensor networks provide useful models that incorporate various important features of holographic duality. A tensor network is usually defined for a fixed graph geometry specified by the connection of tensors. In this paper, we generalize the random tensor network approach to allow quantum superposition of different spatial geometries. We set up a framework in which all possible bulk spatial geometries, characterized by weighted adjacent matrices of all possible graphs, are mapped to the boundary Hilbert space and form an overcomplete basis of the boundary. We name such an overcomplete basis as holographic coherent states. A generic boundary state can be expanded on this basis, which describes the state as a superposition of different spatial geometries in the bulk. We discuss how to define distinct classical geometries and small fluctuations around them. We show that small fluctuations around classical geometries define "code subspaces" which are mapped to the boundary Hilbert space isometrically with quantum error correction properties. In addition, we also show that the overlap between different geometries is suppressed exponentially as a function of the geometrical difference between the two geometries. The geometrical difference is measured in an area law fashion, which is a manifestation of the holographic nature of the states considered.