Title: Holographic coherent states from random tensor networks,
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.