  We propose a theory for coupling matter fields with discrete geometry on
higher-order networks, i.e. cell complexes. The key idea of the approach is to
associate to a higher-order network the quantum entropy of its metric.
Specifically we propose an action having two contributions. The first
contribution is proportional to the logarithm of the volume associated to the
higher-order network by the metric. In the vacuum this contribution determines
the entropy of the geometry. The second contribution is the quantum relative
entropy between the metric of the higher-order network and the metric induced
by the matter and gauge fields. The induced metric is defined in terms of the
topological spinors and the discrete Dirac operators. The topological spinors,
defined on nodes, edges and higher-dimensional cells, encode for the matter
fields. The discrete Dirac operators act on topological spinors, and depend on
the metric of the higher-order network as well as on the gauge fields via a
discrete version of the minimal substitution. We derive the coupled dynamical
equations for the metric, the matter and the gauge fields, providing an
information theory principle to obtain the field theory equations in discrete
curved space.
