  The constituents of a complex system exchange information to function
properly. Their signalling dynamics often leads to the appearance of emergent
phenomena, such as phase transitions and collective behaviors. While
information exchange has been widely modeled by means of distinct spreading
processes -- such as continuous-time diffusion, random walks, synchronization
and consensus -- on top of complex networks, a unified and physically-grounded
framework to study information dynamics and gain insights about the macroscopic
effects of microscopic interactions, is still eluding us. In this article, we
present this framework in terms of a statistical field theory of information
dynamics, unifying a range of dynamical processes governing the evolution of
information on top of static or time varying structures. We show that
information operators form a meaningful statistical ensemble and their
superposition defines a density matrix that can be used for the analysis of
complex dynamics. As a direct application, we show that the von Neumann entropy
of the ensemble can be a measure of the functional diversity of complex
systems, defined in terms of the functional differentiation of higher-order
interactions among their components. Our results suggest that modularity and
hierarchy, two key features of empirical complex systems -- from the human
brain to social and urban networks -- play a key role to guarantee functional
diversity and, consequently, are favored.
