The network density matrix formalism allows for describing the dynamics of information on top of complex structures and it has been successfully used to analyze, e.g., a system's robustness, perturbations, coarse-graining multilayer networks, characterization of emergent network states, and performing multiscale analysis. However, this framework is usually limited to diffusion dynamics on undirected networks. Here, to overcome some limitations, we propose an approach to derive density matrices based on dynamical systems and information theory, which allows for encapsulating a much wider range of linear and nonlinear dynamics and richer classes of structure, such as directed and signed ones. We use our framework to study the response to local stochastic perturbations of synthetic and empirical networks, including neural systems consisting of excitatory and inhibitory links and gene-regulatory interactions. Our findings demonstrate that topological complexity does not necessarily lead to functional diversity, i.e., the complex and heterogeneous response to stimuli or perturbations. Instead, functional diversity is a genuine emergent property which cannot be deduced from the knowledge of topological features such as heterogeneity, modularity, the presence of asymmetries, and dynamical properties of a system.