Statistical complexity and the structure of multi-qubit entangled states

Multi-qubit states and their entanglement structure have been of central importance to recent developments in quantum information science and condensed matter. In this work, we investigate the structure of pure multi-qubit entangled states under the lens of statistical complexity and the entropy-complexity (EC) plane. A concept and a tool well known in the domain of complex systems, in particular, time series analysis. This tool permits the construction of a low-dimensional representation of and, in principle, arbitrarily high-dimensional probability space. At the same time, and for a given value of the entropy, allows for the identification of high and low complexity probability vectors, thus defining a classification for these objects. Exploiting this conceptual framework and methodology, we characterize several well known multi-qubit states based on the EC coordinates of their subsystem marginals. We also investigate the structure of eigenstates of many-body Hamiltonians and their out-of-equilibrium dynamics. In the later case we focus on the comparison between local equilibration and decoherence. Finally, as a complementary application of the method, using the diagonal ensemble, we take a look into the road to equilibrium for a paradigmatic example of a many-body chaotic Hamiltonian and present a clear separation between quantum dynamics and dynamics driven by random Hamiltonians, which, nevertheless capture the system properties at equilibrium.