Characterizing complexity and criticality in quantum systems requires diagnostics that are both computationally tractable and physically insightful. We apply a measure of quantum state complexity for $n$-qubit systems, defined as the divergence between the Shannon or von Neumann entropy of the computational basis distribution and the second-order Rényi entropy. This quantity has already been used earlier termed as structural entropy and it is particularly powerful as the Rényi entropy is directly related to state purity, linear entropy, and the inverse participation ratio, providing a clear physical grounding. While other Rényi orders could be used, the second order offers a deep and established connection to these key physical quantities. We first validate the measure in canonical noise channels, showing it peaks at the boundary between quantum and classical regimes. We then demonstrate its power in many-body physics. For systems exhibiting a many-body localization transition, including deformed random matrix ensembles and a disordered Heisenberg spin chain, the complexity measure reliably signals the crossover from integrable or localized to quantum-chaotic or ergodic phases. Crucially, the maximum complexity occurs in the nonergodic yet extended states at the transition, precisely capturing the critical region where the system is neither fully localized nor thermalized. Furthermore, within the chaotic phase, the measure correlates with the survival probability of local excitations, revealing a spectrum of thermalization properties. Our results establish that the entropic complexity is a simple, versatile, and effective probe for identifying nontrivial quantum regimes and transitions giving a new and alternative insight into such systems.

Close

PrevNext

View figure in article

• Many-body localization
• Quantum chaos
• Qubits