Spectral Rigidity and Complexity from Fisher–Entropy Balance Applications to Open Quantum Systems

We introduce a scalar measure of structural complexity-the rigidity ratio R[P ] = I[P ]/S[P ]-that captures the balance between Fisher information I[P ] and Shannon entropy S[P ] in spectral distributions. Applied to the Brody and Ginibre ensembles, R[P ] detects nontrivial complexity maxima that elude conventional spectral statistics. In the Brody ensemble, we identify a peak at an intermediate level repulsion exponent β ≈ 0.45, indicating a regime of maximal structural sensitivity. In the Ginibre ensemble, R[P ] tracks the breakdown of Hermitian rigidity in open quantum systems. Our results suggest that R[P ] offers a robust, interpretable, and computa-tionally tractable diagnostic of complexity transitions across a range of quantum systems.