Quantum and classical entropic complexity of the thermal state: coherence, decoherence, and the ergodic-to-localized crossover in random-matrix and many-body models

Imre Varga; Onur Aktan

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arXiv · 2607.15798

Quantum and classical entropic complexity of the thermal state: coherence, decoherence, and the ergodic-to-localized crossover in random-matrix and many-body models

Imre Varga·Onur Aktan

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Abstract

Does thermal averaging preserve signatures of eigenstate complexity? We study the entropic complexity C = S_1 - S_2 (Shannon minus second-order Renyi entropy) across the ergodic-to-localized crossover of three disordered models: the Rosenzweig-Porter (RP) ensemble, the power-law banded random matrix (PLBRM) ensemble, and the random-field Heisenberg chain. We compare a wavefunction-level quantity C_eig to the complexity of the thermal (Gibbs) state before and after pointer-basis dephasing, via the trace (C_tr) and diagonal (C_diag) complexities. The answer is mostly no: thermal averaging strongly suppresses, but does not eliminate, eigenstate-complexity signatures. C_eig develops a pronounced mid-phase maximum in RP and, at matching fractal dimension, in the structurally independent PLBRM model; a high-statistics scan resolves a weak (about 10%) but reproducible thermal shadow of this peak in the thermal diagonal complexity. This feature is confined to the two random-matrix models; in the Heisenberg chain the eigenstate complexity instead peaks at the many-body localization transition. A second, genuinely thermal feature, an edge just inside the ergodic phase, has no eigenstate counterpart and, in PLBRM, recedes with system size. Both features are tracked by scale-invariant crossover indicators: the log-ratio of the trace and diagonal crossover temperatures, which vanishes upon localization, and the relative entropy of coherence. Entropic complexity thus cleanly separates thermal-state and eigenstate physics: a sharp wavefunction-level feature, reproducible across unrelated random-matrix constructions, leaves only a faint, structurally distinct imprint on thermal observables.

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Imre Varga, Onur Aktan. 2026-07-17. Quantum and classical entropic complexity of the thermal state: coherence, decoherence, and the ergodic-to-localized crossover in random-matrix and many-body models. https://arxiv.org/abs/2607.15798

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