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Spectral bounds on entropy and ergotropy via statistical effective temperature in classical polarization and quantum thermal states

Tariq Aziz1, Meng-Long Song1, Liu Ye1, Dong Wang1,*, José J. Gil2,†, and Sabre Kais3

  • *Contact author: dwang@ahu.edu.cn
  • †Contact author: ppgil@unizar.es
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Phys. Rev. Research 7, 033117 – Published 4 August, 2025

DOI: https://doi.org/10.1103/fhqs-g7c6

Abstract

We formulate a unified definition of the statistical effective temperature (SET), τd, for finite-dimensional classical and quantum systems, using dimension-dependent indices of purity derived from the eigenvalue spectrum. This spectral approach bypasses the need for Hamiltonians or energy gaps and remains applicable to both quantum density matrices and classical polarization coherency matrices. The SET framework naturally describes the divergence of inverse temperature near pure, nondegenerate states, consistent with the third law. Using entropy-SET (Sd−τd) diagrams, we explore spectral bounds in two-, three-, and four-level systems, which reveal physically realizable entropy regions, rank-dependent constraints, and cusplike features. A Hamiltonian-free parametric spectrum ansatz provides a universal reference curve within these bounds. Furthermore, we derive spectral bounds on ergotropy as a function of entropy and SET, which quantify the maximum extractable work under passive constraints and introduce the notion of structured states, engineered spectral configurations that saturate these bounds. Our analysis shows that SET serves as a thermodynamically meaningful and operationally relevant quantity for bounding entropy and ergotropy in both classical polarization systems and quantum thermal states.

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Phys. Rev. Research 7, 033117– Published 4 August, 2025
  • Received 1 May 2025
  • Accepted 8 July 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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