Spectral Bounds on Entropy and Ergotropy via Statistical Effective Temperature in Classical Polarization and Quantum Thermal States

Abstract

We formulate a unified definition of the statistical effective temperature (SET) 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, non-degenerate states, consistent with the third law. Using entropy-SET diagrams, we explore spectral bounds in two-, three-, and four-level systems, which reveal physically realizable entropy regions, rank-dependent constraints, and cusp-like 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.