Quantum-Coherent Thermodynamics: Leaf Typicality via Minimum-Variance Foliation

Abstract

Equilibrium statistical ensembles commute with the Hamiltonian and thus carry no coherence in the energy eigenbasis. We develop a framework in which energy fluctuations can retain genuinely quantum-coherent contributions. We foliate state space into ``minimum-variance leaves,'' defined by minimizing the average energy variance over all pure-state decompositions, with the minimum set by the quantum Fisher information. On each leaf we construct the least-biased state compatible with normalization and mean energy, defining a leaf-canonical ensemble. The Gibbs ensemble is recovered on the distinguished commuting leaf, while generic states are organized by their leaf label. This structure provides a natural setting to extend eigenstate thermalization beyond equilibrium via a ``leaf typicality'' hypothesis. According to that hypothesis, local observables depend only on the leaf and energy and are reproduced by a representative pure state drawn from the optimal ensemble, whose minimized energy spread reduces the complexity of time evolution.