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

TL;DR

This paper introduces a framework for quantum-coherent thermodynamics using minimum-variance foliations, extending eigenstate thermalization beyond equilibrium by organizing states into leaves based on energy fluctuations.

Contribution

It develops a novel geometric structure of state space with minimum-variance leaves, enabling a new approach to quantum thermodynamics and eigenstate thermalization.

Findings

Defines minimum-variance leaves based on quantum Fisher information.

Constructs leaf-canonical ensembles for each leaf.

Extends eigenstate thermalization hypothesis beyond equilibrium.

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.