Maximum entropy distributions of wavefunctions at thermal equilibrium

TL;DR

This paper introduces a maximum entropy principle for quantum wavefunction ensembles at thermal equilibrium, revealing the importance of Rènyi divergence in defining the equilibrium state.

Contribution

It proposes the 'Scrooge ensemble' framework, establishing a novel maximum entropy principle that incorporates measurement entropy and Rènyi divergence for quantum systems.

Findings

The Rènyi divergence plays a crucial role in quantum thermal equilibrium.

Energy expectation alone is insufficient to define the equilibrium wavefunction distribution.

The proposed framework challenges traditional assumptions about quantum microstate ensembles.

Abstract

Statistical mechanics reveals that the properties of a macroscopic physical system emerge as an average over an ensemble of statistically independent microscopic subsystems, each occupying a specific microstate. In the study of quantum systems, these microstates can be chosen to correspond to the pure state wavefunctions of individual quantum systems. However, the physical principles that govern the distribution of a pure state wavefunction ensemble, even under conditions of thermal equilibrium, are not well established. For instance, the canonical Boltzmann distribution cannot be applied to wavefunctions because they lack a definite energy. In this manuscript, we present a maximum entropy principle for the quantum wavefunction ensemble at thermal equilibrium, the so-called Scrooge ensemble. We highlight that a constraint on the energy expectation value, or even the shape of the associated eigenstate distribution, fails to yield a valid equilibrium state. We find that in addition to these constraints, one must also constrain the measurement entropy to be equal to the R\'enyi divergence of the ensemble with respect to the Gibbs state, indicating that the R\'enyi divergence may have uninvestigated physical importance to thermal equilibrium in quantum systems.