Assigning Temperatures to Eigenstates

TL;DR

This paper investigates various methods for assigning temperatures to eigenstates in finite quantum systems, comparing their consistency with canonical temperature and analyzing their dependence on system size and distance measures.

Contribution

It introduces and compares different temperature definitions for eigenstates, highlighting their relationships and differences in finite quantum systems.

Findings

Temperature from trace distance matches canonical temperature but matrices are not close.

Reduced density matrix temperatures fluctuate with system size.

In certain limits, different temperature definitions converge to the canonical temperature.

Abstract

In the study of thermalization in finite isolated quantum systems, an inescapable issue is the definition of temperature. We examine and compare different possible ways of assigning temperatures to energies or equivalently to eigenstates in such systems. A commonly used assignment of temperature in the context of thermalization is based on the canonical energy-temperature relationship, which depends only on energy eigenvalues and not on the structure of eigenstates. For eigenstates, we consider defining temperature by minimizing the distance between (full or reduced) eigenstate density matrices and canonical density matrices. We show that for full eigenstates, the minimizing temperature depends on the distance measure chosen and matches the canonical temperature for the trace distance; however, the two matrices are not close. With reduced density matrices, the minimizing temperature has fluctuations that scale with subsystem and system size but appears to be independent of distance measure. In particular limits, the two matrices become equivalent while the temperature tends to the canonical temperature.