Assigning temperatures to eigenstates

In the study of thermalization in finite isolated quantum systems, an inescapable issue is the definition of tem-perature. 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.

Automated summary

The study examines different methods of assigning temperature to energies or eigenstates in finite isolated quantum systems. It is found that the commonly used assignment based on the canonical energy-temperature relationship only depends on energy eigenvalues and not on the structure of eigenstates. The definition of temperature for eigenstates is considered by minimizing the distance between (full or reduced) eigenstate density matrices and canonical density matrices. The results show that the minimizing temperature depends on the distance measure chosen and matches the canonical temperature for the trace distance.