Thermal Equilibrium Distribution in Infinite-Dimensional Hilbert Spaces

Abstract

The thermal equilibrium distribution over quantum-mechanical wave functions is a so-called Gaussian adjusted projected (GAP) measure, $GAP({\rho }_{\beta })$, for a thermal density operator ${\rho }_{\beta }$ at inverse temperature $\beta$. More generally, $GAP(\rho )$ is a probability measure on the unit sphere in Hilbert space for any density operator $\rho$ (i.e., a positive operator with trace 1). In this note, we collect the mathematical details concerning the rigorous definition of $GAP(\rho )$ in infinite-dimensional separable Hilbert spaces. Its existence and uniqueness follows from Prohorov's theorem on the existence and uniqueness of Gaussian measures in Hilbert spaces with given mean and covariance. We also give an alternative existence proof. Finally, we give a proof that $GAP(\rho )$ depends continuously on $\rho$ in the sense that convergence of $\rho$ in the trace norm implies weak convergence of $GAP(\rho )$.

12 pages LaTeX, no figures

References in corpus (4)

• Typicality of pure states randomly sampled according to the Gaussian adjusted projected measure
• Universal Probability Distribution for the Wave Function of a Quantum System Entangled with Its Environment
• Smoothness of Wave Functions in Thermal Equilibrium
• Spin and the Thermal Equilibrium Distribution of Wave Functions

Cited by in corpus (2)

• Canonical Typicality For Other Ensembles Than Micro-Canonical
• Long-Time Behavior of Typical Pure States from Thermal Equilibrium Ensembles