How `hot' are mixed quantum states?

Abstract

Given a mixed quantum state $\rho$ of a qudit, we consider any observable $M$ as a kind of `thermometer' in the following sense. Given a source which emits pure states with these or those distributions, we select such distributions that the appropriate average value of the observable $M$ is equal to the average Tr$M\rho$ of $M$ in the stare $\rho$. Among those distributions we find the most typical one, namely, having the highest differential entropy. We call this distribution conditional Gibbs ensemble as it turns out to be a Gibbs distribution characterized by a temperature-like parameter $\beta$. The expressions establishing the liaisons between the density operator $\rho$ and its temperature parameter $\beta$ are provided. Within this approach, the uniform mixed state has the highest `temperature', which tends to zero as the state in question approaches to a pure state.