Thermal reduction of a two-mode squeezed vacuum (${\rho }_1=(1-q){\sum }_{n\ge 0}{q}^n|n⟩⟨n|$)

Tracing out one mode leaves geometric occupation probabilities with $q=|\lambda {|}^2$. In an oscillator basis of frequency $\omega$, they are Boltzmann weights with $k_{B}T=\hslash \omega /\ln (1/q)$. The joint state can be pure while each reduced state is mixed.