Thermal representation of a reduced coupled-oscillator ground state ($\nu =\sqrt{{\omega }_{+}{\omega }_{-}}$)

Tracing one oscillator out of a symmetric coordinate-coupled Gaussian ground state gives a thermal density operator for an effective oscillator of frequency $\nu$. Its geometric ratio is $q=[(\sqrt{{\omega }_{+}}-\sqrt{{\omega }_{-}})/(\sqrt{{\omega }_{+}}+\sqrt{{\omega }_{-}}){]}^2$. It is generally squeezed relative to the original bare-frequency number basis. The covariance ratio fixes the effective frequency rather than permitting an arbitrary thermal-basis identification.