Squeezed thermal state ($V_{1,2}=(2n_{\mathrm{th}}+1)e^{\mp 2r}/4$)

Applying a single-mode squeeze operator to a thermal state gives the displayed principal optical-quadrature variances. Their uncertainty product exceeds the pure-state minimum when $n_{\mathrm{th}}>0$. Sufficient squeezing is needed to put the quiet optical quadrature below the vacuum reference.