Barrier convention

Either start with the horizon state and apply its scattering map, or start with outgoing covariance ${\Gamma }_{\ell }{n}_{\mathrm{H}}$. Route A uses the latter and does not apply the greybody barrier twice.

An additional detector amplitude $d(\omega )$, with an independent vacuum loss environment, would give $q_{\mathrm{detected}}=|d{|}^2{q}_{\mathrm{out}}$. This is separately labelled detector loss. Beamsplitter vacuum dilations are appropriate under those input assumptions Weedbrook,Caruso.