An explicit attenuator and its separate budgets

Let the system and bath have $m$ oscillator modes. A passive beam-splitter unitary has output annihilators

Prepare an independent bath state $\tau$ with $\mathrm{Tr}(\tau b_j)=0$ and finite total number $n_B$. Tracing the bath defines a channel ${\mathcal{A}}_{\eta }$. This is a standard attenuation construction HolevoWerner; the needed identities are proved below.