Specified complete decoder

For $q=\sqrt{\eta }{e}^{\mathrm{i}\alpha }\ne 0$, set $C_{\alpha }=e^{-\mathrm{i}\alpha N_{\mathrm{R}}}$; at $q=0$, fix any phase. Let $W_{\mathrm{e}}$ embed logical $0,1$ as bath words $00,11$, with all unused modes vacuum, and $P_{\mathrm{e}}=W_{\mathrm{e}}{W_{\mathrm{e}}}^{†}$. Define

The retained/off-code Kraus construction makes this map CPTP on every eight-mode bath input.