= Conditional input ensemble after a local measurement
{title2=$\rho_C^{x,y}$}
For a bipartite input <density operator> $\rho^x_{AC}$ and a <POVM> $\{E_y\}$ on $A$, let $q(y|x)=\operatorname{Tr}[(E_y\otimes I)\rho^x]$. For $q(y|x)>0$, the conditioned input on $C$ is
$$
\rho_C^{x,y}=\frac{\operatorname{Tr}_A[(\sqrt{E_y}\otimes I)\rho^x(\sqrt{E_y}\otimes I)]}{q(y|x)}.
$$
Its positivity follows from the positive sandwich before the <partial trace>. A <quantum channel> on $C$ acts on these conditional <density operators> even if the original input is an <entangled state>.
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