Measure-and-prepare channel

A measure-and-prepare channel measures a POVM $\{M_x\}$ and outputs a prescribed state ${\sigma }_x$: $T(\rho )={\sum }_x\mathrm{Tr}(M_x\rho ){\sigma }_x$. Such channels are exactly the entanglement-breaking channels.

A measurement channel writes a POVM's outcome into an orthogonal classical register:For any input orthonormal basis $|j⟩$, the Kraus operators $K_{a,j}=|a⟩⟨j|\sqrt{E_a}$ realize this map and satisfy ${\sum }_{a,j}{K}_{a,j}^{†}{K}_{a,j}=I$. It is therefore a trace-preserving completely positive map.

For a bipartite input density operator ${\rho }_{AC}^x$ and a POVM $\{E_y\}$ on $A$, let $q(y|x)=\mathrm{Tr}[(E_y\otimes I){\rho }^x]$. For $q(y|x)>0$, the conditioned input on $C$ isIts 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.