Rotated readout and POVMs

For $|u_{\kappa }⟩=V|\kappa ⟩$, define the macro

Its law is $⟨u_{\kappa }|{\rho }_S|u_{\kappa }⟩$, and its residual is proportional to $(⟨u_{\kappa }{|}_S\otimes I)\psi$.

For a finite POVM $(E_x{)}_x$, the map

is an isometry because ${\sum }_x{E}_x=I$. After finite qubit encoding and zero-state padding, it extends to a unitary. Reading the outcome register and returning the quantum residual implements

A finite classical decoder handles non-power-of-two label sets. Unused padded labels have zero weight on the encoded input subspace. This is one implementation; the POVM alone does not specify all possible conditional maps.