Typical-subspace compression with a failure flag ($F_e({\pi }^{\otimes n},{\mathcal{N}}_n)\ge [\mathrm{Tr}({\pi }^{\otimes n}\Pi ){]}^2$)

Measure the quantum typical subspace projector $\Pi$. Encode its successful sector isometrically and reserve one orthogonal compressed flag for failure; decode the flag to a fixed state. The composite channel is ${\mathcal{N}}_n(\tau )=\Pi \tau \Pi +\mathrm{Tr}[(I-\Pi )\tau ]|{\phi }_0⟩⟨{\phi }_0|$. It is trace preserving. The Kraus formula for entanglement fidelity gives the displayed lower bound, while convexity of the square gives the same bound for average pure-source fidelity. One flag adds only a vanishing asymptotic rate overhead.