Proof

Proof. The accessible output has orthogonal vacuum and code blocks. Their eigenvalues are $q_N$ and $(1-q_N)$ times the eigenvalues of ${\rho }_C$. The complement has the reversed block weights. The entropy formulas follow, including endpoints by continuity.

For many cells, the emitted subset $S$ is an orthogonal flag with the displayed probability. Its conditional state is an isometric copy of ${\rho }_{C_S}$. The flag entropy is $K{h}_2(q_N)$. The entropy of orthogonal blocks gives (eq:manyentropy). Tracing an external reference first gives the same formulas for the reduced code state.