= Solution
For a block with nonzero typical dimension, write $P=P_\varepsilon^{(n)}$ and $Q=I-P$. Let the compressed register be a copy of the typical space plus a one-dimensional orthogonal failure flag. Choose an isometry $V:P\mathcal H^{\otimes n}\to\mathcal K_n$ onto the successful code sector, extended by zero on $Q$, and a normalized flag $|f\rangle$ orthogonal to its range. Choose any normalized vector $|\varphi\rangle$ in the source <Hilbert space>. Define
$$
\mathcal C_n(X)=VPXPV^\dagger+\operatorname{Tr}(QX)|f\rangle\langle f|,
$$
$$
\mathcal D_n(Y)=V^\dagger YV+\langle f|Y|f\rangle|\varphi\rangle\langle\varphi|.
$$
These are <CPTP maps>: the first performs the projective test and either encodes or prepares a flag; the second embeds the successful sector and prepares a fixed state on failure. Their trace identities follow from $V^\dagger V=P$ and $VV^\dagger+|f\rangle\langle f|=I_{\mathcal K_n}$. Thus <typical-subspace compression with a failure flag> implements
$$
\Lambda_n(X)=PXP+\operatorname{Tr}(QX)|\varphi\rangle\langle\varphi|.
$$
For a source vector $|\Psi_k\rangle$, let $a_k=\langle\Psi_k|P|\Psi_k\rangle=\alpha_k^2$. Its overlap with the decoded state is
$$
f_k=a_k^2+(1-a_k)|\langle\Psi_k|\varphi\rangle|^2\geq a_k^2\geq2a_k-1,
$$
where the final step is $(1-a_k)^2\geq0$. Averaging proves <average pure-source fidelity after typical projection>:
$$
\boxed{F_n\geq\sum_kp_k^{(n)}\alpha_k^4\geq2\sum_kp_k^{(n)}\alpha_k^2-1=2\operatorname{Tr}(\rho^{(n)}P)-1.}
$$
No orthogonality of the emitted vectors was used. The code dimension is $\dim\mathcal T_\varepsilon^{(n)}+1$; the extra flag has vanishing asymptotic rate cost. Exceptional small block lengths with empty typical space can use any channel, since they do not affect reliability.
Back to article page
