Recovery optimization on orthogonal flagged branches

The coordinatewise operations make the tuple space a complex vector space. The pairing in the Statement is linear in its first argument and conjugate symmetric by [F5]. Its value on $(v,v)$ is a finite sum of nonnegative terms, vanishing exactly when every $v_j=0$, so it is positive definite and hence an inner product. Extending a basis of each $K_j$ by zero in the other coordinates gives a finite basis of $K$: these vectors span coordinatewise, and a linear relation vanishes in each coordinate only when all coefficients vanish. Thus $K$ is a finite-dimensional complex inner-product space.

Each branch supremum is attained. Indeed, in fixed bases, [F3] identifies branch decoders with positive semidefinite Choi matrices $J$ whose output partial trace is $I_{K_j}$. This set is nonempty: the replacement channel $Y\mapsto \mathrm{tr}(Y)I_H\mathrm{/}d$ is completely positive and trace-preserving. The set is closed by its matrix inequalities and linear partial-trace equations. It is bounded because $\mathrm{tr}J=\dim K_j$, its diagonal entries are nonnegative, and positivity of each $2\times 2$ principal submatrix gives $\mathrm{\mid }{J}_{ab}{\mathrm{\mid }}^2\le J_{aa}{J}_{bb}$. The matrix coordinates form a finite-dimensional real Euclidean space, so [F4] gives compactness. The Choi formula determines a map by its values on matrix units, and the overlap in [F2] is a linear function of those values; hence the branch objective is continuous. By [F4] it attains a maximum.

Define the coordinate map $q_j:K\to K_j$ by $q_j(v)=v_j$. For $x\in K_j$ and $v\in K$, the pairing gives $⟨{\iota }_{j}x,v{⟩}_K=⟨x,v_j{⟩}_{K_j}$, so $q_j$ is the adjoint ${\iota }_j^{\ast }$ by [F6]; uniqueness follows from [F5], since a vector orthogonal to every vector, including itself, is zero. Also $⟨{\iota }_{j}x,{\iota }_{j}y{⟩}_K=⟨x,y{⟩}_{K_j}$, so each inclusion is an isometry. Coordinate evaluation yields ${\iota }_j^{\ast }{\iota }_j=I_{K_j}$ and ${\iota }_k^{\ast }{\iota }_j=0:K_j\to K_k$ when $k\mathrm{\ne }j$. The map $P_j={\iota }_j{q}_j$ retains exactly coordinate $j$; it is the identity on $\mathrm{im}{\iota }_j$ and zero on the other coordinate subspaces, which are orthogonal to that image by the summed pairing. For every $v\in K$, the vector $P_{j}v$ lies in $\mathrm{im}{\iota }_j$ and the residual $v-P_{j}v$ has zero coordinate $j$, so it is orthogonal to every vector in that image by [F8]. Thus [F9], viewed as an endomorphism of $K$, identifies $P_j$ as the orthogonal projector onto $\mathrm{im}{\iota }_j$, and ${\sum }_j{P}_j=I_K$ because the sum restores every coordinate of every tuple.

For any decoder $\mathcal{R}$ on $K$, define ${\mathcal{R}}_j(Y)=\mathcal{R}({\iota }_{j}Y{\iota }_j^{\ast })$. Conjugation by ${\iota }_j$ is completely positive, since every finite amplification is conjugation by an inclusion, and it preserves trace because ${\iota }_j^{\ast }{\iota }_j=I_{K_j}$. Composition with $\mathcal{R}$ therefore makes each ${\mathcal{R}}_j$ a channel. By linearity of the channel action and of the overlap in [F2], $F_e(\mathcal{R}\mathcal{N})={\sum }_j{p}_j{F}_e({\mathcal{R}}_j{\mathcal{N}}_j)$. Thus every left-side value is at most the sum of the branchwise suprema.

Choose a maximizing channel ${\mathcal{R}}_j$ for each of the finitely many branches and define $\mathcal{R}(Y)={\sum }_j{\mathcal{R}}_j({\iota }_j^{\ast }Y{\iota }_j)$. Every compression is completely positive, and a finite sum of completely positive maps is completely positive. Also $\mathrm{tr}\mathcal{R}(Y)={\sum }_j\mathrm{tr}({\iota }_j^{\ast }Y{\iota }_j)=\mathrm{tr}Y$, since ${\sum }_j{P}_j=I_K$; hence $\mathcal{R}$ is a decoder. The coordinate identities in step 2.1 give ${\iota }_j^{\ast }{\iota }_j=I_{K_j}$ and ${\iota }_k^{\ast }{\iota }_j=0$ for $k\mathrm{\ne }j$, so $\mathcal{R}({\iota }_{j}Y{\iota }_j^{\ast })={\mathcal{R}}_j(Y)$ for every $Y\in \mathcal{L}(K_j)$. The identity in step 3.1 therefore gives the sum of the branch maxima. This achieves the upper bound from step 3.1, proving the equality and the asserted construction.