Theorem: Complete-output restoration

For (eq:interface), choose any ${\tau }_C\in \mathcal{D}(C)$ and put $P=V{V}^{†}$. The map

is CPTP and satisfies

For every operator $X$ on $Q\otimes C$,

Thus all logical matrix units and all referenced state distinctions are preserved on the complete output. For a maximally entangled code–reference input, the complete output has

Proof. The first term of (eq:global-decoder) is a completely positive compression. The second is a positive-effect measurement followed by preparation of ${\tau }_C$, hence is completely positive. Their traces sum to

On $Y=VX{V}^{†}$, the second term vanishes and the first returns $X$. This proves the left-inverse identity on all operators.

An isometry preserves all nonzero singular values of an operator under left and right embedding. The same is true of $I_Q\otimes V$, proving (eq:isometric-trace-norm). The statement about matrix units also follows directly from the left inverse.

Finally, let

The state $(I_Q\otimes V)|{\Phi }_d⟩$ is pure with $d$ equal Schmidt coefficients. Its $Q$ and $RB$ entropies are both ${\log }_{2}d$, and its joint entropy vanishes.