Theorem: Complete-output restoration

For the complete isometry $V$, the map

is CPTP and satisfies

Moreover, for every code–reference operator $X$,

Proof. The compression term is completely positive. The second term is a positive-effect measurement followed by fixed-state preparation and is also completely positive. Their traces sum to $\mathrm{Tr}Y$, because $V^{†}V=I_C$.

On the encoded range, the second term vanishes and $V^{†}(VX{V}^{†})V=X$. This proves the left-inverse identity on every operator, and hence with every reference.

An isometric embedding preserves the nonzero singular values of an operator. Applying this to $I_Q\otimes V$ proves the trace-norm equality.