Lemma: Complete destructive Enter identity

For every finite retained factor $T$,

as linear maps on all operators. The map ${{\mathcal{M}}_{\mathcal{B}}}^T$ is CPTP, and

Consequently ${\mathcal{I}}^T$ is CPTP and

Proof. Let

Then

Thus the map is completely positive and trace preserving.

Expand an arbitrary operator as

Both the $r$-block of ${{\mathcal{M}}_{\mathcal{B}}}^T(X)$ and the $r$-block of ${{\mathcal{J}}_{\mathcal{B}}}^T(X)$ are $X_{rr}$. Hence the two maps agree on every operator, not merely on the source family.

Discarding the record gives ${\sum }_r{X}_{rr}={\mathrm{Tr}}_{B}X$, proving (eq:drop). Lemma lem:step proves that ${\mathcal{I}}^T$ is CPTP. Finally, any local trace-preserving map on $B$ leaves the partial trace over $B$ unchanged. In the present Kraus representation,

which proves (eq:drop-step).