Lemma: Small-access privacy of the connected encoding

If $|A|\le 2$ among the eight exterior legs, then

If only left exterior legs are accessible, their channel is

Proof. The second identity follows from the right trace used in the proof of Theorem thm:patch and

For two accessible legs on one side, block privacy proves the result.

For one leg on each side, first observe that $F^{†}QF=0$ for every nonidentity one-site exterior Pauli $Q$. Indeed, the reduction of $|T⟩⟨T|$ to its bulk leg, joining leg, and that exterior leg is maximally mixed; expanding its matrix elements gives this identity. Expectations of nonidentity Pauli operators supported on the selected two legs therefore vanish in (eq:W). Their Pauli expansion proves the stated replacement channel.