Theorem: Accessible patch recovery

The following sufficient recovery statements hold on all matrices on $C_L\otimes C_R$:

1. If $l\ge 3$, a decoder on $A$ satisfies

   $${D_A}^L{{\mathcal{N}}_{\mathcal{A}}}^W={\mathrm{Tr}}_{C_R}.$$

2. If $r\ge 3$, a decoder on $A$ satisfies

   $${D_A}^R{{\mathcal{N}}_{\mathcal{A}}}^W={\mathrm{Tr}}_{C_L}.$$

3. If $l\ge 3,r\ge 2$, or $l\ge 2,r\ge 3$, a decoder on $A$ recovers the entire two-qubit input:

   $${D_A}^{LR}{{\mathcal{N}}_{\mathcal{A}}}^W={\mathrm{id}}_{C_L\otimes C_R}.$$

All decoders are finite compositions of the locally constructed isometries and decoders. They preserve arbitrary reference correlations, including correlations with the spectator input.

Proof. Use the right-to-left orientation in Proposition prop:patchisometry. The left block is the two-input isometry $F_L$ on $C_L$ and the internal edge. Any three received left outputs recover these two input registers by Lemma lem:oriented. Discarding the recovered edge and all other outputs returns $C_L$. As a channel on the original two-qubit input, this is exactly ${\mathrm{Tr}}_{C_R}$, proving item one. The opposite orientation proves item two.

If three left outputs and two right outputs are received, first invert the left block on the received triple. This returns $C_L$ and the internal edge coherently. The recovered edge and the two right outputs constitute three boundary legs of the original right five-qubit encoding. Apply the corresponding decoder from Theorem thm:idealaccess to recover $C_R$. Return $C_L,C_R$. The other cardinality pattern uses the opposite orientation.

All intermediate identities are channel identities on their full input spaces, and hence remain valid on arbitrary entangled inputs and after independent reference amplification.