Corollary: A unique full residual ground state is additional data

Suppose the complete residual has a positive Hamiltonian $H_B$ with a one-dimensional zero-energy eigenspace spanned by $|b_0⟩$. If

then $\mathcal{E}={\mathcal{T}}_{|b_0⟩⟨b_0|}$ and outgoing recovery is exact. Zero energy of a selected subsystem, or zero energy in a degenerate complete ground space, does not imply this conclusion.

Proof. A positive state of zero expectation for a positive Hamiltonian is supported on its kernel. The one-dimensional kernel therefore forces $\mathcal{E}(\rho )=|b_0⟩⟨b_0|$ for every logical state. States linearly span the logical operator space, so the channel is the stated replacement channel. Theorem thm:exact gives recovery. Proposition prop:insufficiency has a degenerate residual ground space containing $L$ and proves the nonimplication.