Theorem: A constant complement can contain non-local magic

Let $0<\kappa <\pi /4$, and prepare the fixed syndrome

Define

Then

so ${\delta }_{\mathrm{full}}={{\epsilon }_{\mathrm{rec}}}^R=0$.

Nevertheless,

and the complete encoding Choi state has

Proof. Tracing $R_0{R}_1$ gives the displayed replacement channel. Discarding $R_1$ and returning $R_0$ recovers the code exactly.

The two nonzero eigenvalues of the $B$-reduction are unequal. Local unitaries preserve them, whereas a pure stabilizer reduction must have a flat nonzero spectrum by Lemma lem:flat. Thus no local unitaries convert ${\chi }_{\kappa }$ to a stabilizer state. The local unitary groups are compact and the stabilizer set is finite, so the minimum distance is strictly positive.

The Choi state is $|{\Phi }_d{⟩}_{Q{R}_0}\otimes |{\chi }_{\kappa }{⟩}_{R_{1}B}$. For the Choi magic statement, take a qubit code, or more generally $d=2^k$, with its fixed qubit structure. Across $QR:B$, its $B$-reduction has the same two unequal eigenvalues. The same argument applies.