Proposition: Two-preparation leakage witness

For any orthonormal pair $|0⟩,|1⟩$ in the code, define

Every decoder satisfies

Proof. Use a maximally entangled input on that two-dimensional subcode. If the decoder error is $ϵ$, its overlap with the corresponding Bell vector is at least $1-ϵ$. Dilate the decoder and project the global pure state onto that Bell vector. The normalized projected state is a Bell vector times an environmental vector, and its distance from the original pure state is at most $\sqrt{ϵ}$. Partial trace gives

Pinching the two-dimensional reference in the pair basis yields

The triangle inequality bounds the left side below by $\ell /2$.