Theorem: Transfer of the same specified receiver

Suppose, after access-preserving identifications,

and the ideal decoded error is at most $e$. Then

If the complete ideal complement is within $\zeta$ of replacement, the actual complete complement has leakage at most $\min \{1,\zeta +\eta \}$, provided the comparison includes its full residual.

Proof. Partial trace contracts the complete-channel comparison. Insert the ideal accessible channel and ideal decoder in the triangle inequality, and apply CPTP contraction. Apply the same argument to the complete complementary output and the specified ideal replacement state.