Example: High Bell fidelity need not give uniform accuracy

Let

The Choi overlap of ${\mathrm{Ad}}_W$ with identity is

which approaches one as $d$ grows. Nevertheless

Indeed, $W$ maps $(|0⟩+|1⟩)/\sqrt{2}$ to an orthogonal state. Every computational-basis beacon is unchanged. A decoder applying $W^{†}$ repairs the channel, but the uncorrected receiver has maximal worst-case error.