A sharp norm inequality for entanglement-breaking channels

For a channel $Φ$ on $M_d$ written in the generalised Bloch parameterisation $r \mapsto Ar+c$, we prove that entanglement breaking implies $\|A\|_*^2 + \frac{d(d-1)}{2}|c|^2 \le (d-1)^2$, where $\|\cdot\|_*$ denotes the nuclear norm. The bound is attained in every dimension by the completely dephasi