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. Read on the Choi state of the channel, the inequality is an instance of the covariance-matrix criterion of Gittsovich et al.; what is added here is an elementary proof in Holevo form and a complete description of the equality cases. Apart from the constant channel onto a pure state, which shows that the coefficient $d(d-1)/2$ cannot be increased, equality holds exactly for the measure-and-prepare channels with rank-one effects $α_k|e_k\rangle\langle e_k|$ and pure outputs $|ψ_k\rangle\langleψ_k|$ whose overlaps are an affine image of those of the measurement directions, $|\langleψ_j|ψ_k\rangle|^2 = P + (1-P)\,(d|\langle e_j|e_k\rangle|^2 - 1)/(d-1)$ for a common $P = \mathrm{Tr}\,Φ(\mathbf{1}/d)^2$. These include the completely dephasing channel, the re-preparation channel of every rank-one POVM, a one-parameter family interpolating between dephasing and the constant channel, and non-unital non-projective examples already at $d = 2$. The qubit case reads $\|A\|_*^2+|c|^2\le1$ and extends to full rank a rank-two condition of Ruskai. The proof sharpens Ruskai's bound $\|A\|_*\le1$ by retaining the POVM completeness relation $\sum_k f_k=0$, which her argument discards; this recentres $A$ from a second moment into a cross-covariance, and is the entire content of the improvement.