Lemma: Helstrom optimization

For traceless Hermitian $Y$,

The positive-spectral projector $P_{+}(Y)$ attains the positive maximum.

Proof. Write $Y=Y_{+}-Y_{-}$, with positive parts of orthogonal supports. Since $\mathrm{tr}Y=0$, $\mathrm{tr}{Y}_{+}=\mathrm{tr}{Y}_{-}=\frac{1}{2}\Vert Y{\Vert }_1$. For an effect $M$,

Taking $M=P_{+}(Y)$ attains the upper endpoint.