Trine state ensemble ($p_j=1/3,|t_j⟩=(|0⟩+e^{2\pi ij/3}|1⟩)/\sqrt{2}$)

Three equally weighted pure qubit states with coplanar unit Bloch vectors separated by $120$ degrees. One example is $|t_j⟩=(|0⟩+{\omega }^j|1⟩)/\sqrt{2}$, with $\omega =e^{2\pi i/3}$ and $j=0,1,2$. The roots-of-unity identity $1+\omega +{\omega }^2=0$ cancels the average off-diagonal matrix entries, so $\frac{1}{3}{\sum }_j|t_j⟩⟨t_j|=I/2$.

This is an ensemble of preparation states, whereas a trine qubit POVM consists of measurement effects $E_j=\frac{2}{3}|t_j⟩⟨t_j|$. The POVM–ensemble duality for the maximally mixed state relates them but does not identify their operational roles.