Solution to the king’s problem with observables that are not mutually complementary

We investigate the king’s problem of the measurement of operators ${\vec{n}}_k\bullet \vec{\sigma }$ ($k=1$, 2, 3) instead of the three Cartesian components ${\sigma }_x,{\sigma }_y$, and ${\sigma }_z$ of the spin operator $\vec{\sigma }$. Here, ${\vec{n}}_k$ are three-dimensional real unit vectors. We show the condition over three vectors ${\vec{n}}_k$ to ascertain the result for measurement of any one of these operators.