Solution to the King's problem with observables being not mutually complementary

Abstract

We investigate the King's problem of the measurement of operators ${\vec{n}}_k\cdot \vec{\sigma }(k=1,2,3)$ 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.

7 pages