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

Abstract

We investigate the king's problem of the measurement of operators ${\stackrel{P\vec}{n}}_{k}∙\stackrel{P\vec}{\ensuremath{\sigma}}$ ($k=1$, 2, 3) instead of the three Cartesian components ${\ensuremath{\sigma}}_{x},{\ensuremath{\sigma}}_{y}$, and ${\ensuremath{\sigma}}_{z}$ of the spin operator $\stackrel{P\vec}{\ensuremath{\sigma}}$. Here, ${\stackrel{P\vec}{n}}_{k}$ are three-dimensional real unit vectors. We show the condition over three vectors ${\stackrel{P\vec}{n}}_{k}$ to ascertain the result for measurement of any one of these operators.