Solution to the mean king’s problem with mutually unbiased bases for arbitrary levels

The mean king’s problem with mutually unbiased bases is reconsidered for arbitrary $d$-level systems. Hayashi et al. [Phys. Rev. A 71, 052331 (2005)] related the problem to the existence of a maximal set of $d-1$ mutually orthogonal Latin squares, in their restricted setting that allows only measurements of projection-valued measures. However, we then cannot find a solution to the problem when, e.g., $d=6$ or $d=10$. In contrast to their result, we show that the king’s problem always has a solution for arbitrary levels if we also allow positive operator-valued measures. In constructing the solution, we use orthogonal arrays in combinatorial design theory.