### Abstract

The mean king's problem with mutually unbiased bases is reconsidered for arbitrary d -level systems. Hayashi [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.

Original language | English |
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Article number | 050301 |

Journal | Physical Review A - Atomic, Molecular, and Optical Physics |

Volume | 73 |

Issue number | 5 |

DOIs | |

Publication status | Published - 2006 May 15 |

### ASJC Scopus subject areas

- Atomic and Molecular Physics, and Optics

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## Cite this

Kimura, G., Tanaka, H., & Ozawa, M. (2006). Solution to the mean king's problem with mutually unbiased bases for arbitrary levels.

*Physical Review A - Atomic, Molecular, and Optical Physics*,*73*(5), [050301]. https://doi.org/10.1103/PhysRevA.73.050301