## Abstract

For quantum systems having a finite number N of orthogonal states, we investigate a particular relation among different measurements, called "mutual unbiasedness," which we show plays a special role in the problem of state determination. We define two bases {|v_{i}〉} and {|w_{j}〉} to be mutually unbiased if all inner products across their elements have the same magnitude: |〈ν_{1}|w_{j}|= 1 √N for all i, j. Two non-degenerate measurements are defined to be mutually unbiased if the bases comprising their eigenstates are mutually unbiased. We show that if one can find N + 1 mutually unbiased bases for a complex vector space of N dimensions, then the measurements corresponding to these bases provide an optimal means of determining the density matrix of an ensemble of systems having N orthogonal states, in the sense that the effects of statistical error are minimized. We show further that the number of mutually unbiased bases one may find for a given N is at most N + 1. Finally, we show that N + 1 mutually unbiased bases do exist whenever N is a power of a prime, and we construct such bases explicitly.

Original language | English (US) |
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Pages (from-to) | 363-381 |

Number of pages | 19 |

Journal | Annals of Physics |

Volume | 191 |

Issue number | 2 |

DOIs | |

State | Published - May 1 1989 |

Externally published | Yes |

## ASJC Scopus subject areas

- Physics and Astronomy(all)