Mutually unbiased bases

Summary

Two orthonormal bases of $\C^d$ are mutually unbiased if every vector of one has the same overlap with every vector of the other. A measurement in one basis then gives no information about a state prepared in the other. At most $d + 1$ such bases exist, and $d + 1$ exist when $d$ is a prime power. They are used in quantum key distribution and in state tomography.