A new proof for the existence of mutually unbiased bases
We develop a strong connection between maximally commuting bases of orthogonal unitary matrices and mutually unbiased bases. A necessary condition of the existence of mutually unbiased bases for any finite dimension is obtained. Then a constructive proof of the existence of mutually unbiased bases for dimensions which are power of a prime is presented. It is also proved that in any dimension d the number of mutually unbiased bases is at most d+1. An explicit representation of mutually unbiased observables in terms of Pauli matrices are provided for d=2^m.