2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/89406We 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.Revised version. To appear in the special issue of Algorithmica on Quantum Algorithms and Quantum CryptographyQuantum Physicstext