Abstract
A complete set of d + 1 mutually unbiased bases exists in a Hilbert space of dimension d whenever d is power of a prime. We discuss a simple construction of d + 1 disjoint classes (each one having d − 1 commuting operators) such that the corresponding eigenstates form sets of unbiased bases. One of these classes is diagonal and can be mapped to “ladder” operators by means of the finite Fourier transform. Using this idea, we naturally introduce the notion of quantum phase as complementary to the inversion. Relevant examples involving qubits and qutrits are discussed.
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