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Generation of bases with definite factorization for an n-qubit system and mutually unbiased sets construction

We propose a systematic procedure to construct all the possible bases with a definite factorization structure (eigenstates of n commuting monomials constructed as products of Pauli operators) for an n-qubit system, as well as the possibility of collecting them into mutually unbiased sets. We also discuss an algorithm for the determination of basis separability and propose a criteria for complementarity between such bases. The results are applied to generate non-isomorphic complete sets of mutually unbiased bases.