The paper gives a criterion for a system of vectors to satisfy the Welch bounds with equality for any k. By the correspondence between MUBs and complex projective 2-designs described in , this result is applicable (with k = 2) for complete systems of MUBs. Based on it, we have obtained a necessary and sufficient condition on the set of bases to be mutually unbiased. In particular, we use it to obtain new simpler proofs for known families of complete systems of MUBs. Combinatorial problems are stated that are equivalent to the existence of a complete set of MUBs in particular special cases
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