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All complex Hadamard matrices in dimensions two to five are known. We use this fact toderive all inequivalent sets of mutually unbiased (MU) bases in low dimensions. We finda three-parameter family of triples of MU bases in dimension four and two inequivalentclasses of MU triples in dimension five. We confirm that the complete sets of (d + 1)MU bases are unique (up to equivalence) in dimensions below six, using only elementaryarguments for d less than five.

All mutually unbiased bases in dimensions two to five