Orbits of mutually unbiased bases

We express Alltop?s construction of mutually unbiased bases as orbits under the Weyl?Heisenberg group in prime dimensions and find a related construction in dimensions 2 and 4. We reproduce Alltop?s mutually unbiased bases using abelian subgroups of the Clifford group in prime dimensions, in direct analogy to the well-known construction of mutually unbiased bases using abelian subgroups of the Weyl?Heisenberg group. Finally, we prove three theorems relating to the distances and linear dependencies among different sets of mutually unbiased bases.

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