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We provide a construction of sets of (d/2+1) mutually unbiased bases (MUBs)in dimensions d=4,8 using maximal commuting classes of Pauli operators. We showthat these incomplete sets cannot be extended further using the operators ofthe Pauli group. However, specific examples of sets of MUBs obtained using ourconstruction are shown to be strongly unextendible; that is, there does notexist another vector that is unbiased with respect to the elements in the set.We conjecture the existence of such unextendible sets in higher dimensions(d=2^{n}, n>3) as well. Furthermore, we note an interesting connection betweenthese unextendible sets and state-independent proofs of the Kochen-SpeckerTheorem for two-qubit systems. Our construction also leads to a proof of thetightness of a H_{2} entropic uncertainty relation for any set of three MUBsconstructed from Pauli classes in d=4.

Unextendible Mutually Unbiased Bases from Pauli Classes