Complementary decompositions and unextendible mutually unbiased bases

Unextendible sets of Mutually Unbiased Bases (MUBs) are examined from the point of view of complementary subalgebras. We show, that the linear span of less than d+1d+1 factors of Md⊗MdMd⊗Md does not contain pure states, and therefore some complementary decompositions give rise to unextendible sets of MUBs. We provide some new complementary decompositions, and thus prove strong unextendibility of some set of MUBs.