We construct an Abelian algebra built by maximally entangled pure states. The partial transposition maps this algebra for odd dimensions into a full matrix algebra. ppt-states are in one-to-one correspondence to states with a positive definite Wigner function. Special extremal ppt-states correspond to projections of various dimensions. In particular, we recover the projections corresponding to a complete set of mutually unbiased bases in prime dimensions.
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Heide Narnhofer (2006) studied this question.