NEW BOUNDS OF MUTUALLY UNBIASED MAXIMALLY ENTANGLED BASES IN Cd ⊗ Ckd

Mutually unbiased bases which is also maximally entangled bases is called mutually unbiased maximally entangled bases (MUMEBs). We study the construction of MUMEBs in bipartite system. In detail, we construct 2(p(a) - 1) MUMEBs in C-d circle times C-d by properties of Cuss sums for arbitrary odd d. It improves the known lower bound p(a) - 1 for odd d. Certainly, it also generalizes the lower bound 2(p(a) - 1) for d being a single prime power. Furthermore, we construct MUMEBs in C-d circle times C-kd for general k >= 2 and odd d. We get the similar lower bounds as k, b are both single prime powers. Particularly, when k is a square number, by using mutually orthogonal Latin squares, we can construct more MUMEBs in C-d circle times C-kd, and obtain greater lower bounds than reducing the problem into prime power dimension in some cases.