Construction of mutually unbiased maximally entangled bases through permutations of Hadamard matrices

We construct mutually unbiased maximally entangled bases (MUMEBs) in bipartite system C d ⊗ C d ( d ≥ 3 ) with d a power of a prime number. Precisely, by means of permutation matrices and Hadamard matrices, we construct 2 ( d - 1 ) MUMEBs in C d ⊗ C d . It follows that M ( d , d ) ≥ 2 ( d - 1 ) , which is twice the number given in Liu et al. ( 2016 ), where M ( d , d ) denotes the maximal size of all sets of MUMEBs in C d ⊗ C d . In addition, let q be another power of a prime number, we construct MUMEBs in C d ⊗ C q d from those in C d ⊗ C d by the use of the tensor product of unitary matrices.