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

AbstractWe construct mutually unbiased maximally entangled bases (MUMEBs) in bipartite system $${\mathbb {C}}^d\otimes {\mathbb {C}}^d (d\ge 3)$$Cd⊗Cd(d≥3) with d a power of a prime number. Precisely, by means of permutation matrices and Hadamard matrices, we construct $$2(d-1)$$2(d-1) MUMEBs in $${\mathbb {C}}^d\otimes {\mathbb {C}}^d$$Cd⊗Cd. It follows that $$M(d,d)\ge 2(d-1)$$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 $${\mathbb {C}}^d\otimes {\mathbb {C}}^d$$Cd⊗Cd. In addition, let q be another power of a prime number, we construct MUMEBs in $${\mathbb {C}}^d\otimes {\mathbb {C}}^{qd}$$Cd⊗Cqd from those in $${\mathbb {C}}^d\otimes {\mathbb {C}}^d$$Cd⊗Cd by the use of the tensor product of unitary matrices.