Mutually unbiased maximally entangled bases from difference matrices

Based on maximally entangled states, we explore the constructions of mutually unbiased bases in bipartite quantum systems. We present a new way to construct mutually unbiased bases by difference matrices in the theory of combinatorial designs. In particular, we establish $q$ mutually unbiased bases with $q-1$ maximally entangled bases and one product basis in ${\mathbb{C}}^q\otimes {\mathbb{C}}^q$ for arbitrary prime power $q$. In addition, we construct maximally entangled bases for dimension of composite numbers of non-prime power, such as five maximally entangled bases in ${\mathbb{C}}^{12}\otimes {\mathbb{C}}^{12}$ and ${\mathbb{C}}^{21}\otimes {\mathbb{C}}^{21}$, which improve the known lower bounds for $d=3m$, with $(3,m)=1$ in ${\mathbb{C}}^d\otimes {\mathbb{C}}^d$. Furthermore, we construct $p+1$ mutually unbiased bases with $p$ maximally entangled bases and one product basis in ${\mathbb{C}}^p\otimes {\mathbb{C}}^{p^2}$ for arbitrary prime number $p$.