Mutually unbiased maximally entangled bases in $\mathbb {C}^d\otimes \mathbb {C}^d$

Abstract

We study mutually unbiased maximally entangled bases (MUMEB’s) in bipartite system $\mathbb {C}^d\otimes \mathbb {C}^d (d \ge 3)$. We generalize the method to construct MUMEB’s given in Tao et al. (Quantum Inf Process 14:2291–2300, 2015), by using any commutative ring R with d elements and generic character of $(R,+)$ instead of $\mathbb {Z}_d=\mathbb {Z}/d\mathbb {Z}$. Particularly, if $d=p_1^{a_1}p_2^{a_2}\ldots p_s^{a_s}$ where $p_1, \ldots , p_s$ are distinct primes and $3\le p_1^{a_1}\le \cdots \le p_s^{a_s}$, we present $p_1^{a_1}-1$ MUMEB’s in $\mathbb {C}^d\otimes \mathbb {C}^d$ by taking $R=\mathbb {F}_{p_1^{a_1}}\oplus \cdots \oplus \mathbb {F}_{p_s^{a_s}}$, direct sum of finite fields (Theorem 3.3).