Bounds on the number of mutually unbiased entangled bases

We provide several bounds on the maximum size of MU k-Schmidt bases in $$\mathbb {C}^{d}\otimes \mathbb {C}^{d'}$$ . We first give some upper bounds on the maximum size of MU k-Schmidt bases in $$\mathbb {C}^{d}\otimes \mathbb {C}^{d'}$$ by conversation law. Then we construct two maximally entangled mutually unbiased (MU) bases in the space $$\mathbb {C}^{2}\otimes \mathbb {C}^{3}$$ , which is the first example of maximally entangled MU bases in $$\mathbb {C}^d\otimes \mathbb {C}^{d'}$$ when $$d\not \mid d'$$ . By applying a general recursive construction to this example, we are able to obtain two maximally entangled MU bases in $$\mathbb {C}^{d}\otimes \mathbb {C}^{d'}$$ for infinitely many $$d,d'$$ such that d is not a divisor of $$d'$$ . We also give some applications of the two maximally entangled MU bases in $$\mathbb {C}^{2}\otimes \mathbb {C}^{3}$$ . Further, we present an efficient method of constructing MU k-Schmidt bases. It solves an open problem proposed in [Y. F. Han et al., Quantum Inf. Process. 17, 58 (2018)]. Our work improves all previous results on maximally entangled MU bases.