Bounds on the number of mutually unbiased entangled bases

We provide several bounds on the maximum size of MU k-Schmidt bases in C-d circle times C-d'. We first give some upper bounds on the maximum size of MU k-Schmidt bases in C-d circle times C-d' by conversation law. Then we construct two maximally entangled mutually unbiased (MU) bases in the space C-2 circle times C-3, which is the first example of maximally entangled MU bases in C-d circle times C-d when d (sic) d'. By applying a general recursive construction to this example, we are able to obtain two maximally entangled MU bases in C-d circle times 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 C-2 circle times 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.