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.

我们提供了\（\ mathbb {C} ^ {d} \ otimes \ mathbb {C} ^ {d'} \）中MU k -Schmidt基的最大大小的几个界限。我们首先根据会话定律对\（\ mathbb {C} ^ {d} \ otimes \ mathbb {C} ^ {d'} \）中MU k -Schmidt基的最大大小给出一些上限。然后我们在空间\（\ mathbb {C} ^ {2} \ otimes \ mathbb {C} ^ {3} \）中构造两个最大纠缠的互不偏（MU）基，这是最大纠缠的MU基的第一个示例在\（\ mathbb {C} ^ d \ otimes \ mathbb {C} ^ {d'} \）中时，当\（d \ not \ mid d'\）时。通过将通用递归构造应用于此示例，我们能够获得两个最大纠缠的MU基。\（\ mathbb {C} ^ {d} \ otimes \ mathbb {C} ^ {d'} \）无穷多个\（d，d'\），使得d不是\（d'\）的除数。我们还给出了\（\ mathbb {C} ^ {2} \ otimes \ mathbb {C} ^ {3} \）中两个最大纠缠MU基的一些应用。此外，我们提出了一种构建MU k-施密特碱基的有效方法。它解决了[YF Han et al。，量子信息。处理。17，58（2018）]。我们的工作改进了在最大纠缠MU基上的所有先前结果。