Entangled bases with fixed Schmidt number

An entangled basis with fixed Schmidt number $k$ (EBk) is a set of orthonormal basis states with the same Schmidt number $k$ in a product Hilbert space $\mathbb{C}^d\otimes\mathbb{C}^{d'}$. It is a generalization of both the product basis and the maximally entangled basis. We show here that, for any $k\leq\min\{d,d'\}$, EBk exists in $\mathbb{C}^d\otimes\mathbb{C}^{d'}$ for any $d$ and $d'$. Consequently, general methods of constructing SEBk (EBk with the same Schmidt coefficients) and EBk (but not SEBk) are proposed. Moreover, we extend the concept of EBk to multipartite case and find out that the multipartite EBk can be constructed similarly.