Entangled bases with fixed Schmidt number

TL;DR

This paper introduces entangled bases with fixed Schmidt number in bipartite and multipartite systems, proving their existence for all relevant dimensions and proposing methods for their construction.

Contribution

It establishes the existence of entangled bases with fixed Schmidt number for all dimensions and extends the concept to multipartite systems, providing construction methods.

Findings

EBk exists for all dimensions where k ≤ min{d,d'}

Methods for constructing SEBk and EBk are proposed

Multipartite EBk can be constructed similarly

Abstract

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.