Absolutely entangled set of pure states

Abstract: Quite recently, Cai et al. [arXiv:2006.07165v1] proposed a new concept "absolutely entangled set" for bipartite quantum systems: for any possible choice of global basis, at least one state of the set is entangled. There they presented a minimum example with a set of four states in two qubit systems and they proposed a quantitative measure for the absolute set entanglement. In this work, we derive two necessity conditions for a set of states to be an absolutely entangled set. In addition, we give a series constructions of absolutely entangled bases on $\mathbb{C}^{d_1}\otimes \mathbb{C}^{d_2}$ for any nonprime dimension $d=d_1\times d_2$. Moreover, based on the structure of the orthogonal product basis in $\mathbb{C}^2\otimes \mathbb{C}^n$, we obtain another construction of absolutely entangled set with $2n+1$ elements in $\mathbb{C}^2\otimes \mathbb{C}^n$.