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

Absolutely entangled set of pure states