Criteria of absolutely separability from spectrum for qudit-qudits states

Separability from the spectrum is a significant and ongoing research topic in quantum entanglement. In this study, we investigate properties related to absolute separability from the spectrum in qudits-qudits states in the bipartite states space $\mathcal{H}_{mn}=\mathcal{H}_m \otimes \mathcal{H}_n$. Firstly, we propose the necessary and sufficient conditions for absolute separable states in the Hilbert space $\mathcal{H}_{4n}$. These conditions are equivalent to the positive semidefiniteness of twelve matrices resulting from the symmetric matricizations of eigenvalues. Furthermore, we demonstrate that this sufficient condition can be extended to the general $\mathcal{H}_{mn}$ case, improving existing conclusions in the literature. These sufficient conditions depend only on the first few leading and last few leading eigenvalues, significantly reducing the complexity of determining absolute separable states. On the other hand, we also introduce additional sufficient conditions for determining that states in $\mathcal{H}_{mn}$ are not absolutely separable. These conditions only depend on $2m-1$ eigenvalues of the mixed states. Our sufficient conditions are not only simple and easy to implement. As applications, we derive distance bounds for eigenvalues and purity bounds for general absolutely separable states.