We show exactly that the squared entanglement of formation (SEF) obeys a set of hierarchical monogamy relations for an arbitrary
N-qubit mixed state. Based on this set of monogamy relations, we are able to construct the set of hierarchical multipartite entanglement indicators for
N-qubit states, which still work well even when the concurrence-based indicators lose efficacy. As a by-product, an intriguing analytical relation between the entanglement of formation (EOF) and squared concurrence (SC) for an arbitrary mixed state of $2\ensuremath{\bigotimes}d$ systems is derived, making the concurrence calculable via the corresponding EOF. Furthermore, we analyze the multipartite entanglement dynamics in composite cavity-reservoir systems with the present set of hierarchical indicators. Moreover, for multilevel systems, it is illustrated that the SEF can be monogamous even if the SC is polygamous.
The detection of multipartite entanglement in multipartite quantum systems is a fundamental and key issue in quantum information theory.In this paper, we investigate
k-nonseparability and
k-partite entanglement of
N-partitequantum systems from the perspective of the generalized Wigner-Yanase skew information introduced by Yang
et al.[\href{https://doi.org/10.1103/PhysRevA.106.052401}{Phys. Rev. A \textbf{106}, 052401 (2022)}]. More specifically, we develop two different approaches in form of inequalities to construct entanglement criteria, which are expressed in terms of the generalized Wigner-Yanase skew information.Any violation of these inequalities by a quantum state reveals its
k-nonseparability or
k-partite entanglement,so these inequalities present the hierarchic classifications of
k-nonseparability or
k-partite entanglement for all
N-partite quantum states from
N-nonseparability to
2-nonseparability or from
2-partite entanglement to
N-partite entanglement, which are more refined than well-known ways. It is shown that our results reveal some
k-nonseparability and
k-partite entanglement that remain undetected by other methods,and these are illustrated through some examples.