Hierarchical monogamy relations for the squared entanglement of formation in multipartite systems

Abstract

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 $2\otimes d$ 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.