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 ⊗ 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.

Linear monogamy of entanglement in three-qubit systems

“…(9) in ref. 30 , we find that the second-order derivative d 2 E F ( ρ AC )/ dx 2 ≥ 0 and similarly d 2 E F ( ρ AB )/ dp 2 ( x ) ≥ 0 in the region x ∈ [0, 1]. So the second-order derivative d 2 E F ( ρ AB )/ dx 2 ≤ 0 in the same region.…”

“…(7) in ref. 30 , we have the second-order derivative d 2 g ( x )/ dx 2 ≤ 0 in the whole region x ∈ [0, c ].…”

General monogamy relations of quantum entanglement for multiqubit W-class states

“…. We have used in the first and last equalities that the entanglement of formation obeys the relation E(ρ) = f (C 2 (ρ)) for a bipartite 2 ⊗ D, D ≥ 2, quantum state ρ [19]. The second equality is due to the fact that C 2 (|ψ A1...An ) = n i=2 C 2 (ρ A1Ai ).…”

Lower and upper bounds for entanglement of Rényi-α entropy

“…The aim of this paper is to provide computable lower and upper bounds for ERαE of arbitrary dimensional bipartite quantum systems, and these results might be utilized to investigate the monogamy relation [43][44][45][46] in high-dimensional states.…”