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Physical Review A

Comment on “Unification of multiqubit polygamy inequalities”

Wei Song, Junlong Zhao, Long-Bao Yu, Lihua Zhang

Published May 4, 2017• 2 citations
Publisher

Abstract

Recently, Kim established a unified view of polygamy of multiqubit entanglement [Phys. Rev. A 85, 032335 (2012)]. In order to prove their main results, Kim first proposed an important property which is illustrated in Lemma 2. We point out that his proofs of Lemma 2 are flawed because of some errors in his derivations. Furthermore, we present an improved method to prove the original results.

DOI
10.1103/physreva.95.056301

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Recently, Kim established a unified view of polygamy of multiqubit entanglement [Phys. Rev. A 85, 032335 (2012)]. In order to prove their main results, Kim first proposed an important property which is illustrated in Lemma 2. We point out that his proofs of Lemma 2 are flawed because of some errors in his derivations. Furthermore, we present an improved method to prove the original results.
I establish a unified view of polygamy of multiqubit entanglement. I first introduce a two-parameter generalization of the entanglement of assistance, namely, the unified entanglement of assistance for bipartite quantum states, and provide an analytic lower bound in two-qubit systems. I show a broad class of polygamy inequalities of multiqubit entanglement in terms of the unified entanglement of assistance that encapsulates all known multiqubit polygamy inequalities as special cases. I further show that this class of polygamy inequalities can be improved into tighter inequalities for three-qubit systems.
Abstract In an article in 2010, Kim et al introduced the definition of Rényi- α entanglement for bipartite quantum states and established an analytic formula of Rényi- α entanglement for arbitrary two-qubit states with α ⩾ 1 . They also derived a monogamy of entanglement in multi-qubit systems in terms of Rényi- α entanglement for α ⩾ 2 Kim et al (2010 J. Phys. A: Math. Theor . 43 445305). We find the proofs of theorems 2 and 3 contain some errors and we also present an improved derivation to overcome this flaw. The alternative derivation shows that the main conclusions remain valid despite the invalidity of the proofs.
The monogamy and polygamy properties of quantum entanglement characterize fundamental constraints on the distribution of entanglement in multipartite quantum systems. In this paper, we investigate tighter monogamy and polygamy relations for multipartite entanglement. By establishing a new mathematical inequality, we derive a family of improved monogamy and polygamy inequalities for tripartite quantum systems and further extend these results to general multipartite systems. Comparisons with existing results show that the obtained bounds are tighter. Illustrative examples are provided to demonstrate the effectiveness of the proposed relations.
Abstract The monogamy and polygamy properties of quantum entanglement characterize fundamental constraints on the distribution of entanglement in multipartite quantum systems. In this paper, we investigate tighter monogamy and polygamy relations for multipartite entanglement. By establishing a new mathematical inequality, we derive a family of improved monogamy and polygamy inequalities for tripartite quantum systems and further extend these results to general multipartite systems. Comparisons with existing results show that the obtained bounds are tighter. Illustrative examples are provided to demonstrate the effectiveness of the proposed relations.