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arXiv:1604.08077 · Scientific Reports

Monogamy relation of multi-qubit systems for squared Tsallis-q entanglement

Guang-Ming Yuan, Wei Song, Ming Yang, Da-Chuang Li, Junlong Zhao, Zhuo‐Liang Cao

Published Jun 27, 2016• 29 citations
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Abstract

Tsallis-q entanglement is a bipartite entanglement measure which is the generalization of entanglement of formation for q tending to 1. We first expand the range of q for the analytic formula of Tsallis-q entanglement. For , we prove the monogamy relation in terms of the squared Tsallis-q entanglement for an arbitrary multi-qubit systems. It is shown that the multipartite entanglement indicator based on squared Tsallis-q entanglement still works well even when the indicator based on the squared concurrence loses its efficacy. We also show that the μ-th power of Tsallis-q entanglement satisfies the monogamy or polygamy inequalities for any three-qubit state.

DOI
10.1038/srep28719

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Tsallis-q entanglement is a bipartite entanglement measure which is the generalization of entanglement of formation for q tending to 1. We first expand the range of q for the analytic formula of Tsallis-q entanglement. For , we prove the monogamy relation in terms of the squared Tsallis-q entanglement for an arbitrary multi-qubit systems. It is shown that the multipartite entanglement indicator based on squared Tsallis-q entanglement still works well even when the indicator based on the squared concurrence loses its efficacy. We also show that the μ-th power of Tsallis-q entanglement satisfies the monogamy or polygamy inequalities for any three-qubit state.
Unified-(q, s) entanglement [Formula: see text] is a generalized bipartite entanglement measure, which encompasses Tsallis-q entanglement, Rényi-q entanglement, and entanglement of formation as its special cases. We first provide the extended (q; s) region of the generalized analytic formula of [Formula: see text]. Then, the monogamy relation based on the squared [Formula: see text] for arbitrary multiqubit mixed states is proved. The monogamy relation proved in this paper enables us to construct an entanglement indicator that can be utilized to identify all genuine multiqubit entangled states even the cases where three tangle of concurrence loses its efficiency. It is shown that this monogamy relation also holds true for the generalized W-class state. The αth power [Formula: see text] based general monogamy and polygamy inequalities are established for tripartite qubit states.
In this paper, we study the monogamy inequality of Tsallis qqq-entropy entanglement. We first provide an analytic formula of Tsallis qqq-entropy entanglement in two-qubit systems for $\frac{5\ensuremath{-}\sqrt{13}}{2}\ensuremath{\le}q\ensuremath{\le}\frac{5+\sqrt{13}}{2}.$ The analytic formula of Tsallis qqq-entropy entanglement in $2\ensuremath{\bigotimes}d$ system is also obtained and we show that Tsallis qqq-entropy entanglement satisfies a set of hierarchical monogamy equalities. Furthermore, we prove the squared Tsallis qqq-entropy entanglement follows a general inequality in the qubit systems. Based on the monogamy relations, a set of multipartite entanglement indicators is constructed, which can detect all genuine multiqubit entangled states even in the case of NNN-tangle vanishes. Moreover, we study some examples in multipartite higher-dimensional system for the monogamy inequalities.
We prove that the squared R\'enyi-$\ensuremath{\alpha}$ entanglement ($\mathrm{SR}\ensuremath{\alpha}\mathrm{E}$), which is the generalization of entanglement of formation, obeys a general monogamy inequality in an arbitrary NNN-qubit mixed state. Furthermore, for a class of R\'enyi-$\ensuremath{\alpha}$ entanglement, we prove that the monogamy relations of the $\mathrm{SR}\ensuremath{\alpha}\mathrm{E}$ have a hierarchical structure when the NNN-qubit system is divided into kkk parties. As a by-product, the analytical relation between the R\'enyi-$\ensuremath{\alpha}$ entanglement and the squared concurrence is derived for bipartite $2\ensuremath{\bigotimes}d$ systems. Based on the monogamy properties of $\mathrm{SR}\ensuremath{\alpha}\mathrm{E}$, we can construct the corresponding multipartite entanglement indicators, which still work well even when the indicators based on the squared concurrence and EOF lose their efficacy. In addition, the monogamy property of the $\ensuremath{\mu}\mathrm{th}$ power of R\'enyi-$\ensuremath{\alpha}$ entanglement is analyzed.
We prove exactly that the squared entanglement of formation, which quantifies the bipartite entanglement, obeys a general monogamy inequality in an arbitrary multiqubit mixed state. Based on this kind of exotic monogamy relation, we are able to construct two sets of useful entanglement indicators: the first one can detect all genuine multiqubit entangled states even in the case of the two-qubit concurrence and n-tangles being zero, while the second one can be calculated via quantum discord and applied to multipartite entanglement dynamics. Moreover, we give a computable and nontrivial lower bound for multiqubit entanglement of formation.
Monogamy relation is one of the essential properties of quantum entanglement, which characterizes the distribution of entanglement in a multipartite system. By virtual of the unified-(q,s) entropy, we obtain some novel monogamy and polygamy inequalities in general class of entanglement measures. For the multiqubit system, a class of tighter monogamy relations are established in term of the α-th power of unified-(q,s) entanglement for α ≥ 1. We also obtain a class of tighter polygamy relations in the β-th (0 ≤ β ≤ 1) power of unified-(q,s) entanglement of assistance. Applying these results to specific quantum correlations, e.g., entanglement of formation, Renyi-q entanglement of assistance, and Tsallis-q entanglement of assistance, we obtain the corresponding monogamy and polygamy relations. Typical examples are presented for illustration. Furthermore, the complementary monogamy and polygamy relations are investigated for the α-th (0 ≤ α ≤ 1) and β-th (β ≥ 1) powers of unified entropy, respectively, and the corresponding monogamy and polygamy inequalities are obtained.