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arXiv:1401.3205 · Physical Review Letters

General Monogamy Relation for the Entanglement of Formation in Multiqubit Systems

Yan-Kui Bai, Yuan-Fei Xu, Z. D. Wang

Published Sep 5, 2014• 171 citations
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Abstract

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.

DOI
10.1103/physrevlett.113.100503

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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.
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.
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.
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.
We show exactly that the squared entanglement of formation (SEF) obeys a set of hierarchical monogamy relations for an arbitrary NNN-qubit mixed state. Based on this set of monogamy relations, we are able to construct the set of hierarchical multipartite entanglement indicators for NNN-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 monogamy of entanglement means that entanglement cannot be freely shared. In 2014, Oliveira et al. [T. R. de Oliveira, M. F. Cornelio, and F. F. Fanchini, Phys. Rev. A 89, 034303 (2014)] proposed a monogamy relation in the linear version and considered it in terms of entanglement of formation. Here we generalize the above version and consider a multilinear monogamy relation for a multiqubit system in terms of entanglement of formation and concurrence. Based on the above results, we present an entanglement criterion for genuine entangled states; also we consider the absolutely maximally entangled states and present what an absolutely maximally entangled state is for a three-qubit system. Last, we apply our results to a three-qubit pure state in terms of quantum discord.
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.