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arXiv:1603.02760

Generalized entanglement constraints in multi-qubit systems in terms of Tsallis entropy

Jeong San Kim

Published Mar 9, 2016• 0 citations
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Abstract

We provide generalized entanglement constraints in multi-qubit systems in terms of Tsallis entropy. Using quantum Tsallis entropy of order $q$, we first provide a generalized monogamy inequality of multi-qubit entanglement for $q=2$ or $3$. This generalization encapsulates multi-qubit CKW-type inequality as a special case. We further provide a generalized polygamy inequality of multi-qubit entanglement in terms of Tsallis-$q$ entropy for $1 \leq q \leq2$ or $3 \leq q \leq 4$, which also contains the multi-qubit polygamy inequality as a special case.

DOI
10.48550/arxiv.1603.02760

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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.
We characterize the polygamy nature of quantum entanglement in multiparty systems in terms of qqq-expectation value for the full range of $q\ensuremath{\ge}1$. By investigating some properties of generalized quantum correlations in terms of qqq-expectation value and Tsallis qqq-entropy, we establish a class of polygamy inequalities of multiparty quantum entanglement in arbitrary dimensions based on qqq-expected entanglement measure. As Tsallis qqq-entropy is reduced to von Neumann entropy, and qqq-expectation value becomes the ordinary expectation value when qqq tends to 1, our results encapsulate previous results of polygamy inequalities based on von Neumann entropy as special cases.
We establish a unified view to the polygamy of multi-party quantum entanglement in arbitrary dimensions. Using quantum Tsallis-qqq entropy, we provide a one-parameter class of polygamy inequalities of multi-party quantum entanglement. This class of polygamy inequalities reduces to the known polygamy inequalities based on tangle and entanglement of assistance for a selective choice of the parameter qqq. We further provide one-parameter generalizations of various quantum correlations based on Tsallis-qqq entropy. By investigating the properties of the generalized quantum correlations, we provide a sufficient condition, on which the Tsallis-qqq polygamy inequalities hold in multi-party quantum systems of arbitrary dimensions.
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.
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We establish a unified view of the polygamy of multiparty quantum entanglement in arbitrary dimensions. Using quantum Tsallis-qqq entropy, we provide a one-parameter class of polygamy inequalities of multiparty quantum entanglement. This class of polygamy inequalities reduces to the known polygamy inequalities based on tangle and entanglement of assistance for a selective choice of the parameter qqq. We further provide one-parameter generalizations of various quantum correlations based on Tsallis-qqq entropy. By investigating the properties of the generalized quantum correlations, we provide a sufficient condition on which the Tsallis-qqq polygamy inequalities hold in multiparty quantum systems of arbitrary dimensions.