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arXiv:1612.04480

Tsallis entropy and general polygamy of multi-party quantum entanglement in arbitrary dimensions

Jeong San Kim

Published Dec 14, 2016• 0 citations
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Abstract

We establish a unified view to the polygamy of multi-party quantum entanglement in arbitrary dimensions. Using quantum Tsallis-$q$ 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 $q$. We further provide one-parameter generalizations of various quantum correlations based on Tsallis-$q$ entropy. By investigating the properties of the generalized quantum correlations, we provide a sufficient condition, on which the Tsallis-$q$ polygamy inequalities hold in multi-party quantum systems of arbitrary dimensions.

DOI
10.48550/arxiv.1612.04480

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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.
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.
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.
Strong polygamy of multi-party qqq-expected quantum correlations
We show that the polygamous nature of multi-party quantum correlations can be characterized in a {\em stronger} form based on Tsallis qqq-entropy and qqq-expectation value. By considering the amount of entanglement that can be distributed in multi-party systems, we establish a class of strong polygamy inequalities of multi-party entanglement in terms of Tsallis qqq-entropy and qqq-expectation for q≥1q \geq 1q≥1. Our new class of inequalities is in fact tighter than the usual polygamy inequalities of multi-party entanglement, and the tightness is explicitly illustrated by an example. Moreover, our new class of inequalities is concerned with the qqq-expected entanglement distributed between a single party and any possible subsets of the rest parties whereas the usual polygamy inequality only considers the entanglement between a single party and another. We further establish the equivalence between strong polygamy inequalities of quantum entanglement and quantum discord distributed in multi-party quantum systems.
We show that the mutually exclusive nature of classical and quantum correlations distributed in multiparty quantum systems can be characterized in terms of qqq-expectation value. Using Tsallis qqq entropy and qqq-expectation value, we first provide generalized definitions of classical and quantum correlations, and establish their trade-off relations in three-party quantum systems of arbitrary dimension with respect to qqq-expectation value for $q\ensuremath{\ge}1$. We also provide equivalence conditions for monogamy and polygamy inequalities of quantum entanglement and quantum discord distributed in three-party quantum systems of arbitrary dimension with respect to qqq-expectation value for $q\ensuremath{\ge}1$.
We provide generalized entanglement constraints in multi-qubit systems in terms of Tsallis entropy. Using quantum Tsallis entropy of order qqq, we first provide a generalized monogamy inequality of multi-qubit entanglement for q=2q=2q=2 or 333. 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-qqq entropy for 1≤q≤21 \leq q \leq21≤q≤2 or 3≤q≤43 \leq q \leq 43≤q≤4, which also contains the multi-qubit polygamy inequality as a special case.
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.