Unified monogamy relation of entanglement measures

Zhong‐Xi Shen, Kang-Kang Yang, Yu Lu +2 more

Mei-Ming Zhang, Naihuan Jing, Hui Zhao

Yu Chen Guo

A particularly interesting feature of nonrelativistic quantum mechanics is the monogamy laws of entanglement. Although the monogamy relation has been explored extensively in the last decade, it is still not clear to what extent a given entanglement measure is monogamous. We give here a conjecture on the amount of entanglement contained in the reduced states by observing all the known related results at first. Consequently, we propose the monogamy power of an entanglement measure and the polygamy power for its dual quantity, the assisted entanglement, and show that both the monogamy power and the polygamy power exist in any multipartite systems with any dimension, from which we formalize exactly for the first time when an entanglement measure and an assisted entanglement obey the monogamy relation and the polygamy relation respectively in a unified way. In addition, we show that any entanglement measure violates the polygamy relation, which is misstated in some recent papers. Only the existence of monogamy power is conditioned on the conjecture, all other results are strictly proved.

Zhi‐Xiang Jin, Cong‐Feng Qiao

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.

Yue Cao, Naihuan Jing, Kailash C. Misra +1 more

Abstract It is well known that any well-defined bipartite entanglement measure $$\mathcal {E}$$ E obeys $$\gamma $$ γ th-monogamy relations Eq. (1.1) and assisted measure $$\mathcal {E}_{a}$$ E a obeys $$\delta $$ δ th-polygamy relations Eq. (1.2). Recently, we presented a class of tighter parameterized monogamy relation for the $$\alpha $$ α th $$(\alpha \ge \gamma )$$ ( α ≥ γ ) power based on Eq. 1.1. This study provides a family of tighter lower (resp. upper) bounds of the monogamy (resp. polygamy) relations in a unified manner. In the first part of the paper, the following three basic problems are focused: tighter monogamy relation for the $$\alpha $$ α th ( $$0\le \alpha \le \gamma $$ 0 ≤ α ≤ γ ) power of any bipartite entanglement measure $$\mathcal {E}$$ E based on Eq. (1.1); tighter polygamy relation for the $$\beta $$ β th ( $$ \beta \ge \delta $$ β ≥ δ ) power of any bipartite assisted entanglement measure $$\mathcal {E}_{a}$$ E a based on Eq. (1.2); tighter polygamy relation for the $$\omega $$ ω th ( $$0\le \omega \le \delta $$ 0 ≤ ω ≤ δ ) power of any bipartite assisted entanglement measure $$\mathcal {E}_{a}$$ E a based on Eq. (1.2). In the second part, using the tighter polygamy relation for the $$\omega $$ ω th ( $$0\le \omega \le 2$$ 0 ≤ ω ≤ 2 ) power of CoA, we obtain good estimates or bounds for the $$\omega $$ ω th ( $$0\le \omega \le 2$$ 0 ≤ ω ≤ 2 ) power of concurrence for any N-qubit pure states $$|\psi \rangle _{AB_{1}\cdots B_{N-1}}$$

Limin Gao, Fengli Yan, Ting Gao