arXiv:1810.09131

Tighter monogamy and polygamy relations using Rényi-$α$ entropy

Yanying Liang, Zhu‐Jun Zheng, Chuan-Jie Zhu

Published Oct 22, 2018• 0 citations

Abstract

We investigate monogamy relations related to the Rényi-$α$ entanglement and polygamy relations related to the Rényi-$α$ entanglement of assistance. We present new entanglement monogamy relations satisfied by the $μ$-th power of Rényi-$α$ entanglement with $α\in[\sqrt{7}-1)/2,(\sqrt{13}-1)/2]$ for $μ\geqslant2$, and polygamy relations satisfied by the $μ$-th power of Rényi-$α$ entanglement of assistance with $α\in[\sqrt{7}-1)/2,(\sqrt{13}-1)/2]$ for $0\leqμ\leq1$. These relations are shown to be tighter than the existing ones.

DOI: 10.48550/arxiv.1810.09131

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arXiv:1810.09131

Tighter monogamy and polygamy relations using Rényi- $α$ entropyTighter monogamy and polygamy relations using Rényi-$α$ entropy

Yanying Liang, Zhu‐Jun Zheng, Chuan-Jie Zhu

Published Oct 22, 2018• 0 citations

arXiv HTML PDF

Abstract

We investigate monogamy relations related to the Rényi-

α

entanglement and polygamy relations related to the Rényi-

α

entanglement of assistance. We present new entanglement monogamy relations satisfied by the

μ

-th power of Rényi-

α

entanglement with

α\in[\sqrt{7}-1)/2,(\sqrt{13}-1)/2]

for

μ\geqslant2

, and polygamy relations satisfied by the

μ

-th power of Rényi-

α

entanglement of assistance with

α\in[\sqrt{7}-1)/2,(\sqrt{13}-1)/2]

for

0\leqμ\leq1

. These relations are shown to be tighter than the existing ones.

DOI: 10.48550/arxiv.1810.09131

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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 $|\psi \rangle _{AB_{1}\cdots B_{N-1}}$ 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 $$|\psi \rangle _{AB_{1}\cdots B_{N-1}}$$

arXiv:2503.00694The European Physical Journal PlusFeb 5, 20251 citation

Tighter monogamy and polygamy relations using Rényi-

α

entropy

We investigate monogamy relations related to the Rényi-

α

entanglement and polygamy relations related to the Rényi-

α

entanglement of assistance. We present new entanglement monogamy relations satisfied by the

μ

-th power of Rényi-

α

entanglement with

α\in[\sqrt{7}-1)/2,(\sqrt{13}-1)/2]

for

μ\geqslant2

, and polygamy relations satisfied by the

μ

-th power of Rényi-

α

entanglement of assistance with

α\in[\sqrt{7}-1)/2,(\sqrt{13}-1)/2]

for

0\leqμ\leq1

. These relations are shown to be tighter than the existing ones.

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}}