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

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Entropic methodology for entanglement measures

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arXiv:2206.07905 · Quantum Information Processing

Estimating parameterized entanglement measure

Zhi-Wei Wei, Ming-Xing Luo, Shao-Ming Fei

Published Jun 14, 2022• 7 citations

arXiv HTML PDF

DOI: 10.1007/s11128-022-03551-4

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q

-concurrence (

q>1

) and

α

-concurrence (

0<α<1

), or termed the parameterized entanglement monotone together. We obtain, in the light of the informationally complete (

N

M

)-positive operator-valued measure [(

N

M

)-POVM], the lower bounds for the case of

\frac12<α<1

1<q<2

for two-qudit states, and the case of

2\leqslant q<3

for two-qubit states. We list several examples which show that the lower bounds based on (

N

M

)-POVM outperform that of GSIC-POVM and SIC-POVM, and all these measurement based bounds are better then the ones induced by positive partial transpose (PPT) and realignment criteria in literature. In addition, we obtain an analytical formula of the parameterized entanglement monotone with

\frac12<α<1

and

1<q<2

for the isotropic state.

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}}