Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1206.6669 · Physical Review A

Measure of multipartite entanglement with computable lower bounds

Yan Hong, Ting Gao, Fengli Yan

Published Dec 21, 2012• 88 citations
arXivHTMLPDF

Abstract

In this paper, we present a measure of multipartite entanglement ($k$-nonseparable), $k$-ME concurrence ${C}_{k\ensuremath{-}\mathrm{ME}}(\ensuremath{\rho})$, that unambiguously detects all $k$-nonseparable states in arbitrary dimensions, where the special case 2-ME concurrence ${C}_{2\ensuremath{-}\mathrm{ME}}(\ensuremath{\rho})$ is a measure of genuine multipartite entanglement. The measure $k$-ME concurrence satisfies important characteristics of an entanglement measure, including the entanglement monotone, vanishing on $k$-separable states, convexity, subadditivity, and being strictly greater than zero for all $k$-nonseparable states. Two powerful lower bounds on this measure are given. These lower bounds are experimentally implementable without quantum state tomography and are easily computable as no optimization or eigenvalue evaluation is needed. We illustrate detailed examples in which the given bounds perform better than other known detection criteria.

DOI
10.1103/physreva.86.062323

Related papers

Parametrized multipartite entanglement measures

Hui Li, Ting Gao, Fengli Yan

We investigate parametrized multipartite entanglement measures from the perspective of $k$ nonseparability in this paper. We present two types of entanglement measures in $n$-partite systems, $q\text{\ensuremath{-}}k\text{\ensuremath{-}}\mathrm{ME}$ concurrence $(q>1,\phantom{\rule{0.28em}{0ex}}2\ensuremath{\le}k\ensuremath{\le}n)$ and $\ensuremath{\alpha}\text{\ensuremath{-}}k\text{\ensuremath{-}}\mathrm{ME}$ concurrence $(0\ensuremath{\le}\ensuremath{\alpha}<1,\phantom{\rule{0.28em}{0ex}}2\ensuremath{\le}k\ensuremath{\le}n)$, which unambiguously detect all $k$-nonseparable states in arbitrary $n$-partite systems. Rigorous proofs show that the proposed $k$-nonseparable measures satisfy all the requirements for being an entanglement measure including the entanglement monotone, strong monotone, convexity, vanishing on all $k$-separable states, and being strictly greater than zero for all $k$-nonseparable states. In particular, the $q\text{\ensuremath{-}}\mathrm{2}\text{\ensuremath{-}}\mathrm{ME}$ concurrence and $\ensuremath{\alpha}\text{\ensuremath{-}}2\text{\ensuremath{-}}\mathrm{ME}$ concurrence, renamed as $q\text{\ensuremath{-}}\mathrm{GME}$ concurrence and $\ensuremath{\alpha}\text{\ensuremath{-}}\mathrm{GME}$ concurrence, respectively, are two kinds of genuine entanglement measures corresponding to the case where the systems are divided into bipartition $(k=2)$. The lower bounds of the two classes $k$-nonseparable measures are obtained by employing the approach that takes into account the permutationally invariant part of a quantum state. Furthermore, the relations between $q\text{\ensuremath{-}}n\text{\ensuremath{-}}\mathrm{ME}$ concurrence $(\ensuremath{\alpha}\text{\ensuremath{-}}n\text{\ensuremath{-}}\mathrm{ME}$ concurrence) and global negativity are established. In addition, we discuss the degree of separability and elaborate on an effective detection method with concrete examples. Moreover, we compare the $q\text{\ensuremath{-}}\mathrm{GME}$ concurrence defined by us to other genuine entanglement measures.

arXiv:2308.16393•Physical Review A•Jan 16, 2024•23 citations

Measure of genuine multipartite entanglement with computable lower bounds

Zhihao Ma, Zhihua Chen, Jing‐Ling Chen +3 more

We introduce an intuitive measure of genuine multipartite entanglement, which is based on the well-known concurrence. We show how lower bounds on this measure can be derived and also meet important characteristics of an entanglement measure. These lower bounds are experimentally implementable in a feasible way enabling quantification of multipartite entanglement in a broad variety of cases.

arXiv:1101.2001•Physical Review A•Jun 20, 2011•300 citations

Extended negativity measure for multipartitek-nonseparable states

Yinzhu Wang, Danxia Wang, Li Huang

In this paper, an extended negativity entanglement measure for multipartite k-nonseparable states with respect to k-partition was proposed. We show that this measure is well-defined, i.e. it satisfies some basic properties as an entanglement measure. In addition, we give a relation between this measure and k-ME concurrence, and obtain a lower bound of this measure.

International Journal of Quantum Information•Feb 1, 2016•2 citations

Improved lower bounds on genuine-multipartite-entanglement concurrence

Zhihua Chen, Zhihao Ma, Jing‐Ling Chen +1 more

Genuine-multipartite-entanglement (GME) concurrence is a measure of genuine multipartite entanglement that generalizes the well-known notion of concurrence. We define an observable for GME concurrence. The observable permits us to avoid full state tomography and leads to different analytic lower bounds. By means of explicit examples we show that entanglement criteria based on the bounds have a better performance with respect to the known methods.

arXiv:1205.3057•Physical Review A•Jun 22, 2012•58 citations

Genuine multipartite entanglement detection and lower bound of multipartite concurrence

Ming Li, Shao-Ming Fei, Xianqing Li‐Jost +1 more

The problems of genuine multipartite entanglement detection and classification are challenging. We show that a multipartite quantum state is genuine multipartite entangled if the multipartite concurrence is larger than certain quantities given by the number and the dimension of the subsystems. This result also provides a classification of various genuine multipartite entanglements. We present a lower bound of the multipartite concurrence in terms of bipartite concurrences. While various operational approaches are available for providing lower bounds of bipartite concurrences, our results give an effective operational way to detect and classify genuine multipartite entanglement. As applications, genuine multipartite entanglement of tripartite systems is analyzed in detail.

arXiv:1512.08845•Physical Review A•Dec 28, 2015•33 citations

Lower bounds for genuine multipartite entanglement and coherence measures

Yue Dai, Yuli Dong, Zhenyu Xu +3 more

Experimentally quantifying entanglement and coherence are extremely important for quantum resource theory. However, because the quantum state tomography requires exponentially growing measurements with the number of qubits, it is hard to quantify entanglement and coherence based on the full information of the experimentally realized multipartite states. Fortunately, other methods have been found to directly measure the fidelity of experimental states without quantum state tomography. Here we present a fidelity-based method to derive experimentally accessible lower bounds for measures of genuine multipartite entanglement and coherence. On the one hand, the method works for genuine multipartite entanglement measures including the convex-roof extended negativity, the concurrence, the G-concurrence, and the geometric measure for genuine multipartite entanglement. On the other hand, the method also delivers observable lower bounds for the convex roof of the $l_{1}$-norm of coherence, the geometric measure of coherence, and the coherence of formation. Furthermore, all the lower bounds are based on the fidelity between the chosen pure state and the target state, and we obtain the lower bounds of several real experimental states as examples of our results.

•Oct 17, 2018•0 citations
In this paper, we present a measure of multipartite entanglement (kkk-nonseparable), kkk-ME concurrence Ck\ensuremath−ME(\ensuremathρ){C}_{k\ensuremath{-}\mathrm{ME}}(\ensuremath{\rho})Ck\ensuremath−ME​(\ensuremathρ), that unambiguously detects all kkk-nonseparable states in arbitrary dimensions, where the special case 2-ME concurrence C2\ensuremath−ME(\ensuremathρ){C}_{2\ensuremath{-}\mathrm{ME}}(\ensuremath{\rho})C2\ensuremath−ME​(\ensuremathρ) is a measure of genuine multipartite entanglement. The measure kkk-ME concurrence satisfies important characteristics of an entanglement measure, including the entanglement monotone, vanishing on kkk-separable states, convexity, subadditivity, and being strictly greater than zero for all kkk-nonseparable states. Two powerful lower bounds on this measure are given. These lower bounds are experimentally implementable without quantum state tomography and are easily computable as no optimization or eigenvalue evaluation is needed. We illustrate detailed examples in which the given bounds perform better than other known detection criteria.
We investigate parametrized multipartite entanglement measures from the perspective of kkk nonseparability in this paper. We present two types of entanglement measures in nnn-partite systems, $q\text{\ensuremath{-}}k\text{\ensuremath{-}}\mathrm{ME}$ concurrence $(q>1,\phantom{\rule{0.28em}{0ex}}2\ensuremath{\le}k\ensuremath{\le}n)$ and $\ensuremath{\alpha}\text{\ensuremath{-}}k\text{\ensuremath{-}}\mathrm{ME}$ concurrence $(0\ensuremath{\le}\ensuremath{\alpha}<1,\phantom{\rule{0.28em}{0ex}}2\ensuremath{\le}k\ensuremath{\le}n)$, which unambiguously detect all kkk-nonseparable states in arbitrary nnn-partite systems. Rigorous proofs show that the proposed kkk-nonseparable measures satisfy all the requirements for being an entanglement measure including the entanglement monotone, strong monotone, convexity, vanishing on all kkk-separable states, and being strictly greater than zero for all kkk-nonseparable states. In particular, the $q\text{\ensuremath{-}}\mathrm{2}\text{\ensuremath{-}}\mathrm{ME}$ concurrence and $\ensuremath{\alpha}\text{\ensuremath{-}}2\text{\ensuremath{-}}\mathrm{ME}$ concurrence, renamed as $q\text{\ensuremath{-}}\mathrm{GME}$ concurrence and $\ensuremath{\alpha}\text{\ensuremath{-}}\mathrm{GME}$ concurrence, respectively, are two kinds of genuine entanglement measures corresponding to the case where the systems are divided into bipartition (k=2)(k=2)(k=2). The lower bounds of the two classes kkk-nonseparable measures are obtained by employing the approach that takes into account the permutationally invariant part of a quantum state. Furthermore, the relations between $q\text{\ensuremath{-}}n\text{\ensuremath{-}}\mathrm{ME}$ concurrence $(\ensuremath{\alpha}\text{\ensuremath{-}}n\text{\ensuremath{-}}\mathrm{ME}$ concurrence) and global negativity are established. In addition, we discuss the degree of separability and elaborate on an effective detection method with concrete examples. Moreover, we compare the $q\text{\ensuremath{-}}\mathrm{GME}$ concurrence defined by us to other genuine entanglement measures.
The problems of genuine multipartite entanglement detection and classification are challenging. We show that a multipartite quantum state is genuine multipartite entangled if the multipartite concurrence is larger than certain quantities given by the number and the dimension of the subsystems. This result also provides a classification of various genuine multipartite entanglements. We present a lower bound of the multipartite concurrence in terms of bipartite concurrences. While various operational approaches are available for providing lower bounds of bipartite concurrences, our results give an effective operational way to detect and classify genuine multipartite entanglement. As applications, genuine multipartite entanglement of tripartite systems is analyzed in detail.
Experimentally quantifying entanglement and coherence are extremely important for quantum resource theory. However, because the quantum state tomography requires exponentially growing measurements with the number of qubits, it is hard to quantify entanglement and coherence based on the full information of the experimentally realized multipartite states. Fortunately, other methods have been found to directly measure the fidelity of experimental states without quantum state tomography. Here we present a fidelity-based method to derive experimentally accessible lower bounds for measures of genuine multipartite entanglement and coherence. On the one hand, the method works for genuine multipartite entanglement measures including the convex-roof extended negativity, the concurrence, the G-concurrence, and the geometric measure for genuine multipartite entanglement. On the other hand, the method also delivers observable lower bounds for the convex roof of the l1l_{1}l1​-norm of coherence, the geometric measure of coherence, and the coherence of formation. Furthermore, all the lower bounds are based on the fidelity between the chosen pure state and the target state, and we obtain the lower bounds of several real experimental states as examples of our results.