Although many different entanglement measures have been proposed so far, much less is known in the multipartite case, which leads to the previous monogamy relations in literatures are not complete. We establish here a strict framework for defining multipartite entanglement measure (MEM): apart from the postulates of bipartite measure [i.e., vanishing on separable and nonincreasing under local operations and classical communication (LOCC)], a genuine MEM should additionally satisfy the \textit{unification condition} and the \textit{hierarchy condition}. We then come up with a \textit{complete monogamy formula} under the unified MEM (an MEM is called a unified MEM if it satisfies the unification condition). Consequently, we propose MEMs which are multipartite extensions of entanglement of formation (EoF), concurrence, tangle, the convex-roof extension of negativity and negativity, respectively. We show that multipartite extensions of the bipartite measures that are defined by the convex-roof structure are completely monogamous, the extensions of EoF, concurrence and tangle are genuine MEMs (an MEM is called a genuine MEM if it satisfies both the unification condition and the hierarchy condition), and multipartite extensions of both negativity and the convex-roof extension of negativity are unified MEMs but not genuine MEMs.Although many different entanglement measures have been proposed so far, much less is known in the multipartite case, which leads to the previous monogamy relations in literatures are not complete. We establish here a strict framework for defining multipartite entanglement measure (MEM): apart from the postulates of bipartite measure [i.e., vanishing on separable and nonincreasing under local operations and classical communication (LOCC)], a genuine MEM should additionally satisfy the \textit{unification condition} and the \textit{hierarchy condition}. We then come up with a \textit{complete monogamy formula} under the unified MEM (an MEM is called a unified MEM if it satisfies the unification condition). Consequently, we propose MEMs which are multipartite extensions of entanglement of formation (EoF), concurrence, tangle, the convex-roof extension of negativity and negativity, respectively. We show that multipartite extensions of the bipartite measures that are defined by the convex-roof structure are completely monogamous, the extensions of EoF, concurrence and tangle are genuine MEMs (an MEM is called a genuine MEM if it satisfies both the unification condition and the hierarchy condition), and multipartite extensions of both negativity and the convex-roof extension of negativity are unified MEMs but not genuine MEMs.

Aug 22, 20190 citations

Breakthrough: Multipartite Entanglement

Songbo Xie

Springer thesesJan 1, 20250 citations

Efficient multipartite entanglement purification with the entanglement link from a subspace

Fu‐Guo Deng

We present an efficient multipartite entanglement purification protocol (MEPP) for $N$ -photon systems in a Greenberger-Horne-Zeilinger state with parity-check detectors. It contains two parts. One is the conventional MEPP with which the parties can obtain a high-fidelity $N$ -photon ensemble directly, similar to the MEPP with controlled-not gates. The other is our recycling MEPP in which the entanglement link is used to produce some $N$ -photon entangled systems from entangled ${N}^{\ensuremath{'}}$-photon subsystems ($2\ensuremath{\leqslant}{N}^{\ensuremath{'}}<N$) coming from the instances which are just discarded in all existing conventional MEPPs. The entangled ${N}^{\ensuremath{'}}$-photon subsystems are obtained efficiently by measuring the photons with potential bit-flip errors. With these two parts, the present MEPP has a higher efficiency than all other conventional MEPPs.We present an efficient multipartite entanglement purification protocol (MEPP) for $N$-photon systems in a Greenberger-Horne-Zeilinger state with parity-check detectors. It contains two parts. One is the conventional MEPP with which the parties can obtain a high-fidelity $N$-photon ensemble directly, similar to the MEPP with controlled-not gates. The other is our recycling MEPP in which the entanglement link is used to produce some $N$-photon entangled systems from entangled ${N}^{\ensuremath{'}}$-photon subsystems ($2\ensuremath{\leqslant}{N}^{\ensuremath{'}}<N$) coming from the instances which are just discarded in all existing conventional MEPPs. The entangled ${N}^{\ensuremath{'}}$-photon subsystems are obtained efficiently by measuring the photons with potential bit-flip errors. With these two parts, the present MEPP has a higher efficiency than all other conventional MEPPs.

arXiv:1110.0059Physical Review ANov 11, 2011115 citations

Entangled multipartite systems

Springer eBooksJan 1, 20070 citations

Multipartite entanglement measure and complete monogamy relation

Yu Jing Guo, Lin Zhang

Although many different entanglement measures have been proposed so far, much less is known in the multipartite case, which leads to the previous monogamy relations in the literature being not complete. We establish here a strict framework for defining the multipartite entanglement measure (MEM): apart from the postulates of the bipartite measure, i.e., vanishing on separable measures and nonincreasing under local operations and classical communication, a complete MEM should additionally satisfy the unification condition and the hierarchy condition. We then come up with a complete monogamy formula for the unified MEM (an MEM is called a unified MEM if it satisfies the unification condition) and a tightly complete monogamy relation for the complete MEM (an MEM is called a complete MEM if it satisfies both the unification condition and the hierarchy condition). Consequently, we propose MEMs which are multipartite extensions of entanglement of formation (EoF), concurrence, tangle, Tsallis $q$ entropy of entanglement, R\'enyi $\ensuremath{\alpha}$ entropy of entanglement, the convex-roof extension of negativity, and negativity. We show that (i) the extensions of EoF, concurrence, tangle, and Tsallis $q$ entropy of entanglement are complete MEMs; (ii) multipartite extensions of R\'enyi $\ensuremath{\alpha}$ entropy of entanglement, negativity, and the convex-roof extension of negativity are unified MEMs but not complete MEMs; and (iii) all these multipartite extensions are completely monogamous, and the ones which are defined by the convex-roof structure (except for the R\'enyi $\ensuremath{\alpha}$ entropy of entanglement and the convex-roof extension of negativity) are not only completely monogamous but also tightly completely monogamous. In addition, as a byproduct, we find a class of states that satisfy the additivity of EoF. We also find a class of tripartite states one part of which can be maximally entangled with the other two parts simultaneously according to the definition of mixed maximally entangled state (MMES) in Li et al. [Z. Li, M. Zhao, S. Fei, H. Fan, and W. Liu, Quantum Inf. Comput. 12, 0063 (2012)]. Consequently, we improve the definition of maximally entangled state (MES) and prove that the only MES is the pure MES.Although many different entanglement measures have been proposed so far, much less is known in the multipartite case, which leads to the previous monogamy relations in the literature being not complete. We establish here a strict framework for defining the multipartite entanglement measure (MEM): apart from the postulates of the bipartite measure, i.e., vanishing on separable measures and nonincreasing under local operations and classical communication, a complete MEM should additionally satisfy the unification condition and the hierarchy condition. We then come up with a complete monogamy formula for the unified MEM (an MEM is called a unified MEM if it satisfies the unification condition) and a tightly complete monogamy relation for the complete MEM (an MEM is called a complete MEM if it satisfies both the unification condition and the hierarchy condition). Consequently, we propose MEMs which are multipartite extensions of entanglement of formation (EoF), concurrence, tangle, Tsallis $q$ entropy of entanglement, R\'enyi $\ensuremath{\alpha}$ entropy of entanglement, the convex-roof extension of negativity, and negativity. We show that (i) the extensions of EoF, concurrence, tangle, and Tsallis $q$ entropy of entanglement are complete MEMs; (ii) multipartite extensions of R\'enyi $\ensuremath{\alpha}$ entropy of entanglement, negativity, and the convex-roof extension of negativity are unified MEMs but not complete MEMs; and (iii) all these multipartite extensions are completely monogamous, and the ones which are defined by the convex-roof structure (except for the R\'enyi $\ensuremath{\alpha}$ entropy of entanglement and the convex-roof extension of negativity) are not only completely monogamous but also tightly completely monogamous. In addition, as a byproduct, we find a class of states that satisfy the additivity of EoF. We also find a class of tripartite states one part of which can be maximally entangled with the other two parts simultaneously according to the definition of mixed maximally entangled state (MMES) in Li et al. [Z. Li, M. Zhao, S. Fei, H. Fan, and W. Liu, Quantum Inf. Comput. 12, 0063 (2012)]. Consequently, we improve the definition of maximally entangled state (MES) and prove that the only MES is the pure MES.

arXiv:1908.08218Physical Review AMar 2, 202051 citations

We present an efficient multipartite entanglement purification protocol (MEPP) for

N

-photon systems in a Greenberger-Horne-Zeilinger state with parity-check detectors. It contains two parts. One is the conventional MEPP with which the parties can obtain a high-fidelity

N

-photon ensemble directly, similar to the MEPP with controlled-not gates. The other is our recycling MEPP in which the entanglement link is used to produce some

N

-photon entangled systems from entangled ${N}^{\ensuremath{'}}$-photon subsystems ($2\ensuremath{\leqslant}{N}^{\ensuremath{'}}<N$) coming from the instances which are just discarded in all existing conventional MEPPs. The entangled ${N}^{\ensuremath{'}}$-photon subsystems are obtained efficiently by measuring the photons with potential bit-flip errors. With these two parts, the present MEPP has a higher efficiency than all other conventional MEPPs.

q

entropy of entanglement, R\'enyi $\ensuremath{\alpha}$ entropy of entanglement, the convex-roof extension of negativity, and negativity. We show that (i) the extensions of EoF, concurrence, tangle, and Tsallis

q

entropy of entanglement are complete MEMs; (ii) multipartite extensions of R\'enyi $\ensuremath{\alpha}$ entropy of entanglement, negativity, and the convex-roof extension of negativity are unified MEMs but not complete MEMs; and (iii) all these multipartite extensions are completely monogamous, and the ones which are defined by the convex-roof structure (except for the R\'enyi $\ensuremath{\alpha}$ entropy of entanglement and the convex-roof extension of negativity) are not only completely monogamous but also tightly completely monogamous. In addition, as a byproduct, we find a class of states that satisfy the additivity of EoF. We also find a class of tripartite states one part of which can be maximally entangled with the other two parts simultaneously according to the definition of mixed maximally entangled state (MMES) in Li et al. [Z. Li, M. Zhao, S. Fei, H. Fan, and W. Liu, Quantum Inf. Comput. 12, 0063 (2012)]. Consequently, we improve the definition of maximally entangled state (MES) and prove that the only MES is the pure MES.