Multipartite entanglement measure and complete monogamy relation

Abstract

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 unification condition and the hierarchy condition. We then come up with a 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. PACS numbers: 03.67.Mn, 03.65.Db, 03.65.Ud.Introduction.-Entanglement is recognized as the most important resource in quantum information processing tasks [1]. A fundamental problem in this field is to quantify entanglement. Many entanglement measures have been proposed for this purpose, such as the distillable entanglement [2], entanglement cost [2, 3], entanglement of formation [3, 4], concurrence [5][6][7], tangle [8], relative entropy of entanglement [9,10], negativity [11,12], geometric measure [13][14][15], squashed entanglement [16,17], the conditional entanglement of mutual information [18], three-tangle [19], the generalizations of concurrence [20,21], and the α-entanglement entropy [22], etc. However, apart from the α-entanglement entropy, all other measures are either only defined on the bipartite case or just discussed with only the axioms of the bipartite case.

Multipartite uncertainty relation with quantum memory

“…This generalized entropic uncertainty depends on the conditional von-Neumann entropies, Holevo quantities, and the mutual information. We expect that the inequality will bring on more potential applications in quantum information and communication, e.g., entanglement detection 64 , multipartite entanglement-structure detection 65 , witnessing multipartite entanglement 66 , detection of genuine multipartite entanglement in multipartite systems 67 , exploring the efficient multipartite entanglement criteria 68 – 70 , analyzing the monogamy and polygamy relations of multipartite quantum states 71 , 72 , and so on. It means that our multipartite uncertainty relation will have significant applications in entanglement detection and precision measurements.…”

Characterizing entanglement and measurement’s uncertainty in neutrino oscillations

“…Then, one can make an extension to the case of an arbitrary tripartite pure state ρ ABC (t), where the entanglement quantified by EOF can be typically simplified into [60] EOF(ρ…”

Monogamy of quantum discord

“…[40]). Notice that, for these bipartite measures,only the relation between A|BC, AB and AC are revealed, the global correlation in ABC and the correlation contained in part BC is missed [43]. That is, the monogamy relation in such a sense is not "complete".…”

“…Recently, we established a complete monogamy relation for entanglement in Ref. [43]. For a unified tripartite entanglement measure E (3) , it is said to be completely monogamous if for any ρ ABC ∈ S ABC that satisfies [43]…”

“…An important issue closely related to a correlation measure of composite quantum system is to explore the distribution of the correlation throughout a multipartite state . In this context, there are two ways to describe these distribution: one is the monogamy or polygamy relation by the bipartite correlation measure [22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41] and the other one is the relation based on the multipartite correlation measure [43][44][45]. For the bipartite measure of quantum discord, although the square of quantum discord obeys the monogamy relation for three qubit states [19], other systems always display polygamous behaviour [17][18][19][20][21].…”