When Is a Genuine Multipartite Entanglement Measure Monogamous?

A crucial issue in quantum communication tasks is characterizing how quantum resources can be quantified and distributed over many parties. Consequently, entanglement has been explored extensively. However, there are few genuine multipartite entanglement measures and whether it is monogamous is so far unknown. In this work, we explore the complete monogamy of genuine multipartite entanglement measure (GMEM) for which, at first, we investigate a framework for unified/complete GMEM according to the unified/complete multipartite entanglement measure we proposed in 2020. We find a way of inducing unified/complete GMEM from any given unified/complete multipartite entanglement measure. It is shown that any unified GMEM is completely monogamous, and any complete GMEM that is induced by given complete multipartite entanglement measure is completely monogamous. In addition, the previous GMEMs are checked under this framework. It turns out that the genuinely multipartite concurrence is not as good of a candidate as GMEM.


I. INTRODUCTION
Entanglement is a quintessential manifestation of quantum mechanics and is often considered to be a useful resource for tasks like quantum teleportation or quantum cryptography [1][2][3][4], etc.There has been a tremendous amount of research in the literatures aimed at characterizing entanglement in the last three decades [1][2][3][4][5][6][7][8][9].In an effort to contribute to this line of research, however, the genuine multiparty entanglement, which represents the strongest form of entanglement in many body systems, still remains unexplored or less studied in many facets.
A fundamental issue in this field is to quantify the genuine multipartite entanglement and then analyze the distribution among the different parties.In 2000 [10], Coffman et al. presented a measure of genuine three-qubit entanglement, called "residual tangle", and discussed the distribution relation for the first time.In 2011, Ma et al. [11] established postulates for a quantity to be a GMEM and gave a genuine measure, called genuinely multipartite concurrence (GMC), by the origin bipartite concurrence.The GMC is further explored in Ref. [12], the generalized geometric measure is introduced in Refs.[13,14], the average of "residual tangle" and GMC, i.e., (τ + C gme )/2 [15], is shown to be genuine multipartite entanglement measures.Another one is the divergence-based genuine multipartite entanglement measure presented in [16,17].Recently, Ref. [18] introduced a new genuine three-qubit entanglement measure, called concurrence triangle, which is quantified as the square root of the area of triangle deduced by concurrence.Consequently, we improved and supplemented the method in [18] and proposed a general way of defining GMEM in Ref. [19].
The distribution of entanglement is believed to be monogamous, i.e., a quantum system entangled with another system limits its entanglement with the remaining others [20].There are two ways in this research.The first one is analyzing monogamy relation based on bipartite entanglement measure, and the second one is based on multipartite entanglement measure.For the former one, considerable efforts have been made in the last two decades [10,[21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40].It is shown that almost all bipartite entanglement measures we known by now are monogamous.In 2020, we established a framework for multipartite entanglement measure and discussed its monogamy relation which is called complete monogamy relation and tight complete monogamy relation [22].Under this framework, the distribution of entanglement becomes more clear since it displays a complete hierarchy relation of different subsystems.We also proposed several multipartite entanglement measure and showed that they are completely monogamous.
The situation becomes much more complex when we deal with genuine entanglement since it associates with not only multiparty system but also the most complex entanglement structure.The main purpose of this work is to establish the framework of unified/complete GMEM, by which we then present the definition of complete monogamy and tight complete monogamy of unified and complete GMEM respectively.Another aim is to find an approach of deriving GMEM from the multipartite entanglement measure introduced in Ref. [22].In the next section we list some necessary concepts and the associated notations.In Section III we discuss the framework of unified/complete GMEM and give several illustrated examples.Then in Section IV, we investigate the complete monogamy relation and tight complete monogamy relation for GMEM accordingly.A summary is concluded in the last section.

II. PRELIMINARY
For convenience, in this section, we recall the concepts of genuine entanglement, complete multipartite entanglement measure, monogamy relation, complete monogamy relation, and genuine multipartite entanglement measure.In the first subsection, we introduce the coarser relation of multipartite partition by which the following concepts can be easily processed.For simplicity, throughout this paper, we denote by an m-partite Hilbert space with finite dimension and by S X we denote the set of density operators acting on H X .

A. Coarser relation of multipartite partition
, s(i) < s(j) whenever i < j, and s(p) < t(q) whenever s < t for any possible p and q, 1 ≤ s, t ≤ k.For instance, partition if by one or two of the following ways (the coarser relation was also introduced in Ref. [41], but the third case in Ref. [41] is not valid here): the set of all the partitions that are coarser than We take the five-partite system ABCDE for example, Ξ(A|B|CD|E − A|B) = {CD|E, A|CD|E, B|CD|E, A|CD, A|E, B|E, A|C, A|D, B|C, B|D}.
For more clarity, we fix the following notations.Let for the case of (C1) and by, for the case of of (C2).For example, A|B|C|D ≻ a A|B|D ≻ a B|D, A|B|C|D ≻ b AC|B|D ≻ b AC|BD.

B. Multipartite entanglement
An m-partite pure state |ψ ∈ H A1A2•••Am is called biseparable if it can be written as It is clear that whenever a state is k-separable, it is automatically also l-separable for all 1 < l < k ≤ m.An m-partite mixed state ρ is biseparable if it can be written as a convex combination of biseparable pure states ρ = i p i |ψ i ψ i |, wherein the contained {|ψ i } can be biseparable with respect to different bipartitions (i.e., a mixed biseparable state does not need to be separable with respect to any particular bipartition).Otherwise it is called genuinely m-partite entangled (or called genuinely entangled briefly).We dnote by S A1A2•••Am g the set of all genuinely entangled states in S A1A2•••Am .Throughout this paper, for any ρ ∈ S A1A2•••Am and any given k-partition state for which we consider it as a k-partite state with respect to the partition [3,42,43] if it satisfies: An m-partite entanglement measure E (m) is said to be an m-partite entanglement monotone if it is convex and does not increase on average under m-partite stochastic LOCC.For simplicity, throughout this paper, if E is an entanglement measure (bipartite, or multipartite) for pure states, we define and call it the convex-roof extension of E, where the minimum is taken over all pure-state decomposition {p i , |ψ i } of ρ (Sometimes, we use E F to denote E F hereafter).When we take into consideration an m-partite entanglement measure, we need discuss whether it is defined uniformly for any k-partite system at first, k < m.Let E (m) be a multipartite entanglement measure (MEM).If E (k) is uniquely determined by E (m) for any 2 ≤ k < m, then we call E (m) a uniform MEM.For example, GMC, denoted by C gme [11], is uniquely defined for any k, thus it is a uniform GMEM.Recall that, and via the convex-roof extension for mixed states [11].All the unified MEMs presented in Ref. [22] are uniform MEM.That is, a uniform MEM is series of MEMs that have uniform expressions definitely.A uniform MEM E (m) is called a unified multipartite entanglement measure if it also satisfies the following condition [22]: The unification condition should be comprehended in the following sense [22].Let And , where π is a permutation of the subsystems.In addition, where the vertical bar indicates the split across which the entanglement is measured.A uniform MEM E (m) is called a complete multipartite entanglement measure if it satisfies both (E3) above and the following [22]: We need remark here that, although the partial trace is in fact a special trace-preserving completely positive map, we cannot derive Namely, different from that of bipartite case, the unification condition can not induced by the m-partite LOCC.For any bipartite measure E, E(A|BC) ≥ E(AB) for any ρ ABC since ρ AB = Tr C ρ ABC can be obtained by partial trace on part C and such a partial trace is in fact a bipartite LOCC acting on A|BC.But ρ AB can not be derived from any tripartite LOCC acting on ρ ABC .Thus, whether E (3) (A|BC) ≥ E (2) (AB) is unknown.
Several unified tripartite entanglement measures were proposed in Ref. [22]: for pure state |ψ ∈ H ABC , and then by the convex-roof extension for mixed state ρ ABC ∈ S ABC (for mixed state, N (3) is replaced with ) is the Rényi α-entropy.In addition [22], for any ρ ∈ S ABC .E f , C (3) , τ (3) and T (3) q are shown to be complete tripartite entanglement measures while R α , N (3) and N (3) F are proved to be unified but not complete tripartite entanglement measures [22].In Ref. [44], we introduce three unified tripartite entanglement measures (but not complete tripartite entanglement measures) in terms of fidelity: for any pure state |ψ in H ABC , where F is the Uhlmann-Jozsa fidelity F [45,46], which is defined as and the A-fidelity, F A , is the square of the quantum affinity A(ρ, σ) [50,51], i.e., For mixed states, E F ′ ,F , and E AF ,F are defined by the convex-roof extension as in Eq. ( 4).

D. Monogamy relation
For a given bipartite measure Q (such as entanglement measure and other quantum correlation measure), Q is said to be monogamous (we take the tripartite case for example) if [10,26] However, Eq. ( 12) is not valid for many entanglement measures [10,24,52,53] but some power function of Q admits the monogamy relation [i.e., Q α (A|BC) Q α (AB) + Q α (AC) for some α > 0].In Ref. [23], we address this issue by proposing an improved definition of monogamy (without inequalities) for entanglement measure: A bipartite measure of entanglement E is monogamous if for any ρ ∈ S ABC that satisfies the disentangling condition, i.e., we have that E(ρ AC ) = 0, where ρ AB = Tr C ρ ABC .With respect to this definition, a continuous measure E is monogamous according to this definition if and only if there exists 0 < α < ∞ such that for all ρ acting on the state space H ABC with fixed dim H ABC = d < ∞ (see Theorem 1 in Ref. [23]).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 are missed [22].That is, the monogamy relation in such a sense is not "complete".For a unified tripartite entanglement measure E (3) , it is said to be completely monogamous if for any ρ ∈ S ABC that satisfies [22] we have that E (2)  3) is a continuous unified tripartite entanglement measure.Then, E (3) is completely monogamous if and only if there exists 0 < α < ∞ such that [22] for all ρ ABC ∈ S ABC with fixed dim H ABC = d < ∞, here we omitted the superscript (2,3) of E (2,3) for brevity.Let E (3) be a complete MEM.E (3) is defined to be tightly complete monogamous if for any state ρ ABC ∈ S ABC that satisfying [22] we have E (2) (ρ BC ) = 0, which is equivalent to for some α > 0, here we omitted the superscript (2,3) of E (2,3) for brevity.For the general case of E (m) , one can similarly followed with the same spirit.

E. Genuine entanglement measure
A function E (m) g : S A1A2•••Am → R + is defined to be a measure of genuine multipartite entanglement if it admits the following conditions [11]: (ρ) > 0 for any genuinely entangled state ρ ∈ S A1A2•••Am .(Thisitem can be weakened as: E (m) g (ρ) 0 for any genuinely entangled state ρ ∈ S A1A2•••Am .That is, maybe there exists some state which is genuinely entangled such that E (m) g (ρ) = 0.In such a case, the measure is called not faithful.Otherwise, it is called faithful.For example, the "residual tangle" is not faithful since it is vanished for the W state.) is said to be a genuine multipartite entanglement monotone if it does not increase on average under m-partite stochastic LOCC.For example, C gme is a GMEM.

III. COMPLETE GENUINE MULTIPARTITE ENTANGLEMENT MEASURE
Analogous to that of unified/complete multipartite entanglement measure established in Ref. [22], we discuss the unification condition and the hierarchy condition for genuine multipartite entanglement measure in this section.We start out with observation of examples.Let |ψ be an m-partite pure state in H A1A2•••Am .Recall that, the multipartite entanglement of formation E (m) f is defined as [22] where ρ X := Tr X (|ψ ψ|).We define where δ(ρ) = 0 if ρ is biseparable up to some bi-partition and δ(ρ) = 1 if ρ is not biseparable up to any bipartition.For mixed state, it is defined by the convex-roof extension.Obviously, E g−f is a uniform GMEM since 0 for any n [54], where I(A 1 : The following properties are straightforward: For any g (ρ ABC ) for some ρ ABCD ∈ S ABCD g , then the entanglement between part ABC and part D is zero, which means that ρ ABCD is biseparable with respect to the partition ABC|D, a contradiction.In addition, let |ψ ABC be a tripartite genuine entangled state in H ABC , then |ψ ABC |ψ D is not a four-partite genuine entangled state, i.e., E (4)  g (|ψ That is, the genuine multipartite entanglement measure is not necessarily decreasing under discarding of subsystem.However, for the genuine entangled state, it is decreasing definitely.From this observations, we give the following definition.
be a uniform genuine entanglement measure.If it satisfies the unification condition, i.e., and a unified genuine multipartite entanglement measure, where π(•) denotes the permutation of the subsystems.
since 'some amount of entanglement' may be hided in the combined subsystem.For example, the quantity E (3) g (AB|C|D) seems can not report the entanglement contained between subsystems A and B. We thus present the following definition.
then it is said to be a complete genuine multipartite entanglement measure.

By definition, E (m)
g−f is a complete GMEM.But C gme is not a complete GMEM since it does not satisfy the hierarchy condition (21).We take a four-partite state for example.Let 8 .In general, C gme is even not a unified GMEM since we can not guarantee the unification condition (20) hold true.
We now turn to find unified/complete GMEM.E (m) . This motivates us to obtain unified/complete GMEMs from the unified/complete MEMs.Proposition 1.Let E (m) be a unified/complete multipartite entanglement measure (resp.monotone), and define whenever E (m) is not defined by the convex-roof extension for mixed state, where the minimum is taken over all purestate decomposition {p i , |ψ i } of ρ ∈ S A1A2•••Am , δ(ρ) = 1 whenever ρ is genuinely entangled and δ(ρ) = 0 otherwise.Then E (m) g is a unified/complete genuine multipartite entanglement measure (resp.monotone). Proof.
satisfy the unification condition (resp.hierarchy condition) on S A1A2•••Am g whenever E (m) satisfies the unification condition (resp.hierarchy condition) on S A1A2•••Am .
Consequently, according to Proposition 1, we get for pure states and define by the convex-roof extension for the mixed states (for mixed state, N is replaced with the convex-roof extension of N g−F ), and N (3)  g (ρ) = δ(ρ) ρ Ta Tr + ρ T b Tr + ρ Tc Tr − 3 for any ρ ∈ S ABC .These tripartite measures, except for N (3) g are in fact special cases of E F g−123 in Ref. [19].Generally, we can define for pure states and define by the convex-roof extension for the mixed states (for mixed state, N (m) g is replaced with N (m) g−F ), and for any ρ ∈ S A1A2•••Am .According to Proposition 1, together with Theorem 5 in Ref. [22], the statement below is straightforward.
, and T (m) g−q are complete genuine multipartite entanglement monotones while R (m) and E (m) g−AF are unified genuine multipartite entanglement monotones but not complete genuine multipartite entanglement monotones.
Very recently, we proposed the following genuine four-partite entanglement measures [19].Let E be a bipartite entanglement measure and let for any given |ψ ∈ H ABCD , where 2) is a genuine four-partite entanglement measure.Let E (3) be a tripartite entanglement measure, for any given |ψ ∈ S ABCD , where E (3)  3) is a genuine four-partite entanglement measures but not uniform GMEM.
Generally, we can define E F g−1234•••m(2) by the same way and it is a uniform GMEM.We check below E is a complete GMEM whenever E is an entanglement monotone.We only need to discuss the case of m = 4 and the general cases can be argued similarly.
for some decomposition ρ = j p j |φ j φ j |.Then for any j, where ρ ABC j = Tr D (|φ j φ j |).Therefore as desired.In addition, it is clear that is a complete GMEM whenever E is an entanglement monotone.Remark 1.It is clear that, for E F g−1234•••m(2) , the inequality in Eq. ( 21) is a strict inequality, i.e., In addition, according to the proof of Proposition 4 in Ref. [22], Eq. ( 21) holds for E (m) , and T (m) g−q .Namley, in general, there does not exist ρ

IV. COMPLETE MONOGAMY OF GENUINE MULTIPARTITE ENTANGLEMENT MEASURE
We are now ready for discussing the complete monogamy relation of GMEM.By the previous arguments, the genuine multipartite entanglement measure is not necessarily decreasing under discarding of subsystem.However, for the genuine entangled state, it does decrease.We thus conclude the following definition of complete monogamy for genuine entanglement measure.Definition 3. Let E (m) g be a uniform GMEM.We call E (m) g is completely monogamous if for any ρ ∈ S A1A2•••Am g we have holds for all That is, any unified GMEM is completely monogamous.Moreover, according to the proof of Theorem 1 in Ref. [23], we can get the equivalent statement of complete monogamy for continuous genuine tripartite entanglement measure (the general m-partite case can be followed in the same way).
Analogously, for the four-partite case, if E g is a continuous uniform GMEM, then E g is completely monogamous if and only if there exist 0 < α, β < ∞ such that E α g (ρ ABCD ) > E α g (ρ ABC ) + E α g (ρ ABD ) + E α g (ρ ACD ) + E α g (ρ BCD ), for all ρ ABCD ∈ S ABCD g with fixed dim H ABC = d < ∞, here we omitted the superscript (3,4) of E (3,4) for brevity.Since C gme may be not a unified GMEM, we conjecture that C gme is not completely monogamous.
As a counterpart to the tightly complete monogamous relation of the complete multipartite entanglement measure in Ref. [22], we give the following definition.Definition 4. Let E (m) g be a complete GMEM.We call E (m) g is tightly complete monogamous if it satisfies the genuine disentangling condition, i.e., either for any ρ ∈ S A1A2•••Am g that satisfies we have that holds for all Γ ∈ Ξ(X

Definition 2 .
Let E (m) g be a unified GMEM.If E (m) g admits the hierarchy condition, i.e.,
For any genuine entangled pure state |ψ ∈ H ABCD , and any bipartite entanglement monotone E, it is clear thatE g−1234(2) (|ψ ) > E F (ρ XY ) for any {X, Y } ∈ {A, B, C, D}.For any pure state decomposition of ρ ABC , ρ ABC = i p i |ψ i ψ i |, we have E(|ψ A|BCD ) p i E(|ψ i B|AC ) since any ensemble {p i , |ψ i } can be derived by LOCC from |ψ .It follows that E g−1234(2) (|ψ ) > E F g−123(2) (ρ ABC ).By symmetry of the subsystems, we get the unification condition is valid for pure state.For mixed state ρ ∈ S ABCD i p i E(|ψ i A|BC ), E(|ψ AB|CD ) i p i E(|ψ i AB|C ), and E(|ψ B|ACD ) i g , we let