Superactivation of monogamy relations for nonadditive quantum correlation measures
Zhi-Xiang Jin Shao-Ming Fei





Corresponding author: jzxjinzhixiang@126.com
School of Mathematical Sciences,  Capital Normal University,  Beijing 100048,  China

Corresponding author: feishm@cnu.edu.cn
School of Mathematical Sciences,  Capital Normal University,  Beijing 100048,  China
Max-Planck-Institute for Mathematics in the Sciences, Leipzig 04103, Germany


We investigate the general monogamy and polygamy relations satisfied by quantum correlation measures.
We show that there exist two real numbers α and β such that for any quantum correlation measure Q, Q^x is monogamous if x≥α and polygamous if 0≤ x≤β for a given multipartite state ρ. For β <x<α, we show that the monogamy relation can be superactivated by finite m copies ρ^⊗ m of ρ for nonadditive correlation measures.
As a detailed example, we use the negativity as the quantum correlation measure to illustrate such superactivation of monogamy properties. A tighter monogamy relation is presented at last.


Superactivation of monogamy relations for nonadditive quantum correlation measures
    Shao-Ming Fei
    December 30, 2023
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§ INTRODUCTION


Quantum mechanically the more a two-level system is entangled with another two-level system, the less it can be entangled with a third one <cit.>. This behavior, known as entanglement monogamy, has also been found in larger systems <cit.>. Monogamy of entanglement is one of the nonintuitive phenomena of quantum physics that distinguish it from classical physics. Classically, three random bits can be maximally correlated. However, it is not possible to prepare three qubits in a way that any two qubits are maximally entangled <cit.>, i.e., a quantum system entangled with one of other subsystems limits its entanglement with the remaining ones. Moreover, the monogamy property has emerged as the ingredient in the security analysis of quantum key distribution <cit.>.

The monogamy of entanglement has fundamental implications in some quantum information processing. For example, the lack of monogamy is considered a huge obstacle to the implementation of quantum cryptography <cit.>. It also plays roles in detecting phases in many-body physics <cit.>, and provides information that may help us to understand the mysterious behavior of black holes <cit.>. Moreover, the monogamy of quantum correlation is essential for proving that asymptotic cloning is equivalent to state estimation <cit.> and making quantum key distribution secure <cit.>.

For a tripartite system A, B and C, the usual monogamy of a quantum correlation measure Q implies that the correlation Q_A|BC between A and BC satisfies the following inequality,

    Q_A|BC≥ Q_AB +Q_AC,

where Q_AB (Q_AC) stands for the correlation between systems A and B (C) of the corresponding reduced bipartite system.
Inequality (<ref>) was originally proven for arbitrary three-qubit states, adopting the squared concurrence C^2 as the correlation (entanglement) measure <cit.>. Variations of the CKW inequality and its generalizations to N-partite systems have been presented for a number of entanglement measures in discrete as well as continuous variable systems <cit.>.

Dually, the polygamy relation is quantitatively described by

    Q_A|BC≤ Q_AB +Q_AC.

Recently, polygamy inequalities have been given for multiqubit systems under various entanglement measures <cit.>. The monogamy and polygamy inequalities are also proposed in terms of
non-negative power of various entanglement measures <cit.>, and a generalization of the polygamy constraint of multipartite entanglement in arbitrary dimensional quantum systems has been given in Ref. <cit.>.

It has been shown that the concurrence, negativity, and tangle adopt monogamy relations for multiqubit systems <cit.>. However, it is still unclear for high dimensional systems. It has been shown that the entanglement of assistance and the entanglement of assistance associated with the Tsallis-q entropy are polygamous for any multipartite systems <cit.>. The concurrence of assistance, tangle of assistance and negativity of assistance are proved to be polygamous only for multiqubit systems <cit.>. The polygamy problem for other entanglement measures remains open for high dimensional systems.

Due the importance of monogamy relations, it is also interesting to ask whether the monogamy property of a quantum state can be superactivated. Namely, if ρ does not satisfy monogamous relations with respect to some correlation measure, can the many copy state ρ⊗ρ⋯⊗ρ be monogamous?

In this paper, we study the monogamy and polygamy relations for general quantum correlation measures. We show that there always exist two real numbers α and β for any quantum correlation measures and
any given quantum states: Q^x is monogamous if x≥α and polygamous if 0≤ x≤β.
For β <x<α, we find that the monogamy property depends not only on the quantum state, the quantum correlation measure, the power of the quantum corrlation measure, but also on the number of copies of the state. The phenomena is similar to the nonlocality revealed by the violation of Bell inequalities.
For some bipartite states that admit local hidden variable models, their nonlocality can be super activated by m copies of the states <cit.>. Here we show that for a state which does not admit monogamous relations, the m copies of the state could satisfy monogamous relations. We use the negativity as the quantum correlation measure to illustrate such super activation of monogamy properties.
We also present the definition of regularized quantum correlation measures and show that almost all regularized quantum correlation measures satisfy the monogamy relations, except for the measures which are additive. We give a tighter monogamy relation at last and generalize it to the multipartite case.

Throughout this paper, Q(ρ_A|B_1B_2,⋯,B_N-1) denotes the quantum correlation of the state ρ_AB_1B_2,⋯,B_N-1 under bipartite partition  A and B_1B_2,⋯,B_N-1, which keeps invariant under discarding subsystems only for states satisfying  monogamy relations. For simplicity, we denote Q(ρ_AB_i) by Q_AB_i, and Q(ρ_A|B_1B_2,⋯,B_N-1) by Q_A|B_1B_2,⋯,B_N-1.




§ MONOGAMY AND POLYGAMY RELATIONS FOR QUANTUM CORRELATION MEASURES


[Theorem 1]. Let Q be a continuous measure of quantum correlation. For any tripartite state ρ_ABC in discrete finite dimensional Hilbert space, there exist real numbers α
and β such that

(1) if x≥α, then Q^x is monogamous,

    Q^x_A|BC≥ Q^x_AB +Q^x_AC,


(2) if 0≤ y≤β, then Q^y  is polygamous,

    Q^y_A|BC≤ Q^y_AB +Q^y_AC.


[Proof].
As Q is a measure of quantum correlation, it is nonincreasing under partial trace. Therefore, Q_A|BC≥max{Q_AB,Q_AC} for any state ρ_ABC. If Q_A|BC=0, the result is obvious. Therefore, we assume Q_A|BC> 0 and set x_1=Q_AB/Q_A|BC, x_2=Q_AC/Q_A|BC. Clearly, there exists γ such that

    x_1^γ+x_2^γ≤ 1,

since x_1 and x_2 decrease when γ increases. Let f(ρ_ABC) be the smallest value of γ that saturates the inequality (<ref>). As Q is continuous, we obtain
α=sup_ρ_ABCf(ρ_ABC), which proves the case 1 of Theorem 1.

On the other hand, there always exists δ≥0 such that

    x_1^δ+x_2^δ≥ 1,

since x_1 and x_2 increase when δ decreases. Let g(ρ_ABC) be the largest value of δ that saturates the inequality (<ref>). As Q is continuous, we obtain
β=inf_ρ_ABCg(ρ_ABC)>0, which proves the case 2 of Theorem 1.
1ex1ex

Note that the monogamy relation satisfied by Q^α remains for larger power than α <cit.>, i.e., Q^α_A|BC≥ Q^α_AB +Q^α_AC implies that Q^η_A|BC≥ Q^η_AB +Q^η_AC for any η≥α. This power α depends on the correlation measure Q and the dimension of the system <cit.>. Generally, it is hard to compute, especially for higher dimensional systems. The case 1 of Theorem 1 indicates that almost any correlation measures can give rise to monogamous relations for sufficient larger α. Similarly, the polygamy relation of Q^β can be preserved for smaller power <cit.>, i.e., Q^β_A|BC≤ Q^β_AB +Q^β_AC implies that Q^θ_A|BC≤ Q^θ_AB +Q^θ_AC for any 0≤θ≤β. The polygamy power β also depends on the measure Q. The monogamy power α is a dual concept of the polygamy power β. They both reflect the shareability of the correlations Q among the subsystems. In the following, we consider the monogamy relation of Q^x for β≤ x≤α.



§ SUPERACTIVATION OF MONOGAMY RELATIONS


Monogamy characterizes the distribution of quantum correlations, which depends on the quantum states, the quantum correlation measure Q, the dimension of the system, and the power x in the quantum correlation measure Q^x. In Refs. <cit.> the monogamy relations are established for x≥ 2 in multiqubit systems for concurrence and negativity, while for x≥√(2) for the entanglement of formation. However, it has been proved that the monogamy inequality fails for some three-qutrite quantum states under the squared concurrence.

From Theorem 1 it is obvious that there exist states ρ such that for β≤ x≤α, Q^x does not satisfy a monogamy relation. To investigate the monogamy property for such states,
we define, from the inequality (<ref>), the residual quantum correlation as
R(ρ_ABC)=Q_A|BC- Q_AB-Q_AC
for the tripartite case. Such residual correlation increases when Q is replaced by Q^x and x increases <cit.>, a fact that can also be deduced from Theorem 1. In other words, the residual quantum correlation R can be changed from negative to positive by varying x.
This fact enables the super activation of monogamy relations, similar to super activation of nonlocality <cit.>:
A state ρ_ABC does not satisfy the monogamy relation, while the state ρ^⊗ m_ABC is monogamous, i.e., R(ρ^⊗ m_ABC)≥0 while R(ρ_ABC)≤0 for some m>1.




Here, ρ^⊗ m_A_1A_2...A_n=ρ_A_11A_12...A_1mA_21...A_2m...A_n1...A_nm, A_ij denotes the ith party of the jth copy of ρ_A_1A_2...A_n. From the definition of the residual quantum correlation, one gets R^(n)(ρ^⊗ m_A_1A_2...A_n)=Q(ρ^⊗ m_A_1|A_2...A_n)-∑_i=2^nQ(ρ^⊗ m_A_1A_i), where ρ^⊗ m_A_1A_i=ρ_A_11...A_1mA_i1...A_im=tr_A_21...A_(i-1)mA_(i+1)1...A_nm(ρ_A_11A_12...A_1mA_21...A_2m...A_n1...A_nm), Q(ρ^⊗ m_A_1|A_2...A_n)=Q(ρ_A_11A_12...A_1m|A_21...A_2m...A_n1...A_nm) denotes the quantum correlation between the first party and the rest ones after the m copies of ρ_A_1A_2...A_n, i.e., the quantum correlations between A_1 and A̅_̅1̅, view A_11A_12...A_1m and A_21...A_2m...A_n1...A_nm as A_1 and A̅_̅1̅, respectively, and Q(ρ^⊗ m_A_1A_i)=Q(ρ_A_11...A_1m|A_i1...A_im) is the quantum correlation of the first party and the ith party.



In the following, we consider the quantum correlation measures that are nonadditive. If a measure of quantum correlation is additive, i.e., Q(ρ⊗δ)=Q(ρ)+Q(δ), then a nonmonogamous state cannot be changed to a monogamy one by superactivation. We have the following result on superactivation of monogamy relations.

[Theorem 2]. Let Q be a continuous measure of quantum correlation. For any state ρ_ABC that Q^x does not satisfy the monogamy relation (<ref>) for β≤ x≤α, there always exists a positive integer m such that

    Q(ρ^⊗ m_A|BC)≥ Q(ρ^⊗ m_AB) +Q(ρ^⊗ m_AC).


[Proof]. For any state ρ_AB of the composite systems A and B with eigenvalues λ_i
and the corresponding eigenstates |i⟩, let us introduce a third system B' such that |ψ⟩=∑_i√(λ_i)|i⟩|î⟩ is a pure state of the tripartite system ABB', with orthonormal basis |î⟩ of B'.
Then ρ_AB=tr_B'|ψ⟩⟨ψ|, where tr_B' is the partial trace over B'. As tr_B' is a local operation performed on B', Q(ρ_AB)≤ Q(|ψ⟩⟨ψ|). Since ρ_A=tr_BB'|ψ⟩⟨ψ|=tr_Bρ_AB, the Schmidt coefficients of |ψ⟩ are √(λ_i(ρ_A)). Hence, the quantum correlation has the form, Q(|ψ⟩⟨ψ|)=f(λ⃗(ρ_A)), where f is a function of λ⃗(ρ_A) given by the nonzero eigenvalues of the state ρ_A. Thus, Q(ρ_AB) depends on ρ_A only and Q(ρ_AB)≤ f(λ⃗(ρ_A)). Therefore, there exists a positive number 0≤ L≤ 1 such that Q(ρ_AB)=L f(λ⃗(ρ_A)), and Q^k(ρ_AB)=L^k f^k(λ⃗(ρ_A))=g(λ⃗^k(ρ_A)), where k is a positive integer, λ⃗^k(ρ_A)=(λ_1^k(ρ_A),...,λ_r_A^k(ρ_A)), and g is a function of  λ_i^k(ρ_A), i=1,2,...,r_A, r_A is the rank of the reduced density matrix ρ_A.
Let us now consider ρ^⊗ m_AB. Since the eigenvalues of ρ^⊗ m_AB are {λ_i^m}, i=1,2,..., r_AB, r_AB is the rank of density matrix ρ_AB, which are the elements of the function Q^m(ρ_AB). Using the Theorem 1, we have that there always exits a positive integer m such that the inequality (<ref>) holds.1ex1ex

As an example let us consider the three qubits W state, |W⟩_ABC=1/√(3)(|100⟩+|010⟩+|001⟩), which does not satisfy the usual monogamy relation (<ref>) <cit.>.
We consider the superactivation of monogamy relations for W states under the entanglement measure negativity. Given a bipartite state ρ_AB, the negativity is defined by <cit.>
N(ρ_AB)=(||ρ_AB^T_A||-1)/2,
where ρ_AB^T_A is the partial transpose with respect to the subsystem A, ||X|| denotes the trace norm of X, i.e., ||X||=Tr√(XX^†).
Negativity is a computable measure of entanglement, and is a convex function of ρ_AB. For any bipartite pure state |ψ⟩_AB in d⊗ d' (d≤ d') quantum system with Schmidt decomposition |ψ⟩_AB=∑_i=0^d-1√(λ_i)|ii⟩, λ_i≥0, ∑_i=0^d-1λ_i=1, the negativity N(ρ_AB) is given by
N(|ψ⟩_AB)=2∑_i<j√(λ_iλ_j)=(Tr√(ρ_A))^2-1,
where λ_i are the eigenvalues of the reduced density matrix of |ψ⟩_AB. For a mixed state ρ_AB, the convex-roof of entanglement negativity (CREN) is defined by

    N_c(ρ_AB)=min∑_ip_iN(|ψ_i⟩_AB),

where the minimum is taken over all possible pure state decompositions {p_i, |ψ_i⟩_AB} of ρ_AB. From Refs. <cit.>, N^x_c satisfies the monogamy relation if x≥2 for states in 2⊗ 2⊗ 2^n systems.
It is easy to get that x=2 for the W-class states <cit.>.
N_c (the case x=1) itself is not monogamous for the W states.
Consider m copies of the W states.
We have N_c(|W⟩_A|BC^⊗ m)=1/2[(1+4√(2)/3)^m-1], N_c(ρ_AB^⊗ m)=N_c(ρ_AC^⊗ m)=1/2[(1+4/3)^m-1], where ρ_AB and ρ_AC are the reduced states of |W⟩_ABC. The relation between
N_c(|W⟩_A|BC^⊗ m) and the summation of N_c(ρ_AB^⊗ m) and N_c(ρ_AC^⊗ m) is shown in Fig. 1. From Fig. 1 one can see that |W⟩_ABC^⊗ m satisfy the monogamy inequality for m>3.



Similar to violation of Bell inequalities, which can be enhanced by superactivation, the monogamy relations can also be superactivated. The monogamy relation satisfied by the xth power of a quantum correlation measure Q can be revealed by finite copies of a state for β<x<α. We can define the regularized quantum correlation measure of Q as Q^∞(ρ)=lim_m→∞1/mQ(ρ^⊗ m). Thus, from the conclusion of Theorem 2, it is easy to get Q^∞(ρ_A|BC)≥ Q^∞(ρ_AB)+Q^∞(ρ_AC).

It is clear that monogamy is a property of both the quantum state and quantum correlation measure. From the results of Theorem 1 and Theorem 2, the monogamy relation may also depend on the power of the correlation measure and the number of copies. However, not all quantum correlation measures are able to make a quantum state from being nonmonogamous to monogamous by finite copies, such as any quantum correlation measures satisfying the property of additivity, Q(ρ^⊗ m)=m Q(ρ). Moreover, there are special classes of states, which are always monogamous for any quantum correlation measures. For example, the generalized n-partite GHZ-class state admitting the multipartite Schmidt decomposition <cit.>, |ψ⟩_A_1A_2⋯ A_n=∑_i=1λ_i|i_1⟩⊗|i_2⟩⊗⋯⊗|i_n⟩, ∑_iλ_i^2=1, λ_i>0, for which we always have E(|ψ⟩_A_1|A_2⋯ A_n)>0, while E(ρ_A_1A_i)=0 for all i=2,⋯,n, for any quantum entanglement measures E.

Remark. In Ref. <cit.>, the authors have shown that quantum nonlocality can be superactivated. For some bipartite states that admit local hidden variable models, their nonlocality can be superactivated by copies of the states. There are quantum states whose Bell violations can be arbitrarily enlarged by increasing the dimensions. Similarly, for superactivation of monogamy relations, by increasing the number m of the copies, the monogamy relations can be always superactivated for nonadditive quantum correlation, e.g, under the regularized quantum correlation measure. From Fig. 1, one sees that the residual correlation can be made arbitrarily large by increasing the number of copies.



§ SUPER ACTIVATION OF MONOGAMY RELATIONS FOR MULTIPARTITE SYSTEMS


Theorem 1 can be generalized to the multipartite systems. For any n-partite state ρ_A_1A_2⋯ A_n, one can view A_1, A_2, and A_3⋯ A_n as A, B, and C, respectively, in Theorem 1. Then by partitioning the last system C into two subsystems A_3 and A_4⋯ A_n, one can apply Theorem1 repeatedly. We have the following result.

[Theorem 3]. Let Q be a continuous measure of quantum correlation. For any state ρ_A_1A_2⋯ A_n, there exist real numbers α and β such that

(1) if x≥α, then Q^x is monogamous,

    Q^x_A_1|A_2⋯ A_n≥∑_j=2^nQ^x_A_1A_j;


(2) if 0≤ y≤β, then Q^y  is polygamous,

    Q^y_A_1|A_2⋯ A_n≤∑_j=2^nQ^y_A_1A_j.


Similarly concerning Theorem 2, we have for multipartite system:

[Theorem 4]. Let Q be a continuous measure of quantum correlation.
For any state ρ_A_1A_2⋯ A_n that Q^x does not satisfy the monogamy relation (<ref>) for β≤ x≤α, there always exists a positive integer m such that

    Q(ρ^⊗ m_A_1|A_2⋯ A_n)≥∑_j=2^n Q(ρ^⊗ m_A_1A_j).


Let us consider the entanglement measure negativity N_c, and a class of n-qudit quantum states |W^d_n⟩_A_1⋯ A_n=∑_i=1^d-1(a_1i|i0⋯0⟩+⋯+a_ni|00⋯ i⟩), with ∑_s=1^n∑_i=1^d-1|a_si|^2=1.
First, we prove the negativity N_c of |W_n^d⟩_A_1A_2⋯ A_n does not satisfy the usual monogamy inequality (<ref>). Then, we find the positive integer m such that the monogamy relation for |W_n^d⟩^⊗ m_A_1A_2⋯ A_n is established.

The reduced density matrix ρ_A_1 of |W_n^d⟩_A_1A_2⋯ A_n with respect to the subsystem A_1 is given by

    ρ_A_1      =Tr_A_2⋯ A_n(|W_n^d⟩_A_1A_2⋯ A_n⟨ W_n^d|)
          =∑_i,j=1^d-1a_1ia_1j^∗|i⟩_A_1⟨ j|+Ω_1|0⟩_A_1⟨ 0|,

where Ω_1=∑_s=2^n∑_i=1^d-1|a_si|^2=1-∑_j=1^d-1|a_1j|^2.
From the definition of the negativity and Eq. (<ref>), we have

    N_c(|W_n^d⟩_A_1|A_2⋯ A_n)      =(Tr√(ρ_A_1))^2-1
          =2√((1-Ω_1)Ω_1).


The two-qudit reduced density matrix ρ_A_1A_2 of |W_n^d⟩_A_1A_2⋯ A_n,
similar for ρ_A_1 A_s), s=2,⋯, n, is given by

    ρ_A_1A_2      =Tr_A_3⋯ A_n(|W_n^d⟩_A_1A_2⋯ A_n⟨ W_n^d|)
          =∑_i,j=1^d-1(a_1ia_1j^∗|i0⟩_A_1A_2⟨ j0|+a_1ia_2j^∗|i0⟩_A_1A_2⟨ 0j|
          +a_2ia_1j^∗|0i⟩_A_1A_2⟨ j0|+a_2ia_2j^∗|0i⟩_A_1A_2⟨ 0j|)
          +Ω_2|00⟩_A_1A_2⟨ 00|,

where Ω_2=1-∑_j=1^d-1(|a_1j|^2+|a_2j|^2). Note that, by using the following two un-normalized states

    |x̂⟩_A_1A_2=∑_i=1^d-1(a_1i|i0⟩_A_1A_2+a_2i|0i⟩_A_1A_2),
          |ŷ⟩_A_1A_2=√(Ω_2)|00⟩_A_1A_2,

ρ_A_1A_2 in Eq. (<ref>) can be represented as

    ρ_A_1A_2=|x̂⟩_A_1A_2⟨x̂|+|ŷ⟩_A_1A_2⟨ŷ|.


For any pure state decomposition

    ρ_A_1A_2=∑_h|ϕ̂_h⟩_A_1A_2⟨ϕ̂_h|,

where |ϕ̂_h⟩_A_1A_2 is an un-normalized state in two-qudit subsystem A_1A_2, there exists a unitary matrix u with entries u_hl such that

    |ϕ̂_h⟩_A_1A_2=u_h1|x̂⟩_A_1A_2+u_h2|ŷ⟩_A_1A_2,

for each h. For the normalized state |ϕ_h⟩_A_1A_2=|ϕ̂_h⟩_A_1A_2/√(p_h) with p_h=|⟨ϕ̂_h|ϕ̂_h⟩|, the pure state negativity is given by

    N_c(|ϕ_h⟩_A_1A_2)      =2/p_h|u_h2|^2√((1-Ω_1)(Ω_1-Ω_2))
          =2/p_h|u_h2|^2√((1-Ω_1)∑_i=1^d-1|a_2i|^2)

for each h.

From  Eq. (<ref>) and Eq. (<ref>), we have

    N_c(ρ_A_1A_2)      =min_{p_h,|ϕ_h⟩}∑_hp_hN_c(|ϕ_h⟩)_A_1A_2
          =min_{p_h,|ϕ_h⟩}∑_h2|u_h2|^2√((1-Ω_1)∑_i=1^d-1|a_2i|^2)
          =2√((1-Ω_1)∑_i=1^d-1|a_2i|^2),

where the last equality is due to the choice of u_h2 from the unitary matrix u.
Here we note that the minimum average of the CREN in Eq. (<ref>) does not depend on the choice of the pure state decomposition of ρ_A_1A_2, which simplifies the minimization problem.

By using an analogous method, we have

    N_c(ρ_A_1A_s)=2√((1-Ω_1)∑_i=1^d-1|a_si|^2),

for each s=2,⋯,n.

From Eq. (<ref>) and Eq. (<ref>), one can easily obtain that N_c(|W_n^d⟩_A_1|A_2⋯ A_n)≤∑_s=2^nN_c (ρ_A_1A_s). Therefore, N_c is not monogamous for the state |W_n^d⟩_A_1A_2⋯ A_n.

Now let us consider m copies of the state |W_n^d⟩_A_1A_2⋯ A_n, |W_n^d⟩_A_1A_2⋯ A_n^⊗ m.
We get N_c(|W_n^d⟩_A_1|A_2⋯ A_n^⊗ m)=1/2[(1+4√((1-Ω_1)Ω_1))^m-1], and N_c(ρ_A_1A_s^⊗ m)=1/2[(1+4√((1-Ω_1)∑_i=1^d-1|a_si|^2))^m-1] for s=2,⋯,n.
Since 1+4√((1-Ω_1)Ω_1)>1+4√((1-Ω_1)∑_i=1^d-1|a_si|^2)>1, s=2,⋯,n, it is always possible to choose some positive integer m such that the monogamy relation holds.
For example, let d=2, n=5, a_11=⋯=a_51=1/√(5). Then N_c(|W_5^2⟩_A_1|A_2⋯ A_5^⊗ m)=1/2[(13/5)^m-1], and N_c(ρ_A_1A_s^⊗ m)=1/2[(9/5)^m-1] for s=2,⋯,5.
It is easy to prove that the monogamy relation is satisfied for m≥ 4, see Fig. 2.


To illustrate the relations among the superactivation of monogamy property, the number m of the copies and the number n of the subsystems, let us consider the case a_ij=1/√(nd), i=1,2,...,n, j=1,2,...,d. We have N_c(|W_n^d⟩_A_1|A_2⋯ A_n^⊗ m)=1/2[(1+4√(n-1)/n)^m-1], and N_c(ρ_A_1A_s^⊗ m)=1/2[(1+4/n)^m-1] for each s=2,⋯,n.
Therefore, N_c(|W_n^d⟩_A_1|A_2⋯ A_n^⊗ m)-
∑_s=2^n N_c(ρ_A_1A_s^⊗ m)=
1/2[(1+4√(n-1)/n)^m-(n-1)(1+4/n)^m+n-2]≡ f(n,m).
Figure 3 shows that to keep f(n,m)≥ 0, with the increase of the number n of subsystems, the number m of the copies should also increase. When n tends to infinity, f(n,m) is 0 for any m. If m is sufficiently large, the monogamy relation is always satisfied.
Here, from the conclusion of Theorem 2, the regularized quantum correlation measure always satisfies the monogamy relation, Q^∞(ρ^⊗ m_A_1|A_2⋯ A_n)≥∑_j=2Q^∞(ρ^⊗ m_A_1A_j).





§ TIGHTER MONOGAMY RELATION OF MULTIPARTITE SYSTEMS


In the following, we show that these monogamy inequalities satisfied by the quantum correlation measures can be further refined and become tighter.

[Theorem 5].
Let Q be a continuous measure of quantum correlation. For any tripartite state ρ_ABC, if Q_AB≥ Q_AC, then for any real number s≥ 1, we have

    Q^t_A|BC≥ Q^t_AB+(2^s-1)Q^t_AC,

where t=sα, where α is given in Theorem 1.

[Proof]. By using the inequality (1+t)^x≥ 1+(2^x-1)t^x, 0≤ t ≤ 1, x∈ [1,∞), we have

    Q^xs_A|BC      ≥ (Q^x_AB+Q^x_AC)^s
          =Q^xs_AB(1+Q^x_AC/Q^x_AB)^s 
          ≥ Q^xs_AB[1+(2^s-1)(Q^x_AC/Q^x_AB)^s]
          =Q^xs_AB+(2^s-1)Q^xs_AC.

As the subsystems A and B are equivalent in this case, we have assumed that Q_AB≥ Q_AC without loss of generality. Moreover, if Q_AB=0, we have Q_AB=Q_AC=0: The lower bound becomes trivially zero. 1ex1ex

Since for s≥ 1, 2^s-1≥1, (<ref>) in Theorem 5 gives a tighter monogamy relation, with larger lower bounds, than (<ref>) in Theorem 1.
For the multipartite state, we can also have a tighter monogamy relation.

[Theorem 6]. For multipartite state ρ_A_1A_2⋯ A_n∈ and a continuous measure Q of quantum correlation, if
Q_A_1A_i≥ Q_A_1|A_i+1⋯ A_n for i=2, ⋯, m, and
Q_A_1A_j≤ Q_A_1|A_j+1⋯ A_n for j=m+1,⋯,n-1,
∀ 2≤ m≤ n-2, n≥ 4, we have

    Q^t_A_1|A_2⋯ A_n≥
          Q^t_A_1A_2+(2^s-1) Q^t_A_1A_3+⋯+(2^s-1)^m-2Q^t_A_1A_m
          +(2^s-1)^m(Q^t_A_1A_m+1+⋯+Q^t_A_1A_n-1) 
          +(2^s-1)^m-1Q^t_AB_n

for t=sα, s≥1, with α given in Theorem 1.

[Proof]. From the inequality (<ref>), we have

    Q^t_A_1|A_2⋯ A_n
          ≥  Q^t_A_1A_2+(2^s-1)Q^t_A_1|A_3⋯ A_n
          ≥⋯
          ≥ Q^t_A_1A_2+(2^s-1)Q^t_A_1A_3+⋯+(2^s-1)^m-2Q^t_A_1A_m
              +(2^s-1)^m-1 Q^t_A_1|A_m+1⋯ A_n.

Similarly, as Q_A_1A_j≤ Q_A_1|A_j+1⋯ A_n for j=m+1,⋯,n-1, we get

    Q^t_A_1|A_m+1⋯ A_n
          ≥ (2^s-1)Q^t_A_1A_m+1+Q^t_A_1A_m+2⋯ A_n
          ≥ (2^s-1)(Q^t_A_1A_m+1+⋯+Q^t_A_1A_n-1)
              +Q^t_A_1A_n.

Combining (<ref>) and (<ref>), we have Theorem 6. 1ex1ex



§ CONCLUSION

Entanglement monogamy and polygamy are fundamental properties of quantum multipartite states. We have studied the monogamy and polygamy relations related to general measures of quantum correlation. We have shown that there always exist two real numbers α and β for any nonadditive quantum correlation measures and any given quantum state: Q^x is monogamous if x≥α and polygamous if 0≤ x≤β.

For β <x<α, depending on the detailed quantum correlation measures, quantum states may satisfy neither the monogamy nor the polygamy relations.
However, similar to the nonlocality revealed by the violation of Bell inequalities,
where for some bipartite states that admit local hidden variable models, their nonlocality can superactivated by m copies of the states <cit.>, we have shown that for a state which does not admit monogamous relations under Q^x, β <x<α, the m copies of the state could satisfy monogamous relations.
We have used the negativity as the quantum correlation measure to illustrate such superactivation of monogamy properties. Concerning infinitely many copies we have the regularized quantum correlation measures satisfying the monogamy relations. A tighter monogamy relation has also been given to any quantum correlation measures.

Acknowledgments    This work is supported by the Natural Science Foundation of China(NSFC) under Grants No. 11847209 and No. 11675113; Key Project of Beijing Municipal Commission of Education under No. KZ201810028042.

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