Polygamy relations of multipartite entanglement beyond qubits
Zhi-Xiang Jin Shao-Ming Fei





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

Corresponding author: feishm@mail.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 polygamy relations related to the concurrence of assistance for any multipartite pure states. General polygamy inequalities given by the αth (0≤α≤ 2) power of concurrence of assistance is first presented for multipartite pure states in arbitrary-dimensional quantum systems. We further show that the general polygamy inequalities can even be improved to be tighter inequalities under certain conditions on the assisted entanglement of bipartite subsystems. Based on the improved polygamy relations, lower bound for distribution of bipartite entanglement is provided in a multipartite system. Moreover, the βth (0≤β≤ 1) power of polygamy inequalities are obtained for the entanglement of assistance as a by-product, which are shown to be tighter than the existing ones. A detailed example is presented.


Polygamy relations of multipartite entanglement beyond qubits
    Shao-Ming Fei
    December 30, 2023
=============================================================





§ INTRODUCTION

Quantum entanglement <cit.> has been extensively studied due to its importance in quantum communication and quantum information processing in recent years. The study of quantum entanglement from various viewpoints has been a very active area and has led to many impressive results. Monogamy of entanglement is one of the nonintuitive phenomena of quantum physics that distinguish quantum from classical physics. Different from the classical world, it is not possible to prepare three qubits in a way that any two qubits are maximally entangled. Qualitatively, monogamy of entanglement measures the shareability of entanglement in a composite quantum system. Moreover, the monogamy property has emerged as the ingredient in the security analysis of quantum key distribution <cit.>.	

The monogamy relation was first quantified by Coffman, Kundu, and Wootters <cit.> for three qubits, which satisfies ℰ_A|BC≥ℰ_AB +ℰ_AC. The CKW inequality shows the mutually exclusive nature of multipartite quantum entanglement in a quantitative way: more entanglement shared between two qubits (A and B) necessarily implies less entanglement between the other two qubits (A and C). CKW inequality was generalized for multiqubit systems <cit.> and also studied intensively in more general settings <cit.>. However, the CKW inequality fails for higher-dimensional quantum systems.
It is also not generally true for three-qubit systems with other entanglement measures like entanglement of formation <cit.>. Monogamy of multiqubit entanglement and some higher-dimensional quantum systems were later characterized in terms of various entanglement measures <cit.>.

Whereas the monogamy of entanglement shows the restricted sharability of multipartite quantum entanglement,
the distribution of entanglement in multipartite quantum systems was shown to have a dually monogamous property. Using concurrence of assistance <cit.> as the measure of distributed entanglement, the polygamy of entanglement
provides a lower bound for distribution of bipartite entanglement in a multipartite system <cit.>.
Polygamy of entanglement is characterized as a polygamy inequality, E_a_A|BC≤E_a_AB +E_a_AC for a tripartite quantum state ρ_ABC, where E_a_A|BC is the assisted entanglement <cit.>
between A and BC. Polygamy of entanglement was generalized to multiqubit systems <cit.> and arbitrary dimensional multipartite states <cit.>.

The study of quantum entanglement in higher-dimensional quantum systems is of importance in quantum information processing. Monogamy and polygamy of entanglement can restrict the possible correlations between the authorized users and the eavesdroppers, which tightens security bounds in quantum cryptography. And optimized monogamy and polygamy relations give rise to finer characterizations of the entanglement distributions. Furthermore, to optimize the efficiency of entanglement usage as a resource in quantum cryptography, higher-dimensional quantum systems rather than qubits are preferred in some physical systems for stronger security in quantum key distribution <cit.>.

In this paper, we provide a tighter polygamy inequalities for arbitrary dimensional quantum systems.
General polygamy inequalities given by the αth (0≤α≤ 2) power of concurrence of assistance are first presented for multipartite pure states in arbitrary-dimensional quantum systems. We further show that the general polygamy inequalities can even be improved to be tighter inequalities under certain conditions on the assisted entanglement of bipartite subsystems. Based on the improved polygamy relations, lower bound for distribution of bipartite entanglement is provided for multipartite systems. Moreover, the βth (0≤β≤ 1) power of polygamy inequalities are obtained for the entanglement of assistance, which are shown to be tighter than the existing ones.

For a bipartite pure state |ψ⟩_AB∈H_A⊗H_B, the concurrence is given by <cit.>, C(|ψ⟩_AB)=√(2[1-Tr(ρ_A^2)]),
where ρ_A=Tr_B(|ψ⟩_AB⟨ψ|).
 For mixed state, ρ_AB is defined by the convex roof extension,
C(ρ_AB)=min_{p_i,|ψ_i⟩}∑_ip_iC(|ψ_i⟩),
where the minimum is taken over all possible pure state decompositions of ρ_AB=∑_ip_i|ψ_i⟩⟨ψ_i|,
with p_i≥0, ∑_ip_i=1 and |ψ_i⟩∈H_A⊗H_B.

For three qubit pure state |ψ⟩_ABC, the concurrence of assistance defined  as <cit.>,

    C_a(|ψ⟩_ABC)≡ C_a(ρ_AB)=max_{p_i,|ψ_i⟩}∑_ip_iC(|ψ_i⟩),

where the maximum is taken over all possible pure state decompositions of ρ_AB=Tr_C(|ψ⟩_ABC⟨ψ|)=∑_ip_i|ψ_i⟩_AB⟨ψ_i|.
For pure states ρ_AB=|ψ⟩_AB⟨ψ|, one has C(|ψ⟩_AB)=C_a(ρ_AB).

For an N-qubit state ρ_AB_1⋯ B_N-1∈H_A⊗H_B_1⊗⋯⊗H_B_N-1,
the concurrence C(ρ_A|B_1⋯ B_N-1)  viewed as a bipartite state under the partition
A and B_1,B_2,⋯, B_N-1, satisfies the monogamy relation <cit.>,

    C^2(ρ_A|B_1,B_2⋯,B_N-1)≥∑_i=1^N-1C^2(ρ_AB_i),

where ρ_AB_i=Tr_B_1⋯ B_i-1B_i+1⋯ B_N-1(ρ_AB_1⋯ B_N-1).
Further improved monogamy relations are presented in <cit.> and <cit.>.

Different from the concurrence, the concurrence of assistance for N-qubit states have the form <cit.>,

    C^2_a(ρ_A|B_1,B_2⋯,B_N-1)≤∑_i=1^N-1C_a^2(ρ_AB_i).


For a bipartite arbitrary dimensional pure state |ϕ⟩_AB=∑_i=1^d_1∑_k=1^d_2a_ik|ik⟩_AB in C^d_1⊗ C^d_2,
the concurrence is given by <cit.>

    C^2(|ϕ⟩_AB)=2(1-Tr(ρ_A^2))=4∑_i<j^d_1∑_k<l^d_2|a_ika_jl-a_ila_jk|^2.

And for a mixed state ρ_AB=∑_ip_i|ϕ_i⟩_AB⟨ϕ_i|, from (<ref>) its concurrence of assistance satisfies C_a(ρ_AB)=max_{p_i,|ϕ_i⟩}∑_ip_iC(|ϕ_i⟩)≤∑_m=1^D_1∑_n=1^D_2(max∑_ip_i|⟨ϕ_i|(L_A^m⊗ L_B^n)|ϕ_i^∗⟩|)=∑_m=1^D_1∑_n=1^D_2C_a((ρ_AB)_mn):=τ_a(ρ_AB) <cit.>, where D_1=d_1(d_1-1)/2, D_2=d_2(d_2-1)/2, L_A^m=P_A^m(-|i⟩_A⟨ j|+|j⟩_A⟨ i|)P_A^m, L_B^n=P_B^n(-|k⟩_B⟨ l|+|l⟩_B⟨ k|)P_B^n, and P_A^m=|i⟩_A⟨ i|+|j⟩_A⟨ j|,
P_B^n=|k⟩_B⟨ k|+|l⟩_B⟨ l| are the projections onto the subspaces spanned by the local bases
{|i⟩_A, |j⟩_A} and {|k⟩_B, |l⟩_B}, respectively. In Ref. <cit.>, the authors give a general polygamy inequality for any multipartite pure
state |ϕ⟩_A_1⋯ A_n∈ C^d_1⊗⋯⊗ C^d_n

    τ_a^2(|ϕ⟩_A_1|A_2⋯ A_n)≤∑_i=2^nτ_a^2(ρ_A_1A_i),

where ρ_A_1A_k is the reduced density matrix of |ϕ⟩_A_1|A_2⋯ A_n associated with the subsystems A_1A_k, k=2,⋯,n.



§ WEIGHTED POLYGAMY RELATION FOR CONCURRENCE OF ASSISTANCE


Polygamy of entanglement states that if a multipartite state is maximally entangled with respect to a given kind of multipartite entanglement, then it must be pure <cit.>. This observation implies that all maximally entangled states are necessarily uncorrelated with any other systems. One can even propose this condition as another requisite for a good multipartite entanglement quantifier. Furthermore, it is also important to note that this polygamy holds for all kinds of entanglement, that is, whenever a system reaches a maximum amount of entanglement under any partitions, it becomes “free” of its environment.

Therefore, for states that do not reach the maximum amount of entanglement of assistance under any partition, the polygamy inequality of entanglement provides a lower bound for the distribution of bipartite entanglement in a multipartite system. Meanwhile, the bipartite sharability of entanglement in a multipartite system gives an upper bound of the entanglement. Tighter polygamy inequalities give rise to finer characterization of the entanglement distributions, which are tightly related to the security of quantum cryptographic protocols based on entanglement <cit.> (it limits the amount of correlations that an eavesdropper can have with the honest parties). In the following, we give a class of polygamy inequalities that are tighter than existing ones. First, we give the definition of Hamming weight.

For any non-negative integer j and its binary expansion

    j=∑_i=0^n-1j_i2^i,

with log_2j≤ n and j_i∈{0, 1}, i=0,1,⋯,n-1, we can always define a unique binary vector j⃗ associated with j,

    j⃗=(j_0,j_1,⋯,j_n-1).

For the binary vector j⃗ defined in (<ref>), the Hamming weight w_H(j⃗) is defined by the number of 1's in {j_0,j_1,⋯,j_n-1} <cit.>.


[Lemma 1]. For any real numbers x and t, 0≤ t ≤ 1, 0≤ x ≤ 1, we have (1+t)^x≤ 1+(2^x-1)t^x.

[Proof]. Let f(x,y)=(1+y)^x-y^x with 0≤ x≤ 1, y≥ 1. Then ∂ f/∂ y=x[(1+y)^x-1-y^x-1]≤ 0. Therefore, f(x,y) is an decreasing function of y, i.e., f(x,y)≤ f(x,1)=2^x-1. Set y=1/t, 0<t≤ 1, we obtain (1+t)^x≤ 1+(2^x-1)t^x. When t=0, the inequality is trivial. 

The following theorem provides states that a class of polygamy inequalities satisfied by the α-power of τ_a.
For convenience, we denote τ_a(ρ_AB_i)=τ_a_AB_i the concurrence of assistance ρ_AB_i and τ_a(ρ_A|B_0B_1⋯ B_N-1)=τ_a_A|B_0B_1⋯ B_N-1.

[Theorem 1]. For any multiparty pure state ρ_AB_0⋯ B_N-1, we have

    τ_a^α_A|B_0B_1⋯ B_N-1≤∑_j=0^N-1 (2^α/2-1)^w_H(j⃗)τ_a^α_AB_j

for 0≤α≤2, where j⃗=(j_0,j_1,⋯,j_N-1) is the vector from the binary representation of j and w_H(j⃗) is the Hamming weight of j⃗.

[Proof]. Without loss of generality, we can always have

    τ_a_AB_j≥τ_a_AB_j+1≥ 0,

by relabeling the subsystems.
From (<ref>), it is sufficient to show that

    (∑_j=0^N-1τ_a^2_AB_j)^α/2≤∑_j=0^N-1 (2^α/2-1)^w_H(j⃗)τ_a^α_AB_j.

We first prove the inequality (<ref>) for the case that N is a power of 2, N = 2^n, by mathematical induction.
For n = 1, by using Lemma 1 we have

    τ_a^α_A|B_0B_1≤τ_a^α_AB_0+(2^α/2-1)τ_a^α_AB_1,

which is just the inequality (<ref>) for N=2.

Now let us assume that the inequality (<ref>) is true for N=2^n-1 with n ≥ 2, and consider the case that N=2^n.
For an (N+1)-partite quantum state ρ_AB_0⋯ B_N-1 and its bipartite reduced density matrices ρ_AB_j, j=0,1,⋯, N-1, we have

    (∑_j=0^N-1τ_a^2_AB_j)^α/2
          = (∑_j=0^2^n-1-1τ_a^2_AB_j+∑_j=2^n-1^2^n-1τ_a^2_AB_j)^α/2
          = (∑_j=0^2^n-1-1τ_a^2_AB_j)^α/2(1+∑_j=2^n-1^2^n-1τ_a^2_AB_j/∑_j=0^2^n-1-1τ_a^2_AB_j)^α/2.


Due to (<ref>) we have

    ∑_j=2^n-1^2^n-1τ_a^2_AB_j≤∑_j=0^2^n-1-1τ_a^2_AB_j.

By using Lemma 1 we get

    (∑_j=0^N-1τ_a^2_AB_j)^α/2   ≤(∑_j=0^2^n-1-1τ_a^2_AB_j)^α/2
        +(2^α/2-1)(∑_j=2^n-1^2^n-1τ_a^2_AB_j)^α/2.

Here, the induction hypothesis assures that

    (∑_j=0^2^n-1-1τ_a^2_AB_j)^α/2≤∑_j=0^2^n-1-1 (2^α/2-1)^w_H(j⃗)τ_a^α_AB_j.

From above relations we obtain

    (∑_j=2^n-1^2^n-1τ_a^2_AB_j)^α/2≤∑_j=2^n-1^2^n-1 (2^α/2-1)^w_H(j⃗)-1τ_a^α_AB_j.

Taking into account (<ref>), (<ref>) and (<ref>) we have

    (∑_j=0^2^n-1τ_a^2_AB_j)^α/2≤∑_j=0^2^n-1 (2^α/2-1)^w_H(j⃗)τ_a^α_AB_j ,

which proves the inequality (<ref>) for N=2^n.

Now for an arbitrary positive integer N, consider an (N + 1)-partite state ρ_AB_0⋯ B_N-1.
We can always assume that 0≤ N≤ 2^n for some n. Consider a (2^n+1)-partite quantum state

    ρ'_AB_0⋯ B_2^n-1=ρ_AB_0⋯ B_N-1⊗δ_B_N⋯ B_2^n-1,

which is a product of ρ_AB_0⋯ B_N-1 and an arbitrary (2^n -N )-partite
quantum state δ_B_N⋯ B_2^n-1.

Because ρ'_AB_0⋯ B_2^n-1 is a (2^n +1)-partite state, inequality (<ref>) leads to

    τ_a^α(ρ'_A|B_0B_1⋯ B_2^n-1)≤∑_j=0^2^n-1 (2^α/2-1)^w_H(j⃗)τ_a^α(σ_AB_j),

where σ_AB_j is the bipartite reduced density matrix of ρ'_AB_0⋯ B_2^n-1 for j=0,1,⋯,2^n-1.
Since ρ'_AB_0⋯ B_2^n-1 is a separable state with respect to the bipartition between AB_0⋯ B_N-1 and B_N⋯ B_2^n-1, one has

    τ_a(ρ'_A|B_0B_1⋯ B_2^n-1)=τ_a(ρ_A|B_0B_1⋯ B_N-1),

and

    τ_a(σ_AB_j)=0,

for j=N,⋯,2^n-1. Moreover, for j=0,1,⋯,N-1 one has

    σ_AB_j=ρ_AB_j.


From (<ref> - <ref>), we have

    τ_a^α(ρ_A|B_0B_1⋯ B_N-1)
          =τ_a^α(ρ'_A|B_0B_1⋯ B_2^n-1)
          ≤∑_j=0^2^n-1 (2^α/2-1)^w_H(j⃗)τ_a^α(σ_AB_j)
          =∑_j=0^N-1 (2^α/2-1)^w_H(j⃗)τ_a^α(ρ_AB_j).

This completes the proof. 

We have obtained the general polygamy inequality of the αth (0≤α≤ 2) power of concurrence of assistance for arbitrary-dimensional quantum systems. In fact, (<ref>) is a special case of (<ref>) for α=2. Besides, based on the improved polygamy relations, we get a new upper bound for bipartite entanglement in multipartite systems for 0≤α < 2, which is better than (<ref>). To illustrate the advantage of (<ref>), we give an example as follows.

Let us consider the three-qubit state ρ=|ψ⟩⟨ψ| in the generalized Schmidt decomposition form,
where |ψ⟩=λ_0|000⟩+λ_1e^iφ|100⟩+λ_2|101⟩+λ_3|110⟩+λ_4|111⟩, λ_i≥0, i=0,1,2,3,4 and ∑_i=0^4λ_i^2=1. We have
τ_a_A|BC=2λ_0√(λ_2^2+λ_3^2+λ_4^2),
τ_a_AB=2λ_0√(λ_2^2+λ_4^2), and τ_a_AC=2λ_0√(λ_3^2+λ_4^2).
Take λ_0=λ_1=1/2, λ_2=λ_3=λ_4=√(6)/6,
one has τ_a_A|BC=√(2)/2, τ_a_AB=τ_a_AC=√(3)/3, and the marginal quantum relations
is τ_a^2_AB+τ_a^2_AC-τ_a^2_A|BC≈0.167 for α=2.
For α=1, the marginal quantum relations from (<ref>) is τ_a_AB+ (√(2)-1)τ_a_AC-τ_a_A|BC≈0.109,
which is smaller than the one for α=2.

Since 0≤(2^α/2-1)^w_H(j⃗)≤ 1 for any 0≤α≤ 2, we have

    τ_a^α_A|B_0B_1⋯ B_N-1      ≤∑_j=0^N-1 (2^α/2-1)^w_H(j⃗)τ_a^α_AB_j
          ≤∑_j=0^N-1τ_a^α_AB_j,

for any multipartite quantum state ρ_A|B_0B_1⋯ B_N-1. Thus, we have the following corollary.

[Corollary 1]. For any multiparty pure state ρ_AB_0⋯ B_N-1, we have

    τ_a^α_A|B_0B_1⋯ B_N-1≤∑_j=0^N-1τ_a^α_AB_j

for 0≤α≤2.

The class of weighted polygamy inequalities in Theorem 1 can be further tightened under some condition on bipartite quantum relations.

[Theorem 2].  For any multipartite pure state ρ_AB_0⋯ B_N-1, if

    τ_a^2_AB_i≥∑_j=i+1^N-1τ_a^2_AB_j

for i=0,1,⋯ N-2, we have

    τ_a^α_A|B_0B_1⋯ B_N-1≤∑_j=0^N-1 (2^α/2-1)^jτ_a^α_AB_j

for 0≤α≤ 2.

[Proof].
From Lemma 1, we have

    τ_a^α_A|B_0B_1⋯ B_N-1
          ≤τ_a^α_AB_0+(2^α/2-1) (∑_j=1^N-1τ_a^2_AB_j)^α/2
          ≤τ_a^α_AB_0+(2^α/2-1)τ_a^α_AB_1
     +(2^α/2-1)^2 (∑_j=2^N-1τ_a^2_AB_j)^α/2
          ≤⋯
          ≤τ_a^α_AB_0+(2^α/2-1)τ_a^α_AB_1+⋯+(2^α/2-1)^N-1τ_a^α_AB_N-1.



In Theorem 2, the condition (<ref>) are always satisfied by some states. 
Let us consider a four-qubit state ρ=|W⟩_ABCD⟨ W|, where
|W⟩_ABCD=a|1000⟩+b|0100⟩+c|0010⟩+d|0001⟩, and a^2+b^2+c^2+d^2=1. We have τ_a(ρ_A|BCD)=2a√(1-a^2), τ_a(ρ_AB)=2ab, τ_a(ρ_AC)=2ac, τ_a(ρ_AD)=2ad. The condition (<ref>) is satisfied as long as b^2≥ c^2+d^2. For example, we set b=1/√(2), a=c=d=1/√(6). Then the state ρ=|W⟩_ABCD⟨ W| satisfies the condition (<ref>). On the other hand, if b^2≤ c^2+d^2, e.g., 
c=1/√(2) and a=b=d=1/√(6), then ρ does not satisfy the condition (<ref>).

[Remark 1].
For any non-negative integer j and the corresponding binary vector j⃗ in Eq. (<ref>), the Hamming weight w_H(j⃗) is upper bounded by log_2j. Thus, we have
w_H(j⃗)≤log_2j≤ j, which implies that
τ_a^α_A|B_0B_1⋯ B_N-1≤∑_j=0^N-1 (2^α/2-1)^jτ_a^α_AB_j≤∑_j=0^N-1 (2^α/2-1)^w_H(j⃗)τ_a^α_AB_j,
for 0≤α≤ 2. In other words, inequality (<ref>) in Theorem 2 is tighter than the inequality (<ref>) in Theorem 1 for states satisfying the conditions τ_a^2_AB_i≥∑_j=i+1^N-1τ_a^2_AB_j, i=0,⋯,N-2.



§ POLYGAMY RELATIONS FOR ENTANGLEMENT OF ASSISTANCE


Now we study the polygamy relations for entanglement of assistance.
For a tripartite pure state |ψ⟩_ABC, one gets <cit.> E(|ψ⟩_A|BC)≤ E_a(ρ_AB)+E_a(ρ_AC),
where E(|ψ⟩_A|BC)=S(ρ_A) is the entanglement between A and BC based on the von Neumann entropy
S(ρ)=-Trρlnρ, and E_a(ρ_AB)=max∑_ip_iE(|ψ_i⟩_AB), with the maximization taking over all possible pure state
decompositions of ρ_AB=∑_ip_i|ψ_i⟩_AB⟨ψ_i|. Later, in Ref. <cit.>, the authors presented a general polygamy relation
for any multipartite state ρ_A_1|A_2⋯ A_n,

    E_a(ρ_A_1|A_2⋯ A_n)≤∑_i=2^nE_a(ρ_A_1A_i).


Recently, in Ref.<cit.>, the authors presented another class of multipartite polygamy inequalities based on the βth power of entanglement of assistance (EOA).
For any multipartite state ρ_A|B_0B_1⋯ B_N-1 and 0≤β≤ 1,

    E_a^β_A|B_0B_1⋯ B_N-1≤∑_j=0^N-1β^w_H(j⃗)E_a^β_AB_j,

if E_a_AB_i≥E_a_AB_i+1 for i=0,1,⋯,N-2; and

    E_a^β_A|B_0B_1⋯ B_N-1≤∑_j=0^N-1β^jE_a^β_AB_j,

if E_a_AB_i≥∑_j=i+1^N-1E_a_A|B_j for i=0,1,⋯,N-2.
With a similar consideration to τ_AB_0⋯ B_N-1, we have the following result for EOA.

[Theorem 3]. For any multipartite state ρ_AB_0⋯ B_N-1, we have

    E_a^β_A|B_0B_1⋯ B_N-1≤∑_j=0^N-1 (2^β-1)^w_H(j⃗)E_a^β_AB_j

for 0≤β≤1.

To illustrate the tightness of the inequality (<ref>) compared with the  inequality (<ref>) in <cit.>, we consider the three-qubit state ρ_ABC=|W⟩_ABC⟨ W|, where
|W⟩_ABC=1/√(3)(|100⟩+|010⟩+|001⟩). We have E_a(ρ_A|BC)=S(ρ_A)=log_23-2/3 and
E_a(ρ_AB)=E_a(ρ_AC)=2/3. Thus the marginal EOA from inequality (<ref>) is
E_a^β(ρ_AB)+βE_a^β(ρ_AC)-E_a(ρ_A|BC)=(1+β)(2/3)^β+2/3-log_23.
The marginal EOA from inequality (<ref>) is E_a^β(ρ_AB)+(2^β-1)E_a^β(ρ_AC)-E_a(ρ_A|BC)=2^β(2/3)^β+2/3-log_23.
Fig. 1 shows that our inequality gives a smaller upper bound than (<ref>) in <cit.>, namely, our marginal EOA is smaller than inequallity (<ref>) in <cit.>
for 0<β<1.



Since 0≤(2^β-1)^w_H(j⃗)≤ 1 for any 0≤β≤ 1, we have

    E_a^β_A|B_0B_1⋯ B_N-1      ≤∑_j=0^N-1 (2^β-1)^w_H(j⃗)E_a^β_AB_j
          ≤∑_j=0^N-1E_a^β_AB_j,

for any multipartite quantum state ρ_A|B_0B_1⋯ B_N-1. Thus, we have the following corollary.

[Corollary 2]. For any multipartite pure state ρ_AB_0⋯ B_N-1, we have

    E_a^β_A|B_0B_1⋯ B_N-1≤∑_j=0^N-1E_a^β_AB_j

for 0≤β≤1.

With a similar consideration to Theorem 2, we can tighten the class of weighted polygamy inequalities in Theorem 3 under certain conditions on bipartite quantum correlations.

[Theorem 4]. For any multipartite state ρ_AB_0⋯ B_N-1, we have

    E_a^β_A|B_0B_1⋯ B_N-1≤∑_j=0^N-1 (2^β-1)^jE_a^β_AB_j,

conditioned that

    E_a^2_AB_i≥∑_j=i+1^N-1E_a^2_AB_j,

for i=0,1,⋯ N-2, 0≤β≤ 1.

[Remark 2]. For any non-negative integer j, since w_H(j⃗)≤log_2j≤ j, one has
E_a^β_A|B_0B_1⋯ B_N-1≤∑_j=0^N-1 (2^β-1)^jE_a^β_AB_j≤∑_j=0^N-1 (2^β-1)^w_H(j⃗)E_a^β_AB_j
for 0≤β≤ 1. Therefore, inequality (<ref>) in Theorem 4 is tighter than the inequality (<ref>) in Theorem 3 for
states satisfying the conditions E_a^2_AB_i≥∑_j=i+1^N-1E_a^2_AB_j, i=0,⋯,N-2.

In particular, (<ref>) reduces to (<ref>) in <cit.> for β=1. For 0<β<1, (<ref>) is a tighter polygamy inequality compared with (<ref>). Since w_H(j⃗)≤ j, (<ref>) in Theorem 4 is in general tighter than the (<ref>) in Theorem 3. From the example shown in Fig. 1, one can see that (<ref>) is generally tighter than the result in <cit.>. Hence our weighted polygamy relations give finer characterizations of the entanglement distributions among the subsystems, and help better security analysis of quantum key distribution <cit.> in quantum information processing.




§ CONCLUSION



We have investigated the polygamy relations related to the concurrence of assistance. General polygamy inequalities given by the αth (0≤α≤ 2) power of concurrence of assistance have been presented for multipartite states in arbitrary-dimensional quantum systems. We have further shown that the general polygamy inequalities can even be improved to be tighter ones under certain conditions on the assisted entanglement of bipartite subsystems. Based on the improved polygamy relations, lower bound for distribution of bipartite entanglement has been provided for multipartite systems. Moreover, the βth (0≤β≤ 1) power of polygamy inequalities have been obtained for the entanglement of assistance as a by-product, which are shown to be tighter than the existing ones.

The higher-dimensional quantum systems are the key resources in various quantum information and communication processing tasks.
For instance, the qudit (d>2) systems are preferred in some quantum key distributions, where the use of qudits increases
the coding density and provides stronger security compared to qubits <cit.>.
Our results apply to general polygamy relations of multipartite entanglement in arbitrary higher-dimensional quantum systems.
Moreover, our polygamy inequalities provide tighter constraints and finer characterizations of the entanglement distributions
among the multipartite systems. These results may highlight future works on the study of multipartite quantum entanglement.


Acknowledgments    This work is supported by the NSF of China under Grant No. 11847209; 11675113  and Key Project of Beijing Municipal Commission of Education under No. KZ201810028042.

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