Entanglement of formation and monogamy of multi-party quantum entanglement

We provide a sufficient condition for the monogamy inequality of multi-party quantum entanglement of arbitrary dimensions in terms of entanglement of formation. Based on the classical–classical–quantum(ccq) states whose quantum parts are obtained from the two-party reduced density matrices of a three-party quantum state, we show the additivity of the mutual information of the ccq states guarantees the monogamy inequality of the three-party pure state in terms of EoF. After illustrating the result with some examples, we generalize our result of three-party systems into any multi-party systems of arbitrary dimensions.

the classical-classical-quantum (ccq) states whose quantum parts are obtained from the two-party reduced density matrices of the three-party state. By evaluating quantum mutual information of the ccq states as well as their reduced density matrices, we show that the additivity of the mutual information of the ccq states guarantees the monogamy inequality of the three-party quantum state in terms of EoF. We provide some examples of threeparty pure state to illustrate our result, and we generalize our result of three-party systems into any multi-party systems of arbitrary dimensions. This paper is organized as follows. First we briefly review the definitions of classical and quantum correlations in bipartite quantum systems and recall their trade-off relation in three-party quantum systems. After providing the definition of ccq states as well as their mutual information between classical and quantum parts, we establish the monogamy inequality of three-party quantum entanglement in arbitrary dimensional quantum systems in terms of EoF conditioned on the additivity of the mutual information for the ccq states. We also illustrate our result of monogamy inequality in three-party quantum systems with some examples, and we generalize our result of entanglement monogamy inequality into multi-party quantum systems of arbitrary dimensions. Finally, we summarize our results.

Results
Correlations in bipartite quantum systems. For a bipartite pure state |ψ� AB , its entanglement of formation (EoF) is defined by the entropy of a subsystem, E f |ψ� AB = S(ρ A ) where ρ A = tr B |ψ� AB �ψ| is the reduced density matrix of |ψ� AB on subsystem A, and S(ρ) = −trρ ln ρ is the von Neumann entropy of the quantum state ρ . For a bipartite mixed state ρ AB , its EoF is defined by the minimum average entanglement over all possible pure state decompositions of ρ AB = i p i |ψ i � AB �ψ i |.
For a probability ensemble E = {p i , ρ i } realizing a quantum state ρ such that ρ = i p i ρ i , its Holevo quantity is defined as /p x is the state of system A when the outcome was x. The one-way classical correlation (CC) 16 of a bipartite state ρ AB is defined by the maximum Holevo quantity over all possible ensemble representations E of ρ A induced by measurements on subsystem B.
The following proposition shows a trade-off relation between classical correlation and quantum entanglement (measured by CC and EoF, respectively) distributed in three-party quantum systems. Proposition 1. 17 For a three-party pure state |ψ� ABC with reduced density matrices ρ AB = tr C |ψ� ABC �ψ| , ρ AC = tr B |ψ� ABC �ψ| and ρ A = tr BC |ψ� ABC �ψ| , we have Classical-classical-quantum (CCQ) states. In this section, we consider a four-party ccq states obtained from a bipartite state ρ AB , and provide detail evaluations of their mutual information. Without loss of generality, we assume that any bipartite state as a two-qudit state by taking d as the dimension of larger dimensional subsystem.
For a two-qudit state ρ AB , let us consider the reduced density matrix ρ B = tr A ρ AB and its spectral decomposition Let be the probability ensemble of ρ A = tr B ρ AB from the measurement Here we note that the mutual information between the classical and quantum parts of the ccq state in Eq. (14) as well as its reduced density matrices in Eqs. (15) and (16) are and where the detail calculation can be found in "Methods" section.
Monogamy inequality of multi-party entanglement in terms of EoF. It is known that quantum mutual information is superadditive for any ccq state of the form that is, I(� XY :AB ) ≥ I(� X:AB ) + I(� Y :AB ) 18 . The following theorem shows that the additivity of quantum mutual information for ccq states guarantees the monogamy inequality of three-party quantum entanglement in therms of EoF.
(10) Conditioned on the additivity of quantum mutual information for ccq states, Theorem 1 shows that EoF can characterize the monogamous nature of bipartite entanglement shared in three-party quantum systems, which is illustrateed in Fig. 1.     (33), we note that the mutual information of the ccq state Ŵ XYAB obtained from ρ AB is additive as in Eq. (23). Moreover, the symmetry of GHZ state assures that the same is also true for the reduced density matrix ρ AC = tr B |GHZ� ABC �GHZ| . Thus Theorem 1 guarantees the monogamy inequality of the three-qubit GHZ state in Eq. (25) in terms of EoF. In fact, we have E f |GHZ� A(BC) = S(ρ A ) = ln 2 , whereas the two-qubit reduced density matrices ρ AB and ρ AC are separeble. Thus E f (ρ AB ) = E f (ρ AC ) = 0 and this implies the monogamy inequality in (22). Let us consider another example of three-qubit state.

Example 2. Three-qubit W-state is defined as 20
The two-qubit reduced density matrix of |W� ABC on subsystem AB is obtained as (43) I(Ŵ X:AB ) = 1 3 ln 2. We also note that the symmetry of W state in Eq. (34) would imply the nonadditivity of mutual information for the ccq state Ŵ XYAC obtained from the two-qubit reduced density matrix ρ AC of W state in Eq. (34).
As the additivity of mutual information in Theorem 1 is only a sufficient condition for monogamy inequality in terms of EoF, nonadditivity does not directly imply violation of Inequality (22) for the W state in Eq. (34). However, we note that ρ AB in Eq. (35) is a two-qubit state, therefore its EoF can be analytically evaluated as 5 Moreover, the symmetry of the W state assures that the EoF of ρ AC = tr B |W� ABC �W| is the same, whereas As Eqs. (48), (49) and (50) imply the violation of Inequality (22), W state in Eq. (34) can be considered as an example for the contraposition of Theorem 1; violation of monogamy inequality in (22) implies nonadditivity of quantum mutual information for the ccq state. Now, we generalize Theorem 1 for multi-party quantum states of arbitrary dimension.

Theorem 2.
For any multi-party quantum state ρ A 1 A 2 ···A n with two-party reduced density matrices ρ A 1 A i for i = 2, · · · , n , we have conditioned on the additivity of quantum mutual information where Ŵ XYA 1 A i is the ccq state of the form in Eq. (14) obtained by ρ A 1 A i for i = 2, . . . , n.

Discussion
We have considered possible conditions for monogamy inequality of multi-party quantum entanglement in terms of EoF, and shown that the additivity of mutual information of the ccq states implies the monogamy inequality of three-party quantum entanglement in terms of EoF. We have also provided examples of three-qubit GHZ and W states to illustrate our result in three-party case, and generalized our result into any multi-party systems of arbitrary dimensions.
Most monogamy inequalities of quantum entanglement deal with bipartite entanglement measures based on the minimization over all possible pure state ensembles. As analytic evaluation of such entanglement measure is generally hard especially in higher dimensional quantum systems more than qubits, the situation becomes far more difficult in investigating and establishing entanglement monogamy of multi-party quantum systems of arbitrary dimensions. The sufficient condition provided here deals with the quantum mutual information of the ccq states to guarantee the monogamy inequality of entanglement in terms of EoF in arbitrary dimensions. As the sufficient condition is not involved with any minimization process, our result can provide a useful methodology to understand the monogamy nature of multi-party quantum entanglement in arbitrary dimensions. We finally remark that it would be an interesting future task to investigate if the condition provided here is also necessary.

Methods
Evaluation for the quantum mutual information of the ccq states. Here we evaluate the mutual information of the ccq state in Eq. (14) as well as the reduced density matrices in Eqs. (15) and (16); the classical parts of the four-qudit ccq state Ŵ XYAB in Eq. (14) is x,y=0 |x� X �x| ⊗ y Y y , which is the maximally mixed state in d 2 -dimensional quantum system, therefore its von Neumann entropy is We also note that Eq. (17)  x=0 |x� X �x| , we have the mutual information of Ŵ XAB with respect to the bipartition between X and AB as For the von Neumann entropy of Ŵ YAB in Eq. (16), we have where the first and third equalities are due to the joint entropy theorem and the second equality is from the unitary invariance of von Neumann entropy. Thus the mutual information of Ŵ YAB with respect to the bipartition between Y and AB is www.nature.com/scientificreports/ Proof of Theorem 1. Let us first consider the four-qudit ccq state Ŵ XYAB of the form in Eq. (14) obtained by the two-qudit reduced density matrix ρ AB of |ψ� ABC . From Eqs. (18), (19) and (20), the additivity condition of quantum mutual information for Ŵ XYAB in Eq. (23) can be rewritten as where E 0 and E 1 are the probability ensembles of ρ A in Eqs. (7) and (10), respectively. Because E 0 and E 1 can be obtained from measuring subsystem B of ρ AB by the rank-1 measurement , respectively, the definition of CC in Eq. (4) leads us to J ← (ρ AB ) ≥ χ(E 0 ) and J ← (ρ AB ) ≥ χ(E 1 ) , therefore By considering the ccq state Ŵ XYAC obtained by ρ AC as well as Eq. (24), we can analogously have As the trade-off relation of Eq. (5) in Proposition 1 is universal with respect to the subsystems, we also have S(ρ A ) = J ← (ρ AC ) + E f (ρ AB ) for the given two-qudit state |ψ� ABC , therefore Now Inequalities (62), (63) as well as Eq. (64) lead us to where the second equality is due to ρ AC = ρ B and ρ AB = ρ C for three-party pure state |ψ� ABC .
Proof of Theorem 2. We first prove the theorem for any three-party mixed state ρ ABC , and inductively show the validity of the theorem for any n-party quantum state ρ A 1 A 2 ···A n . For a three-party mixed state ρ ABC , let us consider an optimal decomposition of ρ ABC realizing EoF with respect to the bipartition between A and BC, that is, For each i and the two-party reduced density matrices ρ i AB , let us consider its optimal decomposition ρ i AB = j r ij µ i j AB µ i j realizing EoF, that is, E f ρ i AB = j r ij E f µ i j AB

. Now we have
where the inequality is due to ρ AB = i p i ρ i AB = i,j p i r ij µ i j AB µ i j and the definition of EoF. For each i, we also consider an optimal decomposition ρ i AC = l s il ν i l AC ν i l such that E f ρ i AC = l s il E f ν i l AC . We can analogously have due to ρ AC = i p i ρ i AC = i,l p i s il ν i l AC ν i l and the definition of EoF in Eq. (2). From Inequalities (67), (68) and (69), we have which proves the theorem for three-party mixed states.