Entanglement of Formation and Monogamy of Multi-Party Quantum

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OPEN Entanglement of formation and monogamy of multi‑party quantum entanglement Jeong San Kim

We provide a sufcient 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.

Quantum entanglement is a non-classical nature of quantum mechanics, which is a useful resource in many quantum information processing tasks such as quantum teleportation, dense coding and quantum cryptography 1–3. Because of its important roles in the feld of quantum information and computation theory, there has been a signifcant amount of research focused on quantifcation of entanglement in bipartite quantum systems. Entan- glement of formation(EoF) is the most well-known bipartite entanglement measure with an operational meaning that asymptotically quantifes how many bell states are needed to prepare the given state using local quantum operations and classical communications4. Although EoF is defned in any bipartite quantum systems of arbitrary dimension, its defnition for mixed states is based on ‘convex-roof extension’, which takes the minimum average over all pure-state decompositions of the given state. As such an optimization is hard to deal with, analytic evaluation of EoF is known only in two-qubit systems5 and some restricted cases of higher-dimensional systems so far. In multi-party quantum systems, entanglement shows a distinct behavior that does not have any classical counterpart; if a pair of parties in a multi-party quantum system is maximally entangled, then they cannot be entanglement, not even classically correlated, with the rest parties. Tis restriction of sharing entanglement in multi-party quantum systems is known as the monogamy of entanglement(MoE)6,7. MoE plays an important role such as the security proof of quantum key distribution in quantum cryptography 2,8 and the N-representability problem for fermions in condensed-matter physics 9. Mathematically, MoE can be characterized using monogamy inequality; for a three-party quantum state ρABC and its two-party reduced density matrices ρAB and ρAC,

E ρ ( ) E(ρ ) E(ρ ) A BC ≥ AB + AC (1) where E(ρXY ) is an entanglement measure quantifying the amount of entanglement between subsystems X and Y of the bipartite quantum state ρXY . Inequality (1) shows the mutually exclusive nature of bipartite entanglement E(ρAB) and E(ρAC) shared in three-party quantum systems so that their summation cannot exceeds the total entanglement E ρA(BC) . Using tangle10 as the bipartite entanglement measure, Inequality (1) was frst shown to be true for all three- qubit states, and generalized for multi-qubit systems as well as some cases of higher-dimensional quantum systems11,12. However, not all bipartite entanglement measures can characterize MoE in forms of Inequality (1), but only few measures are known so far satisfying such monogamy inequality 13–15. Although EoF is the most natural bipartite entanglement measure with the operational meaning in quantum state preparation, EoF is known to fail in characterizing MoE as the monogamy inequality in (1) even in three-qubit systems; there exists quantum states in three-qubit systems violating Inequality (1) if EoF is used as the bipartite entanglement measure. Tus, a natural question we can ask is ‘On what condition does the monogamy inequality hold in terms of the given bipartite entanglement measure?’. Here, we provide a sufcient condition that monogamy inequality of quantum entanglement holds in terms of EoF in multi-party, arbitrary dimensional quantum systems. For a three-party quantum state, we frst consider

Department of Applied Mathematics and Institute of Natural Sciences, Kyung Hee University, Yongin ‑si, Gyeonggi‑do 446‑701, Korea. email: [email protected]

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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 three- party pure state to illustrate our result, and we generalize our result of three-party systems into any multi-party systems of arbitrary dimensions. Tis paper is organized as follows. First we briefy review the defnitions of classical and quantum correlations in bipartite quantum systems and recall their trade-of relation in three-party quantum systems. Afer providing the defnition 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 , its entanglement of forma- | �AB tion (EoF) is defned by the entropy of a subsystem, Ef ψ S(ρA) where ρA trB ψ ψ is the reduced | �AB = = | �AB� | density matrix of ψ on subsystem A, and S(ρ) trρ ln ρ is the von Neumann entropy of the quantum state | �AB =− ρ . For a bipartite mixed state ρAB , its EoF is defned by the minimum average entanglement

Ef (ρ ) min p Ef ( ψ ), AB = i | i�AB (2) i over all possible pure state decompositions of ρAB pi ψi AB ψi . = i | � � | For a probability ensemble E pi, ρi realizing a quantum state ρ such that ρ piρi , its Holevo quantity ={ } = i is defned as χ E S(ρ) p S(ρ ). = − i i (3) i x Given a bipartite quantum state ρAB , each measurement MB applied on subsystem B induces a probability x { } x ensemble E px, ρA of the reduced density matrix ρA trAρAB in the way that px tr (IA MB)ρAB is ={ } x x = = [ ⊗ ] the probability of the outcome x and ρA trB (IA MB)ρAB /px is the state of system A when the outcome was =16 [ ⊗ ] x. Te one-way classical correlation (CC) of a bipartite state ρAB is defned by the maximum Holevo quantity

J ←(ρAB) max χ E = E (4) over all possible ensemble representations E of ρA induced by measurements on subsystem B. Te following proposition shows a trade-of relation between classical correlation and quantum entanglement (measured by CC and EoF, respectively) distributed in three-party quantum systems.

17 Proposition 1. For a three-party pure state ψ with reduced density matrices ρAB trC ψ ψ , | �ABC = | �ABC� | ρAC trB ψ ψ and ρA trBC ψ ψ , we have = | �ABC� | = | �ABC� | S(ρA) J ←(ρAB) Ef (ρAC). (5) = +

Classical–classical–quantum (CCQ) states. In this section, we consider a four-party ccq states obtained from a bipartite state ρAB , and provide detail evaluations of their mutual information. Without loss of general- ity, 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 ρAB , let us consider the reduced density matrix ρB trAρAB and its spectral = decomposition d 1 − ρB i ei B ei . (6) = | � � | i 0 = Let i E0 i, σ i (7) ={ A} d 1 be the probability ensemble of ρA trBρAB from the measurement ei ei − on subsystem B of ρAB , in a = {| �B� |}i 1 way that = tr[(I e e )ρ ] i = A ⊗ | i�B� i| AB (8) and

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i 1 σA trB[(IA ei B ei )ρAB]. (9) = i ⊗ | � � |

Based on the eigenvectors ej B of ρB , we also consider the d-dimensional Fourier basis elements 1 d 1 jk { } 2πi ej B − ω e for each j 0, ..., d 1 , where ω e d is the dth-root of unity. Let √d k 0 d k B d |˜ � = = | � = − = 1 j E1 , τ , = d A (10) j

be the probability ensemble of ρA trBρAB obtained by measuring subsystem B in terms of the Fourier basis d 1 = ej B ej j −1 where { ˜ ˜ } = 1 tr (IA ej ej )ρAB (11) d = ⊗ ˜ B ˜ and j τ dtrB (IA ej B ej )ρAB . (12) A = ⊗ |˜ � �˜ | Now we defne the generalized d-dimensional Pauli operators based on the eigenvectors of ρB as d 1 d 1 d 1 − j − − j Z ωd ej ej , X ej 1 ej ωd− ej B ej , (13) = B = + B = |˜ � �˜ | j 0 j 0 j 0 = = = and consider a four-qudit ccq state ŴXYAB d 1 1 − x y y x ŴXYAB x x y y (IA X Z )ρAB(IA Z− X− ) , = d2 | �X � | ⊗ Y ⊗ ⊗ B B ⊗ B B (14) x,y 0 = for some d-dimensional orthonormal bases x and y of the subsystems X and Y, respectively. From {| �X } { Y } Eqs. (8), (9), (11), (12) and (13), the reduced density matrices of ŴXYAB are obtained as 1 d 1 d 1 − x − i x ŴXAB x X x IA X σ i ei B ei IA X− , (15) = d | � � | ⊗ ⊗ B A ⊗ | � � | ⊗ B x 0 i 0 = =

d 1 d 1 1 − y − j 1 y ŴYAB y y IA Z τ ej B ej IA Z− , = d Y ⊗  ⊗ B  A ⊗ d |˜ � �˜ | ⊗ B  (16) y 0 j 0 �= � � � � � � �= � � � �     and

IB ŴAB ρA , (17) = ⊗ d

where IA and IB are d-dimensional identity operators of subsystems A and B, respectively. 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

I(ŴXY AB) ln d S(ρA) S(ρAB), (18) : = + −

I(ŴX AB) ln d S(ρB) χ(E0), (19) : = − + and

I(ŴY AB) χ(E1), (20) : = 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 d 1 1 − xy �XYAB x x y y σ , = d2 | �X � | ⊗ Y ⊗ AB (21) x,y 0 = 18 that is, I(�XY AB) I(�X AB) I(�Y AB) . Te 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.

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Figure 1. Te entanglement between A and BC quantifed by Ef ψ A BC ((a) in the fgure) bounds the | � ( ) summation of the entanglement between A and B quantifed by Ef (ρAB) ((b) in the fgure) and the entanglement between A and C quantifed by Ef (ρAC)((c) in the fgure).

Teorem 1. For any three-party pure state ψ with its two-qudit reduced density matrices trC ψ ψ ρAB | �ABC | �ABC� | = and trB ψ ψ ρAC , we have | �ABC� | = Ef ψ Ef (ρ ) Ef (ρ ), | �A(BC) ≥ AB + AC (22) conditioned on the additivity of quantum mutual information

I(ŴXY AB) I(ŴX AB) I(ŴY AB) (23) : = : + : and

I(ŴXY AC) I(ŴX AC) I(ŴY AC) (24) : = : + : where ŴXYAB and ŴXYAC are the ccq states of the form in Eq. (14) obtained by ρAB and ρAC , respectively.

Conditioned on the additivity of quantum mutual information for ccq states, Teorem 1 shows that EoF can characterize the monogamous nature of bipartite entanglement shared in three-party quantum systems, which is illustrateed in Fig. 1.

Example 1. Let us consider three-qubit GHZ state 19, 1 GHZ ( 000 111 ), | �ABC = √2 | �ABC + | �ABC (25) with its reduced density matrices 1 1 1 ρAB ( 00 AB 00 11 AB 11 ), ρA ( 0 A 0 1 A 1 ), ρB ( 0 B 0 1 B 1 ). (26) = 2 | � � | + | � � | = 2 | � � | + | � � | = 2 | � � | + | � � | 1 Te eigenvalues of ρB are 0 1 with corresponding eigenvectors e0 0 and e1 1 respec- = = 2 | �B = | �B | �B = | �B tively. Tus the ensemble of ρA induced by measuring subsystem B of ρAB in terms of the eigenvectors of ρB , that is, M0 0 0 , M1 1 1 is { B = | �B� | B = | �B� |} 1 0 1 1 E0 0 , σ 0 0 , 1 , σ 1 1 . (27) ={ = 2 A = | �A� | = 2 A = | �A� |}

Because the Fourier basis elements of subsystem B with respect to the eigenvectors of ρB are 1 1 e0 B ( 0 1 ), e1 B ( 0 1 ), (28) |˜ � = 2 | �B + | �B |˜ � = 2 | �B − | �B

the ensemble of ρA induced by measuring subsystem B of ρAB in terms of the Fourier basis in Eq. (28) is

1 0 1 1 1 1 E1 , τ IA, , τ IA . (29) ={2 A = 2 2 A = 2 }

Now we consider the additivity of mutual information of the ccq state ŴXYAB obtained from ρAB in Eq. (26). Due to Eq. (18), the mutual information of ŴXYAB between XY and AB is

I(ŴXY AB) ln 2 S(ρA) S(ρAB) ln 2 (30) : = + − = because S(ρA) S(ρAB) ln 2 from Eqs. (26). For the mutual information of ŴXAB between X and AB, Eq. (19) = = leads us to

I(ŴX AB) ln 2 S(ρB) χ(E0) ln 2 (31) : = − + = where the second equality is from S(ρB) ln 2 and =

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1 1 χ(E0) S(ρA) S( 0 0 ) S( 1 1 ) ln 2, (32) = − 2 | �A� | − 2 | �A� | =

for the ensemble E0 in Eq. (27). For the mutual information of ŴYAB between Y and AB, Eq. (20) leads us to 1 1 1 1 I(ŴY AB) χ(E1) S(ρA) S IA S IA 0 (33) : = = − 2 2 − 2 2 = where the second equality is due to the ensemble E1 in Eq. (29). From Eqs. (30), (31) and (33), we note that the mutual information of the ccq state ŴXYAB obtained from ρAB is additive as in Eq. (23). Moreover, the symmetry of GHZ state assures that the same is also true for the reduced density matrix ρAC trB GHZ GHZ . Tus Teorem 1 guarantees the monogamy inequality of the = | �ABC� | three-qubit GHZ state in Eq. (25) in terms of EoF. In fact, we have Ef GHZ S(ρA) ln 2 , whereas the | �A(BC) = = two-qubit reduced density matrices ρAB and ρAC are separeble. Tus Ef (ρAB) Ef (ρAC) 0 and this implies = = the monogamy inequality in (22).

Let us consider another example of three-qubit state.

Example 2. Tree-qubit W-state is defned as 20 1 W ( 100 010 001 ). | �ABC = √3 | �ABC + | �ABC + | �ABC (34) Te two-qubit reduced density matrix of W on subsystem AB is obtained as | �ABC 2 1 ρAB ψ+ ψ+ 00 AB 00 (35) = 3 AB + 3 | � � | 1 where ψ ( 01 10 ) is the two-qubit Bell state. Te one-qubit reduced density matrices of ρAB + AB = √2 | �AB + | �AB are 2 1 2 1 ρA 0 A 0 1 A 1 , ρB 0 B 0 1 B 1 . (36) = 3 | � � | + 3 | � � | = 3 | � � | + 3 | � � | 2 1 From to the spectral decomposition of ρB in Eq. (36) with the eigenvalues 0 , 1 and corresponding = 3 = 3 eigenvectors e0 0 and e1 1 , respectively, it is straightforward to check that the ensemble of ρA | �B = | �B | �B = | �B induced from measuring subsystem B of ρAB by the eigenvectors of ρB is 2 1 1 0 1 E0 0 , σ IA, 1 , σ 0 A 0 . (37) ={ = 3 A = 2 = 3 A = | � � |} Because the Fourier basis of subsystem A is the same as Eq. (28), it is also straightforward to obtain the ensemble of ρA induced by measuring subsystem B of ρAB in terms of the Fourier basis,

1 j E1 , τA j 1,2, (38) ={2 } = where

0 1 1 1 τ (2 0 A 0 0 A 1 1 A 0 1 A 1 ), τ (2 0 A 0 0 A 1 1 A 0 1 A 1 ). (39) A = 3 | � � | + | � � | + | � � | + | � � | A = 3 | � � | − | � � | − | � � | + | � � |

For the mutual information of the ccq state ŴXYAB obtained from ρAB in Eq. (35), Eq. (18) together with Eqs. (35) and (36) lead us to

I(ŴXY AB) ln 2 S(ρA) S(ρAB) ln 2. (40) : = + − = For the mutual information of ŴXAB between X and AB, Eq. (19) leads us to

I(ŴX AB) ln 2 S(ρB) χ(E0), (41) : = − + where Eq. (37) impies 2 1 1 2 χ(E0) S(ρ ) S I S( 0 0 ) S(ρ ) ln 2. = A − 3 2 A − 3 | �A� | = A − 3 (42) Due to Eq. (36), we have S(ρA) S(ρB) , therefore Eqs. (41) and (42) lead us to = 1 I(ŴX AB) ln 2. (43) : = 3

For the mutual information of ŴYAB between Y and AB, Eq. (20) leads us to

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1 0 1 1 I(ŴY AB) χ(E1) S(ρA) S τA S τA , (44) : = = − 2 − 2 0 1 where the second equality is from the ensemble E1 in Eq. (38). Here we note that τA and τA in Eq. (39) have 3 √5 3 √5 0 1 the same eigenvalues, that is µ0 + and µ0 − , therefore we have S τ S τ . From the spectral = 6 = 6 2 A = A decomposition of ρA in Eq. (36), we have S(ρA) ln 3 ln 2 , and this turns Eq. (44) into = − 3 2 0 I(ŴY AB) ln 3 ln 2 S τA . (45) : = − 3 − From Eqs. (40), (43) and (45), we have

4 0 I(ŴXY AB) I(ŴX AB) I(ŴY AB) ln 2 ln 3 S τA , (46) : − : − : = 3 − + 0 where ln 2 0.693147 , ln 3 1.098612 and S τ 0.381264 . Tus we have ≈ ≈ A ≈ I(ŴXY AB) I(Ŵ X AB ) I(ŴY AB) 0.206848 > 0, (47) : − : − : ≈ which implies the nonadditivity of mutual information fo r the ccq state ŴXYAB obtained from ρAB in Eq. (35). We also note that the symmetry of W state in Eq. (34) would imply the nonadditivity of mutual information for the ccq state ŴXYAC obtained from the two-qubit reduced density matrix ρAC of W state in Eq. (34). As the additivity of mutual information in Teorem 1 is only a sufcient condition for monogamy inequality in terms of EoF, nonadditivity does not directly imply violation of Inequality (22) for the W state in Eq. (34). 5 However, we note that ρAB in Eq. (35) is a two-qubit state, therefore its EoF can be analytically evaluated as

Ef (ρ ) 0.3812. AB ≈ (48) Moreover, the symmetry of the W state assures that the EoF of ρAC trB W W is the same, = | �ABC� | Ef (ρ ) 0.3812, AC ≈ (49) whereas

Ef W S(ρ ) 0.6365. | �A(BC) = A ≈ (50) 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 Teorem 1; violation of monogamy inequality in (22) implies nonadditivity of quantum mutual information for the ccq state.

Now, we generalize Teorem 1 for multi-party quantum states of arbitrary dimension.

Teorem 2. For any multi-party quantum state ρA1A2 An with two-party reduced density matrices ρA1Ai for i 2, , n , we have ··· = ··· n f f , E ρA1(A2 An) E ρA1Ai (51) ··· ≥ i 2 = conditioned on the additivity of quantum mutual information

I ŴXY A1Ai I ŴX A1Ai I ŴY A1Ai (52) : = : + : where ŴXYA A is the ccq state of the form in Eq. (14) obtained by ρA A for i 2, ..., n. 1 i 1 i =

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 difcult in investigating and establishing entanglement monogamy of multi-party quantum systems of arbitrary dimensions. Te sufcient 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 sufcient 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 fnally remark that it would be an interesting future task to investigate if the condition provided here is also necessary.

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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 1 d 1 parts of the four-qudit ccq state ŴXYAB in Eq. (14) is ŴXY 2 x,−y 0 x X x y y , which is the maxi- 2 = d | � � | ⊗ Y mally mixed state in d -dimensional quantum system, therefore its von= Neumann entropy is d 1 − 1 1 S(ŴXY ) ln 2 ln d. =− d2 d2 = (53) x,y 0 = We also note that Eq. (17) leads us to S(Ŵ ) S(ρ ) ln d. AB = A + (54) From the joint entropy theorem21,22, we have d 1 1 − x y y x S(ŴXYAB) 2 ln d S (IA X Z )ρAB(IA Z− X− ) 2 ln d S(ρAB), = + d2 ⊗ B B ⊗ B B = + (55) x,y 0 = where the second equality is due to the unitary invariance of von Neumann entropy. Tus Eqs. (53), (54) and (55) give us the mutual information of the four-qudit ccq state ŴXYAB with respect to the bipartition between XY and AB in Eq. (18). For the von Neumann entropy of ŴXAB in Eq. (15), we have d 1 d 1 1 − x − i x S(ŴXAB) ln d S IA X σ i ei ei IA X− = + d ⊗ B A ⊗ | �B� | ⊗ B x 0 i 0 = = d 1 − i ln d S σ i ei B ei (56) = + A ⊗ | � � | i 0 = d 1 − ln d H(�) S σ i , = + + i A i 1 = where the frst equality is from the joint entropy theorem, the second equality is due to the unitary invariance of von Neumann entropy and the last equality is due to the joint entropy theorem together with H(�) i ln i that is the shannon entropy of the spectrum i of ρB in Eq. (6). Tus we can rewrite =− i ={ } the von Neumann entropy of ŴXAB as d 1 − i S(ŴXAB) ln d S(ρB) iS σ . (57) = + + A i 1 = 1 d 1 Because the classical parts of ŴXAB is the d-dimensional maximally mixed state ŴX d x−0 x X x , we = = | � � | have the mutual information of ŴXAB with respect to the bipartition between X and AB as I(ŴX AB) S(ŴX ) S(ŴAB) S(ŴXAB) ln d S(ρB) χ(E0). (58) : = + − = − + For the von Neumann entropy of ŴYAB in Eq. (16), we have

d 1 d 1 1 − y − j 1 y S(ŴXAB) ln d S IA Z τ ej B ej IA Z− = + d  ⊗ B  A ⊗ d |˜ � �˜ | ⊗ B  y 0 j 0 �= � � �= � � d 1     − j 1 ln d S τ ej B ej (59) = +  A ⊗ d |˜ � �˜ | j 0 �=  d 1  1 − j 2 ln d S τ , = + d A j 1 �= � � where the frst and third equalities are due to the joint entropy theorem and the second equality is from the unitary invariance of von Neumann entropy. Tus the mutual information of ŴYAB with respect to the bipartition between Y and AB is

I(ŴY AB) S(ŴY ) S(ŴAB) S(ŴYAB) : = + − d 1 1 − j ln d S(ρA) ln d 2 ln d S τ = + + − − d A (60) j 1 = χ(E1). =

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Proof of Theorem 1. Let us frst consider the four-qudit ccq state ŴXYAB of the form in Eq. (14) obtained by the two-qudit reduced density matrix ρAB of ψ . From Eqs. (18), (19) and (20), the additivity condition of | �ABC quantum mutual information for ŴXYAB in Eq. (23) can be rewritten as

χ(E0) χ(E1) S(ρ ) S(ρ ) S(ρ ) I(ρ ), + = A + B − AB = AB (61) where E0 and E1 are the probability ensembles of ρA in Eqs. (7) and (10), respectively. Because E0 and E1 can be obtained from measuring subsystem B of ρAB by the rank-1 measurement d 1 d 1 ei B ei i −1 and ej B ej j −1 , respectively, the defnition of CC in Eq. (4) leads us to J ←(ρAB) χ(E0) and {| � � |} = { ˜ ˜ } = ≥ J ←(ρAB) χ(E1) , therefore ≥ 1 1 J ←(ρAB) χ(E0) χ(E1) I(ρAB). (62) ≥ 2 + = 2 By considering the ccq state ŴXYAC obtained by ρAC as well as Eq. (24), we can analogously have 1 J ←(ρAC) I(ρAC). (63) ≥ 2 As the trade-of relation of Eq. (5) in Proposition 1 is universal with respect to the subsystems, we also have S(ρA) J (ρAC) Ef (ρAB) for the given two-qudit state ψ , therefore = ← + | �ABC Ef (ρ ) Ef (ρ ) 2S(ρ ) J ←(ρ ) J ←(ρ ) . AB + AC = A − AB + AC (64) Now Inequalities (62), (63) as well as Eq. (64) lead us to 1 Ef (ρ ) Ef (ρ ) 2S(ρ ) (I(ρ ) I(ρ )) AB + AC ≤ A − 2 AB + AC 1 2S(ρ ) (S(ρ ) S(ρ ) S(ρ ) S(ρ ) S(ρ ) S(ρ )) = A − 2 A + B − AB + A + C − AC (65) S(ρ ) = A Ef ψ , = | �A(BC) where the second equality is due to ρAC ρB and ρAB ρC for three-party pure state ψ . = = | �ABC

Proof of Theorem 2. We frst prove the theorem for any three-party mixed state ρABC , and inductively show the validity of the theorem for any n-party quantum state ρA1A2 A . For a three-party mixed state ρABC , ··· n let us consider an optimal decomposition of ρABC realizing EoF with respect to the bipartition between A and BC, that is, ρ p ψ ψ , ABC = i| i�ABC� i| (66) i with Ef ρA(BC) i piEf ψi A(BC) . From Teorem 1, each pure state ψi ABC of the decomposition (66) = | i � i i | �i satisfes Ef ψi A(BC) Ef ρ Ef ρ with ρ trC ψi ABC ψi and ρ trB ψi ABC ψi , therefore, | � ≥ AB + AC AB = | � � | AC = | � � | i i Ef ρ ( ) p Ef ψ p Ef ρ p Ef ρ . A BC = i | i�A(BC) ≥ i AB + i AC (67) i i i i For each i and the two-party reduced density matrices ρAB , let us consider its optimal decomposition i i i i i ρAB j rij µj µj realizing EoF, that is, Ef ρAB j rijEf µj . Now we have = AB = AB i i piEf ρAB pirijEf µj Ef (ρAB), = AB ≥ (68) i i,j i i i where the inequality is due to ρAB i piρAB i,j pirij µj µj and the defnition of EoF. = = AB i i i For each i, we also consider an optimal decomposition ρAC l sil νl νl such that i i = AC Ef ρAC l silEf ν . We can analogously have = l AC i p Ef ρ Ef (ρ ), i AC ≥ AC (69) i i i i due to ρAC piρ pis ν ν and the defnition of EoF in Eq. (2). From Inequalities (67), (68) = i AC = i,l il l AC l and (69), we have Ef ρ ( ) Ef (ρ ) Ef (ρ ), A BC ≥ AB + AC (70) which proves the theorem for three-party mixed states. For general multi-party quantum system, we use the mathematical induction on the number of parties n; let us assume Inequality (51) is true for any k-party quantum state, and consider an k 1-party quantum state + ρA1A2 Ak 1 for k 3 . By considering ρA1A2 Ak 1 as a three-party state with respect to the tripartition A1 , A2 Ak ··· + ≥ ··· + ··· and Ak 1 , Inequality (70) leads us to +

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. Ef ρA1(A2 Ak 1) Ef ρA1(A2 Ak) Ef ρA1Ak 1 (71) ··· + ≥ ··· + +

As ρA1A2 A in Inequality (71 ) is a k-party quantum state, the induction hypothesis assures that ··· k . Ef ρA1(A2 Ak) Ef ρA1A2 Ef ρA1Ak (72) ··· ≥ +···+ Now Inequalities (71) and (72 ) lead us to the monogamy inequality in (51), which completes the proof.

Received: 11 November 2020; Accepted: 14 January 2021

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