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Monogamy of entanglement without inequalitiesGilad Gour1 and Yu Guo2
1 Department of Mathematics and Statistics and Institute for Quantum Science and Technology, University of Calgary, Calgary,Alberta T2N 1N4, Canada
2Institute of Quantum Information Science, Shanxi Datong University, Datong, Shanxi 037009, ChinaAugust 10, 2018
We provide a fine-grained definition ofmonogamous measure of entanglement thatdoes not invoke any particular monogamyrelation. Our definition is given in terms ofan equality, as opposed to inequality, thatwe call the “disentangling condition". Werelate our definition to the more traditionalone, by showing that it generates standardmonogamy relations. We then show that allquantumMarkov states satisfy the disentan-gling condition for any entanglement mono-tone. In addition, we demonstrate thatentanglement monotones that are given interms of a convex roof extension are monog-amous if they are monogamous on purestates, and show that for any quantum statethat satisfies the disentangling condition, itsentanglement of formation equals the entan-glement of assistance. We characterize allbipartite mixed states with this property,and use it to show that the G-concurrenceis monogamous. In the case of two qubits,we show that the equality between entan-glement of formation and assistance holds ifand only if the state is a rank 2 bipartitestate that can be expressed as the marginalof a pure 3-qubit state in the W class.
Monogamy of entanglement is one of the non-intuitive phenomena of quantum physics that dis-tinguish it from classical physics. Classically, threerandom bits can be maximally correlated. For ex-ample, three coins can be prepared in a state inwhich with 50% chance all three coins show “head",and with the other 50% chance they all show “tail".
Gilad Gour: [email protected] Guo: [email protected]
In such a preparation, any two coins are maxi-mally correlated. In contrast with the classicalworld, it is not possible to prepare three qubitsA,B,C in a way that any two qubits are maxi-mally entangled [1]. In fact, if qubit A is max-imally entangled with qubit B, then it must beuncorrelated (not even classically) with qubit C.This phenomenon of monogamy of entanglementwas first quantified in a seminal paper by Coffman,Kundu, and Wootters (CKW) [1] for three qubits,and later on studied intensively in more general set-tings [2, 3, 4, 5, 14, 15, 6, 16, 17, 18, 7, 19, 20, 21,22, 23, 24, 25, 8, 9, 26, 27, 10, 11, 28, 29, 30, 12,13, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40].
Qualitatively, monogamy of entanglement mea-sures the shareability of entanglement in a com-posite quantum system, i.e., the more two subsys-tems are entangled the less this pair has entangle-ment with the rest of the system. This feature ofentanglement has found potential applications inmany quantum information tasks and other areasof physics, such as quantum key distribution [41,42, 43], classification of quantum states [44, 45, 46],condensed-matter physics [47, 48, 49], frustratedspin systems [50], statistical physics [51], and evenblack-hole physics [52, 53].
A monogamy relation is quantitatively displayedas an inequality of the following form:
E(A|BC) ≥ E(A|B) + E(A|C) , (1)
where E is a measure of bipartite entanglement andA,B,C are three subsystems of a composite quan-tum system. It states that the sum of the entan-glement between A and B, and between A and C,can not exceed the entanglement between A andthe joint system BC. While not all measures ofentanglement satisfy this relation, some do. Con-
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sequently, any measure of entanglement that doessatisfy (1) was called in literature monogamous.
Here we argue that this definition of monoga-mous measure of entanglement captures only par-tially the property that entanglement is monoga-mous. This is evident from the fact that manyimportant measures of entanglement do not satisfythe relation (1). Some of these measures are noteven additive under tensor product [54, 55, 56] (infact, some measures [57] are multiplicative undertensor product). Therefore, the summation in theRHS of (1) is clearly only a convenient choice andnot a necessity. For example, it is well known thatif E does not satisfy this relation, it is still possibleto find a positive exponent α ∈ R+, such that Eα
satisfies the relation. This was already apparent inthe seminal work of [1] in which E was taken tobe the square of the concurrence and not the con-currence itself. More recently, it was shown thatmany other measures of entanglement satisfy themonogamy relation (1) if E is replaced by Eα forsome α > 1 [13, 8, 12, 5, 7, 11, 9, 6, 10].
One attempt to address these issues with the cur-rent definition of a monogamous measure of entan-glement, is to replace the relation (1) with a familyof monogamy relations of the form, E(A|BC) ≥f(E(A|B), E(A|C)
), where f is some function of
two variables that satisfy certain conditions [31].However, such an approach is somewhat artificialin the sense that the monogamy relations are notderived from more basic principles.
In this paper we take another approach tomonogamy of entanglement which is more “fine-grained" in nature, and avoid the introduction ofsuch a function f . Therefore, our definition ofmonogamous measure of entanglement (see Defi-nition 1 below) does not involve a monogamy rela-tion, but instead a condition on the measure of en-tanglement that we call the disentangling condition(following the terminology of [28]). Yet, we showthat our definition is consistent with the more tra-ditional notion of monogamy of entanglement, byshowing that E is monogamous according to ourdefinition if and only if there exists an α > 0 suchthat Eα satisfies (1). Consequently, many moremeasures of entanglement are monogamous accord-ing to our definition. We then provide a character-
ization for the disentangling condition in the formof an equality between the entanglement of forma-tion (EoF) associated with the given entanglementmeasure (see Eq. (7) below) and the entanglementof assistance (EoA) [58], and discuss its relationto quantum Markov chains [59]. In addition, wecharacterize all states for which EoF equals to EoAwhen the measure of entanglement is taken to bethe G-concurrence, and use that to show that theG-concurrence is monogamous. Finally, we showthat in the 2-dimensional case, the bipartite 2-qubitmixed states that can be expressed as the marginalof the 3-qubit W -state, are the only 2-qubit entan-gled states for which the EoF equals the EoA withrespect to any measure of entanglement.
Let HA ⊗ HB ≡ HAB be a bipartite Hilbertspace, and S(HAB) ≡ SAB be the set of den-sity matrices acting on HAB. A function E :SAB → R+ is called a measure of entanglement if(1) E(σAB) = 0 for any separable density matrixσAB ∈ SAB, and (2) E behaves monotonically un-der local operations and classical communications(LOCC). That is, given an LOCC map Φ we have
E(Φ(ρAB)
)≤ E
(ρAB
), ∀ ρAB ∈ SAB . (2)
The measure is said to be faithful if it is zero onlyon separable states.
The map Φ is completely positive and trace pre-serving (CPTP). In general, LOCC can be stochas-tic, in the sense that ρAB can be converted toσABj with some probability pj . In this case, themap from ρAB to σABj can not be described ingeneral by a CPTP map. However, by introduc-ing a ‘flag’ system A′, we can view the ensemble{σABj , pj} as a classical quantum state σA′AB ≡∑j pj |j〉〈j|A
′ ⊗ σABj . Hence, if ρAB can be con-verted by LOCC to σABj with probability pj , thenthere exists a CPTP LOCC map Φ such thatΦ(ρAB) = σA
′AB. Therefore, the definition aboveof a measure of entanglement captures also proba-bilistic transformations. Particularly, E must sat-isfy E
(σA′AB
)≤ E
(ρAB
).
Almost all measures of entanglement studied inliterature (although not all [60]) satisfy
E(σA′AB
)=∑j
pjE(σABj ) , (3)
Accepted in Quantum 2018-08-05, click title to verify 2
which is very intuitive since A′ is just a clas-sical system encoding the value of j. In thiscase the condition E
(σA′AB
)≤ E
(ρAB
)becomes∑
j pjE(σABj ) ≤ E(ρAB
). That is, LOCC can not
increase entanglement on average. Measures of en-tanglement that satisfy this property are called en-tanglement monotones, and they can also be shownto be convex [61]. In the following definition wedenote density matrices acting on a finite dimen-sional tripartite Hilbert space HABC by ρA|BC ,where the vertical bar indicates the bipartite splitacross which we will measure the (bipartite) entan-glement.
Definition 1. Let E be a measure of entangle-ment. E is said to be monogamous if for anyρA|BC ∈ SABC that satisfies
E(ρA|BC) = E(ρAB) (4)
we have that E(ρAC) = 0.
The condition in (4) is a very strong one andtypically is not satisfied by most tripartite statesρABC . Following the terminology of [28] we callit here the “disentangling condition". We will seebelow that states that saturate the strong subaddi-tivity inequality for the von Neumann entropy (i.e.quantum Markov states) always satisfy this equal-ity, for any entanglement monotone E.
Note that the condition in (1) is stronger thanthe one given in Definition 1. Indeed, if E satis-fies (1), then any ρA|BC that satisfies (4) must haveE(ρAC) = 0. At the same time, Definition 1 cap-tures the essence of monogamy: that is, if systemA shares the maximum amount of entanglementwith subsystem B, it is left with no entanglementto share with C.
In Definition 1 we do not invoke a particularmonogamy relation such as (1). Instead, we pro-pose a minimalist approach which is not quanti-tative, in which we only require what is essentialfrom a measure of entanglement to be monoga-mous. Yet, this requirement is sufficient to generatea more quantitative monogamy relation:
Theorem 1. Let E be a continuous measure ofentanglement. Then, E is monogamous according
to Definition 1 if and only if there exists 0 < α <∞ such that
Eα(ρA|BC) ≥ Eα(ρAB) + Eα(ρAC) , (5)
for all ρABC ∈ SABC with fixed dimHABC = d <∞.
We call the smallest possible value for α that sat-isfies Eq. (5) in a given dimension d = dim(HABC),the monogamy exponent associated with a measureE, and denote it simply as α(E). In general, themonogamy exponent is hard to compute, and inthe supplemental material we provide (along withthe proofs of the theorems in this paper) a com-prehensive list of known bounds for the monogamyexponent when E is one of the measures of entan-glement that were studied extensively in literature.
It is important to note that the relationgiven in (5) is not of the form E(A|BC) ≥f(E(A|B), E(A|C)
), where f is some function of
two variables that satisfies certain conditions andis independent of d [31]. This is because themonogamy exponent in (5) depends on the dimen-sion d, whereas f as was used in [31] is universal inthe sense that it does not depend on the dimension.Therefore, if a measure of entanglement such as theentanglement of formation is not monogamous ac-cording to the class of relations given in [31], itdoes not necessarily mean that it is not monoga-mous according to our definition. Moreover, sinceour approach allows for dependence in the dimen-sion, it avoids the issues raised in [31] that measuresof entanglement cannot be simultaneously monoga-mous and (geometrically) faithful (see [31] for theirdefinition of faithfulness).
In general, the class of all states ρABC that sat-isfy the disentangling condition (4) depends on thechoice of the entanglement measure E. However,there is a class of states that satisfy this condi-tion for any choice of an entanglement monotoneE. These are precisely the states that saturatethe strong subadditivity of the von-Neumann en-tropy [59]. For such states, the system B Hilbertspace HB must have a decomposition into a directsum of tensor productsHB =
⊕j H
BLj ⊗HB
Rj , such
that the state ρABC has the form
ρABC =⊕j
qjρABL
j ⊗ ρBRj C , (6)
Accepted in Quantum 2018-08-05, click title to verify 3
where qj is a probability distribution [59].
Theorem 2. Let E be an entanglement mono-tone. Then, E satisfies the disentangling condi-tion (4) for all Markov quantum states ρABC ofthe form (6).
Note that a state of the form (6) has a separa-ble marginal state ρAC , and therefore the abovetheorem is consistent with Definition 1. Conse-quently, to check if an entanglement monotone ismonogamous, one has to consider tripartite statesthat satisfy (4) but have a different form than (6).Perhaps such states do not exist for certain en-tanglement monotones. Indeed, partial results inthis direction were proved recently in [28]. Partic-ularly, it was shown that any pure tripartite en-tangled state ψABC with bipartite marginal stateρAB, that satisfies N
(ψA|BC
)= N
(ρAB
), where
N is the negativity measure, must have the formψABC = ψAB
L ⊗ ψBRC . Note that this is preciselythe form (6) when ρABC is pure. Like the Nega-tivity, we will see later that also the G-concurrencehas this property for pure tripartite states.
Typically it is hard to check if a measure ofentanglement is monogamous since the conditionsin (4) involves mixed tripartite states. However, aswe show below, it is sufficient to consider only puretripartite states in (4), if the entanglement mea-sure E is defined on mixed states by a convex roofextension; that is,
Ef(ρAB
)≡ min
n∑j=1
pjE(|ψj〉〈ψj |AB
), (7)
where the minimum is taken over all pure statedecompositions of ρAB =
∑nj=1 pj |ψj〉〈ψj |AB. We
call Ef the entanglement of formation (EoF) asso-ciated with E. In general, it could be that Ef 6= Eon mixed states like the convex roof extended Neg-ativity is different than the Negativity itself.
Theorem 3. Let Ef be an entanglement monotoneas above. If Ef is monogamous (according to Def-inition 1) on pure tripartite states in HABC , thenit is also monogamous on tripartite mixed statesacting on HABC .
From the above theorem and Corollary 5 be-low it follows that the convex roof extended neg-ativity is monogamous, since it is monogamous for
pure states [28]. Similarly, we will use it to showthat the G-concurrence is monogamous. However,monogamy alone does not necessarily imply thata tripartite state is a Markov state if it satisfiesthe disentangling condition. In the following theo-rem, we provide yet another property of a tripartitestate ρABC that satisfies the disentangling condi-tion. The property is that any LOCC protocol onsuch a tripartite state can not increase on aver-age the initial bipartite entanglement between Aand B. The maximum such average of bipartiteentanglement is known as entanglement of collabo-ration [62]. It is defined by:
EAB|Cc
(ρABC
)= max
n∑j=1
pjE(ρABj
), (8)
where E is a given measure of bipartite entangle-ment, and the maximum is taken over all tripartiteLOCC protocols yielding the bipartite state ρABjwith probability pj . This measure of entanglementis closely related to EoA, denoted here by Ea, inwhich the optimization above is restricted to LOCCwith one way classical communication from systemC to systems A and B. Therefore, EAB|Cc ≥ Eaand in [62] it was shown that this inequality can bestrict.
Theorem 4. Let E be an entanglement monotoneon bipartite mixed states, and let ρABC be a (possi-bly mixed) tripartite state satisfying the disentan-gling condition (4). Then,
E(ρAB
)= EAB|Cc
(ρABC
). (9)
The condition in (9) is a very strong one. Par-ticularly, it states that all measurements on systemC yield the same average of bipartite entanglementbetween A and B. Therefore, the entanglement ofthe marginal state ρAB is resilient to any quantumprocess or measurement on system C. In the casethat ρABC is pure, the EoA, Ea, depends only onthe marginal ρAB, and we get the following corol-lary:
Corollary 5. Let E be an entanglement monotoneon bipartite states, and let ρABC be a pure tripar-tite state satisfying the disentangling condition (4).Then,
E(ρAB
)= Ef (ρAB) = Ea(ρAB) , (10)
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where the EoF, Ef , is defined in (7), and EoA,Ea, is also defined as in (7) but with a maximumreplacing the minimum.
The equality of Ef(ρAB
)= Ea
(ρAB
)in corol-
lary 5 implies that any pure state decompositionof ρAB has the same average entanglement whenmeasured by E. Unless ρAB is pure, it is almostnever satisfied. Nonetheless, there are non-trivialstates (of measure zero) that do satisfy it. Suchstates have the following property:
Theorem 6. Let E be a measure of bipartiteentanglement, and let ρAB ∈ SAB be a bipar-tite state with support subspace supp(ρAB). IfEf (ρAB) = Ea(ρAB) then for any σAB ∈ SABwith supp(σAB) ⊆ supp(ρAB) we have Ef (σAB) =Ea(σAB).
Remark 1. The theorem above follows from some ofthe results presented in [33], and for completenesswe provide its full proof in the appendix.
Theorem 6 demonstrates that the equality be-tween the EoF and EoA corresponds to a propertyof the support of ρAB rather than ρAB itself. In thefollowing theorem, we characterize the precise formof the support of ρAB that yields such an equality.The entanglement monotone we are using is a gen-eralization of the concurrence measure known asthe G-concurrence [57].
The G-concurrence is an entanglement mono-tone that on pure bipartite states is equal to thegeometric-mean of the Schmidt coefficients. On a(possibly unnormalized) vector |x〉 ∈ Cd ⊗ Cd, itcan be expressed as:
G(|x〉) = |det(X)|2/d ; |x〉 = X ⊗ I|φ+〉 , (11)
where |φ+〉 =∑di=1 |i〉A|i〉B is the maximally en-
tangled state, and X ∈ Md(C) is a d× d complexmatrix. For mixed states the G-concurrence is de-fined in terms of the convex roof extension; that is,G(ρAB) := Gf (ρAB).
In the following theorem we denote by N ⊂Md(C), the nilpotent cone, consisting of all d × dnilpotent complex matrices (i.e. X ∈ N iff Xk = 0for some 1 ≤ k ≤ d). While the set N is not avector space, it contains subspaces. For example,the set T of all strictly upper triangular matrices
(i.e. upper triangular matrices with zeros on thediagonal) is a d(d − 1)/2-dimensional subspace inN . From Gerstenhaber’s theorem [63] it followsthat the largest dimension of a subspace in N isd(d − 1)/2, and if a subspace N0 ⊂ N has thismaximal dimension, then it must be similar to T(i.e. their exists an invertible matrix S such thatN0 = ST S−1).
Theorem 7. Let ρAB be a bipartite density matrixacting on Cd ⊗ Cd with rank r. Then,
G(ρAB) = Ga(ρAB) > 0 , (12)
if and only if r ≤ 1 + d(d − 1)/2, and there existsa full Schmidt rank state |x〉AB ∈ Cd ⊗ Cd, and asubspace N0 ⊂ N , with dim(N0) = r−1, such that
supp(ρAB) = {I ⊗ Y |x〉AB∣∣ Y ∈ K} , (13)
where K ≡ {I}⊕N0 ≡ {cI+N∣∣ c ∈ C ; N ∈ N0}.
The direct sum above is consistent with the factthat a nilpotent matrix has a zero trace, so that itis orthogonal to the identity matrix in the Hilbert-Schmidt inner product. The theorem above impliesthat the G-concurrence is monogamous on pure tri-partite states:
Corollary 8. Let |ψABC〉 ∈ Cd ⊗ Cd ⊗ Cn be apure tripartite state, with bipartite marginal ρAB.If G(ψA|BC) = G(ρAB) > 0 then |ψ〉ABC =|χ〉AB|ϕ〉C for some bipartite state |χ〉AB ∈ Cd⊗Cdand a vector |ϕ〉C ∈ Cn.
The above result is somewhat surprising since theG-concurrence is not a faithful measure of entangle-ment. Yet, it states that the disentangling condi-tion forces the marginal state ρAC to be a productstate. This is much stronger than G(ρAC) = 0(which can even hold for some entangled ρAC),and in particular, it states that A and C can noteven share classical correlation. Combining theabove corollary with Theorem 3 implies that theG-concurrence is monogamous on any mixed statethat is acting on Cd⊗Cd⊗Cn. Finally, in the qubitcase, Theorem 7 takes the following form:
Corollary 9. Let ρAB be an entangled two qubitstate with rank r > 1, and let E be any injective(up to local unitaries) measure of pure two qubitentanglement. Then,
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1. If Ef (ρAB) = Ea(ρAB) then r = 2, andρAB = TrC |W 〉〈W |ABC is the 2-qubit marginalof |W 〉 = λ1|100〉+λ2|010〉+λ3|001〉+λ4|000〉,λi ∈ C.
2. Conversely, if ρAB is a marginal of a state inthe W-class then Cf (ρAB) = Ca(ρAB), whereC is the concurrence.
Remark 2. The second part of the theorem impliesthat also Ef (ρAB) = Ea(ρAB), for any ρAB thatis a marginal of a W-state, and any measure E,that can be expressed as a convex function of theconcurrence C [64].
In conclusions, we introduced a new definitionfor a monogamous measure of entanglement. Ourdefinition involves an equality (the disentanglingcondition (4)) rather than the inequality (1). Yet,we showed that our notion of monogamy can repro-duce monogamy relations like in (1) with a smallchange that the measure E is replaced by Eα forsome exponent α > 0. We then showed that convexroof based entanglement monotones of the form (7)are monogamous iff they are monogamous on puretripartite states, and showed further that the dis-entangling condition in (4) holds for any entangle-ment monotone if ρABC is a quantumMarkov state.While it is left open if the converse is also true (atleast for some entanglement monotones), we wereable to show that for the G-concurrence, the onlypure tripartite states that satisfy the disentanglingcondition (4) are Markov states. In addition, werelated the disentangling condition to states thathave the same average entanglement for all convexpure state decompositions, and found a character-ization of such states in Theorem 6 (for generalmeasures of entanglement), and a complete char-acterization in Theorem 7 (for the G-concurrence).Clearly, much more is left to investigate along theselines.
Acknowledgements
This work was completed while Y.G was visitingthe Institute of Quantum Science and Technol-ogy of the University of Calgary under the sup-port of the China Scholarship Council under Grant
No. 201608140008. Y.G thanks Professor C. Si-mon and Professor G. Gour for their hospitality,and thanks Sumit Goswami for helpful discussions.G.G research is supported by the Natural Sci-ences and Engineering Research Council of Canada(NSERC). Y.G is supported by the National Nat-ural Science Foundation of China under Grant No.11301312 and the Natural Science Foundation ofShanxi under Grant No. 201701D121001.
References[1] V. Coffman, J. Kundu, and W.
K. Wootters. Distributed entangle-ment. Phys. Rev. A, 61:052306, 2000.doi:10.1103/PhysRevA.61.052306.
[2] R. Horodecki, P. Horodecki, M. Horodecki,and K. Horodecki, Quantum entangle-ment. Rev. Mod. Phys., 81:865, 2009.doi:10.1103/RevModPhys.81.865.
[3] M. Koashi and A. Winter. Monogamyof quantum entanglement and other cor-relations. Phys. Rev. A, 69:022309, 2004.doi:10.1103/PhysRevA.69.022309.
[4] G. Gour, D. A. Meyer, and B. C.Sanders. Deterministic entanglementof assistance and monogamy con-straints. Phys. Rev. A, 72:042329, 2005.doi:10.1103/PhysRevA.72.042329.
[5] T. J. Osborne and F. Verstraete. Generalmonogamy inequality for bipartite qubit en-tanglement. Phys. Rev. Lett., 96:220503, 2006.doi:10.1103/PhysRevLett.96.220503.
[6] Y.-C. Ou and H. Fan, Monogamy in-equality in terms of negativity for three-qubit states. Phys. Rev. A, 75:062308, 2007.doi:10.1103/PhysRevA.75.062308.
[7] J. S. Kim, A. Das, and B. C. Sanders.Entanglement monogamy of multipar-tite higher-dimensional quantum sys-tems using convex-roof extended nega-tivity. Phys. Rev. A, 79:012329, 2009.doi:10.1103/PhysRevA.79.012329.
[8] X. N. Zhu and S. M. Fei. Entangle-ment monogamy relations of qubit sys-
Accepted in Quantum 2018-08-05, click title to verify 6
tems. Phys. Rev. A, 90: 024304, 2014.doi:10.1103/PhysRevA.90.024304.
[9] Y.-K. Bai, Y.-F. Xu, and Z. D. Wang.General monogamy relation for the entan-glement of formation in multiqubit sys-tems. Phys. Rev. Lett., 113:100503, 2014.doi:10.1103/PhysRevLett.113.100503.
[10] J. H. Choi and J. S. Kim. Negativ-ity and strong monogamy of multi-party quantum entanglement beyondqubits. Phys. Rev. A, 92:042307, 2015.doi:10.1103/PhysRevA.92.042307.
[11] Y. Luo and Y. Li. Monogamy of αthpower entanglement measurement in qubitsystems. Ann. Phys., 362:511-520, 2015.doi:10.1016/j.aop.2015.08.022.
[12] X. N. Zhu and S. M. Fei. Entanglementmonogamy relations of concurrence for N -qubit systems. Phys. Rev. A, 92:062345, 2015.doi:10.1103/PhysRevA.92.062345.
[13] A. Kumar. Conditions for monogamy ofquantum correlations in multipartite sys-tems. Phys. Lett. A, 380:3044-3050, 2016.doi:10.1016/j.physleta.2016.07.032.
[14] G. Gour, S. Bandyopadhyay, and B. C.Sanders. Dual monogamy inequality for en-tanglement. J. Math. Phys., 48:012108, 2007.doi:10.1063/1.2435088.
[15] Y.-C. Ou. Violation of monogamy in-equality for higher dimensional ob-jects. Phys. Rev. A, 75:034305, 2007.doi:10.1103/PhysRevA.75.034305.
[16] T. Hiroshima, G. Adesso and F. Illuminati.Monogamy inequality for distributed Gaussianentanglement. Phys. Rev. Lett., 98:050503,2007. doi:10.1103/PhysRevLett.98.050503.
[17] G. Adesso and F. Illuminati. Strongmonogamy of bipartite and genuine mul-tipartite entanglement: The Gaussiancase. Phys. Rev. Lett., 99:150501, 2007.doi:10.1103/PhysRevLett.99.150501.
[18] J. S. Kim and B. C. Sanders. GeneralizedW-class state and its monogamy relation. J.Phys. A, 41:495301, 2008. doi:10.1088/1751-8113/41/49/495301.
[19] J. S. Kim and B. C. Sanders. Monogamy ofmulti-qubit entanglement using Rényi entropy.J. Phys. A, 43:445305, 2010. doi:10.1088/1751-8113/43/44/445305.
[20] X.-J. Ren and W. Jiang. Entanglementmonogamy inequality in a 2 ⊗ 2 ⊗ 4system. Phys. Rev. A, 81:024305, 2010.doi:10.1103/PhysRevA.81.024305.
[21] M. F. Cornelio and M. C. de Oliveira.Strong superadditivity and monogamyof the Rényi measure of entangle-ment. Phys. Rev. A, 81:032332, 2010.doi:10.1103/PhysRevA.81.032332.
[22] A. Streltsov, G. Adesso, M. Piani, andD. Bruß. Are general quantum correlationsmonogamous? Phys. Rev. Lett., 109:050503,2012. doi:10.1103/PhysRevLett.109.050503.
[23] M. F. Cornelio. Multipartite monogamy of theconcurrence. Phys. Rev. A, 87:032330, 2013.doi:10.1103/PhysRevA.87.032330.
[24] S.-Y. Liu, B. Li, W.-L. Yang, and H.Fan. Monogamy deficit for quantumcorrelations in a multipartite quantumsystem. Phys. Rev. A, 87:062120, 2013.doi:10.1103/PhysRevA.87.062120.
[25] T. R. de Oliveira, M. F. Cornelio, and F.F. Fanchini. Monogamy of entanglement offormation. Phys. Rev. A, 89:034303, 2014.doi:10.1103/PhysRevA.89.034303.
[26] B. Regula, S. D. Martino, S. Lee, andG. Adesso. Strong monogamy conjecturefor multiqubit entanglement: the four-qubitcase. Phys. Rev. Lett., 113:110501, 2014.doi:10.1103/PhysRevLett.113.110501.
[27] K. Salini, R. Prabhub, Aditi Sen(De),and Ujjwal Sen. Monotonically increas-ing functions of any quantum correlationcan make all multiparty states monog-amous. Ann. Phys., 348:297-305, 2014.doi:10.1016/j.aop.2014.06.001.
[28] H. He and G. Vidal. Disentangling theo-rem and monogamy for entanglement neg-ativity. Phys. Rev. A, 91:012339, 2015.doi:10.1103/PhysRevA.91.012339.
Accepted in Quantum 2018-08-05, click title to verify 7
[29] C. Eltschka and J. Siewert. Monogamy equal-ities for qubit entanglement from Lorentz in-variance. Phys. Rev. Lett., 114:140402, 2015.doi:10.1103/PhysRevLett.114.140402.
[30] A. Kumar, R. Prabhu, A. Sen(de), andU. Sen. Effect of a large number of par-ties on the monogamy of quantum cor-relations. Phys. Rev. A, 91:012341, 2015.doi:10.1103/PhysRevA.91.012341.
[31] Lancien et al. Should entanglementmeasures be monogamous or faith-ful? Phys. Rev. Lett., 117:060501, 2016.doi:10.1103/PhysRevLett.117.060501.
[32] L. Lami, C. Hirche, G. Adesso, and A. Win-ter. Schur complement inequalities for covari-ance matrices and monogamy of quantum cor-relations. Phys. Rev. Lett., 117:220502, 2016.doi:10.1103/PhysRevLett.117.220502.
[33] Song et al. General monogamy relation of mul-tiqubit systems in terms of squared Rényi-αentanglement. Phys. Rev. A, 93:022306, 2016.doi:10.1103/PhysRevE.93.022306.
[34] B. Regula, A. Osterloh, and G. Adesso.Strong monogamy inequalities for fourqubits. Phys. Rev. A, 93:052338, 2016.doi:10.1103/PhysRevA.93.052338.
[35] Y. Luo, T. Tian, L.-H. Shao, and Y.Li. General monogamy of Tsallis q-entropy entanglement in multiqubit sys-tems. Phys. Rev. A, 93: 062340, 2016.doi:10.1103/PhysRevA.93.062340.
[36] E. Jung and D. Park. Testing themonogamy relations via rank-2 mix-tures. Phys. Rev. A, 94:042330, 2016.doi:10.1103/PhysRevA.94.042330.
[37] S. Cheng and M. J. W. Hall. Anisotropic in-variance and the distribution of quantum cor-relations. Phys. Rev. Lett., 118:010401, 2017.doi:10.1103/PhysRevLett.118.010401.
[38] G. W. Allen and D. A. Meyer. Polynomialmonogamy relations for entanglement nega-tivity. Phys. Rev. Lett., 118: 080402, 2017.doi:10.1103/PhysRevLett.118.080402.
[39] Q. Li, J. Cui, S. Wang, and G.-L. Long. En-tanglement monogamy in three qutrit systems.
Sci. Rep., 7:1946, 2017. doi:10.1038/s41598-017-02066-8.
[40] S. Camalet. Monogamy Inequality for anylocal quantum resource and entanglement.Phys. Rev. Lett., 119: 110503, 2017.doi:10.1103/PhysRevLett.119.110503.
[41] B. M. Terhal. IBM Journal of Researchand Development,48(1):71-78, 2004.doi:10.1147/rd.481.0071.
[42] M. Pawlowski. Security proof for cryp-tographic protocols based only on themonogamy of Bell’s inequality viola-tions. Phys. Rev. A, 82:032313, 2010.doi:10.1103/PhysRevA.82.032313.
[43] N. Gisin, G. Ribordy, W. Tittel,and H. Zbinden. Quantum cryptog-raphy. Rev. Mod. Phys., 74:145, 2002.doi:10.1103/RevModPhys.74.145.
[44] W. Dür, G. Vidal, and J. I. Cirac. Threequbits can be entangled in two inequiv-alent ways. Phys. Rev. A, 62:062314,2000. doi:10.1103/PhysRevA.62.062314.doi:10.1103/PhysRevA.62.062314.
[45] G. L. Giorgi. Monogamy propertiesof quantum and classical correla-tions. Phys. Rev. A, 84: 054301, 2011.doi:10.1103/PhysRevA.84.054301.
[46] R. Prabhu, A. K. Pati, A. Sen(De), and U.Sen. Conditions for monogamy of quantumcorrelations: Greenberger-Horne-Zeilingerversus W states. Phys. Rev. A, 85:040102(R),2012. doi:10.1103/PhysRevA.85.040102.
[47] Ma et al.Quantum simulation of the wave-function to probe frustrated Heisenbergspin systems. Nat. Phys., 7:399, 2011.doi:10.1038/nphys1919.
[48] F. G. S. L. Brandao and A. W. Harrow,in Proceedings of the 45th Annual ACMSymposium on Theory of Computing, 2013.http://dl.acm.org/citation.cfm?doid=2488608.
[49] A. García-Sáez and J. I. Latorre. Renormal-ization group contraction of tensor networksin three dimensions. Phys. Rev. B, 87:085130,2013. doi:10.1103/PhysRevB.87.085130.
Accepted in Quantum 2018-08-05, click title to verify 8
[50] Rao et al. Multipartite quantum correla-tions reveal frustration in a quantum Isingspin system. Phys. Rev. A, 88:022312, 2013.doi:10.1103/PhysRevA.88.022312.
[51] C. H. Bennett, in Proceedings of theFQXi 4th International Conference,Vieques Island, Puerto Rico, 2014,http://fqxi.org/conference/talks/2014.
[52] L. Susskind. Black hole complementar-ity and the Harlow-Hayden conjecture.https://arxiv.org/abs/1301.4505.
[53] S. Lloyd and J. Preskill. Unitarity of blackhole evaporation in final-state projection mod-els. J. High Energy Phys., 08:126, 2014.doi:10.1007/JHEP08(2014)126.
[54] P. W. Shor, J. A. Smolin, and B. M.Terhal. Nonadditivity of bipartite distill-able entanglement follows from a conjec-ture on bound entangled Werner states.Phys. Rev. Lett., 86:2681–2684, 2001.doi:10.1103/PhysRevLett.86.2681.
[55] P. W. Shor. Equivalence of additivity ques-tions in quantum information theory. Com-mun. Math. Phys., 246(3):453-472, 2004.doi:10.1007/s00220-003-0981-7.
[56] K. G. H. Vollbrecht and R. F. Werner.Entanglement measures under sym-metry. Phys. Rev. A, 64:062307, 2001.doi:10.1103/PhysRevA.64.062307.
[57] G. Gour. Family of concurrence monotonesand its applications. Phys. Rev. A, 71:012318,2005. doi:10.1103/PhysRevA.71.012318.
[58] DiVincenzo et al. Entanglement of Assistance.Lecture Notes in Computer Science, 1509:247,1999. doi:10.1007/3-540-49208-9_21.
[59] P. Hayden, R. Jozsa, D. Petz, and A. Winter.Structure of states which satisfy strong sub-additivity of quantum entropy with equality.Commun. Math. Phys., 246(2):359-374, 2004.doi:10.1007/s00220-004-1049-z.
[60] M. B. Plenio. Logarithmic negativity: afull entanglement monotone that is notconvex. Phys. Rev. Lett., 95:090503, 2005.doi:10.1103/PhysRevLett.95.090503. Er-ratum Phys. Rev. Lett., 95:119902, 2005.doi:10.1103/PhysRevLett.95.119902.
[61] G. Vidal. Entanglement mono-tone. J. Mod. Opt., 47:355, 2000.doi:10.1080/09500340008244048.
[62] G. Gour and R. W. Spekkens. Entanglementof assistance is not a bipartite measure nor atripartite monotone. Phys. Rev. A, 73:062331,2006. doi:10.1103/PhysRevA.73.062331.
[63] M. Gerstenhaber. On nilalgebras and linearvarieties of nilpotent matrices (I). Amer. J.Math., 80:614-622, 1958. doi:10.2307/2372773.
[64] W. K. Wootters. Entanglement of for-mation of an arbitrary state of twoqubits. Phys. Rev. Lett., 80:2245, 1998.doi:10.1103/PhysRevLett.80.2245.
Supplementary MaterialMonogamy of entanglement without inequalities
A The Monogamy exponent
Theorem 1. Let E be a continuous measure of entanglement. Then, E is monogamous according toDefinition 1 if and only if there exists 0 < α <∞ such that
Eα(ρA|BC) ≥ Eα(ρAB) + Eα(ρAC) , (14)
for all ρABC ∈ SABC with fixed dimHABC = d <∞.
Proof. Let E be a monogamous measure of entanglement according to Def.1. Since E is a measure ofentanglement, it is non-increasing under partial traces, and therefore E(ρA|BC) ≥ max{E(ρAB), E(ρAC)}for any state ρA|BC ∈ SABC . We assume E(ρA|BC) > 0 and set x1 ≡ E(ρAB)/E(ρA|BC) and x2 ≡E(ρAC)/E(ρA|BC). Clearly, there exists γ > 0 such that
xγ1 + xγ2 ≤ 1 , (15)
since either xγj → 0 when γ increases, or if x1 = 1 then by assumption x2 = 0 and vise versa. We denoteby f(ρABC) the smallest value of γ that achieves equality in (15). Since E is continuous, so is f , and thecompactness of SABC gives:
α ≡ maxρABC∈SABC
f(ρABC) <∞ . (16)
By definition, α satisfies the condition in (5).
As discussed in the paper, the expression for α in (16) is optimal in the sense that it provides thesmallest possible value for α that satisfies Eq. (5). This monogamy exponent is a function of the measureE, and we denote it by α(E). It may depend also on the dimension d ≡ dim(HABC) (see in Table 1), and,in general, is hard to compute especially in higher dimensions [1, 4, 6, 2, 3, 5]. By definition, α(E) can onlyincrease with d (e.g. Table 1). Table 1 indicates that almost any entanglement measure is monogamous atleast for multi-qubit systems. In addition, almost all the entanglement measures studied in the literatureare continuous, and in particular C, N , Ncr, Ef , τ , Tq, Rα and Er are all continuous [27, 28, 29, 30].
B Quantum Markov States and Monogamy of Entanglement
Recall that quantum Markov states are states that saturate the strong subadditivity of the von-Neumannentropy. That is, they saturate the inequality:
S(ρAB) + S(ρBC) ≥ S(ρABC) + S(ρB) , (17)
where S(ρ) = −Tr [ρ log ρ] is the von-Neumann entropy. In [31] it was shown that the inequality above issaturated if and only if the Hilbert space of system B, HB, can be decomposed into a direct sum of tensorproducts
HB =⊕j
HBLj ⊗HB
Rj (18)
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Table 1: A comparison of the monogamy exponent of several entanglement measures. We denote the one-way distillableentanglement, concurrence, negativity, convex roof extended negativity, entanglement of formation (the original onedefined in [7]), tangle, squashed entanglement, Tsallis-q entanglement and Rényi-α entanglement by Ed, C, N ,Ncr, Ef ,τ , Esq, Tq and Rα, respectively.
E α(E) System ReferenceEd ≤ 1 any system [8]C 2 2⊗3 [9]a
≤√
2 2⊗n [10]≤ 2 2⊗ 2⊗ 2m [11]≤ 2 2⊗n [11]> 2 3⊗3 [12]≤ 2 2⊗ 2⊗ 4 [13]> 3 3⊗ 2⊗ 2 [14]
N ≤ 2 2⊗n [15, 16]b
≤ 2 2⊗ 2⊗ 2m [16]b
≤ 2 d⊗ d⊗ d,d = 2, 3, 4 [1]Ncr ≤ 2 2⊗n [17, 16, 18]Ef ≤ 2 2⊗n [19, 20]
≤√
2 2⊗n [10]> 1 2⊗ 2⊗ 2 [20, 21]
τ ≤ 1 2⊗ 2⊗ 4 [13]Esq ≤ 1 any system [8]
Tq, 2 ≤ q ≤ 3 ≤ 1 2⊗n [22]Tq ≤ 2 2⊗ 2⊗ 2m [23]c
Rα, α ≥ 2 ≤ 1 2⊗n [24, 25]Rα, α ≥
√7−12 ≤ 2 2⊗n [26]d
aα(C) ≤ 2 was shown in [9], and the equality α(C) = 2 follows from the saturation by W states (see, for example,Corollary 9).
bFor pure states.cFor mixed states, and q ∈ [ 5−
√13
2 , 2] ∪ [3, 5+√
132 ].
dFor mixed states.
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such that the state ρABC has the form
ρABC =⊕j
qjρABL
j ⊗ ρBRj C , (19)
where qj is a probability distribution.
Theorem 2. Let E be an entanglement monotone. Then, E satisfies the disentangling condition (4) forall entangled Markov quantum states ρABC of the form (19).
Proof. Since local ancillary systems are free in entanglement theory, one can append an ancillary systemB′ that encodes the orthogonality of the subspaces HB
Lj ⊗HB
Rj . This can be done with an isometry that
maps states in HBLj ⊗HB
Rj to states in HBL⊗HBR⊗|j〉〈j|B′ , where systems BL and BR have dimensions
maxj dim(HB
Lj
)and maxj dim
(HB
Rj
), respectively. Therefore, w.l.o.g. we can write the above state as
ρABC =∑j
qj ρABL
j ⊗ |j〉〈j|B′ ⊗ ρBRCj . (20)
Now, note that with any entanglement monotone E, the entanglement between A and BC is measuredby:
E(ρA|BC
)=∑j
qjE(ρA|B
Lj ⊗ ρB
Rj C)
=∑j
qjE(ρAB
Lj
), (21)
where in the first equality we used the property (3) of entanglement monotones. Similarly, the entangle-ment between A and B is measured by
E(ρAB
)=∑j
qjE(ρAB
Lj ⊗ ρB
Rj
)=∑j
qjE(ρAB
Lj
). (22)
We therefore obtain (4) as long as E(ρA|B
Lj
)> 0 for some j for which qj > 0. This completes the
proof.
C Monogamy of entanglement: pure vs mixed tripartite statesAs discussed in the paper, it is typically hard to check if a measure of entanglement is monogamous sincethe condition in (4) involves mixed tripartite states. On the other hand, it is significantly simpler to checkthe disentangling condition if ρABC that appears in the disentangling condition (4) is pure. We say thatE is monogamous on pure states if for any pure tripartite state ρABC that satisfies (4), E(ρAC) = 0. Inthe theorem below we shown that sometimes if E is monogamous on pure states it is also monogamouson mixed states (that is, it is monogamous according to Def. 1).
For any entanglement monotone E on the set of bipartite density matrices, SAB, we define a corre-sponding entanglement of formation measure, Ef which is defined by the following convex roof extension:
Ef(ρAB
)≡ min
n∑j=1
pjE(|ψj〉〈ψj |AB
), (23)
where the minimum is taken over all pure state decompositions of ρAB =∑nj=1 pj |ψj〉〈ψj |AB. Clearly,
E = Ef on pure bipartite states, but on mixed states they can be different, like the convex roof extendedNegativity is different from the Negativity itself. Since we assume that E is entanglement monotone itis convex so E(ρAB) ≤ Ef (ρAB) for all ρAB ∈ SAB. The corresponding entanglement of formation of
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a given entanglement monotone, is itself an entanglement monotone, and has the following remarkableproperty.
Theorem 3. Let E be an entanglement monotone, and let Ef be its corresponding entanglement offormation (23). If Ef is monogamous (according to Definition 1) on pure tripartite states in HABC , thenit is also monogamous on tripartite mixed states acting on HABC .
Proof. Let ρA|BC =∑j pj |ψj〉〈ψj |ABC be a tripartite state acting on HABC with {pj , |ψj〉ABC} being the
optimal decomposition such that
Ef (ρA|BC) =∑j
pjEf(|ψj〉A|BC
). (24)
We also assume w.l.o.g. that pj > 0. Now, suppose Ef (ρA|BC) = Ef (ρAB), and denote ρABj ≡TrC |ψj〉〈ψj |ABC . Since discarding a subsystem can only decrease the entanglement, we get∑
j
pjEf(|ψj〉A|BC
)≥∑j
pjEf(ρABj
)≥ Ef (ρAB) , (25)
where the last inequality follows from the convexity of Ef and the fact that ρAB =∑j pjρ
ABj . However,
all the inequalities above must be equalities since Ef (ρA|BC) = Ef (ρAB). In particular, we get∑j
pjEf(|ψj〉A|BC
)=∑j
pjEf(ρABj
). (26)
This in turn implies that Ef(|ψj〉A|BC
)= Ef
(ρABj
)for each j since Ef
(|ψj〉A|BC
)≥ Ef
(ρABj
)for each
j (i.e. tracing out subsystem cannot increase entanglement). Since we assume that Ef is monogamous onpure tripartite states, we conclude that for each j, Ef (ρACj ) = 0, where ρACj ≡ TrB|ψj〉〈ψj |ABC . Hence,Ef (ρAC) = 0 since ρAC =
∑j pjρ
ACj and Ef is convex.
D Entanglement of Collaboration and Monogamy of EntanglementMonogamy of entanglement is closely related to entanglement of collaboration. Given a measure of bipar-tite entanglement E, its corresponding entanglement of collaboration, EAB|Cc , is a measure of entanglementon tripartite mixed states, ρABC , given by [32]:
EAB|Cc
(ρABC
)= max
n∑j=1
pjE(ρABj
), (27)
where the maximum is taken over all tripartite LOCC protocols yielding the bipartite state ρABj withprobability pj . The following theorem demonstrates the connection between the disentangling conditionand entanglement of collaboration.
Theorem 4. Let E be an entanglement monotone on bipartite mixed states, and let ρABC be a (possiblymixed) tripartite state satisfying the disentangling condition (4). Then,
E(ρAB
)= EAB|Cc
(ρABC
). (28)
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Proof. Let {ρABj , pj} be the optimal ensemble in (8) obtained by LOCC on ρABC . Since E is a bipartiteentanglement monotone, it does not increase on average:
E(ρA|BC
)≥∑j
pjE(ρABj ) = EAB|Cc
(ρABC
). (29)
On the other hand, by definition E(ρAB) ≤ EAB|Cc
(ρABC
), so that together with (4) we get (9).
Entanglement of collaboration is different from entanglement of assistance, Ea, in which the optimizationin (27) is restricted to LOCC of the following form: Charlie (system C) performs a measurement, andcommunicates the outcome j to Alice and Bob. In [32] the following LOCC protocol was considered:Alice performs a measurement, then sending the outcome to Charlie, and then Charlie performs hismeasurement, and sends back the result to Alice and Bob. It was shown that in such a scenario it ispossible to increase the average entanglement between systems A and B to a value beyond the averageentanglement that can be achieved if only Charlie performed a measurement. Therefore, EAB|Cc can bestrictly larger than Ea, and in general, EAB|Cc ≥ Ea. However, if ρABC satisfies the disentangling conditionthen we must have EAB|Cc = Ea. Indeed, if ρABC satisfies (4) we get
Ea(ρABC) ≥ E(ρAB) = E(ρA|BC
)= EAB|Cc
(ρABC
). (30)
Therefore, one can replace EAB|Cc in (28) with Ea, which may be convenient since Ea is somewhat asimpler measure than EAB|Cc . Note however that we left EAB|Cc in (28) since E
(ρA|BC
)= E
AB|Cc
(ρABC
)implies E
(ρA|BC
)= Ea
(ρABC
)but not vice versa.
D.1 When Entanglement of Formation equals Entanglement of Assistance?An immediate consequence of Theorem 4 above is that if ρABC is a pure state that satisfies the disentan-gling condition then the entanglement of formation of ρAB must be equal to its entanglement of assistance.
Corollary 5. Let E be an entanglement monotone on bipartite states, and let ρABC be a pure tripartitestate satisfying the disentangling condition (4). Then,
E(ρAB
)= Ef (ρAB) = Ea(ρAB) , (31)
where the entanglement of formation, Ef , is defined in (23), and the entanglement of assistance, Ea, isalso defined as in (23) but with a maximum replacing the minimum.
Proof. The proof follows straightforwardly from Theorem (4) recalling that
E(ρAB
)≤ Ef (ρAB) ≤ Ea(ρAB) ≤ EAB|Cc
(ρABC
)= E
(ρA|BC
), (32)
where the first inequality follows from the fact that E is an entanglement monotone, and the last equalityfrom Theorem (4). Therefore, all the inequalities above are equalities since we assume the disentanglingcondition E
(ρA|BC
)= E
(ρAB
). This completes the proof.
In the following theorem we show that the equality between the entanglement of formation andentanglement of assistance is a property of the support space of the the bipartite state in question.
Theorem 6. Let E be a measure of bipartite entanglement, and let ρAB ∈ SAB be a bipartite statewith support subspace supp(ρAB). If Ef (ρAB) = Ea(ρAB) then for any σAB ∈ SAB with supp(σAB) ⊆supp(ρAB) we have Ef (σAB) = Ea(σAB).
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Proof. In the proof we use some of the ideas introduced in [33]. Suppose ρAB =∑j pjρ
ABj , where {pj}
are (non-zero) probabilities and ρABj ∈ SAB. Then, the condition Ef (ρAB) = Ea(ρAB) together with theconvexity (concavity) of Ef (Ea) gives∑
j
pjEf (ρABj ) ≥ Ef (ρAB) = Ea(ρAB) ≥∑j
pjEa(ρABj ) . (33)
But since for all j we also have Ef (ρABj ) ≤ Ea(ρABj ), we get that Ef (ρABj ) = Ea(ρABj ) for all j. LetF(ρAB) be the set of all density matrices in SAB that appear in a convex decomposition of ρAB. Inconvex analysis, F(ρAB) is called a face of SAB. Now, from the argument above we have that if σAB ∈F(ρAB) then Ef (σAB) = Ea(σAB). On the other hand, for any σAB ∈ SAB with the property thatsupp(σAB) ⊂ supp(ρAB) there exists a small enough ε > 0 such that ρAB − εσAB ≥ 0. Denotingby γAB ≡
(ρAB − εσAB
)/(1 − ε) we get that γAB ∈ SAB and ρAB can be expressed as the convex
combination ρAB = εσAB + (1 − ε)γAB. We therefore must have Ef (σAB) = Ea(σAB). This completesthe proof.
In the proof above we called F(ρAB) a face. A face F of SAB is a convex subset of SAB that satisfiesthe following property: if tρ1 + (1 − t)ρ2 ∈ F for some ρ1, ρ2 ∈ SAB and 0 < t < 1, then ρ1, ρ2 ∈ F . Tosee that F(ρAB) is a face of SAB, we first show that it is convex. Indeed, let σ ≡ tσ1 + (1− t)σ2 for somet ∈ [0, 1] and σ1, σ2 ∈ F(ρAB). Since σ1, σ2 ∈ F(ρAB) there exists p, q ∈ (0, 1] and γ1, γ2 ∈ SAB such that
ρAB = pσ1 + (1− p)γ1 = qσ2 + (1− q)γ2 . (34)
The first equality implies that tpρ
AB = tσ1 + t(1−p)p γ1, and the second equality gives 1−t
q ρAB = (1− t)σ2 +
(1−t)(1−q)q γ2. Therefore,(
t
p+ 1− t
q
)ρAB = σAB + t(1− p)
pγ1 + (1− t)(1− q)
qγ2 . (35)
After dividing by tp + 1−t
q , we can see that σAB appears in a convex combination of ρAB. Therefore,F(ρAB) is convex. To complete the proof that F(ρAB) is a face, note that if τ ≡ tρ1 + (1− t)ρ2 ∈ F forsome ρ1, ρ2 ∈ SAB and 0 < t < 1 then clearly ρ1, ρ2 ∈ F(ρAB).
Note that the condition supp(σAB) ⊆ supp(ρAB) is equivalent to σAB ∈ F(ρAB). The precise form ofthe support space of a bipartite state ρAB that satisfies Ef (ρAB) = Ea(ρAB) depends on the measure ofentanglement E. In the following sections we find it precisely for the case where E is the G-concurrence,and we use it to show that the G-concurrence is monogamous.
D.2 Monogamy of the G-concurrence
Any pure bipartite state, |x〉 ∈ Cd ⊗ Cd, can be written as:
|x〉 = X ⊗ I|φ+〉 where |φ+〉 =d∑i=1|i〉A|i〉B , (36)
andX is a d×d complex matrix. The relation above between a complex matrixX ∈Md(C) and a bipartitevector |x〉 ∈ Cd⊗Cd defines an isomorphism betweenMd(C) and Cd⊗Cd. Using this isomorphism, in theremaining of this section we will view interchangeably the support of a density matrix both as a subspaceofMd(C) or as a subspace of Cd ⊗ Cd, depending on the context. We will use capital letters X,Y, Z,W
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for matrices inMd(C), and use lower case letters |x〉, |y〉, |z〉, |w〉 to denote their corresponding bipartitevectors.
With these notations, the G-concurrence of |x〉, which is the geometric mean of the Schmidt coefficientsof |x〉, can be expressed as:
G(|x〉) = |det(X)|2/d , (37)
and for mixed states it is defined in terms of the convex roof extension; that is, G(ρAB) := Gf (ρAB) forany ρAB ∈ SAB. Note that the G-concurrence is homogeneous, and in particular, G(c|x〉) = |c|2G(|x〉).We start in proving the following Lemma:Lemma: Let ρAB be a bipartite density matrix acting on Cd ⊗ Cd with G(ρAB) > 0. Then, there existsa pure state decomposition of ρAB with the following properties:
ρAB =r∑j=1|wj〉〈wj | , G(|wj〉) = 0 , ∀ j = 2, ..., r, (38)
where r is the rank of ρAB and |wj〉 are sub-normalized vectors in Cd ⊗ Cd (i.e. vectors with norm nogreater than 1).
Proof. Let ρAB =∑rj=1 |xj〉〈xj | be the spectral decomposition of ρAB with |xj〉 being the sub-normalized
eigenvectors of ρAB. Clearly, G(|xj〉) > 0 for at least one j. Therefore, w.l.o.g. we assume G(|x1〉) > 0.Let K2 ⊂ supp(ρAB) be the two dimensional subspace spanned by X1 and X2. We first show that K2contains a matrix with zero determinant. Indeed, if det(X2) = 0 we are done. Otherwise, for any λ ∈ Cwe get
G(|x1〉+ λ|x2〉) = |det(X1 + λX2)|2/d
= |det(X2)|2/d∣∣∣det(X1X
−12 + λI)
∣∣∣2/d . (39)
Note that det(X1X−12 + λI) is a polynomial of degree d in λ and must have at least one complex root.
Therefore, there exists λ = λ0 6= 0 such that det(X1 + λ0X2) = 0. This completes the assertion that K2contains a matrix with zero determinant.Next, denote a ≡ 1√
1+|λ0|2and b ≡ λ0√
1+|λ0|2, so that |a|2 + |b|2 = 1, and the vectors |w2〉 ≡ a|x1〉+b|x2〉
and |y〉 ≡ b|x1〉 − a|x2〉 satisfy
|x1〉〈x1|+ |x2〉〈x2| = |y〉〈y|+ |w2〉〈w2| (40)
with G(|w2〉) = 0 (by construction, W2 is proportional to X1 +λ0X2 and therefore has zero determinant).We can therefore write
ρAB = |w2〉〈w2|+ |y〉〈y|+r∑j=3|xj〉〈xj | . (41)
Now, if G(|y〉) = 0 then we denote it as |w3〉 and pick from {|xj〉}rj=3 another state that does not havea vanishing G-concurrence. We therefore assume G(|y〉) 6= 0 and from the same arguments as above weconclude that |y〉〈y| + |x3〉〈x3| = |y〉〈y| + |w3〉〈w3| for some vectors |y〉 and |w3〉, with G(|w3〉) = 0. Byreplacing |y〉 and |x3〉 with |y〉 and |w3〉, and repeating the process we arrive at the desired decompositionof ρAB.
In the following theorem we denote by N ⊂Md(C), the nilpotent cone, consisting of all d× d nilpotentcomplex matrices (i.e. X ∈ N iff Xk = 0 for some 1 ≤ k ≤ d). While the set N is not a vector space, itcontains subspaces. For example, the set T of all strictly upper triangular matrices (i.e. upper triangular
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matrices with zeros on the diagonal) is a d(d − 1)/2-dimensional subspace in N . From Gerstenhaber’stheorem [34] it follows that the largest dimension of a subspace in N is d(d − 1)/2, and if a subspaceN0 ⊂ N has this maximal dimension, then it must be similar to T (i.e. their exists an invertible matrixS such that N0 = ST S−1).
Theorem 7. Let ρAB be a bipartite density matrix acting on Cd ⊗ Cd, and suppose it has rank r > 1.Then,
G(ρAB) = Ga(ρAB) > 0 , (42)
if and only if r ≤ 1 + d(d− 1)/2, and there exists a full rank matrix X ∈Md(C) and a subspace N0 ⊂ Nwith dim(N0) = r − 1 such that
supp(ρAB) = XK ≡ {XY∣∣ Y ∈ K} , (43)
where K ≡ {I} ⊕ N0 ≡ {cI +N∣∣ c ∈ C ; N ∈ N0}.
Remark 3. In the statement of Theorem 7 of the main text, we viewed supp(ρAB) as a subspace of Cd⊗Cd.Here, for convenience of the proof, we use the isomorphism (36) betweenMd(C) and Cd ⊗ Cd, and viewthe supp(ρAB) as a subspace of Md(C). The matrix X in (43) is related to the bipartite state |x〉ABin (13) via (36). Recall also that a nilpotent matrix has a zero trace, and is orthogonal to the identitymatrix in the Hilbert-Schmidt inner product. This is consistent with the direct sum in the definition ofK.
Proof. Suppose there exist X and N0 as above such that supp(ρAB) ⊆ XK, and let
ρAB =r∑j=1|wj〉〈wj | , (44)
where |wj〉 are the sub-normalized vectors given in (38); i.e. G(|wj〉) = 0 for j > 1. Since Wj ∈ XK,there exist constants cj ∈ C and matrices Zj ∈ N0 (corresponding to some sub-normalized vectors |zj〉)such that
Wj = cjX +XZj . (45)
For j > 1 we have
0 = G(|wj〉) = |det(cjX +XZj)|2/d = G(|x〉) |det(cjI + Zj)|2/d (46)
so that det(cjI + Zj) = 0 since G(|x〉) > 0. But since Zj is a nilpotent matrix det(cjI + Zj) = (cj)d.Hence, cj = 0 for j > 1 and we denote by c ≡ c1. Note that c 6= 0 since otherwise we will get G(ρAB) = 0.Therefore, the average G-concurrence of decomposition (44) is given by
r∑j=1
G(|wj〉) = G(|w1〉) = |det(cX +XZ1)|2/d = G(|x〉) |c|2 . (47)
Next, let
ρAB =m∑k=1|yk〉〈yk| , (48)
be another pure state decomposition of ρAB with m ≥ r and {|yk〉} some sub-normalized states. Then,there exists an m× r isometry U = (ukj), i.e. U †U = Ir, such that
|yk〉 =r∑j=1
ukj |wj〉 . (49)
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Using the form (45) with c1 ≡ c and c2 = c3 = ... = cr = 0, we get
Yk =r∑j=1
ukjWj = X
r∑j=1
ukj (cδ1jI + Zj)
≡ X (uk1cI +Nk) , (50)
where Nk ≡∑rj=1 ukjZj ∈ N0 are nilpotent matrices. Consequently,
G(|yk〉) =∣∣det
(X (uk1cI +Nk)
)∣∣2/d = |uk1|2|c|2G(|x〉) , (51)
where we used the fact that det(uk1cI +Nk) = (uk1c)d since Nk is nilpotent. Note that since the matrixU is an isometry,
∑mk=1 |uk1|2 = 1. Therefore,
m∑k=1
G(|yk〉) = |c|2G (|x〉) =r∑j=1
G(|wj〉) , (52)
where the last equality follows from (47). Therefore, all pure states decomposition of ρAB have the sameaverage G-concurrence so that G(ρAB) = Ga(ρAB).To prove the converse, suppose G(ρAB) = Ga(ρAB) > 0, and consider the decomposition (38) of
ρAB as in the Lemma above. Consider the unnormalized state σAB ≡ ρAB − |w1〉〈w1|. The stateσAB =
∑rj=2 |wj〉〈wj | has zero G concurrence since G(|wj〉) = 0 for all j = 2, ..., r. Consider another
decomposition of σAB =∑rj=2 |yj〉〈yj |. Since G(ρAB) = Ga(ρAB) we must have that G(|yj〉) = 0 for all
j = 2, ..., r. Otherwise, the decomposition ρAB = |w1〉〈w1| +∑rk=2 |yk〉〈yk| will have a higher average
G-concurrence than the decomposition in (38). But since the {|yj〉} decomposition was arbitrary, weconclude that all the states in the subspace W ≡ span{|w2〉, ..., |wr〉} have zero G-concurrence.We now denote W1 ≡ X and for j = 2, ..., r we set Nj ≡ X−1Wj . Note that with these nota-
tions, supp(ρAB) = span{X,XN2, XN3, ..., XNr}. Since X is invertible, and det(W ) = 0 for any ma-trix W in the span of W2, ...,Wr, we also have det(N) = 0 for any N in the span of N2, ..., Nr. LetN ∈ span{N2, ..., Nr} be a fixed matrix, and consider the two dimensional subspace W2 ≡ {X,XN} ⊂supp(ρAB). We first show that W2 does not contain a matrix with zero determinant that is linearly inde-pendent of W ≡ XN . Indeed, if there exists a normalized matrix Z ∈ W such that det(Z) = det(W ) = 0with W,Z being linearly independent, then W2 = span{W,Z}, and the rank 2 density matrix
σAB = |w〉〈w|+ |z〉〈z| (53)
must have zero G-concurrence (recall that G(|w〉) = G(|z〉) = 0). On the other hand, since |x〉 ∈ W2 =supp(σAB), the density matrix σAB must have a pure state decomposition containing |x〉. Since G(|x〉) > 0this decomposition does not have a zero average G-concurrence. Therefore, 0 = G(σAB) < Ga(σAB).However, from Theorem 6 any density matrix σAB with a support supp(σAB) = W2 ⊂ supp(ρAB) hasthe same average G-concurrence for all pure state decompositions. We therefore get a contradictionwith Theorem 6, and thereby prove the assertion that W2 does not contain another matrix with zerodeterminant that is linearly independent of W .Since W = XN is the only matrix in W2 with zero determinant (up to multiplication by a constant),
we must have that for any λ 6= 0
0 6= det(X + λW ) = det(X−1) det(I + λX−1W ) = λd det(X−1) det( 1λI +N
). (54)
Setting t ≡ 1λ we conclude that the polynomial
f(t) ≡ det (tI +N) (55)
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is never zero for t 6= 0. On the other hand, f(t) is a polynomial of degree d and the coefficient of td is one(note that it is the characteristic polynomial of −N). But since t = 0 is the only root of f(t), we musthave f(t) = td; that is,
det (tI +N) = tn, ∀ t ∈ C . (56)
Therefore, N must be a nilpotent matrix. Since N was arbitrary, it follows that the subspace N0 ≡span{N2, ..., Nr} is a subspace of nilpotent matrices. This completes the proof.
The G-concurrence is defined in (37) on pure bipartite states with the same local dimension. For a purebipartite state |ψ〉AB ∈ HA ⊗HB with dim(HA) < dim(HB) it is defined by:
G(|ψ〉AB) =(det(ρA)
)1/dA, (57)
where ρA ≡ TrB(|ψ〉〈ψ|AB
)is the reduced density matrix, and dA ≡ dim(HA). With this extended
definition, we have the following result:
Corollary 8. Let |ψABC〉 ∈ Cd ⊗ Cd ⊗ Cn be a pure tripartite state, with bipartite marginal ρAB. If
G(ψA|BC) = G(ρAB) > 0 , (58)
then |ψ〉ABC = |χ〉AB|ϕ〉C for some bipartite state |χ〉AB ∈ Cd ⊗ Cd and a vector |ϕ〉C ∈ Cn.
Proof. Since |ψ〉ABC satisfies the disentangling condition, we get from Corollary 5 that G(ρAB) =Ga(ρAB) > 0. Therefore, from the theorem above there exists a pure state decomposition of the marginalstate ρAB consisting of sub-normalized bipartite states {|wj〉} as in (38), with the form
|w1〉 = (cX +XN1)⊗ IB|φ+〉AB ,
|wj〉 = XNj ⊗ IB|φ+〉AB , ∀ j = 2, ..., r , (59)
where X is a full rank matrix, Nj ∈ N0, and c ∈ C. Since all decompositions have the same averageG-concurrence, we have
G(ρAB) =r∑j=1
G(|wj〉) = | det(X)|2/dc . (60)
On the other hand, using the property that TrB|φ+〉〈φ+|AB = IA, we get from (59) that the marginal ofρAB =
∑rj=1 |wj〉〈wj |AB is given by:
ρA = X(cI +N1)(cI +N †1)X† +Xr∑j=2
NjN†jX† . (61)
Therefore,
G(|ψA|BC〉) =(det(ρA)
)1/d= | det(X)|2/d
det
|cI +N1|2 +r∑j=2|Nj |2
1/d
, (62)
where we used the notation |A| ≡√AA† for any d × d matrix A. Hence, the condition G(|ψA|BC〉) =
G(ρAB) = | det(X)|2/d|c|2 gives
|c|2 =
det
|cI +N1|2 +r∑j=2|Nj |2
1/d
. (63)
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Since c 6= 0 the matrix cI +N1 is invertible. Denoting
A ≡ |cI +N1| and B ≡ A−1
r∑j=2|Nj |2
A−1 , (64)
we get that (63) can be expressed as
|c|2 = [det (A(I +B)A)]1/d =[det
(A2)]1/d
[det(I +B)]1/d . (65)
But since N1 is nilpotent, we have[det
(A2)]1/d
= [det (|cI +N1|)]2/d = |det(cI +N1)|2/d = |c|2 . (66)
We therefore conclude thatdet(I +B) = 1 . (67)
However, since B ≥ 0, Eq. (67) can hold only if B = 0. This in turn is possible only if∑rj=2 |Nj |2 = 0;
i.e. Nj = 0 for all j = 2, ..., d. We therefore conclude that ρAB is a pure state which implies that|ψ〉ABC = |w1〉AB|ϕ〉C , where |ϕ〉C is some pure state.
Note that by combining the above corollary with Theorem 3 we get that the G-concurrence is fullymonogamous (even for mixed tripartite states). This is to our knowledge the fist example of a monogamousmeasure of entanglement that is highly non-faithful. We conclude now with the 2-dimensional version ofTheorem 7. Recall that the concurrence is the two dimensional G-concurrence.
Corollary 9. Let ρAB be an entangled two qubit state with rank r > 1, and let E be any injective (up tolocal unitaries) measure of pure two qubit entanglement. Then,
1. If Ef (ρAB) = Ea(ρAB) then r = 2, and
ρAB = TrC |W 〉〈W |ABC
is the 2-qubit marginal of the W-class state
|W 〉 = λ1|100〉+ λ2|010〉+ λ3|001〉+ λ4|000〉 , (68)
where λj ∈ C and∑4j=1 |λj |2 = 1.
2. Conversely, if ρAB is a marginal of a state in the W-class then Cf (ρAB) = Ca(ρAB), where C is theconcurrence.
Proof. Part 1: In the two qubit case, we can also write E(|ψ〉AB) = g(C(|ψ〉AB)
), where C(|ψ〉AB) is the
concurrence measure of entanglement, which is itself an injective measure. Moreover, since E is injectiveso is the function g. Now, in [35], Wootters showed that there always exists a pure state decompositionof ρAB with each element in the decomposition being equal to Cf (ρAB). Denoting by {pj , |ψj〉AB} thisdecomposition, we have that
Ef(ρAB
)≤∑j
pjg(C(|ψj〉AB)
)= g
(Cf (ρAB)
). (69)
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Similarly, there exists a pure state decomposition {qk, |φk〉AB} of ρAB such that for each k, C(|φk〉AB
)=
Ca(ρAB
)[36]. Hence,
Ea(ρAB
)≥∑k
qkg(C(|φk〉AB)
)= g
(Ca(ρAB)
). (70)
Therefore, the equality Ef (ρAB) = Ea(ρAB) implies that Cf (ρAB) = Ca(ρAB). so that we can ap-ply Theorem 7. There is only one 1-dimensional subspace of N (up to similarity), given by N0 =
span{S
(0 10 0
)S−1
}, where S is a 2× 2 invertible matrix. Now, from (13) together with the decompo-
sition (38) we conclude that ρAB can be expressed as:
ρAB = |x〉〈x|AB + |y〉〈y|AB , (71)
where |x〉AB = X ⊗ IB|φ+〉 for some invertible matrix X, and
|y〉 ≡ I ⊗ S(
0 10 0
)S−1∣∣x〉AB . (72)
Therefore, ρAB is a marginal of the tripartite state
|ψ〉ABC = |x〉AB|0〉C + |y〉AB|1〉C . (73)
Multiplying the above 3-qubit state by the SLOCC element STX−1⊗S−1⊗ IC , and using the symmetryST ⊗ S−1|φ+〉 = |φ+〉 for any invertible matrix S, gives
STX−1 ⊗ S−1 ⊗ IC |ψ〉ABC =(ST ⊗ S−1|φ+〉AB
)|0〉C +
(ST ⊗
(0 10 0
)S|φ+〉AB
)|1〉C
= |φ+〉AB|0〉C +(IA ⊗
(0 10 0
)|φ+〉AB
)|1〉C
= |000〉+ |110〉+ |101〉 . (74)
note that the state above is the W state after a local flip of the first qubit. Therefore, |ψ〉ABC is in theW class.Proof of Part 2: We use for this part Wootters’ formula for the concurrence. Let R =
√ρ1/2ρρ1/2
be Wootters matrix [35], with ρ ≡ ρAB and ρ ≡ σy ⊗ σyρ∗σy ⊗ σy. Wootters showed in [35] that forentangled states Cf (ρAB) = λ1 − λ2 − λ3 − λ4, where λ1, ..., λ4 are the eigenvalues of R in decreasingorder. Furthermore, in [37, 36] it was shown that Ca(ρAB) = Tr[R] = λ1 + λ2 + λ3 + λ4. Therefore,Cf (ρAB) = Ca(ρAB) if and only if R is a rank one matrix. A straightforward calculation shows that forthe bipartite marginals, of the W-class states (68), Tr[R2] = (Tr[R])2 so that the rank of R is one (recallthat R is positive semidefinite). This completes the proof.
References[1] H. He and G. Vidal. Disentangling theorem and monogamy for entanglement negativity. Phys. Rev.
A, 91:012339, 2015. doi:10.1103/PhysRevA.91.012339.
[2] C. Eltschka and J. Siewert. Monogamy equalities for qubit entanglement from Lorentz invariance.Phys. Rev. Lett., 114:140402, 2015. doi:10.1103/PhysRevLett.114.140402.
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[3] K. M. R. Audenaert. On a block matrix inequality quantifying the monogamy of the negativity ofentanglement. Lin. Multilin. Alg., 63(12):2526-2536, 2015. doi/full/10.1080/03081087.2015.1024193.
[4] Lancien et al. Should entanglement measures be monogamous or faithful? Phys. Rev. Lett.,117:060501, 2016. doi:10.1103/PhysRevLett.117.060501.
[5] S. Cheng and M. J. W. Hall. Anisotropic invariance and the distribution of quantum correlations.Phys. Rev. Lett., 118:010401, 2017. doi:10.1103/PhysRevLett.118.010401.
[6] G. W. Allen and D. A. Meyer. Polynomial monogamy relations for entanglement negativity. Phys.Rev. Lett., 118: 080402, 2017. doi:10.1103/PhysRevLett.118.080402.
[7] S. Hill and W. K. Wootters. Entanglement of a pair of quantum bits. Phys. Rev. Lett., 78:5022, 1997.doi:10.1103/PhysRevLett.78.5022.
[8] M. Koashi and A. Winter. Monogamy of quantum entanglement and other correlations. Phys. Rev.A, 69:022309, 2004. doi:10.1103/PhysRevA.69.022309.
[9] V. Coffman, J. Kundu, and W. K. Wootters. Distributed entanglement. Phys. Rev. A, 61:052306,2000. doi:10.1103/PhysRevA.61.052306.
[10] X. N. Zhu and S. M. Fei, Phys. Rev. A 90, 024304 (2014).
[11] T. J. Osborne and F. Verstraete. General mnogamy inequality for bipartite qubit entanglement. Phys.Rev. Lett., 96:220503, 2006. doi:10.1103/PhysRevLett.96.220503.
[12] Y.-C. Ou. Violation of monogamy inequality for higher dimensional objects. Phys. Rev. A, 75:034305,2007. doi:10.1103/PhysRevA.75.034305.
[13] X.-J. Ren and W. Jiang. Entanglement monogamy inequality in a 2 ⊗ 2 ⊗ 4 system. Phys. Rev. A,81:024305, 2010. doi:10.1103/PhysRevA.81.024305.
[14] J. S. Kim and B. C. Sanders. Generalized W-class state and its monogamy relation. J. Phys. A,41:495301, 2008. doi:10.1088/1751-8113/41/49/495301.
[15] Y.-C. Ou and H. Fan, Monogamy inequality in terms of negativity for three-qubit states. Phys. Rev.A, 75:062308, 2007. doi:10.1103/PhysRevA.75.062308.
[16] Y. Luo and Y. Li. Monogamy of αth power entanglement measurement in qubit systems. Ann. Phys.,362:511-520, 2015. doi:10.1016/j.aop.2015.08.022.
[17] J. S. Kim, A. Das, and B. C. Sanders. Entanglement monogamy of multipartite higher-dimensional quantum systems using convex-roof extended negativity. Phys. Rev. A, 79:012329, 2009.doi:10.1103/PhysRevA.79.012329.
[18] J. H. Choi and J. S. Kim. Negativity and strong monogamy of multiparty quantum entanglementbeyond qubits. Phys. Rev. A, 92:042307, 2015. doi:10.1103/PhysRevA.92.042307.
[19] A. Kumar. Conditions for monogamy of quantum correlations in multipartite systems. Phys. Lett. A,380:3044-3050, 2016. doi:10.1016/j.physleta.2016.07.032.
[20] Y.-K. Bai, Y.-F. Xu, and Z. D. Wang. General monogamy relation for the entanglement of formationin multiqubit systems. Phys. Rev. Lett., 113:100503, 2014. doi:10.1103/PhysRevLett.113.100503.
[21] T. R. de Oliveira, M. F. Cornelio, and F. F. Fanchini. Monogamy of entanglement of formation. Phys.Rev. A, 89:034303, 2014. doi:10.1103/PhysRevA.89.034303.
[22] J. S. Kim. Tsallis entropy and general polygamy of multiparty quantum entanglement in arbitrarydimensions. Phys. Rev. A, 94:062338, 2016. doi:10.1103/PhysRevA.94.062338.
[23] Y. Luo, T. Tian, L.-H. Shao, and Y. Li. General monogamy of Tsallis q-entropy entanglement inmultiqubit systems. Phys. Rev. A, 93: 062340, 2016. doi:10.1103/PhysRevA.93.062340.
Accepted in Quantum 2018-08-05, click title to verify 13
[24] J. S. Kim and B. C. Sanders. Monogamy of multi-qubit entanglement using Rényi entropy. J. Phys.A, 43:445305, 2010. doi:10.1088/1751-8113/43/44/445305.
[25] M. F. Cornelio and M. C. de Oliveira. Strong superadditivity and monogamy of the Rényi measureof entanglement. Phys. Rev. A, 81:032332, 2010. doi:10.1103/PhysRevA.81.032332.
[26] Song et al. General monogamy relation of multiqubit systems in terms of squared Rényi-α entangle-ment. Phys. Rev. A, 93:022306, 2016. doi:10.1103/PhysRevE.93.022306.
[27] Y. Guo, J. Hou, and Y. Wang. Concurrence for infinite-dimensional quantum systems. Quant. Inf.Process., 12:2641-2653, 2013. doi:10.1007/s11128-013-0552-6.
[28] Y. Guo and J. Hou. Entanglement detection beyond the CCNR criterion for infinite-dimensions. Chin.Sci. Bull., 58(11):1250-1255, 2013. doi:10.1007/s11434-013-5738-x.
[29] M. J. Donald and M. Horodecki. Continuity of relative entropy of entanglement. Phys. Lett. A,264:257, 1999. doi:10.1016/S0375-9601(99)00813-0.
[30] The continuty of the convex roof extended entanglement measure can be checked according to Propo-sition 2 in [27], the continuty of partial trace and partial transpose is proved in [28], the continuty ofthe realtive entropy entanglement is proved in [29].
[31] P. Hayden, R. Jozsa, D. Petz, and A. Winter. Structure of states which satisfy strong subadditivityof quantum entropy with equality. Commun. Math. Phys., 246(2):359-374, 2004. doi:10.1007/s00220-004-1049-z.
[32] G. Gour and R. W. Spekkens. Entanglement of assistance is not a bipartite measure nor a tripartitemonotone. Phys. Rev. A, 73:062331, 2006. doi:10.1103/PhysRevA.73.062331.
[33] A. Uhlmann. Roofs and Convexity. Entropy, 12:1799-1832, 2010. doi:10.3390/e12071799.
[34] M. Gerstenhaber. On nilalgebras and linear varieties of nilpotent matrices (I). Amer. J. Math., 80:614-622, 1958. doi:10.2307/2372773.
[35] W. K. Wootters. Entanglement of formation of an arbitrary state of two qubits. Phys. Rev. Lett.,80:2245, 1998. doi:10.1103/PhysRevLett.80.2245.
[36] G. Gour, D. A. Meyer, and B. C. Sanders. Deterministic entanglement of assistance and monogamyconstraints. Phys. Rev. A, 72:042329, 2005. doi:10.1103/PhysRevA.72.042329.
[37] T. Laustsen, F. Verstraete, and S. J. van Enk. Local vs. joint measurements for the entanglement ofassistance. Quant. Inf. Comput., 3:64, 2003. arXiv:0206192.
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