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General Monogamy Inequality for Bipartite Qubit Entanglement

Tobias J. Osborne 1, * and Frank Verstraete 2,1 Department of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, United Kingdom

2 Institute for Quantum Information, California Institute of Technology, Pasadena, California 91125, USA(Received 19 April 2005; published 7 June 2006)

We consider multipartite states of qubits and prove that their bipartite quantum entanglement, as

quantied by the concurrence, satises a monogamy inequality conjectured by Coffman, Kundu, andWootters. We relate this monogamy inequality to the concept of frustration of correlations in quantum spinsystems.

DOI: 10.1103/PhysRevLett.96.220503 PACS numbers: 03.65.Ud

Quantum mechanics, unlike classical mechanics, allowsthe existence of pure states of composite systems for whichit is not possible to assign a denite state to two or moresubsystems. States with this property are known as en-tangled states. Entangled states have a number of remark-able features, a fact which has inspired an enormous

literature in the years since their discovery. These proper-ties have led to suggestions that the propensity of multi-partite quantum systems to enter nonlocal superpositionstates might be the dening characteristic of quantummechanics [1,2].

It is becoming clear that entanglement is a physicalresource. The exploration of this idea is a central goal inthe burgeoning eld of quantum information theory. As aconsequence, the study of the mathematics underlyingentanglement has been a very active area and has led tomany operational and information-theoretic insights. Asfor now, only the pure-state case of entanglement sharedbetween two parties is thoroughly understood and quanti-ed; progress on the multipartite setting has been muchslower.

A key property, which may be as fundamental as the no-cloning theorem, has been discovered recently in the con-text of multipartite entanglement: entanglement is mo-nogamous [3,4]. More precisely, there is an inevitabletrade-off between the amount of quantum entanglementthat two qubits A and B, in Alices and Bobs possession,respectively, can share and the quantum correlation thatAlices same qubit A can share with Charlie, a third party,C [3]. In the context of quantum cryptography, such amonogamy property is of fundamental importance becauseit quanties how much information an eavesdropper couldpotentially obtain about the secret key to be extracted. Theconstraints on shareability of entanglement lie also at theheart of the success of many information-theoretic proto-cols, such as entanglement distillation.

In the context of condensed-matter physics, the monog-amy property gives rise to the frustration effects observedin, e.g., Heisenberg antiferromagnets. Indeed, the perfectground state for an antiferromagnet would consist of sin-glets between all interacting spins. But, as a particle can

only share one unit of entanglement with all its neighbors(this immediately follows from the dimension of its localHilbert space), it will try to spread its entanglement in anoptimal way with all its neighbors leading to a stronglycorrelated ground state. The tools developed in this Letterwill allow us to turn such qualitative statements into quan-

titative ones.The problem of fully quantifying the constraints on

distributed entanglement should be seen as analogous tothe N representability problem for fermions [5]. This isbecause, just as is the case for fermions, if the constraintson distributed entanglement were known explicitly thenthis would render trivial [6] the task of computing theground-state energy of condensed-matter systems. Theresults of this Letter represent the rst step towards thefull quantication of the constraints on distributedentanglement.

The main result of this Letter is a proof of the long-standing conjecture of Coffman, Kundu, and Wootters(CKW) [3] that the distribution of bipartite quantum en-tanglement, as measured by the tangle , amongst n qubitssatises a tight inequality:

A1 A2 A1 A3 A1 An A1 A2 A3 ... An ;(1)

where A1 A2 A3 ... An denotes the bipartite quantum entan-glement measured by the tangle across the bipartition A1: A2 A3 . . . An . This CKW inequality has been establishedin the case of three qubits. However, the case of n qubitswas still open [7]. In this Letter we establish Eq. (1) forarbitrary numbers n of qubits.

The outline of this Letter is as follows. We begin byintroducing and dening the quantum correlation measureswe study throughout. Following this, we reduce the CKWinequality to a statement pertaining to quantum correlationmeasures for a pure tripartite system consisting of twoqubits and a four-level quantum system. Such a systemis, up to local unitaries, completely determined by its two-qubit reduced density operator. The proof will then becompleted by showing that the one-way correlation mea-

PRL 96, 220503 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending

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sure [9,10] of a mixed state of two qubits is always larger orequal than its tangle.

In our proof we have utilized a number of techniques.We derive a computable formula for the linear Holevo quantity for all qubit maps and also for the one-waycorrelation measure [10] for all two-qubit states.

To quantify mixed-state bipartite quantum correlationswe study two measures. We also study one channel ca-

pacity measure. The rst measure we consider is the tangle AB , which is the square of the concurrence [3,1114], AB C2 AB . The tangle measure pertains to bipar-tite quantum states AB of a qubit A and a D-level quantum

sytem B. To dene the tangle we introduce the followingentropic measure, the linear entropy S2, for single-qubitstates [15]:

S2 4

2 1 tr 2 4 det :The linear entropy S2 is concave and unitarily invariant.

The tangle is now dened for any state AB of the 2D system via the roof construction (for operational moti-vations and further discussion of this construction see[11,16])

AB 4

inf fp x; xgXx p xS2 trB x ; (2)

where the inmum runs over all pure-state decompositions

fp x; xg of AB, AB Pxp x x.The second correlation measure we need is closelyrelated to a one-way correlation measure [9,10]. For anymixed state AB of a 2 D bipartite quantum system wedene

I 2 AB 4

max

fM x

g

S2 A

Xxp xS2 x ; (3)

where the maximum runs over all positive operator-valuedmeasures (POVMs) fM xg on Bobs system, px tr I B M x AB is the probability of outcome x, and x trB I B M x AB =px is the posterior state in Alices subsystem.

The third measure we will need, the linear Holevo capacity, is a capacity measure for qubit channels . Thismeasure is related to the one-shot Holevo quantity and isdened by

2 ; maxfp x; xgS2 Xx p xS2 x ; (4)

where is a qubit ensemble, is an arbitrary qubitchannel (a trace-preserving completely-positive map),and the maximum runs over all pure-state decompositions

fp x; xg of , Pxp x x.We now turn to the CKW inequality. Our strategy forproving Eq. (1) will be to prove it for states ABC of twoqubits AB, and a 2n 2-dimensional qudit C. The next stepwe use is to proceed via induction by successively parti-tioning the last qudit C into two subsystems, a qubit C1,and a 2n 3-dimensional qudit C2 , and establishing Eq. (1)for the (typically mixed) state AC1C2 . Thus, the formula we

will try to prove is the following

A BC AB AC ; (5)for arbitrary states of a 2 2 2n 2 system ABC.

We begin by trying to prove Eq. (5) for pure states. Inthis case we can use the local-unitary invariance of ACto rotate the basis of subsystem C into the local Schmidtbasis

ju j

i, j

1; . . . ; 4, given by the eigenvectors of C . In

this way we can regard the 2n 2-dimensional qudit C as aneffective four-dimensional qudit. Therefore, it is sufcientto establish Eq. (5) for a 2 2 4 system ABC.

Supposing we have proved the inequality Eq. (5) forpure states we can extend Eq. (5) to mixed states .Consider the minimizing decomposition fp x;j xig for

A BC , and apply the inequality Eq. (5) to each term,

A BC Xx p x x A BC ;Xx p x x AB x AC ; AB AC ;

(6)

where x A BC j xihxj, and we have used the convexityof to arrive at the third line.

Now all that is required to establish the inequalityEq. (5) for an arbitrary system of n qubits is to successivelyapply Eq. (5) to partitions of C according to the inductiverecipe outlined above. We illustrate this procedure for purestates of four qubits ABC1C2. Let C C1C2 be a com-bined pair of qubits and apply Eq. (5),

A BC AB AC ; AB AC1 AC2 ;

(7)

where we have applied the mixed-state version of theinequality Eq. (5) in the second line. It is straightforwardto generalize this procedure to n qubits.

We have now reduced the CKW inequality to an inequal-ity for the tangle for pure states of a tripartite system ABCof two qubits Aand B and a four-level system C. In the caseof pure states, AB and AC contain the same information(up to local unitaries); all possible POVM measurents atBobs side induce all possible pure-state decompositions of

AC, and therefore the following monogamy relation holds(see also Koashi and Winter [9]):

S2 A A BC I 2 AB AC : (8)By comparing Eqs. (5) and (8) we see that in order to

establish Eq. (5) it is sufcient to establish the inequality

AB I 2 AB ; (9)for all two-qubit states AB. As a rst step toward provingthis inequality, we will now derive a computable formulafor I 2 AB .

Any bipartite quantum state AB may be written as

AB I B jr B0Bihr B0Bj ; (10)


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where jr B0Bi is the symmetric two-qubit purication of thereduced density operator B on an auxiliary qubit systemB0 and is a qubit channel from B0 to A. It can nowreadily be seen that the one-way correlation measureI 2 AB is equal to the one-shot channel capacity measure

2 B; : all possible POVM measurements induce allconvex decompositions of B0.

The action of a qubit channel on a single-qubit state

I r2 , where is the vector of Pauli operators, may bewritten as

I Lr l

2 ; (11)

where L is a 3 3 real matrix and l is a three-dimensionalvector. In this Pauli basis, the possible decompositions of

B into pure states are represented by all possible sets of probabilities fp jgand unit vectors fr jgfor which Pj p j r j r B, where I r B2 B. The linear entropy S2, written interms of the Bloch vector r of a two-qubit state, is given byS2 I

r

2 1 j r j2 . In this way we see thatQ r S2

I r2

1 Lr l T Lr l ; (12)

which is a quadratic form in the Bloch vector r .Substituting r j r B x j , one can easily check that the

calculation of 2 B; reduces to determining fp j ; x j gsubject to the conditions Pj p j x j 0 and k r B x j k 1maximizing

maxfp j ;x j gXj p j x

T j L

T Lx j : (13)

Let us, without loss of generality, assume that L T L isdiagonal with diagonal elements x y z. The con-straints k r B x j k 1 lead to the identities

x xj2 1 k r B k2 2r T Bx j x

yj

2 x zj2:

Substituting this into (13), we get

2 B; x 1 k r B k2 maxfp j ;x j gXj p j y x xyj

2

z x xzj

2 :

This expression is obviously maximized by choosing x zj x yj

0 for all j ; the x xj then have to correspond to the roots

of the equation k r B x j k 1. There are exactly two suchroots, showing that the maximum x 1 k r B k2 can bereached by an ensemble of two states.As S2 B 1 k r B k2 , we therefore obtain the fol-lowing computable expression for the linear Holevo

capacity for qubit channels:

2 B; max L T L S2 B : (14)From this expression we also obtain an expression forI 2 AB via the correspondence Eq. (10).

Now that we have a formula for I 2 AB , we want toprove that it is always larger than or equal to AB . Firstof all, we note that a local ltering operation of the form

0 AB I B AB I B ytr I ByB AB leaves

L invariant and transforms

S2 0B det B 2

tr I ByB AB 2S2 B :

It happens that I 2 transforms in exactly the same way asthe tangle does [17] (recalling that the tangle is the squareof the concurrence). As there always exists a lteringoperation for which 0B / I 2, we can assume, withoutloss of generality, that S2 B 1.So let us consider AB with Tr A AB 12 I . As

max LT L 2max L where max L is the largest sin-gular value of L , we want to prove that max L C AB ,where C AB denotes the concurrence of AB. It has been

proven in [18] that any mixed state of two qubits withassociated 3 3 matrix L jk Tr j k can be writ-ten as a convex decomposition of rank-2 density operatorsall having the same L jk . As the concurrence is convex, themaximum concurrence for a given L will certainly beachieved for a rank-2 density operator 2. Next noticethat any rank-2 matrix 2 can, up to local unitaries, bewritten as

2 pj00ih00j 1 p j ihj:Given the concurrence of C j ihj C, then obviouslyC 2 1 p C. Let us now consider max L ; this isthe largest singular value of the sum of two matrices, onehaving singular values p; 0; 0 and the other one having1 p C;C; 1 (corresponding to j00iand j i). Up to leftand right multiplication by unitaries, L is then given by

L 1 pC 0 00 C 00 0 1

0@

1A

pcos

0sin

0@

1A

u T ;

where u is a unit vector. Obviously, the (2,2) element of this matrix is 1 p C, which is certainly a lower boundfor max L . This therefore implies that I 2 AB ABfor all two-qubit states AB, hence proving the CKWinequality Eq. (1).

The CKW inequality is likely to be useful in a number of contexts, allowing simplied proofs of no-broadcastingbounds and constraints for qubit multitap channel capaci-ties. Perhaps the most interesting open problem at thisstage is to generalize Eq. (1) to systems other than qubitsand to the case where A1 consists of more than one qubit. Inboth these cases the available generalizations of the tanglemeasure for quantum entanglement provably cannot yieldentanglement sharing inequalities. It is an interesting openproblem to work out an easily computable measure of quantum entanglement which will yield concrete usefulbounds on the distribution of private correlations.

The CKW inequality may be immediately applied tostudy the entanglement for a wide class of complex quan-


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tum systems. Let us, for example, consider a translation-invariant state of a quantum spin 1=2 system on a latticewith coordination number d. The CKW inequality impliesthat the concurrence C of the reduced density operator

of two nearest neighbors satises C 1hSn i2 = dp , where hSn i is the magnetization in the directionn . Hence the CKW inequality provides a quantitative toolof assessing how far the mean-eld energy will be from the

exact one. Let us, e.g., consider the HeisenbergHamiltonian. As the overlap of a state with a singlet isbounded above by 1 C =2 [18] and as the mean-eldenergy per bond is given by 1=2, the gap between mean-eld theory and the exact ground-state density [19] isbounded above by 1 hSn i2 = 2 dp . The classical result[20] that mean-eld theory becomes exact, i.e., is sepa-rable, when d ! 1 is a limiting case of this inequality.In a similar context, several investigations of the con-straints on distributed entanglement have been carried outrecently. We mention, for example, [21,22]. The validity of some results of these papers were conditioned on the truthof the CKW inequality. As a consequence of this Letter it isnow possible to regard these results as true.

In this Letter we have proved that the distribution of bipartite quantum entanglement is subject to certain share-ability laws. It is tempting to think that such shareabilityconstraints might hold for other quantum correlation quan-tities, such as the Bell violation of a bipartite Bell inequal-ity. This is in fact the case; it has recently been discovered[23] that bipartite Bell violations cannot be distributedarbitrarily.

Finally, it is worth highlighting some classes of quantumstates which saturate the CKW inequality. The classicexample of a quantum state saturating the CKW inequality

is the W state

jW i 1

np j0 . . . 0 1i j0 . . . 1 0i j 1 . . . 0 0i :The W state has the property that the entanglement of anytwo spins is equal, but the entanglement of the spin A1 isnot maximal. One might ask if there are any states whichsaturate the CKW inequality which have the property thatthe spin A1 at the focus is maximally entangled with therest. In this way we could regard such a state as sharing outa full unit of entanglement with its neighbors. Such a statedoes indeed exist and is given by

j i 1

2p j0ij0 . . . 0i 1

2p j1ijW i:In conclusion, we proved the Coffman-Kundu-Wootters

monogamy inequality which quanties the frustration of entanglement between different parties. The unique featureof this inequality is that it is valid for any multipartite stateof qubits, irrespective of the underlying symmetries, whichmakes it much more general than de Finetti type bounds[24]. We also discussed the relevance of the monogamous

nature of entanglement in quantum cryptography and infrustrated quantum spin systems.

We would particularly like to thank Michael Nielsen forintroducing T. J. O. to this problem and for providing atremendous amount of encouragement and suggestions.Also, we are deeply indebted to Andreas Winter for ex-tensive helpful suggestions and comments. We would alsolike to thank Bill Wootters for many encouraging discus-

sions. Finally, T. J. O. is grateful to the EU for support forthis research under the IST project RESQ and also to theUK EPSRC through the grant QIPIRC. F. V. is grateful tothe Gordon and Betty Moore Foundation.

*Electronic address: [email protected] Electronic address: [email protected]

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[6] F. Verstraete and J. I. Cirac, conn-mat/0505140.[7] A CKW-like inequality has been established for n qubits

for an entanglementlike quantity which is a lower boundfor tangle [8]. That result does not imply the CKWinequality.

[8] C.-s. Yu and H.-s. Song, Phys. Rev. A 71 , 042331 (2005).[9] M. Koashi and A. Winter, Phys. Rev. A 69, 022309 (2004).

[10] L. Henderson and V. Vedral, J. Phys. A 34 , 6899 (2001).[11] C. H. Bennett, D.P. DiVincenzo, J. A. Smolin, and W. K.

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literature. While the linear entropy is equivalent to theRenyi 2-entropy, R log tr 2 , the reader shouldnote that they are not the same quantity.

[16] A. Uhlmann, Open Syst. Inf. Dyn. 5, 209 (1998).[17] F. Verstraete, J. Dehaene, and B. De Moor, Phys. Rev. A

64 , 010101(R) (2001).[18] F. Verstraete and H. Verschelde, Phys. Rev. A 66 , 022307

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[19] M.R. Dowling, A. C. Doherty, and S. D. Bartlett, Phys.Rev. A 70 , 062113 (2004).[20] R. F. Werner, Lett. Math. Phys. 17 , 359 (1989).[21] T. Roscilde, P. Verrucchi, A. Fubini, S. Haas, and

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