VOLUME 78, NUMBER 26 P H Y S I C A L R E V I E W L E T T E R S 30 JUNE 1997

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Entanglement of a Pair of Quantum Bits

Scott Hill and William K. WoottersDepartment of Physics, Williams College, Williamstown, Massachusetts 01267

(Received 3 March 1997)

The “entanglement of formation” of a mixed stater of a bipartite quantum system can be definedas the minimum number of singlets needed to create an ensemble of pure states that representsr. Wefind an exact formula for the entanglement of formation for all mixed states of two qubits havingmore than two nonzero eigenvalues, and we report evidence suggesting that the formula is valid fstates of this system. [S0031-9007(97)03443-1]

PACS numbers: 89.70.+c, 03.65.Bz

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Entanglement is the potential of quantum states to ehibit correlations that cannot be accounted for classicaFor decades, entanglement has been the focus of mwork in the foundations of quantum mechanics, being asociated particularly with quantum nonseparability and tviolation of Bell’s inequalities [1]. In recent years, however, it has begun to be viewed also as a potentially useresource. The predicted capabilities of a quantum coputer, for example, rely crucially on entanglement [2and a proposed quantum cryptographic scheme convshared entanglement into a shared secret key [3].both theoretical and potentially practical reasons, it hbecome interesting to quantify entanglement, just asquantify other resources such as energy and informatiIn this Letter we adopt a recently proposed quantitatidefinition of entanglement and derive an explicit formufor the entanglement of a large class of states of a pairbinary quantum systems (qubits).

The simplest kind of entangled system is a pair of qubin a pure but nonfactorizable state. A pair of spin-1

2

particles in the singlet state1p2sj "# l 2 j #" ld is perhaps the

most familiar example, but one can also consider mogeneral states such asaj "# l 1 bj #" l, which may be lessentangled. For any bipartite system in a pure state, Benet al. [4] have shown that it is reasonable to define thentanglement of the system as the von Neumann entrof either of its two parts. That is, ifjcl is the stateof the whole system, the entanglement can be definas Escd ­ 2Tr r log2 r, where r is the partial traceof jcl kcj over either of the two subsystems. (It doenot matter which subsystem one traces over; the resis the same either way.) What Bennettet al. showedspecifically is the following. Considern pairs, each inthe statejcl. Let an observer Alice hold one member oeach pair and let Bob, whom we imagine to be spatiaseparated from Alice, hold the other. Then ifjcl hasE“ebits” of entanglement according to the above definitiothe n pairs can be reversibly converted by purely locoperations and classical communication intom pairs ofqubits in the singlet state, wheremyn approachesE forlargen and the fidelity of the conversion approaches 100

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This interconvertibility is strong justification for the abovedefinition ofE and characterizes it uniquely.

It is somewhat harder to define the entanglementmixed states [5], though again one can use the singletthe basic unit of entanglement and relate the given mixstate to singlets. The new feature in the case of mixestates is that the number of singlets required tocreatethestate is not necessarily the same as the number of singone canextractfrom the state [6]. In this paper we focuson the former quantity, which leads to the followingdefinition of “entanglement of formation” [7]. Given amixed stater of two quantum systemsA andB, considerall possible ways of expressingr as an ensemble of purestates. That is, we consider statesjcil and associatedprobabilitiespi such that

r ­X

i

pijcil kcij . (1)

The entanglement of formation ofr, Esrd, is defined asthe minimum, over all such ensembles, of the averaentanglement of the pure states making up the ensemb

E ­ minX

i

piEscid . (2)

Entanglement of formation has the satisfying property thit is zero if and only if the state in question can beexpressed as a mixture of product states. For easeexpression, we will refer to the entanglement of formatiosimply as “entanglement.”

Peres [8] and Horodeckiet al. [9] have found elegantcharacterizations of states with zero and nonzeroE, andBennett et al. [7] have determined the value ofE formixtures of Bell states. (These are a particular setorthogonal, completely entangled states of two qubits; wwill refer to other sets of such states as “generalized Bestates.”) But the value ofE for most states, even of twoqubits, is not known, and in fact it has not been evidethat one can even expressE in closed form as a functionof the density matrix. The exact formula we derive inthis Letter is proved for all density matrices of two qubithaving only two nonzero eigenvalues, but it appears likethat it applies toall states of this system.

© 1997 The American Physical Society

VOLUME 78, NUMBER 26 P H Y S I C A L R E V I E W L E T T E R S 30 JUNE 1997

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Our starting point is a curious and useful fact about tpure states of a pair of qubits. For such a system,define a “magic basis” consisting of the following foustates (they are the Bell states with particular phases)

je1l ­12 sj "" l 1 j ## ld ,

je2l ­12 isj "" l 2 j ## ld ,

je3l ­12 isj "# l 1 j #" ld ,

(3)

je4l ­12 sj "# l 2 j #" ld ,

where we have used spin-12 notation for definiteness

When a pure statejcl is written in this particular basisas jcl ­

Pi aijeil, its entanglement can be expressed

a remarkably simple way [7] in terms of the componenai : Define the function

E sxd ­ Hs 12 1

12

p1 2 x2 d for 0 # x # 1 , (4)

where H is the binary entropy functionHsxd ­2fx log2 x 1 s1 2 xd log2s1 2 xdg. Then the entangle-ment ofjcl is

Escd ­ E sssCscdddd , (5)

whereC is defined by

Cscd ­Å X

i

a2i

Å. (6)

The quantityC, like E for this system, ranges from zerto one, and it is monotonically related toE, so thatCis a kind of measure of entanglement in its own rigIt is sufficiently useful that we give it its own nameconcurrence. As we look for a pure-state ensemble wiminimum averageentanglement for a given mixed statour plan will be to look for a set of states that all have tsameentanglement, which is to say that they all have tsame concurrence.

Two other facts about the magic basis are wohighlighting. (i) The set of states whose density matricarereal when expressed in the magic basis is the samethe set of mixtures of generalized Bell states (Horodeet al. have called such mixtures “T states” [10]). (ii) Theset of unitary transformations that are real when expresin the magic basis (or real except for an overall phafactor) is the same as the set of transformations thatindependently on the two qubits.

It happens that our formula forE is convenientlyexpressed in terms of a matrixR, which is a function ofrdefined by the equation

Rsrd ­q

prrp

pr . (7)

Here rp is the complex conjugate ofr when itis expressed in the magic basis; that is,rp ­P

ij jeil kejjrjeil kejj. To get some sense of the meaninof R, note that TrR, ranging from 0 to 1, is a measure othe “degree of equality” [11] betweenr and rp, whichin turn measures how nearlyr approximates a mixtureof generalized Bell states. Note also that the eigenval

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of R are invariant under local unitary transformationsthe separate qubits, a fact that makes these eigenvaparticularly eligible to be part of a formula for entanglement, since entanglement must also be invariant unsuch transformations. We now state our main result.

Theorem.—Let r be any density matrix of two qubitshaving no more than two nonzero eigenvalues. Letlmaxbe the largest eigenvalue ofRsrd. Then the entanglemenof formation ofr is given by

Esrd ­ E scd; c ­ maxs0, 2lmax 2 Tr Rd . (8)

[The quantityc can thus be called the concurrence of thdensity matrixr. If r is pure, thenc reduces to theconcurrence defined in Eq. (6).]

Proof.—Let jy1l and jy2l be the two eigenvectors ofr corresponding to its two nonzero eigenvalues. Defithe 2 3 2 matrix t such thattij ­ yi ? yj, where thedot product is taken in the magic basis with no complconjugation: yi ? yj ;

Pkkek j yil kek j yjl. Consider

an arbitrary pure statejcl that can be written in theform jcl ­ ajy1l 1 bjy2l. If jcl is expressed as a fourvector in the magic basis, we can rewrite Eq. (6)Cscd ­ jc ? cj, and

C2scd ­ sc ? cd sc ? cdp ­ Tr fsptstpg , (9)

wheres ­ s ab d sa bdp is the density matrix ofjcl in the

(y1,y2) basis.Let us define the function

fsvd ­ Trfvptvtpg (10)

for any density matrixv expressed in the (y1,y2) basis.From Eq. (9),fsvd ­ C2svd if v represents a pure stateNow, v is a 2 3 2 density matrix, and as such can bwritten as a real linear combination of Pauli matricev ­

12 sI 1 $r ? $sd where rj ­ Trfsjvg. Substituting

this form into Eq. (10) gives us an expression

fsvd ­14 Tr ftptg 1

Xj

rjLj 1Xi,j

rirjMij , (11)

with

Lj ­12 Tr fsjtptg (12)

and

Mij ­14 Tr fsp

i tsjtpg . (13)

Thus f is defined on the surface and interior of a unsphere in three dimensions, the domain of$r.

M is a real, symmetric matrix with eigenvalue6

12 j dettj and 1

4 Tr ftptg, andL is the eigenvector ofMcorresponding to the eigenvalue1

4 Tr ftptg. SinceM hastwo positive eigenvalues and one negative eigenvalfsvd is convex along two directions and concave alona third. For the purpose of this proof, we would like thave a function that is equal tofssd for pure statess, butconvex in all directions. With this in mind we define

gsvd ­ fsvd 112 j dettjsj$rj2 2 1d , (14)

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VOLUME 78, NUMBER 26 P H Y S I C A L R E V I E W L E T T E R S 30 JUNE 1997

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which is identical tof for pure states (j$rj ­ 1). The extraterm added tofsvd in effect adds a constant tof and amultiple of the identity matrix toM. If we define a matrix

N ­ M 112 j dettjI (15)

and a constant

K ­14 Tr ftptg 2

12 j dettj , (16)

then we can write

gsvd ­ K 1X

j

rjLj 1Xi,j

rirjNij . (17)

The added term in Eq. (15) makes all the eigenvaluof N non-negative, one of them being zero. Thusgis a convex function. SinceL is an eigenvector ofNassociated with a positive eigenvalue, and is orthogoto the eigenvector with zero eigenvalue, the functiongis constant along the latter direction. We can imagithe functiong (suppressing one dimension) as a sheetpaper curved upward into a parabolic shape; it achievits minimum value along a straight line. Moreover, oncan show by direct calculation that the minimum valueg is zero. In Fig. 1, we indicate surfaces along whichgis constant, for a generic choice ofjy1l and jy2l. Thesurfaces appear as cylinders with elliptical cross sectioThe mixed stater that we are considering lies on onof these cylinders and can be decomposed into two pstates lying on the same cylinder; that is, having the sa

FIG. 1. The surface of the sphere represents the set ofpure superpositions ofjy1l and jy2l, the eigenvectors ofr.The interior represents all mixed states formed from susuperpositions. The elliptical cylinders are surfaces of constg, and their intersections with the spherical surface are therefcurves of constant entanglement.r itself lies on the verticalaxis, betweenjy1l and jy2l which are at the poles. Itsminimum-entanglement decomposition consists of two pustates lying on the same cylinder asr.

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value of g. (These two states are connected tor by astraight line parallel to the cylinders’ axis.) The next twparagraphs show that no other decomposition ofr has asmaller average entanglement than this one.

Any decomposition ofr into pure states can be viewedas a collection of weighted points on the surface of thsphere in Fig. 1 whose “center of mass” is the poirepresentingr. The average entanglement of such aensemble is the average ofE sss

pgssdddd over the ensemble,

sinceE sssp

gssdddd is equal to entanglement for pure statess.If we can show thatE sss

pgsvdddd, regarded as a function of

v, is convex over the interior of the sphere, then it wifollow that this average cannot be less thanE sss

pgsrdddd.

But we have just seen thatr can be decomposed intotwo pure statess for which gssd is the same asgsrd,so this will prove that the entanglement ofr is equal toE sss

pgsrdddd.

In fact it is not hard to prove the desired convexityThe function gsvd is parabolic with minimum valuezero. Its square root is therefore a kind of cone ais also convex. The functionE sxd is a convex andmonotonically increasing function ofx. It follows, then,from the transitive property of convex functions [12] thaE sss

pgsvdddd is a convex function ofv.

We have thus found the entanglement ofr and needonly express it in a simpler form. Replacingv with r

in Eq. (10) and using the fact thatr is diagonal in thesy1, y2d basis, we obtain

fsrd ­ TrsR2d ­ l21 1 l2

2 , (18)

where l1 and l2 are the nonzero eigenvalues ofR[Eq. (7)]. Similarly, one finds that for the other term inEq. (14),

12 j dettjsj$rj2 2 1d ­ 22l1l2 , (19)

so that gsrd ­ l21 1 l

22 2 2l1l2. Taking the square

root, we arrive at the result

Esrd ­ E scd; c ­ jl1 2 l2j . (20)

The expression (20) is equivalent to Eq. (8) for the caof two nonzero eigenvalues. This completes the proofthe theorem.

Although we have proved our result only for densitmatrices with just two nonzero eigenvalues, we can repthree pieces of evidence suggesting that the formulamay hold quite generally for a system of two qubits.

(1) For a mixture of Bell states, mixed with probabilities p1, . . . , p4, Bennett et al. [7] have shown thatthe entanglement is equal toE scd, with c given bymaxs0, 2pmax 2 1d. But in this caseR is equal tor, sothat our expression is equal to theirs. Thus our formuapplies also to this class of density matrices, mostwhich are not covered by the above theorem.

VOLUME 78, NUMBER 26 P H Y S I C A L R E V I E W L E T T E R S 30 JUNE 1997

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(2) Peres [8] and Horodeckiet al. [9] have provideda test, based on partial transposition, for determiniwhether a given state of two qubits has zero or nonzeE. We have applied both the Peres-Horodecki test andown formula to several thousand randomly chosen densmatrices and have found agreement between them in evcase. That is, Eq. (8) gaveE . 0 if and only if the Peres-Horodecki test indicated the presence of entanglemewhich happened in roughly one-third of the cases.

(3) For each of 25 randomly chosen density matricwith nonzero entanglement, we have explored the spaof all decompositions of the density matrix into purstates, limiting ourselves to ensembles of four state(The example of Bell mixtures [7] suggests that four-staensembles may be sufficient.) In each case, the resulnumerically minimizing the average entanglement of thensemble agrees with the result predicted by our formu

If the formula turns out to be correct for all statesit will considerably simplify studies of entanglementQuestions such as whether the “distillable entanglemeis equal to the entanglement of formation [6,7], that iwhether one can extract as much entanglement as oneinto the state, will presumably be easier to answer if theis an explicit formula for the latter quantity. It is alsoconceivable that our result can be generalized to systewith larger state-spaces, such as an entangled pairn-level atoms, though it is not clear whether there is astructure in such spaces that would play quite the sarole that the magic basis plays in the two-qubit case.imagining possible generalizations, it is interesting to nothat the form ofR has much in common with the “mixed-state fidelity” [11] of Bures, Uhlmann, and Jozsa, whicis in no way special to two-qubit systems.

We would like to thank Charles Bennett, David DiVincenzo, and John Smolin for helpful and stimulatindiscussions.

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[1] For a guide to some of the literature, see L. E. BallentinAm. J. Phys.55, 785 (1986).

[2] See, for example, D. P. DiVincenzo, Science270, 255(1995).

[3] A. K. Ekert, Phys. Rev. Lett.67, 661 (1991); C. H.Bennett, G. Brassard, and N. D. Mermin, Phys. Rev. Le68, 557 (1992).

[4] C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Scmacher, Phys. Rev. A53, 2046 (1996).

[5] Some alternatives to the definition we adopt, basedother criteria, are given in V. Vedral, M. B. Plenio, M. ARippin, and P. L. Knight, Phys. Rev. Lett.78, 2275 (1997).

[6] C. H. Bennett, G. Brassard, S. Popescu, B. SchumacJ. Smolin, and W. K. Wootters, Phys. Rev. Lett.76,722 (1996); A. Peres, Phys. Rev. A54, 2685 (1996);D. Deutsch, A. Ekert, R. Jozsa, C. MacchiavellS. Popescu, and A. Sanpera, Phys. Rev. Lett.77, 2818(1996); M. Horodecki, P. Horodecki, and R. HorodeckPhys. Rev. Lett.78, 574 (1997); see Ref. [7].

[7] C. H. Bennett, D. P. DiVincenzo, J. Smolin, and W. KWootters, Phys. Rev. A54, 3824 (1996). The interpre-tation of Eq. (2) in terms of the number of singlets required is actually rather subtle. It depends, for exampon whetherEsr ≠ rd is always equal to2Esrd, a ques-tion whose answer is not yet known. In this Letter wtake Eq. (2) as the definition ofE and focus only on itsevaluation.

[8] A. Peres, Phys. Rev. Lett.76, 1413 (1996).[9] M. Horodecki, P. Horodecki, and R. Horodecki, Phy

Lett. A 223, 1 (1996).[10] R. Horodecki and M. Horodecki, Phys. Rev. A54, 1838

(1996); R. Horodecki, M. Horodecki, and P. HorodeckPhys. Lett. A222, 21 (1996).

[11] D. Bures, Trans. Am. Math. Soc.135, 199 (1969);A. Uhlmann, Rep. Math. Phys.9, 273 (1976); R. Jozsa,J. Mod. Optics41, 2315 (1994).

[12] See, for example, A. W. Roberts and D. E. VarberConvex Functions(Academic Press, New York, 1973)p. 16.

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