PhysRevLett.80.2245.pdf

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VOLUME 80, NUMBER10 P H Y S I C A L R E V I E W L E T T E R S 9 MARCH1998

Entanglement of Formation of an Arbitrary State of Two Qubits

William K. Wootters

Department of Physics, Williams College, Williamstown, Massachusetts 01267(Received 12 September 1997)

The entanglement of a pure state of a pair of quantum systems is defined as the entropy of eithermember of the pair. The entanglement of formation of a mixed state r is the minimum average

entanglement of an ensemble of pure states that represents r . An earlier paper conjectured an explicitformula for the entanglement of formation of a pair ofbinary quantum objects (qubits) as a functionof their density matrix, and proved the formula for special states. The present paper extends the proofto arbitrary states of this system and shows how to construct entanglement-minimizing decompositions.[S0031-9007(98)05470-2]

PACS numbers: 03.67.a, 03.65.Bz, 89.70.+c

Entanglement is the quantum mechanical property thatSchrdinger singled out many decades ago as the char-acteristic trait of quantum mechanics [1] and that hasbeen studied extensively in connection with Bells in-equality [2]. A pure state of a pair of quantum systemsis called entangled if it is unfactorizable, as is the case,

for example, for the singlet state of two spin-1

2 particles,1

p2 j"# 2j#". A mixedstate is entangled if it can-

not be represented as a mixture of factorizable pure states.In the last couple of years a good deal of work has beendevoted to finding physically motivated quantitative mea-sures of entanglement, particularly for mixed states of abipartite system [3 5]. Perhaps the most basic of thesemeasures is the entanglement of formation, which is in-tended to quantify the resources needed to create a givenentangled state [5].

Having a well justified and mathematically tractablemeasure of entanglement is likely to be of value in anumber of areas of research, including the study of de-

coherence in quantum computers [6] and the evaluationof quantum cryptographic schemes [7]. Unfortunately,most proposed measures of entanglement involve extrem-izations which are difficult to handle analytically, the en-tanglement of formation being no exception to this rule.However, in the special case of entanglement between two

binaryquantum systems such as the spin of a spin-1

2 par-ticle or the polarization of a photonsystems that aregenerically called qubitsan explicit formula for theentanglement of formation has recently been conjecturedand has been proved for a special class of density matri-ces [8]. In this Letter we prove the formula for arbitrarystates of two qubits.

The entanglement of formation is defined as follows[5]. Given a density matrix r of a pair of quantumsystems A and B, consider all possible pure-state decom-positions of r, that is, all ensembles of states jci withprobabilitiespi , such that

r Xi

pijci cij . (1)For each pure state, the entanglement E is defined as theentropy of either of the two subsystems A and B [3]:

Ec 2TrrAlog2 rA 2TrrBlog2 rB . (2)

Here rA is the partial trace ofjc cj over subsystemB, and rB has a similar meaning. The entanglementof formation of the mixed state r is then definedas the average entanglement of the pure states of thedecomposition, minimized over all decompositions ofr :

Er minXi

piEci . (3)

The basic equation (2) is justified by the physical inter-convertibility of a collection of pairs in an arbitrary purestatejc and a collection of pairs in the standard singletstate, the asymptotic conversion ratio being given by Ec[3]. The central claim of this Letter is that for a pair ofqubits, the minimum value specified in Eq. (3) can be ex-pressed as an explicit function ofr , which we develop inthe next few paragraphs. For ease of expression we willusually refer to the entanglement of formation simply asthe entanglement.

Our formula for entanglement makes use of what can becalled the spin flip transformation, which is a functionapplicable to states of an arbitrary number of qubits. For apure state of a single qubit, the spin flip, which we denoteby a tilde, is defined by

jc syjc , (4)where jc is the complex conjugate of jc when itis expressed in a fixed basis such as j", j#, and syexpressed in that same basis is the matrix 0 2i

i 0. For a

spin-1

2 particle this is the standard time reversal operationand indeed reverses the direction of the spin [9]. Toperform a spin flip on n qubits, one applies the abovetransformation to each individual qubit. For example, fora general state r of two qubitsthe object of interest inthis Letterthe spin-flipped state is

r sy syrsy sy , (5)where again the complex conjugate is taken in the

standard basis, which for a pair of spin-1

2 particles isj"",j"#, j#", j##. In this case the spin flip is equivalent [10] to

0031-90079880(10)2245(4)$15.00 1998 The American Physical Society 2245


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complex conjugation in the magic basis, which appearsin Ref. [8].

Though we have introduced the spin flip primarily todeal with mixed states, the concept is also convenient forexpressing the entanglement of a purestate of two qubits.One can show that this entanglement, defined in Eq. (2),can be written as [8]

Ec E

Cc , (6)

where the concurrence C is defined as

Cc jcj cj , (7)and the functionE is given by

EC h1 1

p1 2 C2

2

;

hx 2x log2x 2 1 2 x log21 2 x . (8)

EC is monotonically increasing and ranges from 0 to1 as C goes from 0 to 1, so that one can take theconcurrence as a measure of entanglement in its own

right. For example, the singlet statejc 1p2 j"# 2j#"is left unchanged by a spin flip (except for an overallnegative sign), so that its concurrencejcj cjis equal to1. At the other extreme, an unentangled pure state such asj"# is always mapped by the spin flip transformation intoan orthogonal state, so that its concurrence is zero. Laterwe will use another fact aboutEC, namely, that it is aconvex function (that is, curving upward).

Having defined the spin flip and the functionEC, wecan now present the promised formula for the entangle-ment of formation of a mixed state r of two qubits:

Er ECr , (9)where

Cr max0, l1 2 l2 2 l3 2 l4 , (10)

and the lis are the eigenvalues, in decreasing order, ofthe Hermitian matrix R

pprr

pr. Alternatively, one

can say that the lis are the square roots of the eigenvaluesof the non-Hermitian matrix rr. Note that eachli is anon-negative real number.

The formula (9) was shown in Ref. [8] to be correct forall density matrices of two qubits having no more than twononzero eigenvalues. More recently, Smolin has testedthe formula numerically on several thousand randomlychosen two-qubit density matrices and has found completeagreement [11]. We now prove the formula for all statesof this system by constructing an entanglement-minimizingdecomposition of any given density matrix.

We begin with the fact that every decomposition of adensity matrix r can be obtained via the following pre-scription [12]. First, find a complete set of orthogonaleigenvectorsjyi corresponding to the nonzero eigenval-ues ofr, and subnormalize these vectors so that yi j yiis equal to theith eigenvalue. Then a general decomposi-

tion jwiofr is given by

jwi nX

j1

Uijjyj, i 1, . . . , m . (11)

Heren is the rank ofr, that is, the number of vectorsjyi,and U is an m 3 m unitary matrix, m being greater thanor equal to n . (The complex conjugation is included onlyfor later convenience.) Alternatively, since only the first

n columns ofUare used, we can take Uto be an m 3 nmatrix whose columns are orthonormal vectors. Thestatesjwi in Eq. (11) are automatically subnormalizedso that wi jwi is equal to the probability ofjwi in thedecomposition. We can thus write r

Pijwi wij. In

what follows, we express all decompositions ofrin termsof such subnormalized vectors.

It is helpful to consider separately two classes ofdensity matrix: (i) Those for whichl1 2 l2 2 l3 2 l4is positive or zero, and (ii) those for which the samecombination is negative. We consider class (i) first.

For any density matrix r in this class, we will definesuccessively three specific decompositions of r, the last

of which is the optimal decomposition that we seek. Eachof these decompositions consists of exactly n pure states,n being the rank ofr as above. For the system we areconsidering,n is always less than or equal to 4.

The first decomposition consists of states jxi, i 1 , . . . , n, satisfying

xi j xj lidij. (12)We obtain such a decomposition as follows. First notethat if the set jxiis defined via an n 3 n unitary matrixU as in Eq. (11), then the tilde inner products xij xjcan be written as

xi j xj UtUT

ij, (13)where tij yi j yj is a symmetric but not necessarilyHermitian matrix. (The statesjyiare the eigenstates ofrdefined earlier.) In order that condition (12) be satisfied,we want UtUT to be diagonal. It happens that for anysymmetric matrix t , one can always find a unitary U thatdiagonalizes t in this way [13]. Moreover, the diagonalelements of UtUT can always be made real and non-negative, in which case they are the square roots of theeigenvalues oft t. (To see how this works, note that Umust diagonalize t t in the usual sense; that is,UttUyis diagonal.) The square roots of the eigenvalues oftt

are the same as the eigenvalues ofR, so that condition(12) is fulfilled as long as the diagonalizing matrix U ischosen in such a way that the numbers li appear in theirproper order. Thus one can always find a decompositionwith the desired property.

Our second decomposition ofr , which we labelj yi,i 1, . . . , n, is hardly different from the first:

j y1 jx1 ;j yj ijxj, forj fi 1 .

(14)

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It is indeed physically equivalent to the first decomposi-tion, but the phase factors will become important shortlywhen we take linear combinations of these vectors.

The decomposition jyi has the following specialproperty. Define the preconcurrence c of a pure statejc(possibly subnormalized) to be

cc cj cc

jc

; (15)

that is, the preconcurrence is the same as the concurrenceof Eq. (7) but without the absolute value sign. Thenthe average preconcurrence of jyi has the value Crspecified in Eq. (10):

c Xi

yi j yiyij yiyij yi

Xi

yij yi l1 2 l2 2 l3 2 l4 C r .(16)

Here we have used the fact that ifn , 4, the numbers liwith i . n are all zero.

We would like to find a decomposition that, like jyi,has c Cr, but which also has the property thatthe preconcurrence (and hence the concurrence) of eachindividual state is equal to Cr. It would then followimmediately that the average entanglement isEC r,since this would be the entanglement of each state in thedecomposition.

To accomplish this, note first that any decompositionwith n elements can be written in terms of the statesjyivia the equation

jzi nX

j1

Vijjyj , (17)

where V is an n 3 n unitary matrix. The averagepreconcurrence of the ensemble jziis

c Xi

zi j zi Xi

VYVTii TrVYVT , (18)

where Y is the real diagonal matrix defined by Yij yi j yj. Thus the average preconcurrence is unchangedby any real unitary matrix V (that is, any orthogonalmatrix), since in that case VT V21 and the trace inEq. (18) is preserved.

Let us now restrict ourselves to such orthogonal matri-ces, so as to preserve the average, and use them to try toequalize the preconcurrences. One way to do this is asfollows. First, select the two statesjyi with the largestand smallest values of the preconcurrence. Unless all thepreconcurrences are already equal, the largest one mustbe too large and the smallest one too small (typicallynegative). In the latter case, consider the set of positive-determinant orthogonal transformations that act only onthese two extreme states as in Eq. (17), changing theminto new states that we calljza andjzb and leaving theother statesjyi unchanged. Among this set of transfor-

mations is one that simply interchanges the two extremestates and thus interchanges their preconcurrences. There-fore, by continuity there must exist an intermediate trans-formation that makes the preconcurrence ofjza equal toCr. Perform this transformation, thereby fixing one ele-ment of the ensemble to have the correct concurrence.By repeating this procedure on the remaining elementsof the ensemble, one finally arrives at a set of states all

having concurrence equal to Cr. This we take to beour final decomposition jzi, which, as we have arguedabove, achieves the claimed minimum average entangle-mentECr. Thus the value of entanglement given inour formula (9) can always be attained, at least for thecase in which l1 2 l2 2 l3 2 l4 $ 0.

We now show that no decomposition of r has asmalleraverage entanglement. For this it is enough to showthat no decomposition has a smaller averageconcurrence:The average entanglement cannot be less thanECbecause of the convexity of the functionE. Now, theaverage concurrence of a general decomposition is givenby an equation similar to Eq. (18) but with an absolute

value sign:

C Xi

jVYVTiij . (19)

Here V is an m 3 n matrix whose n columns areorthonormal vectors. The dimension m of these vectorscan be arbitrarily large, since the decomposition mayconsist of an arbitrarily large number of pure states(though prior results guarantee that one need not considervalues of m larger than sixteen [14]). In terms of thecomponents of V and Y, we can write the averageconcurrence as

C Xi

Xj

Vij2Yjj

. (20)

To obtain the desired lower bound on this sum, weneed use only the fact that

Pi jVij2j 1. That is, we

can show that for any complex numbers aij such thatPijaijj 1, we haveX

i

Xj

aijYjj

$ l1 2 l2 2 l3 2 l4. (21)

The proof is straightforward: first note that there is noloss of generality in taking each ai1 to be real and

positive. (The phases of the other aijs can be changedto compensate.) Then we can say

Xi

Xj

aijYjj

$

Xij

aijYjj

l1 2

nXj2

Xi

aij

!lj

(22)

$ l1 2 l2 2 l3 2 l4 Cr .

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Thus no decomposition of r can achieve an averageconcurrence lower than C ror an average entanglementlower thanEC r.

There remains one case to consider, namely, densitymatrices for which l1 2 l2 2 l3 2 l4 , 0. For thesedensity matrices our formula predicts that the entangle-ment should be zero; that is, that there should be a de-composition ofr into unentangled pure states. To show

that this is indeed the case, we again start with the decom-position jxi, i 1, . . . , n, of Eq. (12). Ifn is equal to3the values n 1 and n 2 are not possible for thecase we are now consideringit is convenient to supple-ment this set with a dummy statejx4 equal to the zerovector. From the complete set we directly form our finaldecompositionjzi:jz1 12eiu1 jx1 1 e iu2 jx2 1 eiu3 jx3 1 e iu4 jx4 ,jz2 12eiu1 jx1 1 e iu2 jx2 2 eiu3 jx3 2 e iu4 jx4 ,jz3 12eiu1 jx1 2 e iu2 jx2 1 eiu3 jx3 2 e iu4 jx4 ,jz4 12eiu1 jx1 2 e iu2 jx2 2 eiu3 jx3 1 e iu4 jx4 ,

(23)

where the phase factors are chosen so thatXj

e2iujlj 0 . (24)

Such phase factors can always be found when l1 ,l2 1 l3 1 l4 (l1 being the largest of the four numbersas always). The condition (24) together with the property(12) of the set jxi guarantee that each statejzi haszero concurrence and hence zero entanglement. Thiscompletes the proof of the formula (9).

Our formula makes possible the easy evaluation of en-

tanglement of formation for a pair of qubits, and shouldthus facilitate the investigation of any number of ques-tions concerning entanglement. However, there remainsa basic question concerning the interpretation of the en-tanglement of formation that has not yet been resolved.For any pure statejcof a bipartite system, the entangle-ment Ec defined in Eq. (2) can be interpreted roughlyas the number of qubits that must have been exchangedbetween two observers in order for them to share the statejc [15]. It seems likely that this interpretation appliesalso to the entanglement of formation of amixedstate [5],but this conclusion depends on a property ofErthat hasnot yet been demonstrated [16]. The question is whether

Eris additive,that is, whether the entanglement of for-mation of two pairs of quantum systems is the sum of theentanglements of formation of the individual pairs and notless. If it is determined thatEris indeed additive, then

this finding will considerably strengthen the physical in-terpretation of our formula.

I thank a number of colleagues whose comments andsuggestions have been of great help in this work: ValerieCoffman, Scott Hill, Joydip Kundu, Hideo Mabuchi,Michael Nielsen, David Park, Eric Rains, John Smolin,Ashish Thapliyal, and especially Chris Fuchs.

[1] E. Schrdinger, Proc. Cambridge Philos. Soc. 31, 555(1935).

[2] J. S. Bell, Physics 1, 195 (1964).[3] C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schu-

macher, Phys. Rev. A 53, 2046 (1996); S. Popescu andD. Rohrlich, quant-ph/9610044.

[4] A. Shimony, in Fundamental Problems in QuantumTheory, edited by D. M. Greenberger and A. Zeilinger(New York Academy of Sciences, New York, 1995);C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher,J. Smolin, and W. K. Wootters, Phys. Rev. Lett. 76, 722

(1996); V. Vedral, M. B. Plenio, M. A. Rippin, and P. L.Knight, Phys. Rev. Lett. 78, 2275 (1997); N. J. Cerf andC. Adami, quant-ph/9605039; V. Vedral and M. B. Plenio,quant-ph/9707035; M. Lewenstein and A. Sanpera, quant-ph/9707043.

[5] C. H. Bennett, D. P. DiVincenzo, J. Smolin, and W. K.Wootters, Phys. Rev. A 54, 3824 (1996).

[6] See, for example, D. P. DiVincenzo, Science 270, 255(1995).

[7] See, for example, C. A. Fuchs, N. Gisin, R. B. Griffiths,C.-S. Niu, and A. Peres, Phys. Rev. A 56, 1163 (1997),and references therein.

[8] S. Hill and W. K. Wootters, Phys. Rev. Lett. 78, 5022(1997).

[9] J. J. Sakurai, Modern Quantum Mechanics, edited by SanFu Tuan (Benjamin/Cummings, Menlo Park, CA, 1985),p. 277.

[10] The equivalence was pointed out to the author by V.Coffman and J. Kundu.

[11] J. Smolin (private communication).[12] E. Schrdinger, Proc. Cambridge Philos. Soc. 32, 446

(1936); N. Hadjisavvas, Lett. Math. Phys. 5, 327 (1981);L. P. Hughston, R. Jozsa, and W. K. Wootters, Phys. Lett.A 183, 14 (1993).

[13] R. A. Horn and C. R. Johnson, Matrix Analysis (Cam-bridge University Press, New York, 1985), p. 205.

[14] A. Uhlmann, quant-ph/9701014. Evidence that four statesare sufficient can be found in F. Benatti, H. Narnhofer,

and A. Uhlmann, Rep. Math. Phys. 38, 123 (1996).[15] This interpretation follows directly from the results in

Ref. [3].[16] S. Popescu (private communication).

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