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arXiv:quant-ph/9801069v1

29Jan1998

Mixed-state entanglement and distillation: is there a bound entanglement in

nature?

Michal HorodeckiInstitute of Theoretical Physics and Astrophysics

University of Gdansk, 80952 Gdansk, Poland

Pawel Horodecki

Faculty of Applied Physics and MathematicsTechnical University of Gdansk, 80952 Gdansk, Poland

Ryszard Horodecki

Institute of Theoretical Physics and Astrophysics

University of Gdansk, 80952 Gdansk, Poland

It is shown that if a mixed state can be distilled to the sin-glet form, it must violate partial transposition criterion [A.Peres, Phys. Rev. Lett. 76, 1413 (1996)]. It implies thatthere are two qualitativelydifferent types of entanglement: free entanglement which is distillable, and bound entan-

glement which cannot be brought to the singlet form useful forquantum communication purposes. Possible physical mean-ing of the result is discussed.

Pacs Numbers: 03.65.Bz

Since the famous Einstein, Podolsky and Rosen [1] andSchrodinger [2] papers quantum entanglement still re-mains one of the most striking implications of quantumformalism. In recent years, a great effort was made tounderstand a role of entanglement in nature and funda-mental applications were found in the field of quantum

information theory [36]. The most familiar example ofpure entangled state is the singlet state [7] of two spin- 12

particles

=1

2(| | ), (1)

which cannot be reduced to direct product by any trans-formation of the bases pertaining to each one of the par-ticles.

In practice, due to decoherence effects, we usually dealwith mixed states [8]. A mixed state of quantum systemconsisting of two subsystems is supposed to represent en-tanglement if it is inseparable [9] i.e. cannot be written

in the form =

i

piAi Bi , pi 0,

i

pi = 1. (2)

were Ai andBi are states for the two subsystems. How-

ever, to use the entanglement for quantum informationprocessing, we must have it in pure singlet form. Theprocedure of converting mixed state entanglement to thesinglet form is called distillation [10]. It amounts to ex-traction of pairs [11] of particles in singlet state from

an ensemble described by some mixed state by means oflocal quantum operations and classical communication[10].

The process can be described as follows : the two ob-servers, Alice and Bob, each have N quantum systems

coming from entangled pairs prepared in a given state. Each one can perform local operations with her/hisN particles, and exchange classical information with theother one. The question is whether they can in this wayobtain a pair of entangled qubits (the rest of the quan-tum systems being discarded). They need not succeedevery time, but at least they know when they have beensuccessful. If they managed to do this, one says thatthey have distilled some amount of pure entanglementfrom the state . Subsequently, the distilled singlet pairscan be used e.g. for reliable transmission of quantuminformation via teleportation [5].

Recently, it has been shown [12] that any inseparable

two-qubit state [13] represents the entanglement which,however small, can be distilled to a singlet form. Theresult was obtained by use of the necessary [14] and suf-ficient [15] condition of separability for two-qubit states,local filtering [17,16] and Bennett et al. distillation pro-tocol [10]i.

In this context it seems very natural to make the fol-lowing conjecture:

Conjecture - Any inseparable state can be distilled tothe singlet form.

Surprisingly enough, this conjecture is wrong. In thepresent Letter we will show that there are inseparablestates that cannot be distilled. More specifically, we first

show that any state which can be distilled must violatePeres separability criterion [14]. Then the result fol-lows from the fact [18] that there are inseparable statesthat satisfy the criterion. It shows that there are twoqualitatively different types of entanglement. The first,free entanglement, can be distilled to the singlet form.The second type of entanglement is not distillable andis considered here in analogy with thermodynamics as abound entanglement which cannot be used to perform

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a useful informational work like reliable transmissionof quantum data via teleportation.

Now, let us first shortly describe the Peres criterion.A state satisfies the criterion, if all eigenvalues of itspartial transposition TB are nonnegative (i.e. if TB isa positive operator). Here the partial transposition TB

associated with an arbitrary product orthonormal eifjbasis is defined by the matrix elements in this basis:

TBm,n em f | TB |en f = m,n. (3)Clearly, the matrix TB depends on the basis, but itseigenvalues do not. Thus given a state, one can checkwhether it violates the criterion performing the partialtransposition in an arbitrary product basis. In particular,it implies that violates the criterion if and only if anyN-fold tensor product N = . . .

N

does [14].

Peres showed that the criterion must be satisfied byany separable state [14]. It has been also shown [15] thatfor two-qubit (and qubit-trit) states the criterion is also

sufficient condition for separability. This does not holdfor higher dimensions. The explicit examples of insepa-rable mixtures satisfying criterion were constructed [18].

Now we are in position to present the main result ofthis Letter. Suppose Alice and Bob have a large numberN of pairs each in a state acting on the Hilbert spaceH = HA HB . Then the joint state of N pairs is givenby N. Suppose now that the state is distillable. Thismeans, that Alice and Bob are able to obtain pure singlettwo-qubit pairs for N tending to infinity. This howeverimplies, that for some finite N, they are able to obtainan inseparable two-qubit state 2q. The most generaloperation producing a two-qubit pair they can perform

over the initial amount of N pairs can be written in thefollowing form [19]

2q =1

M

i

Ai Bi NAi Bi , (4)

where M = Tr

i Ai Bi NAi Bi is the normaliza-tion factor and Ai and Bi map the large Hilbert spacesHNA,B into C2. For convenience, we will use unnormalizedstates, as the property of separability as well as satisfyingthe Peres criterion do not depend on the positive factor.Then, for unnormalized states, we omit the condition

ipi = 1 in the definition of separability (2). Conse-

quently let 2q =

i

Ai Bi NAi Bi (5)

and

i = Ai Bi NAi Bi . (6)Since 2q is inseparable then at least for some i = i0the state i0 must be inseparable. Indeed, by summingseparable states we cannot get inseparable one.

Note that the operators Ai0 and Bi0 act into two-dimensional space C2, hence they can be written in theform

Ai0 = |0A| + |1A|, Bi0 = |0B | + |1B |, (7)where |1 and |0 constitute orthonormal basis in C2 andA, A HNA , B , B HNB are arbitrary (possi-bly unnormalized) vectors. Let us now consider two-dimensional projectors PA and PB which project ontothe spaces spanned by A, A and B, B respectively.Then we have

i0 = Ai0 Bi0

PA PB NPA PB

Ai0

Bi0 . (8)Now, since a product action cannot convert separablestate into inseparable one, we obtain that also the state

= PA PB NPA PB (9)is inseparable. Let us write this state in basis |fi |gk, i = 1, 2,...,dimHNA , k = 1, 2,...,dimHNB with fourvectors |f1, |f2 (|g1, |g2) spanning the subspaces de-fined by projectors PA, PB . The only nonzero matrixelements are due to products of those vectors and theydefine a 4 4 matrix M2q which can be thought as two-qubit state. The operation of partial transposition on

affects only those elements (as the remaining ones areequal to zero). IfM2q were positive after partial transpo-sition, then, due to the sufficiency of the partial transpo-sition test for two-qubit case [15], M2q would represent aseparable two-qubit state. Hence, if embedded into thewhole space HN, it would still remain separable. Con-sequently, the state would be separable, which is thecontradiction. Thus partial transposition of M2q must

be negative. Now, since M2q is formed by all nonzeroelements of , then we obtain that also the state mustviolate the Peres criterion, i. e. TB must have a negativeeigenvalue. Now let be the eigenvector correspondingto the eigenvalue. As the vector belongs to the subspaceH2q it follows that the matrix elements | TB | and|( N)TB | are equal. Hence we obtain

|( N)TB | < 0. (10)Thus the state N violates the partial transposition cri-terion. However, as it was mentioned, this implies thatalso does. All the above consideration can be formally

summarised as follows. If the output state of this actionappears to have negative partial transposition, then thebasic component of input state N must have had alsonegative partial transposition . This means nothing butthat any action of type (4) on (including collecting Npairs) preserves positivity of partial transposition. Thisresult can be generalized [20]: any action of the form1

M

i Ai Bi NAi Bi producing an arbitrary two-

component system (not necessarily 2 2 one) preservespositivity of partial transposition.

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Thus we showed that if a state is distillable, itmust violate the Peres separability criterion. It is animportant result as it implies that there are insepara-ble states which cannot be distilled! Indeed, quite re-cently one of us [18] constructed inseparable states whichdo not violate the criterion. Some of those peculiarstates are density matrices for two spin-1 particles (thetwo-trit case). Using the standard basis for this case (

|1|1, |1|2, |1|3, |2|1, |2|2, and so on ... ) thosematrices can be written in the form:

a =1

8a + 1

a 0 0 0 a 0 0 0 a0 a 0 0 0 0 0 0 00 0 a 0 0 0 0 0 00 0 0 a 0 0 0 0 0a 0 0 0 a 0 0 0 a0 0 0 0 0 a 0 0 0

0 0 0 0 0 0 1+a2

01a22

0 0 0 0 0 0 0 a 0

a 0 0 0 a 01a22

0 1+a2

, (11)

with 0 < a < 1. It has been shown [18] by means ofindependent separability criterion that those states areinseparable despite they have positive partial transposi-tion. However, as we have shown above, that the den-sity matrices with positive partial transposition cannotbe distilled to the singlet form. Consequently, any stateof the form (11) cannot be distilled.

It is remarkable, that the question whether a state isdistillable or not has been reduced to the one whetherthere is a two-qubit entanglement in a collection of Npairs for some N. Thus the latter condition is the nec-essary and sufficient condition for any given state to bedistilled. Indeed, as shown above, if a state is distill-able then there exist two-dimensional projections PA andPB so that the state

given by eq. (9) is inseparable.Conversely, if the latter condition is satisfied then canbe distilled by projecting N locally by means of PAand PB and then applying the protocol proposed in [12]which is able to distill any two-qubit inseparable state.There is an open question, whether the condition impliessatisfying Peres criterion. Then the latter would acquirethe physical sense: it would be equivalent to distillability.

Let us now discuss shortly possible physical meaningof our result. As a matter of fact, we have revealed akind of entanglement which cannot be used for send-

ing reliably quantum information via teleportation. Us-ing an analogy with thermodynamics [22], we can con-sider entanglement as a counterpart of energy, and send-ing of quantum information as a kind of informationalwork. Consequently we can consider free entangle-ment (Efree) which can be distilled, and bound en-tanglement (Ebound). In particular, the free entangle-ment is naturally identified with distillable entanglementD as the latter says us how much qubits can we reliablyteleport via the mixed state. This kind of entanglement

can be always converted via distillation protocol to theactive singlet form.

To complete the analogy, one could consider theasymptotic number of singlets which are needed to pro-duce a given mixed state as internal entanglement Eint(the counterpart of internal energy) [21]. Then the boundentanglement can be quantitatively defined by the follow-ing equation

Eint = Efree + Ebound. (12)

In particular, for pure states we have Eint = Efree andEbound = 0. Indeed, pure states can be converted in alossless way into active singlet form [16]. In the presentletter we showed that there exist inseparable states hav-ing reciprocal properties. Namely for the states of type(11) we have Eint = Ebound and Efree = 0.

Now the question arises: is it that Eint = Ebound = 0or = 0? Both cases are curious. In the first case, wewould have inseparable states which can be producedfrom asymptotically zero number of singlet pairs. This

would imply, in turn, that entanglement of formation isnot additive state function [23], as by the very defini-tion it does not vanish for any inseparable states. In thesecond case, we would have curious states which absorbentanglement in an irreversible way. To produce suchstates, one needs some amount of entanglement. Butonce the states were produced, there is no way to recoverany, however little, piece of the initial entanglement. Thelatter is entirely lost.

A natural problem which arises in the context of thepresented result is: what is the physical reason for whichthe partial transposition is connected with distillability?Our conjecture is that it is time which links intimately the

two things. Indeed, transposition can be interpreted asthe operation of time-reversal [24]. Also in the context ofdistillation, there appeared the problem of time. Namely,distillation is inherently connected with the quantum er-ror correction for quantum noisy channel supplementedby two-way classical channel [6]. The quantum capacityof such channels can be strictly larger than without theclassical channel. However, the price we must pay is thatthe error correction with two-way classical communica-tion cannot be used to store the quantum informationin noisy environment [6] because one cannot send signalbackward in time. Needless to say deeper investigationof the connection among the distillation, partial transpo-sition and time reversal seems to be more than desirable.

Finally, it is perhaps worth to mention about the circledescribed by story of the nonlocality of mixed states, be-ginning with the work of Werner [9]. The latter suggestedthat there are curious inseparable states which do notexhibit nonlocal correlations. Then Popescu [25] showedthat there is a subtle kind of pure quantum correlationswhich is exhibited by Werner mixtures. The distillabilityof all two-qubit states [12] proved that all they are also

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nonlocal. One could suspect that the story will end byshowing that all inseparable states can be distilled, hencethey are nonlocal. Here we showed that it is not true. So,one is now faced with the problem similar to the initialone i.e. are the inseparable states with positive partialtransposition nonlocal? Now, in view of the above resultit follows that the problem certainly cannot be solved bymeans of distillation concept.

We would like to thank Asher Peres for helpful com-ments and discussion.

E-mail address: michalh@iftia.univ.gda.pl E-mail address:pawel@mifgate.mif.pg.gda.pl E-mail address: fizrh@univ.gda.pl

[1] A. Einstein, B. Podolsky and N. Rosen, Phys. Rev. 47777 (1935).

[2] E. Schrodinger, Proc. Cambridge Philos. Soc. 31 555

(1935).[3] A. Ekert, Phys. Rev. Lett. 67 (1991) 661.[4] C. H. Bennett and S. J. Wiesner, Phys. Rev. Lett. 69

(1992) 2881.[5] C. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres

and W. K. Wootters, Phys. Rev. Lett. 70, 1895 (1993).[6] C. H. Bennett, D. P. Di Vincenzo, J. Smolin and W. K.

Wootters, Phys. Rev. A 54, 3814 (1997).[7] D. Bohm, Quantum theory (Prentice-Hall, Englewood

Cliffs, N. J., 1951).[8] In this Letter we identify states with their density matri-

ces.[9] R. F. Werner, Phys. Rev. A 40, 4277 (1989).

[10] C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher,J. Smolin and W. K. Wootters, Phys. Rev. Lett. 76, 722(1996).

[11] Recently distillation of multi-particle system was alsoconsidered, see M. Murrao, M. B. Plenio, S. Popescu,V. Vedral and P. L. Knight, Multi-particle entanglementpurification protocols, Report No. quant-ph/9712045.

[12] M. Horodecki, P. Horodecki and R. Horodecki, Phys.Rev. Lett. 78 (1997) 574.

[13] In analogy to classical bits, a two-level (spin-12

) quantumsystem is usually referred to as qubit. Similarly, a threelevel (spin-1) system is called trit.

[14] A. Peres, Phys. Rev. Lett. 76, 1413 (1996);[15] M. Horodecki, P. Horodecki and R. Horodecki, Phys.

Lett. A 223, 1 (1996).[16] C. H. Bennett, H. J. Bernstein, S. Popescu and B. Schu-

macher, Phys. Rev. A 53, 2046 (1996).[17] N. Gisin, Phys. Lett. A 210, 151 (1996).[18] P. Horodecki, Phys. Lett. A, 232, 333 (1997).[19] V. Vedral, M. B. Plenio, M. A. Rippin and P. L. Knight,

Phys. Rev. Lett. 78 (1997) 2275.[20] A. Peres, private communication. It follows from the

simple fact that for any operators A,B,C,D we have(A B

C D)TB = (A DT TBC BT) where T is

usual transposition.

[21] One of the important problems is finding whether theso defined internal entanglement is equal to the entan-glement of formation [23]. The latter is defined [6] byE( ) = inf

ipiS( i), where S( ) = Tr log2 . The

infimum is taken over all ensembles of pure states i sat-isfying i = TrB|ii|, and =

ipi|ii|. It can be

seen that Eint( ) = limN1

NE( N) (we could say that

Eint is additification ofE).

[22] S. Popescu and D. Rohrlich, Phys. Rev.56

, 3219 (1997);M. Horodecki and R. Horodecki, Are there basic lawsof quantum information processing? Report No. quant-ph/9705003.

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

[24] S. L. Woronowicz, Rep. Math. Phys., 10, 165 (1976);P. Busch and J. Lahti, Found. Phys. 20, 1429 (1990); A.Sanpera, R. Tarrach and G. Vidal, Quantum separability,time reversal and canonical decomposition, Report No.quant-ph/9707041.

[25] S. Popescu, Phys. Rev. Lett. 72, 797 (1994); ibid 74,2619 (1995).

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