China

PHYSICAL REVIEW A 69, 024302 ~2004!

Mixture of multiple copies of maximally entangled states is quasipure

Dong Yang* and Yi-Xin ChenZhejiang Institute of Modern Physics and Department of Physics, Zhejiang University, Hangzhou 310027, People’s Republic of

~Received 25 July 2003; revised manuscript received 6 November 2003; published 9 February 2004!

Employing the general bilateral controlled-NOT operation and local state discrimination, the mixed state ofthe formrd

(k)5(1/d2)(m,n50d21 (ufmn&^fmnu) ^ k is proved to be quasipure, where$ufmn&% is the canonical set of

mutually orthogonal maximally entangled states ind3d. Therefore irreversibility does not occur in the processof distillation for this family of states. Also, the distillable entanglement is calculated explicitly.

DOI: 10.1103/PhysRevA.69.024302 PACS number~s!: 03.67.Mn, 03.65.Ud

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Entanglement distillation plays a crucial role in quantuinformation theory@1#. As well known, entanglement is responsible for many quantum tasks and pure entangled sare required in most cases. Unfortunately, due to the intetion with the environment, pure entanglement is fragile aeasy to be blurred by the noise. So distillation of entangment is of importance. Though many distillation protocoand upper bounds are known, distillable entanglemenknown in few nontrivial cases@2#. Actually, entanglemendistillation is closely related to local state discriminatioTwo counterintuitive facts are found about local state dcrimination. One is that there exist product orthogonal stathat could not be discriminated exactly by local operatand classical communication~LOCC! @3#. The other is thatany two orthogonal multipartite states could be discriminawith certainty by only LOCC operations@4#. Based on theresults of Ref.@4#, the distillable entanglement@1# is calcu-lated for a class of mixed states comprising Bell basis@5,6#.For generic bipartite mixed states, entanglement distillatis an irreversible process in the sense that more pure-entanglement is required to create it than can be distifrom it in the asymptotic limit@7#. However, the process odistillation is reversible in the so-called quasipure mixstate @8#. In this paper, the mixture of multiple copies omaximally entangled states with equal probability@9# isproved to be quasipure. So the distillation process is revible which is not so obvious at first glance. Furthermore,distillable entanglement is calculated explicitly.

We consider the mixed state of the form,

rd(k)5

1

d2 (m,n50

d21

~ ufmn&^fmnu! ^ k, ~1!

where $ufmn&% is the canonical set of mutually orthogonmaximally entangled states~MES! in d3d defined as,

ufmn&51

Ad(j 50

d21

e2p i jn /du j &u j % m&, n,m50,1, . . . ,d21,

~2!

where% means addition modulod.

*Electronic address: dyang@zju.edu.cn

1050-2947/2004/69~2!/024302~3!/$22.50 69 0243

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The situation occurs where Alice and Bob sharek pairs ofqudits, each pair in the same maximally entangled stateorder to share entanglement, they should store the clasinformation in classical memory that identifies which stathe k pairs of qudits are in. Now imagine that by some mfortune, the data in the classical memories are deleted,many ebits can be distilled? As is well known, it is difficuto calculate the distillable entanglement of the mixed staThe main result of the paper is that the mixed state of Eq.~1!is quasipure@8#. For quasipure states, irreversibility does noccur in the entanglement distillation. And the distillable etanglement is easy to calculate. Before the proof of the mresult, let us explain what is the meaning of ‘‘quasipure.’’

A mixed state is quasipure if it can be reversibly tranformed to the following form under LOCC operations,

r5( pir i ^ uf i&^f i u, ~3!

wherer i are separable and orthogonal states that can becriminated exactly by LOCC operations. The physical meing of the quasipure state is that every pure entangled staidentified by a separable ‘‘tag state’’ that can be distinguishby LOCC operations. As the transformation is reversible,the entanglement in the original mixed state is containedthe pure states and can be distilled. As Bennettet al. @3#showed that the set of separable operations is strictly lathan the set of LOCC operations, it is necessary to introdthe tag state distinguishable by LOCC operations. Similawe can define separable-quasipure and PPT-quasipurerespect to the operations allowed. For instance, considerconstructed stater51/9(uc i&^c i u ^ uf i&^f i u, whereuc i& isthe separable pure orthogonal basis in 333 considered inRef. @3# and $uf i&% is the canonical set of MES in 333.Explicitly, this state is separable-quasipure. And the distition is a reversible process if separable operations arelowed. Is it also LOCC-quasipure? Does irreversibility occin this mixed state? If it does, we can conclude thatirreversibility originates from the nonlocal property of thseparable basis. Though we conjecture it is the case, theproof escapes from us. However, we will prove that the stof Eq. ~1! is quasipure though it looks unlike.

Main Result. The mixed state of maximally entanglestates in Eq.~1! is quasipure.

First we introduce the operations that are employed. Tcontrolled-NOT operationC is defined asCu i & ^ u j &5u i & ^ u j

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BRIEF REPORTS PHYSICAL REVIEW A69, 024302 ~2004!

%i&, and the bilateral controlled-NOT operation~BXOR! Bis Bu i &A1ur &B1^ u j &A2us&B25u i &A1ur &B1^ u j % i &A2us% r &B2,where $u i &% is the computational basis. DenoteB(m,n) asthe BXOR operation performed on themth pair ~source! andthe nth pair ~target!, andB act onk copies offmn as

Bfmn^ k5B~1,k!B~2,k!•••B~k21,k!fmn

^ k5fm,0^ (k21)f % km,n .

~4!

For brevity, we denote the projectorufmn&^fmnu as Pmnand omit the normalization for the state unless confusoccurs. Three important properties aboutufmn& are utilized.First, (mPmn and (nPmn are separable. For example, thsubspace spanned by$f0n% can also be spanned by$u i i &,i50, . . . ,d21%. For the subspace spanned by$fm0%, it isnot so obvious, but can be proved to be separable in thebasis$uek

A&%, $uekB&%, whereukA&51/Ad( j 50

d21e2p i jk /duejA& and

uekB&51/Ad( j 50

d21e2p i jk /du j B&. Second, with varying n,(mPmn are orthogonal to each other and can be distinguisexactly by LOCC. So do(nPmn with varying m. Third, forfixed n and varyingm, ufmn& can be transformed touf0m& bylocal basis transformation.

Proof.

rd(k)→

B(m

Pm0^ (k21)(

nP% km,n . ~5!

Denote the maximal common factor betweenk and d asg(k,d). When g(k,d)51, we know that $ % km,m50,1, . . . ,d21% is a permutation of the number$0,1, . . . ,d21%. Thed mixed states(nP% km,n are separableand orthogonal to each other. Sord

(k) is quasipure. Wheng(k,d).1, the problem is a little complicated. Supposek

5gk̃,d5gd̃, where k̃,d̃ are integers and satisfyg( k̃,d̃)51. For m50,1, . . . ,d21, it can be written asm5sd̃

1t,s50,1, . . . ,g21,t50,1, . . . ,d̃21. Then

% d~km!5 % (gd̃)gk̃~sd̃1t !5 % (gd̃)gk̃t5g% d̃k̃t. ~6!

Becauseg( k̃,d̃)51, $ % d̃k̃t,t50,1, . . . ,d̃21% is a permuta-tion of the numbers$0,1, . . . ,d̃21%. For brevity, setT(t)5g% d̃k̃t that is different fort50,1, . . . ,d̃21. So

(m

Pm0^ (k21)(

nP% km,n5(

tH(

sPsd̃1t,0

^ k21 J H(n

PT(t),nJ .

~7!

For everyt, (nPT(t),n is separable and orthogonal and candiscriminated by local operations. Now, we show th(sPsd̃,0

^ (k21) is quasipure as for the other casestÞ0,

(sPsd̃1t,0^ (k21) can be transformed to(sPsd̃,0

^ (k21) by permutationof the basis states. Note that a MES ind3d can be analyzedinto two terms as

Psd̃,0~d!5Ps,0~g!P00~ d̃!, ~8!

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wherePs,0(g) represents the projector of the MES ing3g.So

(s

Psd̃,0^ (k21)

5(s

Ps,0^ (k21)~g!P00

^ (k21)~ d̃!. ~9!

Under the dual basis,(sPs,0^ (k21)(g) can be written as

(sP0,s^ (k21)(g). Further employing the operationB,

(s

P0,s^ (k21)~g!→

BP0,s

^ (k22)~g!(s

P0,s~g!. ~10!

So (sPsd̃,0^ (k21) is quasipure. As all the operations are loc

reversible operations, the original state is quasipure. Thomeasurements are performed during the process, theynondemolation measurements and do not destroy the sFurthermore, the entanglement ofrd

(k) can be obtained fromEqs.~7!, ~9!, and~10! as

E~rd(k)!5~k21!ln d2 ln g~k,d!. ~11!

Here the distillable entanglement is equal to the entanment cost as the state is quasipure. An interesting resuthat E(rd

(d21))5E(rd(d))5(d22)lnd,E(rd

(d11))5d ln d. Thestorage ofd copies of MES ind3d is inefficient if the clas-sical information is lost. Storingk copies of MES is moreefficient whenk is coprime to the dimensiond.

In the following, we give two examples for illuminatingthe result.

Example 1 (d53) as shown in Table I. We just show thar3

(3) is quasipure. Other cases can be proved similarly. Unthe operationB,

r3(3)→

B(m

Pm0^ 2(

nP0n . ~12!

Next (mPm0^ 2 is shown to be quasipure.fm0 can be trans-

formed to f0m in the dual basis. PerformingB again,

(mP0m^ 2→

BP00(mP0m that is quasipure. It is explicit tha

E(r3(2))5 ln 3.

Example 2 (d56) as shown in Table II. Now we explainr6

(3) is quasipure. Under the operationB,

TABLE I. Under the operationB, what happens tok copies ofMES in 333. The last column shows the entanglement ofr3

(k) .

k f0n^ k f1n

^ k f2n^ k E(r3

(k))

1 f0n f1n f2n 02 f00^ f0n f10^ f2n f20^ f1n ln 33 f00

^ 2^ f0n f10

^ 2^ f0n f20

^ 2^ f0n ln 3

4 f00^ 3

^ f0n f10^ 3

^ f1n f20^ 3

^ f2n 3ln 3

2-2

BRIEF REPORTS PHYSICAL REVIEW A69, 024302 ~2004!

TABLE II. Under the operationB, what happens tok copies of MES in 636. The last column shows theentanglement ofr6

(k) .

k f0n^ k f1n

^ k f2n^ k f3n

^ k f4n^ k f5n

^ k E(r6(k))

1 f0n f1n f2n f3n f4n f5n 02 f00^ f0n f10^ f2n f20^ f4n f30^ f0n f40^ f2n f50^ f4n ln 62ln 23 f00

^ 2^ f0n f10

^ 2^ f3n f20

^ 2^ f0n f30

^ 2^ f3n f40

^ 2^ f0n f50

^ 2^ f3n 2ln 62ln 3

4 f00^ 3

^ f0n f10^ 3

^ f4n f20^ 3

^ f2n f30^ 3

^ f0n f40^ 3

^ f4n f50^ 3

^ f2n 3ln 62ln 25 f00

^ 4^ f0n f10

^ 4^ f5n f20

^ 4^ f4n f30

^ 4^ f3n f40

^ 4^ f2n f50

^ 4^ f1n 4ln 6

6 f00^ 5

^ f0n f10^ 5

^ f0n f20^ 4

^ f0n f30^ 5

^ f0n f40^ 5

^ f0n f50^ 5

^ f0n 4ln 67 f00

^ 6^ f0n f10

^ 6^ f1n f20

^ 6^ f2n f30

^ 6^ f3n f40

^ 6^ f4n f50

^ 6^ f5n 6ln 6

iverect

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rkng

r6(3)→

B~P00

^ 21P20^ 21P40

^ 2!(n

P0n1~P10^ 21P30

^ 2

1P50^ 2!(

nP3n . ~13!

The reason thatP00^ 21P20

^ 21P40^ 2 is quasipure is

f005~ u00&1u11&1u22&)~ u00&1u11&),

f205~ u01&1u12&1u20&)~ u00&1u11&),

f405~ u02&1u10&1u21&)~ u00&1u11&),

P00^ 21P20

^ 21P40^ 25„P00

^ 2~3!1P10^ 2~3!1P20

^ 2~3!…P00^ 2~2!.

~14!

From the d53 case,(mPm0^ 2(3) is quasipure.P10

^ 21P30^ 2

1P50^ 2 can be proved similarly. Sor6

(3) is quasipure andE(r6

(2)) is calculated to be 2ln 62ln 3.

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.

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Note that different conclusion is obtained@10# for k52,whererd

(2) is demonstrated to be separable. And the relatentropy of entanglement is calculated based on this incorconclusion in a few papers. However, we prove thatrd

(2) isdistillable whend.2.

As is well known, irreversibility is generic in the entanglement distillation@7#. In this paper, we show that thmixture of multiple copies of maximally entangled statesquasipure and prove that the entanglement distillation ireversible process for this nontrivial class at first glanWhether only in the quasipure states is distillation reversideserves to be considered. Another important questiohow to decide the quasipure mixed states.

D. Yang thanks S. J. Gu for helpful discussion. The wois supported by the NNSF of China, the NSF of ZhejiaProvince ~Grant No. 602018!, and the Guang-Biao CaoFoundation at Zhejiang University.

t

pa-thect

,

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.

@7# G. Vidal and J.I. Cirac, Phys. Rev. Lett.86, 5803~2001!; Phys.Rev. A 65, 012323~2002!; G. Vidal, W. Dur, and J.I. Cirac,Phys. Rev. Lett.89, 027901~2002!.

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@9# S. Hamieh and H. Zaraket, e-print quant-ph/0304107. Thisper calculates the distillable entanglement of the mixture ofform Eq.~1!. However, the calculation is based on an incorreresult of Ref.@10# as pointed out in Ref.@6#. As a consequenceit is incorrect.

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