PHYSICAL REVIEW A, VOLUME 63, 042306

Additivity of the entanglement of formation

Fabio Benatti1 and Heide Narnhofer2

1Department of Theoretical Physics, University di Trieste, Strada Costiera 11, I-34100, Trieste, Italy2Institute of Theoretical Physics, University of Vienna, Boltzmanngasse 5, A-1090, Wien, Austria

~Received 22 September 2000; published 16 March 2001!

We address the question of whether the entanglement of formation is additive over tensor products of fullmatrix algebras, and derive a necessary and sufficient condition for optimality of vector states. By discussingtwo particular cases, we suggest that such a condition may be used either to prove or to find a counterexampleto additivity.

DOI: 10.1103/PhysRevA.63.042306 PACS number~s!: 03.67.2a

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I. INTRODUCTION

Entanglement plays a central role in quantum informattheory, the sharing of entanglement between sender anceiver allowing for quantum teleportation@1#, quantum su-perdense coding@2# and more in general for reliable tranmission of quantum information @3–5#. Quantifyingentanglement in a satisfactory way is thus an important is

If sender and receiver share two spin-1/2 particles ipure state,rÞr1^ r2 over the algebra of observablesM1^ M2. The restrictionsr�M1,2

of the stater to the two factoralgebras have the same von Neumann entropy, and a nameasure of its entanglement content is

Ef~r!ªS~r�M1,2!52Tr~r�M1,2

ln r�M1,2!. ~1!

By means of local quantum operations and classical cmunication, entanglement can be distilled@5# and diluted@4#.If sender and receiver shareN pure nonmaximally entanglestatesr of pairs of spin-1/2 particles, then there is a purication protocol providing a number of singlet states asymtotically equal toNEf(r) ~the von Neumann entropy involving base 2 logarithms!. Conversely, in order to createN (Nlarge! copies of the pure stater, sender and receiver musshare approximatelyNEf(r) singlet states beforehand.

When the shared stater is not a pure state, the entanglments of formation and distillation do not coincide@6–8#;one has to distinguish between the maximal numberEf(r) ofsinglets needed to produce the stater by means of localquantum operation and classical communication, andmaximal numberEd(r) of singlets that can be produced bthe same means out of the stater.

In this paper we will be concerned with the entanglemof formation whenr is a mixed state over the tensor produM1^ M2 of two finite-dimensional full matrix algebras. Ithis caseEf(r) is defined by@1#

Ef~r!5minH(i PI

piS~s i�M1!J , ~2!

the minimum being computed over all decompositionsr5( i PI pis i of r into pure statess i over M1^ M2.

We now consider a factor stater ^ s, with r a state overthe algebraM1^ M2 and s a state on the algebraM3^ M4, where theMi ’s are not necessarily identical full ma

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trix algebras. For the sake of clarity, we indicate explicitlyEf(r) the subalgebras to which the states have to bestricted. The question we want to address is that of the ativity of the entanglement of formation, namely, whether

Ef~r ^ s;M1^ M3!5Ef~r;M1!1Ef~s;M3!. ~3!

The request above expresses a strong form of additivityweaker additivity condition requires Eq.~3! with r5s andMi5M , wherei 51, 2, 3, and 4.

Additivity in either form is guaranteed when the states apure, but there is only numerical evidence that it should hfor nonpure states. On the other hand, it seems reasonabhave this quality among the requirements to be compwith by any entanglement measure@4,6#.

In Sec. III we provide an inequality necessary to be sisfied by optimal projectors at which the minimum in Eq.~2!is attained. In Sec. IV we use this to show~strong! additivityin one special case and affinity in another case, which poto a general strategy for showing additivity in general. Tnecessary techniques@9–11# will be presented in Sec. III.They refer to the so-called entropy of a subalgebra@9–12#,which will be introduced in Sec. II.

II. ENTROPY OF A SUBALGEBRA

The additivity of the entanglement of formation or its faure have a quantum information theoretical counterpthere is indeed a connection between Eq.~2! and the maxi-mal accessible informationI (r) of a quantum source described by a mixed stater on a matrix algebraM @14#. If r5( l PLqlr l , thenI (r)ªsupB I B(r), where

I B~r!52(i PI

„Tr~rbi !…In„Tr~rbi !…

1(l PL

ql(i PI

„Tr~r lbi !…In„Tr~r lbi !…, ~4!

the maximum being computed over all choicesB5$bi% ofpositivebiPM such that( i PIbi51M . Optimal choices cor-respond to an optimal detection of the classical informatcarried by the quantum statesr l .

©2001 The American Physical Society06-1

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FABIO BENATTI AND HEIDE NARNHOFER PHYSICAL REVIEW A63 042306

We assumer.0. Then, fromqlr l<r, it follows thatthere exists a unique choice of operators 0,alPM , l PL,with ( l PLal51, such that@15#

qlr l5AralAr, ql5Tr~ral !. ~5!

Let A be a commutativeL-dimensional algebra with identity1A and orthogonal projectorsAl , with ( lAl51A . The mapgA :A→M , obtained by linear extension ofAl→gA(Al)5al , is positive, and gA(1A)51M . Therefore, givenany state s on M, the linear functionals+gA :A→C,s+gA(Al)5Tr(s al) defines a state onA. Using Eq.~5!,

Tr~r lbi !5Tr~rbi !

Tr~ral !Tr~s i

Bal !, s iBª

ArbiAr

Tr~rbi !. ~6!

SettingpiB5Tr(rbi), Eq. ~4! becomes

I B~r!5S~r+gA!2(i PI

piBS~s i

B+gA!. ~7!

Therefore,I (r) is the maximum of Eq.~7! over all possibledecompositions ofr into pure states:

I ~r!5S~r+gA!2 minr5(i pis i

(i

piS~s i+gA!. ~8!

If N is a subalgebra ofM, substituting the restrictionsr�N , s i�N for r+gA or s i+gA , respectively we obtain theso-calledentropy of a subalgebra@9–12#:

Hr~N!ªS~r�N!2 minr5( i pis i

(i

piS~s i�N!. ~9!

The latter quantity is the building block of an extensionthe Kolmogorov-Sinai dynamical entropy~or entropy perunit time! to the quantum realm@13#. According to theabove, the link to the entanglement of formation is as flows: Ef(r;M1)5S(r�M1

)2Hr(M1) @14#.Note that the above expression can be rewritten by me

of the relative entropyS(r;s)ªTr„s(logs2logr)… of twostatesr ands over M:

Hr~N!5 supr5( i pis i

H(i

piS~r�N ;s i�N!J .

If N5M , choosing s i to be the eigenprojectors or, Hr(M )5S(r), the von Neumann entropy of a state cbe expressed in terms of the relative entropy@12#. Thus thenotion of relative entropy might play as useful a role for tentanglement of formation as that of the relative entropythe entanglement in the case of entanglement of distilla@8,9#.

From a more abstract and speculative perspective, asvon Neumann entropy is additive over tensor productsadditivity fails for the entanglement of the formation it alsfails for the entropy of a subalgebra. Then, interpretingentropy of a subalgebra as a measure of its information ctent with respect to a given state, we would be led to c

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clude that there is more information contained in the tenproduct of two independent subalgebras, with respect totensor product of two states over them, than the sum ofinformation contents of the two factors@15,16#.

III. OPTIMAL PROJECTORS

In the following we try to use some of the propertiesHr(N) to investigate the general question as whether eqity ~3! holds. We shall call ‘‘optimal’’ those projectorsuc&^cu appearing in an optimal decomposition at which tminimum in Eq.~9! is attained, as well as the vectors onwhich they project.

If Ef(r;M1) and Ef(s;M3) are achieved at optimal decompositionsr5( l plr l ands5( jqjs j , the factorized de-composition r ^ s5( j ,lqj plr l ^ s j contributes withEf(r1 ;M1)1Ef(s;M3) to Ef(r ^ s;M1^ M3). However,the latter need not be optimal, and the strict inequaEf(r ^ s;M1^ M3),Ef(r1 ;M1)1Ef(s;M3) is not ex-cluded. In fact, a decomposition

r ^ s5(i

a i uc i&^c i u, a i.0, (i

a i51, ~10!

might be optimal with thec i entangled states overM1^ M3.

The difficulty of the task might be grasped by considerithe following argument; consider the Schmidt decomposituc i&5( jb i j uf i j

12& ^ uf i j34&, zuc i uz51, b i j .0, where, for

fixed i the uf i j12& ’s and uf i j

34& ’s form orthonormal bases oveM1^ M2, respectivelyM3^ M4. If it held that

S~ uc i&^c i u�M1^ M3!>(

jb i j

2„S~ uf i j

12&^f i j12u�M1

!

1S~ uf i j34&^f i j

34u�M3!…,

additivity would follow because tensor-product states wothen never be worse than correlated ones. However, a cterexample to the previous sufficient inequality has befound @16#. It turns out that some results@9–11# concerninggeneral properties of optimal decompositions, once adapto the entanglement of formation can be used to successaddress the additivity issue.

Proposition 1. Let r be a state onM1^ M2 , Ef(r;M1)that is achieved atr5( l plr l , andU be a unitary operator onM1^ M2 such thatUM1U†5M1. Then Ef(U

†rU;M1) isachieved at the optimal decompositionU†rU5( l plU

†r lU.Proposition 2. Let r be a state onM1^ M2, and

Ef(r;M1) be achieved atr5( l plr l , That is, Ef(r;M1)5( l plS(r l�M1

); then

Ef~s;M1!5(j

qjS~r j�M1!, ~11!

wheres5( jqjr j is any linear convex combination of optmal states ofr.

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ADDITIVITY OF THE ENTANGLEMENT OF FORMATION PHYSICAL REVIEW A 63 042306

To the above, we add a new property. For the sakesimplicity, we will denote the minimum in Eq.~9! byEf(r;N) even when the subalgebraN is not ~embedded into!a factor of an algebraic tensor product.

Proposition 3. Let uc l&^c l u, l 51,2, . . . ,L, be optimalprojectors over optimal, linearly independent, normalizvectorsuc l& contributing toEf(r;N). Let g lPC, 1< l<L,be complex parameters, and denote

s lªuc l&^c l u�N , s i jªuc i&^c j u�N ,

sovi j ~g!ªg ig j* s i j 1g j* g is j i ,

s~gW !ª(l 51

L

ug l u2s l1 (iÞ j 51

L

g ig j* s i j , s~gW !ªs~gW !

Tr„s~gW !….

Then,

(l 51

L

ug l u2S~s l !21

2 (iÞ j 51

L

Tr„sovi j ~gW !ln s i…

<Tr„s~gW !…S„s~gW !…. ~12!

Conversely, if Eq.~12! holds for all choices ofg l , thengiven any density matrix r5( l 51

L l l uc l&^c l u, 1>l l

>0, ( l 51L l l51, one obtainsEf(r;N)5( l 51

L l lS(s l).Proof of Necessity: With «.0, let

r«,gWª(l 51

L

ug l u2~11«2!uc l&^c l u.

Note that Trr«,gÞ1. However, the expression in Eq.~9!makes sense forr>0 with TrrÞ1, too and, of courseEf(r;N)5(Tr r)Ef(r/Tr r;N). Therefore,c l ’s, being opti-mal and normalized by the assumption, Proposition 2 yie

Ef~r«,gW ;N!5(l 51

L

ug l u2~11«2!S~s l !.

Furthermore, with uf i j &ªg i uc i&2«g j uc j&, i , j 51,2, . . . ,L, one can check thatr«,gW can also be decomposed as

r«,gW 5 (i , j 51

L1

Luf i j &^f i j u1

2«

Ls~gW !.

The latter decomposition cannot contribute more thEf(r«,gW ;N): Ef(r«,gW ;N)< f («), where

f ~«!ª (i , j 51

L zuf i j uz2

LS„r i j ~«!…1

2«

L„Tr s~gW !…S„s~gW !…,

with

r i j ~«!ªuf i j &^f i j u�N

zuf i j uz25

ug i u2s i1«2ug j u2s j2«sovi j ~gW !

ug i u21«2ug j u22« Tr„sovi j ~gW !…

.

Note thatr i j («50)5s i , while

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r i jªdr i j ~«50!

d«5„Tr sov

i j ~gW !…s i2sovi j ~gW !

ug i u2.

Then, expanding

S„r i j ~«!…5S~s i !2« Tr~ r i j ln s j !1O~«2!,

inequality ~12! must then hold at first order in«.Proof of sufficiency: By assumption, inequality~12! holds

for all possible choices of setsG j5$g lj : l 51,2, . . . ,L% of

complex parameters. Let us chooseG j and real parametera j>0 such that( ja j ug l

j u25l l , l 51,2, . . . ,L, ( l 51L l l51

and( ja jg lj (gk

j )* 50 for all lÞk. From Eq.~12! it then fol-lows that

(l 51

L

l lS~s l !<(j

a j„Tr s~gW l !…S„s~gW l !…. ~13!

In the above inequality, the left-hand side is the contributto Ef(r;N) of the decompositionr5( l 51

L l l uc l&^c l u(* ),whereas the right-hand side is the contribution toEf(r;N) ofthe decompositionr5( ja j„( lg j

l uc l&…„(k(gkj )* ^cku…(** ).

As the density matrixr acts on theL-dimensional subspacspanned by the linearly independent vectorsc l , l51,2, . . . ,L, (** ) represents the most general decompotions of r. Thus, from Eq.~13!, it follows that (* ) is anoptimal decomposition forr and uc l& optimal vectors.

Remark. With the choicesg i51, g j5dPR and g25 id, while g l50 whenlÞ i , j , by expanding Eq.~12! withrespect tod, we obtain6d Tr„(s i j 6s j i ) (ln si2ln sj)…<0as a first-order condition ind, from which

Tr„sovi j ~gW !~ ln s i2 ln s j !…50, ;g lPC,

l 51,2, . . . ,L. ~14!

Therefore, the statess i and s j are interchangeable in Eq~12!.

IV. APPLICATIONS

We discuss two cases where the necessary and sufficondition @Eq. ~14!# proves to be useful.

Case 1: The state in Eq.~10! factorizes over M3^ M4 : s5s3^ s4. In this case the additivity ofEf is animmediate consequence of the fact thatEf cannot be in-creased by local operations, like appending and discardsystems locally. However, we include the following proofan illustration of how condition~14! can be used, and at thsame time as a hint how to generalize its application.

Let Ef(r ^ s;M1^ M3) be achieved at an optimal decomposition made of statesuc i& entangled overM1^ M3.Let uc i&5( j ci j uf i j

13& ^ uf i j24& be Schmidt decomposition

with ci j >0 and uf i j13& and uf i j

24& forming, for each fixedi,orthonormal bases in the first and third factors, respectivThus

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FABIO BENATTI AND HEIDE NARNHOFER PHYSICAL REVIEW A63 042306

r ^ s3^ s45(i

a i uc i&^c i u

5(i jk

a ici j cikuf i j13&^f ik

13u ^ uf i j24&^f ik

24u. ~15!

Further, letux l3& be eigenvectors ofs3 and consider the uni

tary operator (PM3)

U l512~12 i !Pl , Plªux l3&^x l

3u.

From Proposition 1 it follows that the vectorsUl uc i&, andUl511^ 12^ U l ^ 14 also giveEf(r ^ s;M1^ M3). Let usconcentrate onuc1&; together withUl uc1&, these have tosatisfy Eq.~12! for all g. Then, according to Propositionwith L22, g151, andg25g,

s15(j

c1 j2 uf1 j

13&^f1 j13u�M1^ M3

~16!

s25Uls1Ul† , sov~g!5gUls11g* s1Ul

† . ~17!

Taking g51, it follows that s(1)5Pls1Pl /Tr(Pls1)and sov(1)52s12(12 i )Pls12(11 i )s1Pl . Inequality~12! thus becomes

22 Tr~Pls1 ln s1!<Tr~Pls1!S„s~1!…. ~18!

We developuf1 j13&5(pbpl

j uxp1& ^ ux l

3&, along an orthonormabasis for the factorM1, then, by means of the spectral dcomposition~16!, settingD j lª^f j

13uPl uf j13&5(pubpl

j u2, we

obtain Pls1Pl5(( j c1 j2 D j l Qjl ) ^ Pl , where Qjlªux j l

1 &^x j l1 u

and ux j l1 &5(p(bpl

j /D j l )uxp1&.

Insertion into Eq.~18! leads to

0>(j

c1 j2 D j l ln

D j l

c1 j2 Tr~Pls!

>(j

c1 j2„D j l 2c1 j

2 Tr~Pls!…,

~19!

the latter inequality coming fromx ln x/y>x2y and holdingfor all orthogonal projectorsPl . Since( j c1 j

2 51 and( lD j l

51, summing overl we obtain thatc1 j51 for one j andc1k50 if kÞ j . Thus, the supposed optimal vectorsuc i& mustbe of the formuc i&5uf i

13& ^ uf i24&, and the supposed optima

decomposition@Eq. ~15!# must reduce to

r ^ s3^ s45(i

a i uf i13&^f i

13u ^ uf i24&^f i

24u. ~20!

Introducing the Schmidt decompositionsuf i13&

5( jd i j13uf i j

1 & ^ uf i j3 &, uf i

24&5( ld i l24uf i l

2 & ^ uf i l4 & and tracing

with respect to them overM2^ M4, orthogonality yields

r5(i

a i(j l

~d i j13!2~d i l

24!2uf i j1 &^f i j

1 u ^ uf i l2 &^f i l

2 u. ~21!

We thus conclude that a decomposition ofr ^ s3^ s4 as inEq. ~15! can be optimal with respect toM1^ M3 only if r is

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not entangled overM1^ M2, in which case Ef(r ^ s3^ s4 ;M1^ M3)50 is obviously additive. Ifr is entangledover M1^ M2, the contradiction is avoided only if the optmal decompositions have the form

r ^ s3^ s45(i

a i uf i12&^f i

12u ^ uf i34&^f i

34u. ~22!

Thus the optimal states cannot carry any entanglementM1^ M3, and additivity follows.

The second case we want to discuss is somewhat theposite of the previous one, where we proved that optimprojectors for the tensor products are products of optimprojectors for the factors. Instead, we want to show thputting together pairs of optimal projectors for the factowe obtain optimal decompositions.

Case 2: We consider the statesrªuf12&^f12u^ uf34&^f34u and rªuf12&^f12u ^ uf34&^f34u over M1^ M2

^ M3^ M4, with uf12& and uf12& optimal vectors for someEf(s;M1) anduf34&5uf3& ^ uf4&, uf34&5uf3& ^ uf4& overM3^ M4. Then we construct the mixture

rl5lr1~12l!r. ~23!

Because of the assumptions, it follows that

Ef~r;M1^ M3!5S~ uf12&^f12u�M1!,

Ef~ r;M1^ M3!5S~ uf12&^f12u�M1!. ~24!

The contribution toEf(rl ;M1^ M3) of decomposition~23!is thus

ElªlS~ uf12&^f12u�M1!1~12l!S~ uf12&^f12u�M1

!

~25!

and we want to prove that this is the best we can have.We proceed as follows: as for Eq.~13!, a general decom-

position of rl is of the formrl5( ia i uc i&^c i u where uc i&5uf12

^ f34&1g i uf12^ f34&, with a i.0 and

(i

a i5l, (i

a i ug i u2512l, (i

a ig i50. ~26!

We now setbª^f4uf4&, aªA12ubu2 and constructuc4&ªuf4&2buf4&/a such thatzuc4uz51 and ^c4uf4&50. Wecan thus rewrite

uc i&5ai uf i123

^ f4&1ag i uf12^ f3

^ c4&, ~27!

where

uf i123&ª

uf12^ f3&1bg i uf12

^ f3^ f4&

ai,

ai2ª11ubu2ug i u212 Re~bg i^f

12uf12&^f3uf3&!.

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ADDITIVITY OF THE ENTANGLEMENT OF FORMATION PHYSICAL REVIEW A 63 042306

With uc i&ªuc i&/Ad i and d iªai21a2ug i u2, decomposition

~23! readsrl5( ia id i uc i&^c i u. The contribution of the latterto the entanglement of formationEf(rl ;M1^ M3) is

Eª(i

a id iS~ uc i&^c i u�M1^ M3!. ~28!

From the orthogonality ofc4 andf4, it follows that

uc i&^c i u�M1^ M35

ai2

d is i

1231a2ug i u2

d is123,

where

s i123

ªuf i123&^f i

123u�M1^ M3,

s123ªuf12&^f12u�M1

^ uf3&^f3u�M3. ~29!

The concavity of the von Neumann entropy yields

E>(i

a i$ai2S~s i

123�M1^ M3!1a2ug i u2S~ uf12&^f12u�M1

!%.

~30!

As done before, we construct

uc3&ªuf3&2duf3&

c,

such that^c3uf3&50, where dª^f3uf3&, cªA12udu2,and

uf i123&ª

bi uc i12

^ f3&1bcg i uf12^ c3&

ai,

uc i12&ª

uf12&1bdg i uf12&bi

, ~31!

bi2ª11ubu2udu2ug i u212 Re~bdg i^f

12uf12&!.

Introducing the Schmidt decompositions overM1^ M2 :uc i

12&5( j ci j uf i j1 & ^ uf i j

2 &, uf12&5( ldl uf l1& ^ uf l

2&, and set-

ting r1ªuc i12&^c i

12u�M1, r2ªuf12&^f12u�M1

, because of the

orthogonality off3 and c3, the states i123 in Eq. ~29!, re-

stricted toM1^ M3, can be represented as

s i1235

1

ai2S bi

2r1 cbib* g i* Ar1VAr2

cbibg iAr2V†Ar1 ubu2c2ug i u2r2D

51

ai2S biAr1V 0

cbg iAr2 0D S biV

†Ar1 cb* g i* Ar2

0 0D ,

where Vª( j ,l^f l2uf i j

12&uf i j1 &^f l

1u is a unitary operator andAr1VAr25uc i

12&^f12u�M1. Since s i

1235A†A has the sameentropy as

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AA†51

ai2S bi

2V†r1V1ubu2c2ug i u2r2 0

0 0D ,

the concavity and invariance under unitary transformatioof the von Neumann entropy yield

S~s i123!5S~AA†!>

bi2

ai2 S~r1!1

c2ubu2ug i u2

ai2 S~r2!;

thus Eq.~30! becomes

E>(i

a i$bi2S~r1!1ug i u2~a21c2ubu2!S~r2!%. ~32!

From the assumed optimality of the statesuf12& anduf12&in Eq. ~31! for some state onM1^ M2 when restricted toM1, we can use the necessary condition~12!. According tothe notation of Proposition 3, we haves15r2 , s2

5uf12&^f12u�M1, g5g i* b* d* , s(g)5r1, and

sov~g!52b* d* g i* Ar2V†Ar12bdg iAr1VAr2.

Thus, Eqs.~12! and ~26! imply that

E>(i

a i$S~ uf12&^f12u�M1!1ug i u2S~ uf12&^f12u�M1

!

2Tr„sov~b* d* g i* !ln r2…%

5El , ~33!

whereEl is the contribution@Eq. ~25!# to the entanglemenof formationE(rl ;M1^ M3) of decomposition~23!, whichturns out then to be already optimal.

V. CONCLUSIONS

In this paper we have derived a necessary and sufficcondition for the optimality of vector states relative to thentropyHr(N) of a subalgebraN. We have related the latteto the entanglement of formation, and applied it to two cocrete cases.

~i! In case 1, the stater ^ s was chosen separable ovM3^ M4, and we proved additivity of the entanglementformation.

~ii ! In case 2, the staterl5lr1(12l) r was not itself atensor product, but a mixture of two statesr andr separableover (M1^ M2) ^ (M3^ M4) with the components of bothover M3^ M4 being pure and separable and those overM1^ M2 optimal for the entropyHn(M1) of some staten onM1^ M2 relative to the subalgebraM1. In this case, weproved not the additivity of the entanglement of formatiobut the affinity of it, in the sense that@compare Eq.~24!#

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FABIO BENATTI AND HEIDE NARNHOFER PHYSICAL REVIEW A63 042306

Ef„lr1~12l!r;M1^ M3…

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