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Entanglement Optimization for Pairs of Qubits Juliane Strassner and Christopher Witte Institut fu¨r Theoretische Physik,Technische Universita¨t Berlin, D-10623 Berlin, Germany (February 1, 2008) Local Operations enhancing the entanglement of bipartite quantum states are of great interest in quantum information processing. Subject of this paper are local selective operations acting on single copies of states. Such operations can lead to larger entanglement with respect to a certain measure as studies before by the Horodeckis and A. Kent et al. [1–3]. We present a complete characterisation of all local operations yielding optimal entanglement for pairs of qubits,extending formerresultsofA.Kentetal. Weintroduceanewtechniquefortheclassificationofstatesaccording to theirbehaviourunderentanglement optimizing operations, using theentanglement properties of the support of density matrices. 1 0 0 The key resource of quantum information theory is part of this letter. 2 quantum entanglement. It is of vital importance to find We use the following facts (see [6,2,3]): ameasureforentanglementandinvestigatehowitcanbe n a increased for a given state. Many applications of quan- • The considered distillation protocols involve J tum information theory, like teleportation, dense coding LQCC’s of a special form: after the operation the 2 or quantum cryptography, require maximally entangled state ρ takes the form 2 states of two qubits shared by two distinct parties, tra- A BρA B ditionally called Alice and Bob. Maximally entangled Θ(ρ)= ⊗ ⊗ (1) 2 Tr(A BρA B) v Pairs, however, have to be prepared in a common quan- ⊗ ⊗ 6 tum process at a certain place. In order to share them 2 between Alice and Bob at least one of the two particles The entanglement of formation (EF) of a state of • 1 mustbe sentthroughaquantumchannel(e.g.anoptical two qubits can be calculated by the formula of 2 fibre). During this process interaction with the environ- Wootters as 1 mentleadstoalossofentanglement,the stateevolvesto 0 1+ 1 C2(ρ) 0 a non-maximally entangled mixed state. EF(ρ)=H( − ) 2 / Thereforepurification,alsocalleddistillation,becomes p h necessary: aprocesswhichincreasestheentanglementof p with H(p)=plog p (1 p)log (1 p) - given pairs by local operations and classical communi- 2 − − 2 − t and the concurrence C, which is given by C = n cation (LQCC) performed by Alice and Bob. The first max 0,λ λ λ λ , the difference of the a purificationprotocolwas presented by Bennett et al. [5]. { 1 − 2 − 3 − 4} u It involves local operations acting on many pairs, called eigenvalues λi of ρ˜ρ=σ2⊗σ2ρ¯σ2⊗σ2ρ, where the q eigenvalues are taken in decreasing order. collective operations. Using such a collective scheme one : v can obtain pure maximally entangled particles from any Linden,MassarandPopescuhaveshownin[7]that Xi given two-spin-1-state, even from mixed ones. • under operationsofthe form(1)the eigenvaluesλ 2 i r Recent publications raised the question, if it is also of the matrix ρ˜ρ transform as a possibletopurifyarbitrarytwo-spin-1-statesifonlynon- 2 ′ collective operations are allowed, i.e. operations that act λi =c1λi (2) oneachpairindividually. Itwasshownthatthisispossi- ble only for pure states. Mixed states of two qubits can- with a factor c that is independent of i. 1 not be purified to maximally entangled states by local As a consequence the ratio of the eigenvalues is operations. Nevertheless there are mixed states whose constant under local operations. entanglement can be increased by such protocols. For some of these states it is even possible to reach maxi- Wewillshowthatthemainqualitativefeatureofdensity mal entanglement, but only with vanishing probability, matrices with respectto entanglement optimizationpro- i.e. there is no limit for the entanglement that can be cedures is the number of product vectors in the support distilled from these states. This process was introduced of this matrix. by Horodecki et al. [1] and called quasi-distillability. Thesupport ofadensitymatrixistheorthogonalcom- In this letter we classify states of two qubits with re- plementofitskernel,i.e. forselfadjointmatricesidentical specttotheirmaximaldistillableentanglement. Firstwe with its range. We can also think of the support as the presenttheentanglementpropertiesofthesupportofthe linear spanofthe eigenvectorswith non-vanishing eigen- considereddensitymatrices,whichwillleadtotheclassi- values. This subspace of C2 C2 determines a subset ⊗ fication givenbelow. The proofs will be given in the last of the state space containing all states having support 1 included in this subspace. Such a subset of state space Let ρ be an entangled state on C2 C2. Set ⊗ is called a face in convex analysis. The different faces of a state space over a tensor product Hilbert space can be Mρ :=sup EF(Θ(ρ)):Θ local operation . { } characterised firstly by the dimension of the respective If ρ is Bell-diagonal, it has been shown in [2] that no subspace (i.e. the rank of the generic elements) and sec- localoperationcanincreasethe EntanglementofForma- ondly by the structure of the subset of product vectors tion E (ρ) of the state, i.e. M = E (ρ). Thus we in this subspace. The first property is invariant under F ρ F assume here that ρ be not Bell-diagonal. Then one of all linear isomorphisms of the Hilbert space. The second the following holds true: one is invariant under factorising linear isomorphisms. In the simplest case of two qubits a complete classifi- I. rankρ = 4. As seen in [2] there exists a local op- cation can be done. For that purpose we have a look at eration Ξ, such that E (Ξ(ρ))=M <1 and Ξ(ρ) F ρ all possible subspaces U C2 C2 (see also [4] for some is Bell-diagonal. We call this property of the state ⊆ ⊗ details): incomplete distillability. A. The case dimU = 4 is trivial. There is only one II. rankρ = 3. M < 1. Either (i.) the kernel of ρ such subspace: C2 C2 itself. ρ contains no product vector and a local opera- ⊗ tion Ξ exists, such that E (Ξ(ρ)) = M and Ξ(ρ) F ρ B. For dimU = 3 there are two cases, easily charac- is Bell-diagonal, i.e. ρ is incompletely distillable. terised by their one dimensional orthogonal com- Or (ii.) the kernel of ρ is the linear span of a plement (the kernel of the respective matrices). productvectorandthereisnolocaloperationwith Either the complement is factorising and the sub- E (Ξ(ρ))=M . Inthiscasewecanfindasequence F ρ space contains two factorising hyperplanes, or the of local operations Ξ , with σ =lim Ξ (ρ) n n n n complement is entangled and the subspace con- Bell-diagonal and E{ (σ}) = M . These s→ta∞tes will F ρ tains a cone-shaped set of factorising vectors. be called incompletely quasi-distillable. III. rankρ = 2. Either (i.) the support of ρ contains exactly one linear independent product vector. In this case M = 1, but there is no local operation ρ with E (Ξ(ρ)) = 1. Instead we find a sequence F of local operations Ξn n N, with limn Ξn(ρ) a Bell-state (i.e. ρ{is }qu∈asi-distillable).→∞Or (ii.) the support of ρ contains exactly two linear independent product vectors and it exists a local operation Ξ, such that E (Ξ(ρ)) = M and Ξ(ρ) is F ρ Bell-diagonal, i.e. the state is incompletely distillable. C. dimU = 2 offers three possibilities: a) the whole subspace is factorising, b) exactly two lin- IV. rankρ = 1, i.e. the state is pure: M = 1 Then ρ ρ ear independent vectors are factorising and c) ist distillable, i.e. there is a localoperationΞ, such exactly one linear independent vector factorises. that Ξ(ρ) is a Bell-state. In order to prove our programme we have a look at the previously existing results. Kent et al. have shown in [3] that states can be brought to a Bell-diagonal form with optimal entanglement, as long as the quan- tity Tr(A BρA B ) cannot be zero for those states. † † ⊗ ⊗ Nevertheless for states that do not fulfil this assump- tion, like quasi-distillable states, an essential continuity argumentfails. For that reasonwe will now characterize D. dimU = 1 is trivial again. Either it factorises, or quasi-distillableandincompletelyquasi-distillablestates, not. which will lead to the characterizationgiven above. Lemma: A quasi-distillable or incompletely quasi- These properties of the support of density matrices will distillable state has a factorising vector in its kernel. now be shown to be the only features determining the behaviour under local transformations with respect to Proof: If ρ is quasi-distillable or incompletely quasi- entanglement optimization. distillable, then the probability for reaching the product We state our results as follows: Ξ (ρ) tends to 0 as n . n →∞ 2 In the following we will use the operators A˜ B˜ := Since ρ is quasi-distillable, there is a factorising n n An Bn , which produce the same state Ξ (ρ⊗). The vector φ φ in its kernel. In the tensor basis An⊗Bn n 3 ⊗ 4 pkrob⊗abiklity of getting the state Ξ (ρ) is given by (00 , 01 , 10 , 11 ) we can assume without loss of gen- n | i | i | i | i eralitythatφ φ havethe form 11 . Thematrix form 3 4 pn :=Tr(Añ⊗BñρA˜†n⊗Bñ†). of ρ is in that c⊗ase | i Since ρ is quasi-distillable, it follows, that 1 a d e c 0 − − e¯ a b 0 nlim Tr(Añ⊗BñρA˜†n⊗Bñ†)=nlim Tr(ρA˜†nAñ⊗Bñ†Bñ)=0. ρ= c¯ ¯b d 0. (3) →∞ →∞  0 0 0 0   Since Añ Bñ = 1 the set of these operators is com-   ptoacataliknmditth⊗oepreereaxktiisotnsaA˜subBs˜eq:u=enlicmeAñi⊗A˜Bñi thBãttesuncdhs Suienscoefρthiestmpaotsriitxivρ˜eρ, awreeh(0a,v0e,(a√,dad≥ 0ba)n2d,(t√haede+igebnv)2a)l-. that A˜ B˜ =1. Then⊗it follows tih→a∞t ni ⊗ ni Since ρ˜ρ is of rank 1, we get √ad=−|b||.|This leads|to| k ⊗ k limi→∞Tr(Añi ⊗BñiρA˜†ni ⊗Bñ†i) 1e−iδac¯−ad eiδca ad eiδ√cad 00 = Tr(A˜ B˜ρA˜ B˜ ) ρ= − d p  = Tr(ρA˜⊗†A˜ B˜††⊗B˜)=† 0  0c¯p e−iδ0√ad d0 00  ⊗     TheoperatorsA˜†A˜,B˜†B˜ arepositiveandcaninorderof where the condition (1+a+d)d c2 (1+a+d)d+ the spectral theorem be decomposed by d has to be made for ρ to be≤p|os|iti≤ve. 4(a+d) The supportof ρ is therefore the subspace spannedby A˜†A˜= aiPi, B˜†B˜ = bjQj the vectors 00 and 1 (√a01 +√d10 ). i j | i √a+d | i | i X X These states are indeed quasi-distillable,which can be with a ,b R+ and projectors P = φ φ ,Q = shown by using the following protocol: i j i i i j ∈ | ih | ψ ψ . The above equation then takes the form | jih j| I. First Alice applies the local filtration A = diag(√d,√a). Tr(ρA˜ A˜ B˜ B˜)=0 † † ⊗ II. NowAliceandBobapplythelocaloperationgiven Tr(ρa b P Q )=0 i j i j ⇔ ⊗ by i,j X a b φ ψ ρφ ψ =0 1 0 1 0 ⇔ i,j i jh i⊗ j| | i⊗ ji An = n0 1 ,Bn = n0 1 . (4) X (cid:18) (cid:19) (cid:18) (cid:19) φ ψ ρφ ψ =0 for all a ,b , a b =0 i j i j i j i j ⇒h ⊗ | | ⊗ i 6 ρ21 φi ψj =0 for all ai,bj, aibj =0. This operation produces the state ⇒ | ⊗ i 6 ′ allTahie,bj|φwi i⊗thψajiibjar6=e 0thaenndpaalrstoopfarttheofktehrneekleornfeρl12offoρr. ρn = TrA(Ann⊗⊗BBnnρρA′Ann⊗⊗BBnn) Since the operation A˜ B˜ cannot be zero there has to 1 a d eiδc c√ad 0 bpreoadupcrtovdeuccttorai|φbji⊗6=ψ0j⊗ianlidestihnertehfeorkeertnheelcoofrρr.espondin2g =c1 1(−1−ana−ec¯−−−√dd+ia−δ+d2c¯2aann2)) (11−−nanea−2−−daidδ++a22aan) (1−1−nan−ae2−diaδd+a+2a2na)d 00  Proposition: A density matrix ρ on C2 C2 is quasi-  (1−na−d+2an)d 1−na2−d+2a 1−na2−d+2a  ⊗  0 0 0 0  distillable if and only if it has rank two and the support     oofrtρhoigsosnpaalnennetdanbgyleda vfaeccttoorrisψin.g vector φ1 ⊗φ2 and an with c1 = d(1+1a d). It can be show−n that Proof: For any quasi-distillable state ρ there exists lim ρ =ρ = 01 +eiλ 10 . n a sequence of operations Ξn n N with limn Ξn(ρ)= n→∞ ∞ | i | i ρ where ρ is a Bell-st{ate.}S∈ince the vec→to∞r ~λ con- This is a maximally entangled state and thus ρ is quasi- ta∞ining the ∞eigenvalues of ρ˜ ρ is given by (1,∞0,0,0) distillable. 2 ∞ ∞ and by (2) the ratio of eigenvalues cannot change under the local operation Ξ , the vector ~λ belonging to Proposition: A density matrix ρ on C2 C2 is in- n 0 ⊗ the originalstate ρ mustbe ofthe form(λ ,0,0,0),with completely quasi-distillable if and only if the kernel of 0 λ =0, i.e. the rank of ρ˜ρ is 1. ρ is spanned by a factorising vector φ φ . The best 0 1 2 6 ⊗ 3 distillation product of ρ is in that case a Bell diagonal and ψ ψ . 1 2 ⊗ state of rank two. Proof: Since ρ is entangled, there must be an en- Proof: Again we assume without loss of generality, tangled vector in its support. If there would be only that the state can be written as (3). We know from [2] one suchvector,ρ wouldbe quasi-distillable(see above). thattheentanglementofformationofastatecanbe fur- For that reason there must be exactly two linear inde- ther increased if and only if it is not Bell diagonal. An pendent factorising vectors in its support, i.e. no linear optimal distillation product must be thus Bell diagonal. combination of these two factorises . Choosing with- Fromthis andthefactthatthe distillationproductmust out loss of generality φ φ = 0 1 we see that 1 2 ⊗ | i| i still havea factorisingvectorin its kernel,it followsthat ψ ψ = (α0 +β 1 ) (γ 0 + δ 1 ), where β = 0 1 2 ⊗ | i | i ⊗ | i | i 6 such a final state must be of the form and γ =0. The invertible local operation 6 0 0 0 0 1 α 1 0 σ = 00 1f/¯2 1f/2 00, Ψ7→(cid:18)0 −ββ1 (cid:19)⊗ −γγδ 1!Ψ 0 0 0 0   leaves 01 invariant, but transforms ψ1 ψ2 into 10 .   | i ⊗ | i Thismeanswecanassumeψ ψ = 10 withoutchang- which yields for σσ˜ eigenvalues (0,0,(1/2+ f )2,(1/2 1⊗ 2 | i | | − ingthepropertiesofthestatewithrespecttolocaltrans- f )2). Since by [3]the ratio of these eigenvalues cannot | | formationsqualitatively. In this case,following the nota- change under local operations, we see that tion of the preceding proof, the density matrix reads: √ad+ b 1/2+ f | | = | | 0 0 0 0 √ad b 1/2 f −| | −| | 0 a b 0 ρ = 0 ¯b d 0. must be fulfilled. From this we get 0 0 0 0   b   | | = f . 2√ad | | The final argument follows the preceding proof, yielding adistillationproductσasseenabove. Neverthelessthere The rank of σ is two whereas ρ has rank three, because is a minor difference: since ρ has rank two, we can now itisneitherdistillablenorquasi-distillable. Forthatrea- find a direct transformation ρ σ, which clearly must 7→ sonnolocaloperationcantransformρintotheentangled be given by the operation elements: state σ, since such a local operation would have to de- creasethe rank(thus being notone to one)and haveen- 1 1 0 0 0 A B := . tangled vectors in its range (see [2]). Nevertheless there ⊗ √2(cid:18)0 √1d(cid:19)⊗(cid:18)0 √1a(cid:19) is a sequence of operations which with probability tend- 2 ing to zero yields σ in the limit. For this purpose we use the operation elements We have seen that the appropriate method to study 1 0 1 0 single-copy distillation procedures is looking at the en- A := n B := n n 0 1 n 0 1 tanglementpropertiesofthestate’ssupport. Theseprop- (cid:18) √d(cid:19) (cid:18) √a(cid:19) ertiesdecidewhetherastatecanbebroughtdirectlyinto getting Bell-diagonal form having optimal entanglement of for- mationorwhetherthiscanbeachievedonlyinalimiting ρ :=A B ρA B (5) n n n n n process with vanishing probability in the limit. It is an ⊗ ⊗ 1 a d e c 0 obvious task to generalize these ideas to higher dimen- −n4− n3√a n3√d e¯ 1 b 0 sion. It must be mentioned, nevertheless, that concepts =  n3c¯√a n¯b2 n2√1ad 0. (6) likeunextendible product baseswillplayamajorroleand  n3√d n2√ad n2  will make such an analysis far more sophisticated.  0 0 0 0     We easily see that the limit of the normalized operation (only leaving those terms proportional to 1/n2) is the state σ. 2 Proposition: AnentangleddensitymatrixρonC2 C2 [1] M.Horodecki,P.Horodecki,andR.Horodecki,Phys.Rev. ⊗ of rank two is incompletely distillable if and only if the A 60, 1888 (1998). supportofρisspannedbytwofactorisingvectorsφ1 φ2 [2] A. Kent,Phys.Rev.Lett. 81, 2839 (1998). ⊗ 4 [3] A. Kent, N. Linden, and S. Massar, Phys. Rev. Lett. 83, Phys. Rev.A 53, 2046 (1996). 2656 (1999). [6] W. Wootters, Phys. Rev.Lett. 80, 2245 (1998). [4] M. Lewenstein, J.I. Cirac, and S. Karnas, quant- [7] N. Linden, S. Massar, and S. Popescu, Phys. Rev. Lett. ph/9903012 (1999) 81, 3279 (1998). [5] C.Bennett,H.Bernstein,S.Popescu,andB.Schumacher, 5