Table Of Content

Entanglement detection via tighter local uncertainty relations Cheng-Jie Zhang1, Hyunchul Nha2, Yong-Sheng Zhang1,∗ and Guang-Can Guo1 1Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China 2Department of Physics, Texas A & M University at Qatar, Doha, Qatar We propose an entanglement criterion based on local uncertainty relations (LURs) in a stronger form than the original LUR criterion introduced in [H. F. Hofmann and S. Takeuchi, Phys. Rev. A 68, 032103 (2003)]. Using arbitrarily chosen operators {Aˆk} and {Bˆk} of subsystems A and B, thetighterLURcriterion,whichmaybeusednotonlyfordiscretevariablesbutalsoforcontinuous variables, can detect more entangled states than the original criterion. 0 PACSnumbers: 03.67.Mn,03.65.Ta,03.65.Ud 1 0 2 I. INTRODUCTION et al. proposed nonlinear witnesses to improve arbitrary n linear witnesses, which is strictly stronger than the orig- a inal linear witnesses [18]. In continuous variable sys- J Entangledstatesareusuallyrecognizedasessentialre- tems, Simon proposed a continuous variable version of 8 sourcesinquantumcomputationandcommunication[1], the partial transpositioncriterionin two-mode Gaussian 2 and more and more experimental realizations of entan- states [34]. At the same time, Duan et al. also intro- glement sources have become available [2–4]. However, ducedacriterion[35]. Bothofthese twocriteriaarenec- ] there are still a number of important, yet open, prob- h essary and sufficient conditions for two-mode Gaussian lems concerning quantum entanglement. In particular, p states. Werner and Wolf improved Simon’s result, and - the separability problem to determine both theoretically they found bound entangled Gaussian states [36]. Fur- nt andexperimentally whetheragivenstate isentangledor thermore, Giedke et al. provided a necessary and suffi- a not is of crucial importance in quantum information sci- cient condition for Gaussian states of bipartite systems u ence [5]. In the past years, a great deal of efforts have of arbitrarily many modes [37]. q been made to solve the separability problem [6–38]. [ In this paper, we propose an entanglement criterion There are many efficient methods proposed for entan- 3 based on LURs in a tighter form than the original LUR glementdetectioninbothfinitedimensionalsystemsand v continuous variable systems. For example, in finite di- criterion. Using arbitrarily chosen operators {Aˆk} and 4 Bˆ of subsystems A and B, the stronger LUR crite- mensional systems, there are the partial transposition k 1 { } rioncan generallydetect more entangled states than the 2 criterion [7], local uncertainty relations (LUR) [8, 9], co- originalLURcriterionduetoanewlyaddednonnegative 4 variancematrixcriterion(CMC)[10,11],thecomputable term, similarto the nonlinear witnesses. Our tighter cri- . cross-normorrealignment(CCNR)criterion[12,13],the 8 terioncanalsobeusedbothfordiscretevariablesandfor 0 permutation separability criteria [14, 15], the criterion continuous variables. 9 based on Bloch representations [16], entanglement wit- 0 nesses [17, 18], and Bell-type inequalities. On the one The paper is organized as follows. In Sec. II we v: hand,the partialtranspositioncriterionisnecessaryand propose an entanglement criterion based on the tighter i sufficient for certain low dimensional systems, but it is LURs(TLURs)andillustrateitsutilitybyanexampleof X known to be only necessary for higher dimensions [7]. Horodecki 3 3 bound entangled states [39]. In Sec. III r The LUR criterion provides only a necessary condition the relations×hips between the TLUR criterion and other a forarbitrarydimensionalsystems,butitcandetectmany entanglement criteria are discussed, and in Sec. IV, a bound entangled states where the partial transposition brief discussion and a summary of our results are given. criterion fails [8]. Moreover, it is shown in Ref. [10, 11] that the LUR criterion is equivalent to the symmetric CMC using orthogonal observables and that the CCNR criteriontogetherwithitsextension[19]andthecriterion based on Bloch representation are their corollaries. The LUR, the symmetric CMC criteria, and their corollaries II. TIGHTER LOCAL UNCERTAINTY are usually considered as complementary to the partial RELATIONS transpositioncriterion. Ontheotherhand,entanglement witnesses and Bell-type inequalities are usually used for In Ref. [8], Hofmann and Takeuchi introduced an en- entanglementdetectioninexperiments. Recently,Gu¨hne tanglement criterion based on the local uncertainty relations. Consider the set of local observables Aˆ N { k}k=1 and Bˆ N forsubsystemsAandB,respectively. Sup- { k}k=1 ∗Electronicaddress: [email protected] pose that the sum uncertainty relations have bounds for 2 arbitrary local states as Proof.– Using Lemma 1, we can obtain that δ(Aˆ + Bˆ )2 δAˆ2 U , (1) k⊗1 1⊗ k ρAB k ≥ A Xk Xk = δ(Aˆ )2 + δ(Bˆ )2 δBˆ2 U , (2) k ρA k ρB k ≥ B Xk Xk Xk +2 ( Aˆ Bˆ Aˆ Bˆ ) k k k k h ⊗ i−h ⊗1ih1⊗ i k X whereUA andUB arenonnegativevalues. Then,forsep- δ(Aˆ )2 + δ(Bˆ )2 arable states, the following inequality holds [8], ≥ k ρA k ρB k k X X 2 [ δ(Aˆ )2 U ][ δ(Bˆ )2 U ] δ(Aˆ + Bˆ )2 U +U . (3) − k ρA − A k ρB − B k⊗1 1⊗ k ρAB ≥ A B sXk Xk Xk = U +U +M2, A B It has been proven that the LUR criterion is equivalent where M = δ(Aˆ )2 U δ(Bˆ )2 U . tothesymmetricCMCusingorthogonalobservables,and k k ρA − A− k k ρB − B (cid:3) thatmanyothercriteria,suchastheCCNRcriterionto- qP qP Remark. It is worth noting that both Lemma 1 and getherwithitsextensionandthecriterionbasedonBloch Theorem1canbeusedforentanglementdetection. Com- representation, are their corollaries [11]. The LUR cri- pared with the original LUR criterion, the tighter criterion is an efficient method to detect bound entangled terion added a squared, thus nonnegative, term M2. states, but is it possible to improve the LUR criterion? Therefore, for a given set of observables Aˆ N and Our idea comes from the nonlinear witnesses that im- { k}k=1 proved the linear witnesses [18]. In the following, the {Bˆk}Nk=1 ateachparty,ourcriterionis strongerthan the LUR criterionwill be indeed developed in a tighter form original LUR and can generally detect more entangled to improve the power of entanglement detection. states. Actually, we can also prove that δ(Aˆ + Before embarking on our criterion, a lemma will be k k ⊗ 1 given. We again consider the sets of local observables 1 ⊗ Bˆk)2ρAB ≤ UA + UB + ( kδP(Aˆk)2ρA −UA + s{aAˆtkis}fNky=t1heanbdou{nBˆdks}Nko=f1thfeorsusmubsuynscteermtasinAtyarnedlaBti,onwshaicph- kδ(Bˆk)2ρB −UB)2. Itisa duaqliPnequalityofEq. (5). pearinginEqs. (1)and(2). Wefirstobtainthefollowing qEPxample 1.– To compare with the original LUR criterion,we consider the same example which has been used lemma. in Ref. [9]. P. Horodecki introduced a 3 3 bound en- Lemma 1. Forbipartiteseparablestates,thefollowing × tangledstateinRef. [39],anditsdensitymatrixρisreal inequality must hold, and symmetric as a ρ = 1;0 1;0 + 1;+1 1;+1 [ δ(Aˆ )2 U ][ δ(Bˆ )2 U ] 1+8a |− ih− | |− ih− | k ρA − A k ρB − B + 0; (cid:0)1 0; 1 + 0;+1 0;+1 + +1;0 +1;0 sXk Xk | − ih − | | ih | | ih | 3a 1 ( Aˆk Bˆk Aˆk Bˆk ) 0. (4) + Emax Emax + Π Π , (cid:1) ± h ⊗ i−h ⊗1ih1⊗ i ≥ 1+8a | ih | 1+8a | ih | k X where Proof.– The proof is given in the Appendix. (cid:3) 1 Emax = ( 1; 1 + 0;0 + +1;+1 ) Theorem 1. (Tighter LURs) For bipartite separable | i √3 |− − i | i | i states,considerthesetsoflocalobservables{Aˆk}Nk=1 and and Bˆ N for subsystems A and B, respectively. If they {satkis}fky=t1he bounds of the sum uncertainty relations in 1+a 1 a Π = +1; 1 + − +1;+1 . Eqs. (1)and(2),thenthefollowinginequalitymusthold, | i 2 | − i 2 | i r r The real parameter a covers the range 0 < a < 1. Xk δ(Aˆk⊗1+1⊗Bˆk)2ρAB ≥UA+UB aIfndon{eλˆkch(2o)o}s8ke=s1tihnetrsoedtuscoefdlioncaRleof.bs[9e]r,vtahbelebso{uλˆnkd(1e)n}t8ka=n1- 2 gledstateviolatesboththe originalLURandthe tighter + δ(Aˆ )2 U δ(Bˆ )2 U .(5) LUR (TLUR) criterion. On the other hand, if we define "sXk k ρA − A−sXk k ρB − B# CTLUR =1−[ kδ(Aˆk⊗1+1⊗Bˆk)2−M2]/(UA+UB) P 3 are several corollaries that can be obtained in a form 0.00200 reduced to some other criteria or improved versions of 0.00175 (a) them. One of them is derived from local orthogonal ob- TLUR 0.00150 LUR servables(LOOs)GÂk andGˆBk whichareorthogonalbases of the observable spaces ( ) and ( ) and satisfy A B 0.00125 Tr(GÂGÂ)=Tr(GˆBGˆB)=B δH . B H C k l k l kl 0.00100 Corollary 1. A stronger witness can be obtained from TLUR using complete sets of LOOs as the set of local 0.00075 observables, 0.00050 1 1 GÂ GˆB GÂ GˆB 2 0.00025 − h k ⊗ ki− 2 h k ⊗1−1⊗ ki k k 0.00000 X X 2 0.0 0.2 0.4 0.6 0.8 1.0 1 a −2 1−Trρ2A− 1−Trρ2B ≥0, (6) 1.000 (cid:18)q q (cid:19) (b) Eq. (6) holds for all bipartite separable states. 0.999 Proof.–OnecanchooseAˆ =GÂ,Bˆ = GˆB,anduse k k k − k (GÂ)2 = d , GÂ 2 = Trρ2, (GˆB)2 = d , 0.998 kh k i A kh ki A kh k i B and GˆB 2 = Trρ2, which have been shown in Ref. p 0.997 P[23], tokohbtkaiin CoroPllaBry 1. P (cid:3) TLUR RePmark. Corollary 1 is an improved version of the 0.996 LUR nonlinearwitnessintroducedinRef. [23]. Anyentangled states, which can be detected by the original nonlinear 0.995 witness [23], 1 GÂ GˆB 1 GÂ − kh k ⊗ ki− 2 kh k ⊗1−1⊗ GˆB 2 0, can also be detected by Corollary 1; the con- 0.994 ki ≥ P P 0.0 0.2 0.4 0.6 0.8 1.0 verse is not true in general. a It is worth noticing that Corollary 1 can be eas- FIG. 1: (color online). (a) Describing the amount of entan- ily realized in experiments, since the left hand side of glementforHorodecki3×3boundentangledstateviaCTLUR Eq. (6) can be directly measured ( GÂ 2 = Trρ2, (solid line) and CLUR (dashed line). CTLUR is always larger GˆB 2 =Trρ2). Toshowthis,wewikllhprkoviideashoArt thanCLUR. (b)DetectingtheentanglementofHorodecki3×3 exakmh pkleiin the foBllowing. P bound entangled state with white noise. The regions above PExample 2.– To compare with the original nonlinear the curves can be detected as entangled states by the TLUR witness, we consider the same example which has been criterion (solid line) and the original LUR criterion (dashed used in Ref. [23]. Let us consider a noisy singlet state of line), respectively. the form ρ(p)=pψ ψ +(1 p)ρ , (7) s s sep | ih | − as Ref. [8] defined CLUR, where CTLUR and CLUR pro- where ψ = (01 10 )/√2 and ρ = 2/300 00 + s sep vide quantitative estimates of the amount of entangle- | i | i−| i | ih | 1/301 01. Actually, ρ(p) is entangled for any p > 0 ment verified by the violation of LURs, it can be dis- [23]|. Niohw w| e choose GÂ and GˆB as covered that C is always larger than C , which k k TLUR LUR has been demonstrated in Fig. 1(a). Furthermore, let σ σ σ us consider a mixture of this state with white noise, {GÂk}4k=1 = − √x2,−√y2,−√z2,√12 , ρ(p) = pρ+(1 p) /9. Taking the TLUR in Eq. (5) (cid:26) (cid:27) aρn(pd)tahsetLheURsetino−fElqo.1ca(l3)obwsietrhvatbhleesS[c2h3m],idotnemfiantdriscetshaotf {GˆBk}4k=1 = (cid:26)√σx2,√σy2,√σz2,√12(cid:27). (8) more entangledstates can be detected by the TLUR cri- It can be seen that ρ(p) voilates the original nonlinear terion than by the LUR criterion. Detailed results are witness for all p > 0.25 [23]. Using Eq. (6) with these shown in Fig. 1(b). LOOs, one finds that ρ(p) is entangled at least for p > 0.221. Besides Corollary1, we can also obtainthe conclusion III. RELATION WITH OTHER CRITERIA that the CCNR criterion, the criterion based on Bloch representations,andtheextensionofCCNRcriterionare In this section, we discuss the relation between The- the corollariesof Theorem1. This is because these three orem 1 and some other entanglement conditions that criteria are the corollaries of the symmetric CMC cri- have been proposed in the past. Actually, if we choose terion and that the symmetric CMC using orthogonal somespecialsetsoflocalobservablesinTheorem1,there observables is equivalent to the LUR criterion. 4 It is worth noticing that Theorem 1 can also be used IV. DISCUSSION AND CONCLUSION forcontinuousvariables. Ifwechoose axˆ1, apˆ1 A and {| | | | } xˆ2/a, pˆ2/a B as the sets of local observables, and use {(∆xˆ )2− + (}∆pˆ )2 [xˆ ,pˆ ] = 1 for j = 1,2, the There are still several questions about the TLUR. j j j j h i h i ≥ | | First,Theorem1andtheoriginalLURcriterionarecon- following corollary will be obtained from Theorem 1. sidered for bipartite systems. Is it possible to generalize Corollary 2. For continuous variable systems, define them to multipartite systems? Second, we have shown uˆ = axˆ1 +xˆ2/a and vˆ = apˆ1 pˆ2/a with xˆj and pˆj′ | | | |′ − that Theorem 1 is stronger than the LUR when the set satisfying [xˆj,pˆj′]=iδjj′ (j,j =1,2). The inequality of local observables is chosen. However, if one chooses 1 all possible sets of local observables, is Theorem 1 still 2 2 2 2 h(∆uˆ) i+h(∆vˆ) i≥a + a2 +M (9) stronger than or just equivalent to the LUR criterion? Finally, for discrete variable systems, Theorem 1 can be holds for all separable states, where used for detecting bound entangled states. Is it then M = a (∆xˆ1)2 + (∆pˆ1)2 1 possible to detect bound entangled states for continuous | | h i h i− − (∆xˆ2)2 + (∆pˆ2)2 1/a. variables? These questions are interesting and worth for h i h i−p | | further research. Remark. Corollary 2 is an improved version of the p entanglement criterion introduced in Ref. [35], which In summary, we have proposed an entanglement cri- provided an inequality (∆uˆ)2 + (∆vˆ)2 a2 +1/a2 terion based on the TLUR, which can be viewed as an h i h i ≥ for separable states. Since a squared nonnegative term extensionoftheoriginalLURcriterion. Usingarbitrarily M2 has been added in the right hand side of Eq. (9), chosenoperators Aˆ and Bˆ ofsubsystemsAandB, k k corollary 2 is strictly stronger than the criterion shown the TLURcriterio{n,w}hichm{ ay}be usednotonly fordis- in Ref. [35]. crete variables but also for continuous variables, can de- There is another interesting relation between the tect more entangled states than the LUR criterion since TLURcriterionandthesymmetricCMCusingarbitrary a nonnegative term has been added, similar to the non- observables. Notice that Refs. [10, 11] mainly discussed linear witnesses. the symmetric CMC using orthogonal observables and concluded that the LUR criterion is equivalent to the symmetric CMC using orthogonal observables. Interest- ingly, if arbitrary local observables Aˆ and Bˆ are k k { } { } used, the TLUR can be obtained from the symmetric ACKNOWLEDGMENTS CMC [40]. Proposition 1. TheTLURcriterionisacorollaryofthe symmetric CMC using arbitrary local observables [40]. We would like to thank Otfried Gu¨hne for helpful dis- Proof.— Using Eq. (43) of Ref. [11] cussions,andanonymousrefereeforvaluablesuggestions. kCk2Tr ≤ kA − κAkTrkB − κBkTr where k · kTr is This work was funded by the National Fundamental Re- the trace norm (i.e., the sum of the singular val- search Program (Grant No. 2006CB921900), the Na- ues), κA = ipiγS(ρAi ), κB = ipiγS(ρBi ), and γS tional Natural Science Foundation of China (Grants No. stands for the symmetric covariance matrix, one can 10674127, No. 60621064 and No. 10974192), the In- ((oTTbrrtACai)n2th≤aTtrκ[PkPC)(kkT(2ThrrABˆk≤⊗ BˆkTkAriκ−−)hκAÂPk=k⊗Trk1B[ih1−⊗δκ(ABBˆˆkk)iT2)r]2 == naannovdaNttPihoRenPKFu.gCnra.dnsWtfor1on-m7g-7Ft-ho6eufnrCdohamitnioeQnsae.tAaHrcNaNdaeitsmioysnuaoplfpRSorcetiseeendacrebcsyh, − A − B k k ρA − p δ(Aˆ )2 ][ δ(Bˆ )2 p δ(Bˆ )2 ] Funds. i i k k ρAi k k ρB − i i Pk k ρBi ≤ [ δ(Aˆ )2 U ][ δ(Bˆ )2 U ], where Pk Pk ρA − PA k k ρBP − PB δ(Aˆ )2 U and δ(Bˆ )2 U have Pk k ρAi ≥ A P k k ρBi ≥ B been used. Therefore, Lemma 1 and Theorem 1 can P P be obtained from the symmetric CMC using arbitrary APPENDIX observables. (cid:3) Remark. Refs. [10,11]showthattheLURcriterionus- ing arbitrary observables is equivalent to the symmetric Here we prove Lemma 1. The density matrix for CMC using orthogonal observables. Obviously, the sym- bipartite separable states can be expressed as ρAB = metric CMC using orthogonal observables is a corollary ipiρAi ⊗ρBi . Notice that the lemma is equivalent to ofthesymmetricCMCusingarbitrary observables. From P Proposition 1, TLUR criterion is also a corollary of the symmetric CMC using arbitrary observables. However, [ δ(Aˆk)2ρA −UA][ δ(Bˆk)2ρB −UB] whether the TLUR criterion is equivalent to the LUR k k X X criterion (the symmetric CMC using orthogonal observ- [ ( Aˆ Bˆ Aˆ Bˆ )]2. (10) k k k k ables) is unknown. ≥ h ⊗ i−h ⊗1ih1⊗ i k X 5 The right hand side (RHS) and the left hand side (LHS) where we have used p ( (A )2 of Eq. (10) can be written as A 2) = p ( (A )2 kA 2i)i h kU ii an−d h kii p ( (B k)2 i i hB k2)ii=−Ph Pkii p≥( (BA)2 RHS Bk 2)i iUh P. k Pii − h kii k i i h k ii − = [ pi Aˆk i Bˆk i ( pi Aˆk i)( pi′ Bˆk i′)] 2 Ph kiPi ≥ B P P { h i h i − h i h i } k i i i′ WiththehelpoftheCauchy-Schwarzinequality,itcan XX X X = pi (Aˆk pi′ Aˆk i′) i (Bˆk pj′ Bˆk j′) i b2e obtained that { h − h i i h − h i i } k i i′ j′ XX X X p A B 2, i k i k i ≡ { h i h i } Xk Xi LHS ( p A 2)( p B 2) ≥ ih kii ih kii where we have defined hAˆkii =hAˆkiρAi , hBˆkii =hBˆkiρBi , Xk Xi Xk Xi Ak = Aˆk − i′pi′hAˆkii′ and Bk = Bˆk − j′pj′hBˆkij′ ≥ ( pihAkiihBkii)2 for convenience. Therefore, k i P P XX = RHS. LHS = ( p (A )2 U )( p (B )2 U ) i k i A i k i B h i − h i − k i k i XX XX ( p A 2)( p B 2), Therefore, Lemma 1 has been proved. (cid:3) ≥ ih kii ih kii k i k i XX XX [1] M. A. Nielsen and I. L. Chuang, Quantum Computation Clarisse, P. Wocjan, Quantum Inf. Comput. 6, 277 andQuantumInformation (CambridgeUniversityPress, (2006). Cambridge, 2000). [16] J.I.deVicente,QuantumInf.Comput.7,624(2007);J.I. [2] D. Leibfried, E. Knill, S. Seidelin, J. Britton, R. B. de Vicente, J. Phys.A 41, 065309 (2008). Blakestad, J. Chiaverini, D. B. Hume, W. M. Itano, [17] B.Terhal,Phys.Lett.A271,319(2000);G.TóthandO. J. D. Jost, C. Langer, R. Ozeri, R. Reichle, and D. J. Gu¨hne,Phys.Rev.Lett.94,060501(2005);F.A.Bovino, Wineland, Nature438, 639 (2005). G. Castagnoli, A. Ekert, P. Horodecki, C. M. Alves and [3] H.Häffner,W.Hänsel,C.F.Roos,J.Benhelm,D.Chek- A. V. Sergienko, Phys. Rev. Lett. 95, 240407 (2005); al-kar, M. Chwalla, T. Körber, U. D. Rapol, M. Riebe, O. Gu¨hne, G. Tóth, P. Hyllus, and H. J. Briegel, ibid. P. O. Schmidt, C. Becher, O. Gu¨hne, W. Du¨r, and R. 95, 120405 (2005); F. Mintert, Phys. Rev.A 75, 052302 Blatt, Nature 438, 643 (2005). (2007); R. Augusiak, M. Demianowicz, P. Horodecki, [4] C.-Y. Lu, X.-Q. Zhou, O. Gu¨hne, W.-B. Gao, J. Zhang, ibid. 77, 030301(R) (2008). Z.-S. Yuan, A. Goebel, T. Yang and J.-W. Pan, Nature [18] O. Gu¨hne and N. Lu¨tkenhaus, Phys. Rev. Lett. 96, Phys.3, 91 (2007). 170502 (2006); [5] R.F. Werner, Phys.Rev.A 40, 4277 (1989). [19] C.-J. Zhang, Y.-S. Zhang, S. Zhang, and G.-C. Guo, [6] D. Bruß, J. Math. Phys. 43, 4237 (2002); M. B. Ple- Phys. Rev.A 77, 060301(R) (2008). nio, S. Virmani, Quantum Inf. Comput. 7, 1 (2007); [20] O. Gu¨hne, Phys. Rev.Lett. 92, 117903 (2004). R. Horodecki, P. Horodecki, M. Horodecki, and K. [21] G. Tóth, C. Knapp, O. Gu¨hne, and H. J. Briegel, Phys. Horodecki, Rev. Mod. Phys. 81, 865 (2009); O. Gu¨hne Rev. Lett. 99, 250405 (2007); G. Tóth, C. Knapp, O. and G. Tóth, Phys. Rep.474, 1 (2009). Gu¨hne, and H. J. Briegel, Phys. Rev. A 79, 042334 [7] A.Peres,Phys.Rev.Lett.77,1413(1996);M.Horodecki, (2009). P. Horodecki, and R. Horodecki, Phys. Lett. A 223, 1 [22] G. Tóth,and O. Gu¨hne, Phys. Rev. Lett. 102, 170503 (1996). (2009); G. Tóth, W. Wieczorek, R. Krischek, N. Kiesel, [8] H.F.HofmannandS.Takeuchi,Phys.Rev.A68,032103 P. Michelberger, and H. Weinfurter, New J. Phys. 11, (2003). 083002 (2009). [9] H.F. Hofmann, Phys. Rev.A 68, 034307 (2003). [23] O. Gu¨hne, M. Mechler, G. Tóth, and P. Adam, Phys. [10] O.Gu¨hne, P.Hyllus, O.Gittsovich, and J. Eisert, Phys. Rev. A 74, 010301(R) (2006). Rev.Lett. 99, 130504 (2007). [24] S.YuandN.-L.Liu,Phys.Rev.Lett.95,150504(2005). [11] O.Gittsovich, O.Gu¨hne, P. Hyllus,and J. Eisert, Phys. [25] P. Aniello and C. Lupo, J. Phys. A: Math. Theor. 41, Rev.A 78, 052319 (2008). 355303 (2008); C. Lupo, P. Aniello, and A. Scardicchio, [12] O.Rudolph,arXiv:quant-ph/0202121. ibid. 41, 415301 (2008). [13] K. Chen and L.-A. Wu, Quantum Inf. Comput. 3, 193 [26] O. Gu¨hne amd M. Seevinck,arXiv:0905.1349. (2003). [27] R. Augusiak and J. Stasin´ska, New J. Phys. 11, 053018 [14] M. Horodecki, P. Horodecki, and R. Horodecki, Open (2009). Syst.Inf. Dyn.13, 103 (2006). [28] W. K. Wootters, Phys. Rev. Lett. 80, 2245 (1998); A. [15] H. Fan, arXiv:quant-ph/0210168; P. Wocjan, M. Uhlmann, Phys. Rev. A 62, 032307 (2000); P. Rungta, Horodecki, Open Syst. Inf. Dyn. 12, 331 (2005); L. V. Buˇzek, C. M. Caves, M. Hillery and G. J. Milburn, 6 Phys.Rev.A 64, 042315 (2001). Phys. Lett. A 373, 1616 (2009); Z.-H. Ma, F.-L. Zhang, [29] H.Fan,K.MatsumotoandH.Imai,J.Phys.A36,4151 and J.-L. Chen, Phys.Rev. A 78, 064305 (2008). (2003);H.Fan,V.Korepin,andV.Roychowdhury,Phys. [34] R. Simon, Phys.Rev. Lett.84, 2726 (2000). Rev.Lett. 93, 227203 (2004). [35] L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, Phys. [30] F.Mintert,M.Ku´sandA.Buchleitner,Phys.Rev.Lett. Rev. Lett.84, 2722 (2000). 92, 167902 (2004); H. P. Breuer, J. Phys. A 39, 11847 [36] R.F.WernerandM.M.Wolf,Phys.Rev.Lett.86,3658 (2006); J.I. deVicente,Phys.Rev.A75, 052320 (2007). (2001). [31] K. Chen, S. Albeverio and S.-M. Fei, Phys. Rev. Lett. [37] G. Giedke, B. Kraus, M. Lewenstein, and J. I. Cirac, 95, 040504 (2005); K. Chen,S. Albeverioand S.-M. Fei, Phys. Rev.Lett. 87, 167904 (2001). ibid.95, 210501 (2005). [38] H.Nha,andM.S.Zubairy,Phys.Rev.Lett.101,130402 [32] M.Li,S.-M.Fei,andZ.-X.Wang,J.Phys.A41,202002 (2008); Q. Sun, H. Nha, and M. S. Zubairy, Phys. Rev. (2008); M.-J. Zhao, Z.-X. Wang, and S.-M. Fei, Rep. A 80, 020101(R) (2009). Math. Phys. 63, 409 (2009); S.-M. Fei, M.-J. Zhao, K. [39] P. Horodecki, Phys. Lett. A 232, 333 (1997). Chen,andZ.-X.Wang,Phys.Rev.A80,032320 (2009). [40] O. Gu¨hne, Private communication. [33] Z.-H. Ma, F.-L. Zhang, D.-L. Deng, and J.-L. Chen,