Local copying of orthogonal entangled quantum states

Transcription

1 Local copying of othogonal entangled quantum states Fabio Anselmi 1, Anthony Chefles 1,3 and Matin B Plenio 2 1 School of Physics, Astonomy and Mathematics, Univesity of Hetfodshie, Hatfield AL10 9AB, Hetfodshie, UK 2 QOLS, The Blackett Laboatoy, Impeial College London, Pince Consot Rd, London SW7 2BW, UK anselmi.fabio@hotmail.com, A.Chefles@hets.ac.uk and m.plenio@impeial.ac.uk New Jounal of Physics 6 (2004) 164 Received 21 July 2004 Published 12 Novembe 2004 Online at doi: / /6/1/164 Abstact. In classical infomation theoy one can, in pinciple, poduce a pefect copy of any input state. In quantum infomation theoy, the no cloning theoem pohibits exact copying of non-othogonal states. Moeove, if we wish to copy multipaticle entangled states and can pefom only local opeations and classical communication (LOCC), then futhe estictions apply. We investigate the poblem of copying othogonal, entangled quantum states with an entangled blank state unde the estiction to LOCC. Thoughout, the subsystems have finite dimension D. We show that if all of the states to be copied ae non-maximally entangled, then novel LOCC copying pocedues based on entanglement catalysis ae possible. We then study in detail the LOCC copying poblem whee both the blank state and at least one of the states to be copied ae maximally entangled. Fo this to be possible, we find that all the states to be copied must be maximally entangled. We obtain a necessay and sufficient condition fo LOCC copying unde these conditions. Fo two othogonal, maximally entangled states, we povide the geneal solution to this condition. We use it to show that fo D = 2, 3, any pai of othogonal, maximally entangled states can be locally copied using a maximally entangled blank state. Howeve, we also show that fo any D which is not pime, one can constuct pais of such states fo which this is impossible. 3 Pesent addess: Depatment of Mathematical Physics, National Univesity of Ieland, Maynooth, Co. Kildae, Ieland. New Jounal of Physics 6 (2004) 164 PII: S (04) /04/ $30.00 IOP Publishing Ltd and Deutsche Physikalische Gesellschaft

2 2 DEUTSCHE PHYSIKALISCHE GESELLSCHAFT Contents 1. Intoduction 2 2. The poblem of LOCC copying Geneal consideations Catalytic copying LOCC copying of a pue othogonal set including a maximally entangled state Fom of the local Kaus opeatos Condition fo LOCC copying LOCC copying of two othogonal maximally entangled states A spectal copying condition Consequences Discussion 23 Acknowledgments 23 Refeences Intoduction The no cloning theoem of Woottes and Zuek [1] and Dieks [2] pohibits the ceation of pefect copies of non-othogonal quantum states. This famous esult has pofound implications fo quantum communications, e.g. the secuity of quantum cyptogaphy [3]. It is also well known that any set of othogonal states can be pefectly copied in pinciple. Howeve, it is not known how well this can be achieved if thee ae estictions on the set of possible quantum opeations. A common scenaio in quantum infomation pocessing and communications is whee a multipaticle, possibly entangled state is distibuted among a numbe of spatially sepaated paties. Each of these paties can pefom abitay local opeations on the subsystems they possess. Howeve, they can only send classical infomation to each othe. When this is the case, the paties ae esticted to pefoming local (quantum) opeations and classical communication (LOCC). Thee has been a consideable amount of activity devoted to undestanding the popeties of LOCC opeations. Cetain specific quantum infomation pocessing tasks, such as entanglement distillation and, moe ecently, state discimination, have been the focus of a paticulaly lage amount of attention with espect to the LOCC constaint. In this pape, we investigate the poblem of copying othogonal, entangled, quantum states unde these conditions. Quantum copying and quantum state discimination ae closely elated opeations [4]. In the study of state discimination unde the LOCC constaint, it has been found that any pai of othogonal, entangled, pue, bipatite states can be pefectly disciminated by LOCC [5]. This is not geneally possible fo moe than two states. Also, it has been found that any two nonothogonal, entangled, pue, bipatite states can be optimally disciminated by LOCC [6] [8]. Again, this is not geneally possible fo moe than two states [9]. We see that the LOCC constaint imposes estictions on the numbe of states fo which cetain discimination tasks ae possible. Given that copying is closely elated to state discimination, we might imagine that the LOCC constaint could also affect the numbe of states fo which cetain copying pocedues ae possible. We will show in this pape that this is indeed the case. New Jounal of Physics 6 (2004) 164 (

3 3 DEUTSCHE PHYSIKALISCHE GESELLSCHAFT In fact, we shall see that some of the estictions on LOCC copying ae, if we wish to use entanglement efficiently, moe sevee than those on LOCC state discimination. This will tun out to be a consequence of the fact that, when copying states by LOCC, thee ae cetain factos we must take into account that do not apply to LOCC state discimination. In LOCC state discimination, the oiginal state is typically destoyed. This is of no concen, since we only wish to know the state, not peseve it. Howeve, in copying the state, not only do we wish to peseve the oiginal state, we also wish to impint it onto anothe system initialized in a blank state. If we estict ouselves to pefoming LOCC copying and the states we wish to copy ae entangled, then the blank state must be entangled also. If this wee not the case, then the copying pocedue would ceate entanglement, which is well known to be impossible unde LOCC [10, 11]. It was ecently discoveed by Ghosh et al [12] in an independent wok, that some sets of othogonal, maximally entangled states can be copied by LOCC and with a maximally entangled blank state. These authos consideed LOCC copying of Bell states. The Bell states, which ae maximally entangled states of two qubits, each contain one ebit of entanglement. These authos showed that LOCC copying of any two Bell states is possible with a blank state containing one ebit of entanglement. They found, howeve, that to copy all fou Bell states equies one futhe ebit of entanglement. This is still less than the two futhe ebits that would be equied to pefom an abitay opeation on the fou qubits compising the states ψ j and b by LOCC [13] [15]. In this pape, we obtain numeous esults which elate to the poblem of copying pue, bipatite, othogonal, entangled states by LOCC. Thoughout, we ae inteested in making pefect copies deteministically. In section 2, we set up the copying poblem in geneal tems. In doing so, we acknowledge the fact that an LOCC opeation may, in pinciple, involve an unlimited numbe of ounds of classical communication. As such, the opeation may become unwieldy in fomal tems. Rathe than deal with this possibility diectly, we take an altenative appoach based on the fact that LOCC opeations fom a subset of the set of sepaable opeations. The fom of a geneal sepaable opeation is well known and moe convenient to wok with. Fo the easons we gave above, the LOCC copying pocedue must use an entangled blank state. Entanglement is a pecious esouce in quantum infomation pocessing. Consequently, it is highly desiable that entanglement is used efficiently and, if at all possible, conseved by the opeation. To investigate this matte fully, we equie a measue of entanglement. The poblem of quantifying entanglement is cental to quantum infomation theoy. Fo pue, bipatite states in the asymptotic limit, whee many copies of the states ae available, a unique measue of entanglement can be povided [16, 17]. This is the entopy of entanglement. Howeve, in the scenaio consideed in this pape, we only have one copy of each of the states to be pocessed: the state to be copied and the blank state. In this one-shot scenaio, thee exist pais of incompaable states, fo which one cannot unambiguously decide whethe the entanglement of one state is geate than, less than o equal to that of the othe. Ideally, we would like the entanglement of the blank state to equal that of the most entangled of the states to be copied, as this would epesent the most efficient use of entanglement. Howeve, ou desie to use entanglement efficiently leads us to, in geneal, account fo the possibility of incompaability of the blank state and some of the states to be copied. We show that when all of the states to be copied ae non-maximally entangled, accounting fo this possible incompaability leads to scenaios whee, although the LOCC copying pocedue is possible, the blank state cannot be diectly tansfomed into the state to be copied by LOCC. Instead, the oiginal copy of the state seves as an entanglement catalyst [18] which facilitates New Jounal of Physics 6 (2004) 164 (

4 4 DEUTSCHE PHYSIKALISCHE GESELLSCHAFT the copying pocedue. This point is illustated in the simple case whee we wish to copy just one state. In section 3, we analyse in detail the poblem of locally copying N othogonal, entangled states with D-dimensional subsystems. The blank state is also taken to be an entangled state whose subsystems ae also D-dimensional. We focus in paticula on the situation whee one of the states to be copied is maximally entangled. This simplifies the poblem in many espects. Fistly, the possibility of catalytic copying, with its attendant complications, does not aise, since a maximally entangled state cannot seve as an entanglement catalyst [18]. Consequently, the blank state must be maximally entangled also. Secondly, we show that the local Kaus opeatos fo a sepaable copying opeation must be popotional to unitay opeatos if they ae to copy a maximally entangled state. This is vey helpful, since any sepaable opeation whose Kaus opeatos have this popety can be pefomed by LOCC. Indeed, we find that we may, without loss of geneality, take the entie copying opeation to consist of just two local unitay opeations, with one being caied out by each paty, and no classical communication. This implies that if one of the states to be copied is maximally entangled, then they must all be maximally entangled. We then use the convenient fom of these opeatos to obtain a geneal necessay and sufficient condition fo LOCC copying of ND-dimensional maximally entangled states with a maximally entangled blank state. This condition is difficult to solve fo abitay N and D. Howeve, it can be solved exactly fo N = 2 and all D. In section 4, we pesent this solution in detail and descibe a numbe of its consequences. In paticula, we find that fo D = 2, 3, any pai of maximally entangled, bipatite pue states can be copied using the same LOCC opeation and a maximally entangled blank state. Howeve, we also show that fo any D which is not pime, thee exist such pais fo which this copying opeation is impossible. We conclude in section 5 with a discussion of ou esults. 2. The poblem of LOCC copying 2.1. Geneal consideations Let us conside the following scenaio, depicted in figue 1. We have two paties, Alice and Bob, occupying spatially sepaated laboatoies α and β espectively. Alice and Bob each have two D-dimensional quantum systems. Alice s systems will be labelled 1 and 3 while Bob s will be labelled 2 and 4. Associated with each of these systems is a copy of the D-dimensional Hilbet space H. The tenso poduct Hilbet spaces of Alice s and Bob s pais will be denoted by H α and H β espectively. Alice and Bob also possess ancillay quantum systems enabling them to cay out abitay local quantum opeations. They also shae a two-way classical channel allowing unlimited classical communication between them. Conside now a set of entangled, bipatite, pue states { ψ j }, whee j {1,...,N}. Thoughout this aticle, when N>1, we shall take the ψ j to be othogonal. This implies that, without the LOCC estiction, the states could be pefectly copied. Paticles 1 and 2 ae initially pepaed in one of these states although Alice and Bob do not know which one. Paticles 3 and 4 ae initially pepaed in the known, bipatite, blank state b. Alice and Bob aim to pefom the tansfomation ψ 12 j b34 ψ 12 j ψ34 j (2.1) by LOCC. Hee, the supescipts indicate the paticles that have been pepaed in each state. New Jounal of Physics 6 (2004) 164 (

5 5 DEUTSCHE PHYSIKALISCHE GESELLSCHAFT Figue 1. Depiction of the scenaio consideed in this pape. Laboatoies α and β ae spatially sepaated. These laboatoies contain the pais of paticles (1,3) and (2,4) espectively. Paticles 1 and 2 ae initially pepaed in one of the entangled states ψ j. Paticles 3 and 4 ae initially pepaed in the entangled blank state b. The aim is to pefom the copying tansfomation in equation (2.1) by LOCC. Geneal quantum state tansfomations ae descibed using the quantum opeations fomalism [19, 20].A quantum opeation on a quantum system with Hilbet space S is epesented mathematically by a completely positive, linea, tace non-inceasing map fom the set of linea opeatos on S to itself (when the input and output Hilbet spaces ae identical, which is the case in the pesent context). Let us denote such a map by E and conside a quantum system whose initial state is descibed by a density opeato ρ. This map tansfoms the density opeato accoding to ρ E(ρ) T(E(ρ)). (2.2) A paticulaly useful epesentation of quantum opeations is the opeato-sum epesentation: E(ρ) = K F k ρf k, (2.3) k=1 whee K is some positive intege. Fo E to be a physically ealizable quantum opeation, the F k, which ae known as the Kaus opeatos, must be linea opeatos that satisfy K F k F k 1, (2.4) k=1 whee 1 is the identity opeato on S. The equality holds when the map is tace peseving fo all states, in which case the quantum opeation is deteministic fo all states. If the opeation is not tace peseving fo a paticula initial state, then it can only be implemented with pobability equal to the tace of the final state. Whethe o not the opeation has been implemented can be always be detemined in pinciple, and this may be viewed as a genealized measuement. Moe geneally, any expeiment implements a tace peseving sum of tace non-inceasing New Jounal of Physics 6 (2004) 164 (

6 6 DEUTSCHE PHYSIKALISCHE GESELLSCHAFT quantum opeations. The opeation that has actually been caied out can always, in pinciple, be detemined, and it fomally coesponds to a paticula outcome of a genealized measuement. Thee ae many paticula kinds of quantum opeations of special inteest. In the pesent context, two kinds ae paticulaly impotant. These ae the sepaable opeations [21] and the LOCC pocedues. In a sepaable opeation acting on two systems in spatially sepaated laboatoies α and β, the F k may be witten as F k = A k B k. (2.5) Hee, A k and B k ae local Kaus opeatos acting on H α and H β espectively. In this context, we may efe to the F k as the global Kaus opeatos. LOCC pocedues ae sequences of tace peseving local quantum opeations caied out in the individual laboatoies, intespesed with ounds of classical communication. The infomation eceived at each laboatoy is used to contol the subsequent local opeation at the same location. The numbe of ounds of classical communication can be abitaily lage and, consequently, LOCC pocedues can be difficult to wok with. Howeve, the set of such pocedues is a subset of the set of sepaable opeations. It follows that sepaability is only a necessay condition fo a quantum opeation to be implementable by LOCC [10]. It is not sufficient. Still, the fact that the global Kaus opeatos fo sepaable opeations have the simple fom shown in (2.5) often makes such opeations a useful stating point fo investigating poblems elating to LOCC. See, fo example, [9] Catalytic copying Due to the limitations on the LOCC manipulation of entanglement, it is a non-tivial matte to detemine the set of blank states which enable one to copy, by LOCC, even a single, known state ψ. In pinciple, the conditions unde which this is possible can be obtained using Nielsen s theoem [22]. This esult specifies the conditions unde which one pue, bipatite, entangled state can be tansfomed into anothe by deteministic LOCC. Nielsen s theoem involves the concept of majoization, which we will biefly eview. Conside two eal, R-component vectos v = (v 1,...,v R ) and w = (w 1,...,w R ). Futhemoe, let v and w be the vectos obtained fom v and w by aanging thei components in noninceasing ode. The vecto w is said to majoize the vecto v if w i, (2.6) i=1 v i i=1 fo all {1,...,R} and with the equality holding fo R =. This majoization elation is usually witten as w v o v w. Conside now two pue bipatite states φ 1 and φ 2. These may be witten in Schmidt decomposition fom as φ s = R i=1 λsi x si y si, whee s {1, 2} and whee the maximum subsystem Hilbet space dimension is R. The Schmidt vectos λ s = (λ s1,...,λ sr ) may, without loss of geneality, be taken to have eal, non-negative components. Nielsen s theoem states that φ 1 can be tansfomed by deteministic LOCC into φ 2 if and only if λ 1 λ 2. (2.7) New Jounal of Physics 6 (2004) 164 (

7 7 DEUTSCHE PHYSIKALISCHE GESELLSCHAFT Retuning to the poblem of LOCC copying, let the states ψ and b have the Schmidt vectos λ ψ and λ b. We wish to implement the tansfomation ψ 12 b 34 ψ 12 ψ 34, (2.8) by LOCC. Nielsen s theoem implies that this will be possible if and only if λ ψ λ b λ ψ λ ψ. (2.9) Clealy, this copying tansfomation will be possible if the tansfomation b ψ is possible by LOCC, i.e., if λ b λ ψ. Howeve, what if b ψ is impossible by LOCC? When b cannot be tansfomed into ψ by deteministic LOCC, thee appea, at fist sight, to be two cases to conside, coesponding to whethe o not ψ b is possible by LOCC. Howeve, we shall now show that the possibility of the LOCC tansfomation ψ b, when combined with ou assumptions that the copying tansfomation in (2.8) is possible by LOCC and that the tansfomation ψ b is not, leads to a contadiction. To do this, it is useful to intoduce anothe elation between two vectos, the tumping elation. Conside two eal vectos v and w. If thee exists a eal vecto u such that u v u w, (2.10) then we say that w tumps v and wite this elation as v T w o w T v. Fom (2.9), we clealy see that λ b T λ ψ. (2.11) The tumping elation is weake than the majoization elation: that is, if v w then v T w,but not necessaily vice vesa. We ae assuming that ψ b is possible by LOCC, which implies that λ ψ λ b. Theefoe, λ ψ T λ b. (2.12) We shall now use the following theoem due to Jonathan and Plenio [18]: if v T w and w T v, then v = w. Combining this esult with (2.11) and (2.12), we see that λ ψ = λ b. When this is so, it follows that λ b λ ψ and, by Nielsen s theoem, that the tansfomation b ψ is actually possible by LOCC, which contadicts ou pemise. The emaining possibility is that both b ψ and ψ b ae impossible to pefom by LOCC. When this is the case, the states ψ and b ae said to be incompaable. Even though incompaable states cannot be tansfomed into each othe by LOCC, thee is the possibility that the tansfomation in (2.8) is possible. When this is so, ψ 12, which is unchanged by the copying pocedue, is said to act as a catalyst fo the tansfomation b 34 ψ 34. The poblem of finding, fo a geneal state ψ, the set of blank states b fo which ψ can be copied by entanglement catalysis is a challenging task. This is due to the fact that no analytical way of odeing the Schmidt coefficients of a geneal tenso poduct of two states has yet been discoveed. Nevetheless, by numeical methods, one can easily check fo paticula states whethe o not the majoization elation in (2.7) is satisfied. One can then seach fo pais of pue, bipatite entangled states such that one cannot be tansfomed into anothe diectly but fo which the tansfomation is possible with a catalyst. A specific example of catalytic copying, which we obtained in this way, is as follows. Conside the case of D = 5 and a state ψ with Schmidt coefficients 0.39, 0.26, 0.18, 0.17 and 0. Conside also a blank state b with Schmidt coefficients 0.32, 0.28, 0.24, and Fo these two states, one can New Jounal of Physics 6 (2004) 164 (

8 8 DEUTSCHE PHYSIKALISCHE GESELLSCHAFT eadily veify using Nielsen s theoem that the tansfomation b ψ is impossible by LOCC while the tansfomation ψ 12 b 34 ψ 12 ψ 34 can be caied out this way. The main focus of this pape is on LOCC copying of multiple quantum states with efficient use of entanglement. Even fo a single, known state, the poblem is complicated by the possibility of catalytic copying as we have just demonstated. To genealize this to multiple states, we would equie an undestanding of multi-state catalytic entanglement tansfomations, about which little, if anything, is cuently known. Fotunately, thee is a lage class of states sets that we can conside fo which the issue of catalysis does not aise. These ae sets whee at least one of the states to be copied is maximally entangled. Thei pefeential status is a consequence of the fact that maximally entangled states cannot seve as catalysts fo pue, bipatite entanglement tansfomations [18]. Such sets will be the focus of ou attention fo the emainde of this pape. 3. LOCC copying of a pue othogonal set including a maximally entangled state 3.1. Fom of the local Kaus opeatos Retuning to the poblem of locally copying the N states ψ j, ecall that we equie the copying opeation to be sepaable. This implies that the global Kaus opeatos will have the fom shown in (2.5), whee the A 13 k and Bk 24 act on H α and H β espectively. In tems of these opeatos, the copying tansfomation will have the fom A 13 k B24 k ψ12 j b34 =σ jk ψ 12 j ψ34 j. (3.1) Hee, the supescipts on the opeatos indicate the paticles on which they act. Also, the σ jk ae some complex coefficients that satisfy K k=1 σ jk 2 = 1. Sepaability of the copying opeation is, as we have noted above, only a necessay and not a sufficient condition fo LOCC copying. Howeve, the combination of the sepaability condition with specific featues elating to paticula sets of states can lead us to exact necessay and sufficient conditions fo LOCC copying. The emainde of this pape is devoted to investigating the LOCC copying poblem fo a class of such sets. These ae sets whee at least one of the states to be copied is maximally entangled. Fo the sake of definiteness, let the state ψ 1 be maximally entangled. It is known [18] that a maximally entangled state cannot seve as a catalyst. Theefoe, the tansfomation b ψ 1 must be possible by LOCC. Since we ae esticting ouselves to blank states of a pai of D dimensional paticles, it follows fom Nielsen s theoem that the blank state is necessaily maximally entangled also. This section is devoted to detemining the conditions unde which the ψ j can be copied by LOCC when both ψ 1 and the blank state b ae maximally entangled. In the fist pat of this section, we will see how the equiements of ou opeation have inteesting implications fo the fom of the local Kaus opeatos in equation (3.1). We will then obtain the geneal necessay and sufficient conditions unde which ou desied opeation is physically possible. To begin, let { x i } be an othonomal basis fo the single paticle Hilbet space H. We will fequently wok with the following efeence maximally entangled state in H 2 : ψ s max = 1 D D i=1 x i xs i. (3.2) New Jounal of Physics 6 (2004) 164 (

9 9 DEUTSCHE PHYSIKALISCHE GESELLSCHAFT We will also fequently encounte the poduct states xi xs j, fo paticles, s whee, s {1,...,4}. As such, it is convenient to adopt a simple notation fo these states. Define X s µ = x i xs j, (3.3) whee µ = µ(i, j) {1,...,D 2 }. Each value of µ must coespond to unique values of i and j. This can be achieved, fo example, by letting µ = i + D(j 1) with i, j {1,...,D}. Moe geneally, we will use Geek subscipts to index elements of this basis accoding to the same fomula as fo µ. The fact that the state ψ 1 is maximally entangled implies that thee exists a unitay opeato U 1 on H such that ψ 12 1 =(U ) ψ 12 max. (3.4) When the paticle pai (3,4) is in this state, we eplace the supescipts 1 and 2 with 3 and 4 espectively. The blank state b is also maximally entangled, so thee exists a unitay opeato U b on H such that b 34 =(U 3 b 1 4 ) ψ 34 max. (3.5) We now poceed to show that, without loss of geneality, the A 13 k and Bk 24 in (3.1) may be taken to be, up to multiplicative coefficients, unitay. To do this, we note that the most geneal LOCC pocedue consists of an abitaily long sequence of local opeations in Alice s and Bob s laboatoies intespesed with ounds of classical communication. The entie LOCC opeation is initiated by one paty. Fo the sake of definiteness, and without loss of geneality, let this paty be Alice. Alice implements a deteministic local opeation on he system. This opeation, which is tace peseving, may be a sum of tace non-inceasing opeations in which Alice obtains (classical) infomation about which of these opeations was caied out. The entie opeation is then a genealized measuement. If it is, then the measuement esult is communicated to Bob. Upon eceiving this, Bob implements a local opeation coesponding to this esult. He then communicates a desciption of his opeation to Alice (if she does not aleady know the opeation he will pefom given the classical infomation she sent him) togethe with any measuement esults and the pocess can epeat an abitaily lage numbe of times. The cucial point is the fact that if Alice and Bob begin with the state ψ1 12 b34, which is a maximally entangled state of the pais (1,3) and (2,4), then the LOCC copying pocedue will poduce the state ψ1 12 ψ34 1, which is also a maximally entangled state of these pais of paticles. No LOCC pocedue can tansfom a maximally entangled state into a non-maximally entangled state, and then into anothe maximally entangled state. It follows that each step in thei LOCC copying pocedue can do no moe than tansfom one maximally entangled state of these pais of paticles into anothe. So, let χ 1 and χ 2 be maximally entangled states of the pais (1,3) and (2,4). We may wite these states as χ =(V ) ψ 12 max ψ34 max, (3.6) whee {1, 2} and the V 13 ae unitay opeatos acting on H α. We will now investigate the popeties of a local opeation in one laboatoy that tansfoms χ 1 into χ 2. Fo the sake of definiteness, we let this opeation be caied out by Alice in he laboatoy α. The following New Jounal of Physics 6 (2004) 164 (

10 10 DEUTSCHE PHYSIKALISCHE GESELLSCHAFT agument applies equally well if the opeation wee to be caied out by Bob. Alice caies out a local opeation, which we shall denote by E 13. This takes the fom of a completely positive, linea, tace non-inceasing map on the space of linea opeatos on H α. Intepeting this opeation as coesponding to a genealized measuement outcome, whose pobability is p fo the initial state χ 1, this opeation must poduce the state χ 2 accoding to E ( χ 1 χ 1 ) = p χ 2 χ 2. (3.7) Let us now define the following opeation on paticles 1 and 3 whose action on an abitay density opeato ρ 13 of these paticles is Fom (3.6) (3.8) we see that Ẽ 13 (ρ 13 ) = V 13 2 E 13 (V 13 1 ρ13 V 13 1 )V (3.8) Ẽ ( ψ 12 max ψ12 max ψ34 max ψ34 max ) = p ψ12 max ψ12 max ψ34 max ψ34 max. (3.9) We will now poceed to show that the above tansfomation implies that Ẽ 13 ( ) = p1 13 ( )1 13. (3.10) To do so, let us expand (3.9) in tems of the X µ basis states, which gives D 2 µ,ν=1 Ẽ 13 ( X 13 µ X13 ν ) X24 µ X24 ν D2 =p µ,ν=1 X 13 µ X13 ν X24 µ X24 ν, (3.11) whee we have omitted the oveall facto of 1/D 2. Acting on the (2,4) states to the left with X 24 and to the ight with Xδ 24 and making use of thei othonomality, we obtain Ẽ 13 ( X 13 γ X13 δ ) = p X13 γ X13 δ. (3.12) An abitay linea opeato acting on H 2 may be witten as = D 2 γ,δ=1 ω γδ X γ X δ, (3.13) having the matix elements ω γδ in the X γ basis. Fom (3.12) and the lineaity of Ẽ 13, it eadily follows that Ẽ 13 ( 13 ) = p 13. (3.14) Since this is tue fo any linea opeato on H 2, we equie that (3.10) is tue. Combining this with (3.8), we see that Alice s opeation has the fom Ẽ 13 (ρ 13 ) = p(v 2 V 1 )13 ρ 13 (V 1 V 2 )13. (3.15) γ New Jounal of Physics 6 (2004) 164 (

11 11 DEUTSCHE PHYSIKALISCHE GESELLSCHAFT In this local opeation, the Kaus opeatos may be taken to be p(v 2 V 1 )13, which ae clealy popotional to unitay opeatos. Futhemoe, Alice s oveall local Kaus opeatos A 13 k ae simply the poducts of the local Kaus opeatos coesponding to the elementay steps she caies out in the entie LOCC pocedue. These must also be popotional to unitay opeatos, since the poduct of any numbe of unitay opeatos is also a unitay opeato. Clealy, the above agument also applies if the elementay step is caied out by Bob. We ae theefoe led to the following conclusion: the local Kaus opeatos A 13 k and Bk 24 fo the entie LOCC pocedue ae, up to oveall multiplicative coefficients, unitay. These coefficients ae eal and non-negative since they ae, fom ou above definition of the elementay step local Kaus opeatos, poducts of the squae oots of pobabilities. We may then wite A 13 k = f k à 13 k, (3.16) B 24 k = g k B 24 k. (3.17) Hee, à 13 k and B 24 k ae unitay opeatos on H α and H β espectively and the f k,g k ae the eal, non-negative coefficients which satisfy K (f k g k ) 2 = 1, (3.18) k=1 as a consequence of (2.4) and the fact that ou LOCC pocedue is tace peseving. This has seveal impotant consequences that we can take advantage of. The fist is the fact that any sepaable quantum opeation whose local Kaus opeatos have this popety can be caied out by LOCC. This can be done in the following way. At one of the laboatoies, say α, a andom vaiable Y with K possible values y k and pobability distibution p k = (f k g k ) 2 is geneated. On obtaining the esult y k, Alice caies out the local unitay opeation à 13 k. She also communicates the value of Y to Bob, who then poceeds to implement the tansfomation B 24 k. The fact that the global Kaus opeatos F k ae, up to multiplicative coefficients, unitay implies that each one can be implemented deteministically. Futhemoe, they must each cay out the desied LOCC copying tansfomation, fo each of the states to be copied. Othewise, the final state would be mixed. This implies that a necessay and sufficient condition fo implementing the copying tansfomation is that the copying pocedue can be implemented by a single global Kaus opeato F = A 13 B 24, whee A 13 and B 24 ae unitay. When this is the case, the complex coefficients σ jk in (3.1), whee we may dop the index k, have unit modulus. Implementing these obsevations, (3.1) becomes A 13 B 24 ψ 12 j b34 =e iθ j ψ 12 j ψ34 j, (3.19) fo some angles θ j. The fact that A and B ae unitay implies that the states ψ j must all be maximally entangled. The eason fo this is that, if any non-maximally entangled state ψ j could be pefectly copied, then paticles 3 and 4, initially pepaed in the maximally entangled state b, would be left in the non-maximally entangled state ψ j. This is impossible to achieve with a pai of local unitay opeatos. In the emainde of this section, we shall use the above findings to obtain a geneal necessay and sufficient condition fo LOCC copying, with a maximally entangled blank state, of the states ψ j when they ae othonomal and maximally entangled. New Jounal of Physics 6 (2004) 164 (

12 12 DEUTSCHE PHYSIKALISCHE GESELLSCHAFT 3.2. Condition fo LOCC copying We saw above that, if ψ 1 is maximally entangled, then the ψ j ae all maximally entangled. Consequently, we may wite all of these states in the same fom as we did fo ψ 1 in (3.4), that is, as ψ 12 j =(U1 j 1 2 ) ψ 12 max, (3.20) fo some unitay opeatos U j on H. Again, when consideing the paticle pai (3,4) in one of these states, we will change the supescipts 1 and 2 to 3 and 4 espectively. Let us now define the following unitay opeatos on H α : C 13 j = (U 1 j U 3 j )A13 (U 1 j U3 b ). (3.21) With a small amount of algeba, it is easily seen that (3.19) is equivalent to C 13 j B 24 ψ 12 max ψ34 max =eiθ j ψ 12 max ψ34 max. (3.22) In tems of the two-paticle basis set { X µ }, this can be witten as C 13 j B 24 D 2 µ=1 X 13 µ X24 µ =eiθ j D 2 X 13 µ=1 µ X24 µ. (3.23) Notice that the Xν 13 X24 τ fom a basis fo the total Hilbet space H α H β. Acting to the left thoughout with Xν 13 X24 τ we obtain, D 2 µ=1 This can be witten as X 13 ν C13 j X13 µ X24 τ B24 X 24 µ =eiθ j D 2 µ=1 δ νµ δ τµ = e iθ j δ ντ. (3.24) C j B T = e iθ j 1. (3.25) Hee, 1 is the identity opeato on H 2 and T denotes the tanspose in the { X µ } basis. Solving fo B and making use of unitaity, we find that B = e iθ j C j, (3.26) whee denotes complex conjugation in the { X µ } basis. Fom this, we see that the opeatos e iθ j C j ae independent of j. Using the explicit expession fo Cj 13 in (3.21), we see that e iθ j (U 1 j U 3 j )A13 (U 1 j U3 b ) = eiθ j (U 1 j U 3 j )A13 (U 1 j U 3 b ), (3.27) fo all j, j {1,...,N}. Acting thoughout to the left with Uj 1 U3 j U 1 j U 3 b we obtain and to the ight with e iθ j A 13 [(U j U j )1 1 3 ] = e iθ j [(U j U j )1 (U j U j )3 ]A 13. (3.28) New Jounal of Physics 6 (2004) 164 (

13 13 DEUTSCHE PHYSIKALISCHE GESELLSCHAFT Pio to poceeding, we shall make a bief digession. Fom this point onwads, we will be concened with opeato equations involving just two paticles in a shaed entangled state. Consequently, it will be convenient to dop the paticle supescipts. We do this because the paticles involved will follow the tenso poduct odeing convention we established fo such paticle pais in (3.4) and the subsequent paagaph. Also, the analysis that follows in the next section will be quite inticate and will not benefit fom unnecessay notation. Fo the sake of notational convenience, define the unitay opeatos T jj = U j U j. (3.29) Using this and the unitaity of A, we find that (3.28) is equivalent to A(T jj 1)A = e i(θ j θ j ) (T jj T jj ). (3.30) Fom the above agument, it follows that the existence of a unitay opeato A on H 2 which satisfies this equation, fo some angles θ j and θ j, is both necessay and sufficient fo the existence of an LOCC copying pocedue which, with a maximally entangled blank state b, copies all of the ψ j. The next section will be devoted to the case of N = 2. Pio to addessing this case, we shall make some futhe geneal obsevations. Having defined the opeatos U j in tems of the efeence maximally entangled state ψ max in (3.20), one might suspect that the T jj also make implicit efeence to this state. Howeve, this is not so. We can, in fact, wite these opeatos solely in tems of the states to be copied, ψ j, and D, the dimensionality of H. To do so, conside ψ j ψ j = 1 D D U j x i x i U j x i x i, (3.31) i,i =1 whee we have used (3.2). Denoting by PT the patial tace with espect to the second system, we find D D PT( ψ j ψ j ) = U j x i x i U j T( x i x i ) = i,i =1 D i=1 U j x i x i U j = T jj. (3.32) Hee we have used equation (3.29) and the completeness of the x i. We see that the copying condition in (3.30) can be expessed solely in tems of the states to be copied and the dimensionality of the single-paticle Hilbet space. Notice that, fom (3.32), if we take the full tace of ψ j ψ j we obtain ψ j ψ j = 1 D T(T jj ). (3.33) It is known fom the oiginal no cloning theoem that, fo pefect copying to be possible, we equie the states ψ j and ψ j to be eithe othogonal o, up to a phase, identical. It is inteesting to see how this fact also follows fom (3.30). Taking the full tace thoughout equation (3.30) and making use of the unitaity of A, we obtain DT(T jj ) = e i(θ j θ j ) [T(T jj )] 2. (3.34) New Jounal of Physics 6 (2004) 164 (

14 14 DEUTSCHE PHYSIKALISCHE GESELLSCHAFT This is a simple quadatic equation in T(T jj ), whose oots ae 0 and De i(θ j θ j ). Fom (3.33), we easily see that these oots coespond to ψ j and ψ j being othogonal and, up to a phase, identical espectively. The poblem of detemining when a unitay opeato A on H 2 satisfying (3.30) exists appeas to be quite challenging fo abitay N and D. Howeve, fo N = 2, the poblem can be solved exactly fo all D. We will pesent the detailed solution to this poblem and exploe some of its consequences in the next section. 4. LOCC copying of two othogonal maximally entangled states 4.1. A spectal copying condition Fom the above discussion, it follows that a necessay and sufficient condition fo LOCC copying of two maximally entangled states ψ 1 and ψ 2 with a maximally entangled blank state is that thee exists a two-paticle unitay opeato A which implements the tansfomation in (3.30) fo j, j {1, 2} and some angles θ 1 and θ 2. Notice fom the definition of the T jj in (3.29) that T jj = 1, the identity opeato on H. Consequently, fo j = j,(3.30) is tivially satisfied by any unitay opeato A and any angles θ j. Also, the equations fo T 12 and T 21 ae simply the Hemitian adjoints of each othe, so if one is tue then so is the othe. It follows that fo the case of N = 2, we need only conside one of these equations. Fo the sake of definiteness, we will focus on the opeato T 12, which we will wite simply as T. We also wite θ = θ 1 θ 2. Fo suitable choices of θ 1 and θ 2, this can take any eal value. Ou condition then becomes A(T 1)A = e i θ (T T), (4.1) whee 1 is again the identity opeato on H. We can simplify this expession futhe by emoving the phase facto in the following way: define T = e i θ T. (4.2) Then by simple substitution we find that (4.1) is equivalent to A( T 1)A = T T. (4.3) A unitay opeato A satisfying this equation exists if and only if T 1 and T T have the same eigenvalues, with the same multiplicities. So, we may wite ou condition fo LOCC copying of the two states as spec( T T) = spec( T 1), (4.4) whee spec denotes the spectum. Thoughout this section, it will be convenient to goup the eigenvalues accoding to multiplicity. So, let M D be the numbe of distinct eigenvalues. We shall wite these as λ, whee {1,...,M}. It is easy to see fom (4.4) that, fo evey intege R 2, we have spec( T R ) = spec( T 1 (R 1) ). (4.5) New Jounal of Physics 6 (2004) 164 (

15 15 DEUTSCHE PHYSIKALISCHE GESELLSCHAFT This implies that λ 1 λ 2...λ R spec( T) (4.6) fo all j {1,...,M} and j {1,...,R}. To detemine which pais of maximally entangled states can be simultaneously locally copied with a maximally entangled blank state, we must find out which unitay opeatos satisfy (4.4). The cuent section will focus on solving this poblem and exploing some of the consequences of its solution. Pio to giving this solution, we make the following intiguing obsevation. The physical poblem of LOCC copying leads to the mathematical poblem expessed in (4.4), whee physical consideations equie that T is unitay. Howeve, if we ae inteested in this equation fom a puely mathematical pespective, then thee is the question of what popeties a geneal linea opeato T must have in ode to solve (4.4). We will now show that the eigenvalues of any linea opeato, if they ae all non-zeo, must have unit modulus in ode to satisfy (4.4). To pove this, we make use of the fact that we may, without loss of geneality, take the λ to be aanged in non-inceasing ode in tems of thei moduli: 0 < λ 1 λ 2 λ M. (4.7) Let us notice that equation (4.4) implies that λ 2 1 spec( T). We now assume that λ 1 = min { λ } < 1. It immediately follows that λ 2 1 = λ 1 2 < min { λ } fo λ 1 0, contadicting this assumption. Ou assumption must theefoe be false. Similaly, we see that (4.4) implies that λ 2 M spec( T). Let us assume that λ M =max { λ } > 1. We then obtain λ 2 M = λ M 2 > max { λ }, which also leads to a contadiction. This agument implies that λ =1 and leads to the conclusion that the non-zeo eigenvalues must be of the fom λ = e iφ (4.8) fo some angles φ [0, 2π). Without loss of geneality, we may take these angles to be odeed accoding to 0 φ 1 φ 2 φ M < 2π. (4.9) We will now pove that a unitay opeato T, whose eigenvalues ae of couse all non-zeo, satisfies (4.4) if and only if the following two conditions ae satisfied: (i) The distinct eigenvalues of T ae the Mth oots of unity, fo some positive intege M which is a facto of D and which may be equal to D itself. (ii) The distinct eigenvalues of T have equal degeneacy. We will fist pove the necessity of condition (i), following which we will see that when this condition is satisfied, condition (ii) is necessay and sufficient fo (4.4) to hold. Ou poof of the necessity of (i) begins by establishing that, fo each, thee is a positive intege k {1,...,M} such that λ k = 1. To pove this, notice that, fom (4.8), we obtain λ n = einφ (4.10) (4.11) New Jounal of Physics 6 (2004) 164 (

16 16 DEUTSCHE PHYSIKALISCHE GESELLSCHAFT fo evey intege n. When n is non-negative, we see fom (4.6) that we must have λ n spec( T). Howeve, the spectum of T is finite. In view of this, conside a paticula eigenvalue λ and two abitay positive integes n and n. Fom (4.6), we see that λn,λn spec( T). The spectum of T has pecisely M distinct eigenvalues. So, fo fixed n, let us define n = n + k, whee k {1,...,M}. Thee clealy must be at least one value of k fo which λ n = λ n. When these ae equal, we have e in φ = e in φ. This implies that e i(n n )φ = e ik φ = λ k = 1, (4.12) as equied. One impotant consequence of (4.10) is the fact that 1 spec( T). (4.13) This follows fom (4.6), which tells us that any poduct of eigenvalues of T is also an eigenvalue of T. We simply apply this to (4.10), taking R = k and 1,..., M =. Fom this, we see that the odeing of the angles in (4.9) implies that φ 1 = 0. We can then update (4.9) in the light of (4.13) to obtain 0 = φ 1 φ 2 φ M < 2π. (4.14) Anothe consequence of (4.6) is the fact that, fo each {1,...,M}, λ 1 = λ spec( T). (4.15) We obtain this in the following way. We know fom (4.6) and, in the case of k = 1, equation (4.13), that λ k 1 spec( T). Howeve, it follows fom (4.10) that λ k 1 = λ 1,soweget(4.15). Let us now use the above obsevations to pove that the λ must be the Mth oots of unity. Fom (4.6) and (4.15), we easily obtain λ λ spec( T), (4.16) fo all, {1,...,M}. We now set = ( mod M) + 1. We also wite the angula spacings between neighbouing eigenvalues as δ = { φ+1 φ : {1,...,M 1}, 2π + φ 1 φ M : = M. (4.17) Combining these definitions and making use of (4.16), we obtain e iδ spec( T). (4.18) The mean value of the δ is 2π/M. Conside now the smallest of these angula spacings, which we shall denote by δ min, which must be non-zeo because we ae woking with distinct eigenvalues. To fit the M distinct eigenvalues aound the unit cicle, we equie that δ min 2π/M. Howeve, we know fom (4.10) that e ikδ min = 1 fo some k {1,...,M}. It is impossible to satisfy this equiement fo non-zeo δ min unless δ min 2π/M. Combining these two New Jounal of Physics 6 (2004) 164 (

17 17 DEUTSCHE PHYSIKALISCHE GESELLSCHAFT inequalities gives δ min = 2π/M. (4.19) It is now easy to see that the λ must be the Mth oots of unity. Given that e iδ min is an eigenvalue of T, which we know to be the case fom (4.18), we can apply (4.6) to conclude that the e iδ min, fo all {1,...,M}, ae also eigenvalues of T. These M complex numbes, which ae distinct, ae the Mth oots of unity. Since T has exactly M distinct eigenvalues, we conclude that the spectum of T consists pecisely of these Mth oots of unity. This completes the poof of the necessity of condition (i). Let us now show that when condition (i) is satisfied, condition (ii) is necessay and sufficient fo T to satisfy (4.4). We will begin by poving its necessity. The eigenvalues λ of T have been gouped accoding to thei multiplicity. So, let us denote the degeneacy of λ,asan eigenvalue of T,byd T. Combining the fact that the λ ae the Mth oots of unity fo some intege facto M of D with the phase odeing in (4.14), we see that the distinct eigenvalues of T ae given by [ ] 2πi( 1) λ = exp. (4.20) M Futhemoe, must have d T = D. (4.21) =1 Of couse, the λ ae also the eigenvalues of T T. Howeve, they will have diffeent degeneacies. So, let us denote by d T T the degeneacy of λ as an eigenvalue of T T.Fo these degeneacies, we have =1 As a consequence of (4.4), we see that d T T = D 2. (4.22) d T T = Dd T. (4.23) Making use of (4.20), we find that the d T T can be explicitly expessed in tems of the d T in the following way: define { 1: (s + s ) mod M = 1, G ss = 0: (s + s ) mod M 1, (4.24) whee s, s {1,...,M}. Afte some algeba, we find that we may wite d T T = s,s =1 G ss d T s d T s. (4.25) New Jounal of Physics 6 (2004) 164 (

18 18 DEUTSCHE PHYSIKALISCHE GESELLSCHAFT Combining (4.23) and (4.25), we see that the degeneacies d T must satisfy s,s =1 G ss d T s d T s T = Dd. (4.26) This is a necessay and sufficient condition fo the λ to satisfy (4.4). It is evident fom this expession that, fo each, the left-hand side is a quadatic fom. Fo example, fo = 1, we have ( ) d T 1 d T M d T 1. d T M = Dd T 1. (4.27) The coesponding quadatic foms fo = 2,...,M ae obtained fom (4.27) by cyclically shifting the elements of each column in this matix down by 1 places. Let us define σ(, s) = ( s) mod M +1. (4.28) Using this and equation (4.24), one can eadily veify that s=1 G ss d T s = d T σ(,s ), (4.29) fom which we obtain s,s =1 G ss d T s δ s 1 = d T. (4.30) Hee, δ s 1 is the usual Konecke delta. Combining this equation with (4.26), we get s,s =1 G ss d T T s (ds Dδ s 1) = 0. (4.31) Making use of (4.29), we find that this equation leads to s =1 d T σ(,s ) d T s = d T s =1 d T s. (4.32) We will now use this expession to show that the degeneacies d T must all be equal to D/M. Notice, fom (4.21), that D/M is the aveage of the d T s. They must all be equal if the maximum degeneacy is equal to this aveage degeneacy. Let max be a value of such that d T max is the maximum degeneacy. As a consequence of the positivity of the d T, the following inequality New Jounal of Physics 6 (2004) 164 (

19 19 DEUTSCHE PHYSIKALISCHE GESELLSCHAFT must be satisfied: d T σ( max,s ) d T s s =1 d T max s =1 d T s (4.33) with the equality holding only if d T σ(max,s ) = d T max fo all s. Now, fo any fixed, σ(, s ) meely pemutes the integes s {1,...,M}, so that all degeneacies must, fom (4.32), be equal to the maximum degeneacy. This completes the poof of necessity. Let us finally pove that when the distinct eigenvalues of T ae the Mth oots of unity, it is also sufficient that they have equal degeneacies d T = D/M to satisfy (4.4). This is simple to show. Fo λ givenby(4.20), (4.26) is equivalent to the spectal copying condition in (4.4). When d T = D/M,(4.26) is equivalent to G ss = M. (4.34) s,s =1 To show that this equation is satisfied, we note that, when the d T ae all equal, then (4.29)gives G ss = 1. (4.35) s=1 Summing this expession ove the index s and making use of (4.24) leads to (4.34), completing the poof of sufficiency. Let us take the oppotunity hee to discuss the above esults, in thei physical context, pio to exploing some of thei consequences. Fo two othogonal, maximally entangled bipatite states ψ 1 and ψ 2, having D-dimensional subsystems, to be locally copyable with a D- dimensional maximally entangled blank state, it is necessay and sufficient that the eigenvalues of the associated unitay opeato T, defined though (3.29) and (4.2) ae, fo some intege facto M of D, the Mth oots of unity and that these eigenvalues ae equally degeneate. We defined the opeato T in (4.2) in tems of the opeato T which contains all of the infomation about the elationship between ψ 1 and ψ 2.This definition amounted to the emoval of the phase facto e i θ in (4.2). This facto was emoved in ode to simplify the above poofs of the LOCC copying conditions. Howeve, fo a paticula pai of states, it is T, athe that T, that aises natually. As such, it is impotant to fomulate these LOCC copying conditions in tems of the spectum of the T opeato also. This is easily done. The incopoation of this abitay phase facto is equivalent to an abitay otation of the spectum in the complex plane. So, LOCC copying of ψ 1 and ψ 2 is possible if and only if the eigenvalues of T ae, up to an oveall otation, equally degeneate Mth oots of unity fo some intege facto M of D. In othe wods, they must have equal angula spacing and be equally degeneate. Clealy, fo any paticula pai of othogonal maximally entangled states ψ 1, ψ 1 and a paticula maximally entangled blank state b fo which the LOCC copying opeation is possible, it is impotant to have an explicit pesciption fo caying out this pocedue. This amounts to knowing two suitable local unitay opeatos A and B fo which (3.19) is satisfied. Fom the esults we have obtained hee and in the peceding section, it is possible to obtain specific opeatos which cay out the equied task. New Jounal of Physics 6 (2004) 164 (

20 20 DEUTSCHE PHYSIKALISCHE GESELLSCHAFT Ou stating point is the thee states involved in the copying pocedue, and also the abitay efeence maximally entangled state ψ max. These ae pesumably known. Fom these, we deduce the opeato T using (3.32) and the fact that T = T 12. The opeato T is obtained using (4.2) and by setting θ equal to the smallest among the aguments of the eigenvalues of T. Fom (4.3) and the unitaity of A, we see that we may wite T 1 = T T = λ P, (4.36) =1 λ Q. (4.37) =1 Hee, P and Q ae the pojectos onto the eigenspaces of λ, which is an Mth oot of unity given by (4.20), as an eigenvalue of T 1 and T T espectively. Let us denote these eigenspaces by H T 1 and H T T. These spaces have dimension Dd T. Using these notions, we can obtain a unitay opeato A that satisfies (4.3) in the following way. Let { ξ l } and { η l } be othonomal bases fo H T 1 and H T T espectively. We clealy have l {1,...,Dd T }. Now conside the unitay opeato A = Dd T ξ l η l. (4.38) =1 l=1 One can easily show that AP A = Q, which implies that A satisfies (4.3) as equied. We must now find a suitable opeato B. To do so, we ae equied to know the opeato U b. This can be deduced fom (3.5)tobe U b = D PT( b ψ max ). (4.39) If we now combine (3.21) and (3.26), we find that B is given by B = e iθ j (U j U b ) T A T (U j U b )T, (4.40) fo eithe j = 1, 2 and whee T again denotes the tanspose in the X µ. We may neglect the phase facto hee entiely as it has no effect on the physical natue of the tansfomation. We shall now exploe some of the consequences of the local copying condition in (4.4), paying paticula egad to the elationship between othogonality and local copyability of two maximally entangled states with a maximally entangled blank state Consequences Having established the LOCC copying condition fo a pai of othogonal, maximally entangled, bipatite, pue states with a maximally entangled blank state, it is natual to enquie as to when this condition is satisfied. We shall find that the dimensionality D of the single paticle Hilbet space H plays a pominent ole hee. We will show that fo D = 2, 3, evey pai of othogonal, maximally entangled, bipatite, pue states can be locally copied with a maximally entangled blank state. Howeve, we will then New Jounal of Physics 6 (2004) 164 (