Unified Universal Quantum Cloning Machine and Fidelities
Yi-Nan Wang, Han-Duo Shi, Zhao-Xi Xiong, Li Jing, Xi-Jun Ren, Liang-Zhu Mu, Heng Fan
aa r X i v : . [ qu a n t - ph ] A p r Unified Universal Quantum Cloning Machine and Fidelities
Yi-Nan Wang , Han-Duo Shi , Zhao-Xi Xiong , Li Jing , Xi-Jun Ren , Liang-Zhu Mu ∗ , and Heng Fan † School of Physics, Peking University, Beijing 100871, China School of Physics and Electronics, Henan University, Kaifeng 4750011, China Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China (Dated: November 4, 2018)We present a unified universal quantum cloning machine, which combines several different existinguniversal cloning machines together including the asymmetric case. In this unified framework, theidentical pure states are projected equally into each copy initially constituted by input and one half ofthe maximally entangled states. We show explicitly that the output states of those universal cloningmachines are the same. One importance of this unified cloning machine is that the cloning processionis always the symmetric projection which reduces dramatically the difficulties for implementation.Also it is found that this unified cloning machine can be directly modified to the general asymmetriccase. Besides the global fidelity and the single-copy fidelity, we also present all possible arbitrary-copy fidelities.
PACS numbers: 03.67.Ac, 03.65.Aa, 03.67.Lx, 03.65.Ta
Introduction .— No-cloning theorem is fundamental forquantum mechanics and quantum information sciencethat states an unknown quantum state can not be clonedperfectly[1]. However, we can try to clone a quantumstate approximately with the optimal quality [2], or in-stead, we can try to clone it perfectly with the largestprobability [3]. So various quantum cloning machineshave been designed for different quantum informationtasks [4-18]. Experimentally, quantum cloning machineshave been realized in optics system [19–22], nuclearmagnetic resonance system [23, 24], diamond nitrogen-vacancy center system [25], etc.The universal quantum cloning machine is first pro-posed by Buˇzek and Hillery [2] which can copy optimallyone arbitrary qubit equally well to two copies. Later moregeneral cases have been studied, see Ref.[17] for a review.So far there exists two universal quantum cloning ma-chines which can clone N identical d -level pure quantumstates to M copies, M ≥ N . One is proposed by Wernerin Ref. [5], and the other is proposed by Fan et al. in Ref.[7]. Both have advantages from different points of view.Also we know some limited cases of asymmetric cloning[14]. It seems that all those cloning machines are quitedifferent and no simple connection exists. In this Letter,we will present a simple and unified cloning transforma-tion which combines all those cloning machines together.In the framework of the unified quantum cloning, theidentical pure states are projected equally into each copywhere the output is initially constituted by input and onehalf of the maximally entangled states, the left half of themaximally entangled states act as the ancillary states.We will show explicitly that the density operators fromthose cloning machines are the same. Since the symmet- ∗ [email protected] † [email protected] ric operator is constituted by SWAP gates, the quantumcircuit corresponding to this unified cloning machine canthus be designed accordingly. Also importantly, this uni-fied universal cloning machine can be easily modified tothe general asymmetric case. As we know, only a fewlimited results of asymmetric cloning are known whilethe general case is still absent.The optimality of the cloning machine is judged gener-ally by whether the obtained fidelity of the cloning out-put state achieves its optimal bound. So far the optimalglobal fidelity and single-copy fidelity have already beenobtained [5, 13]. It is, however, necessary to find optimalgeneral-copy fidelities since it is possible that differentoptimal fidelities give different criteria, as happened inphase-covariant cloning machine [26]. With the unifiedand optimal universal cloning machine, we will presentall possible arbitrary-copy fidelities so that the optimal-ity of the cloning machine can be quantified from differentaspects. Equivalence of two universal quantum cloningmachines .—For a pure state, | ϕ i = P j x j | j i , in d -dimensional Hilbert space H with density operator σ ≡ | ϕ ih ϕ | , d =dim H , P j | x j | = 1, we refer it to aqudit. The N → M Werner cloning machine is presentedas [5], ρ out = d [ N ] d [ M ] s M (cid:16) σ N ⊗ I ⊗ ( M − N ) (cid:17) s M , (1)where d [ N ] = C Nd + N − = ( d + N − N !( d − , it is the dimension ofthe symmetric subspace of N -fold Hilbert space, d [ N ] =dim H ⊗ N + , I is the identity on H , s M is the symmetricprojector which maps states in H ⊗ M onto its symmetricsubspace H ⊗ M + . Explicitly, s M = P M~m | ~m ih ~m | , wherestate | ~m i ≡ | m , m , ..., m d i is a completely symmetricstate with m j states in | j i , and the summation is assumedto run all possible values with constraint, P j m j = M .In Werner cloning machine, the information of the in-put is equally projected to each copy. The initial M − N identities are also understandable since the assumptionof the universal cloning machine is that the input stateis arbitrary thus a completely mixed state, I /d , shouldbe a suitable candidate before the cloning procession isapplied.Following Buˇzek-Hillery 1 → N → M for qubit byGisin and Massar [4], Fan et al. proposed the followingtransformation for qudit case [7], see Fig.1, U | ~n i ⊗ R = η M − N X ~k vuutY j ( n j + k j )! n j ! k j ! | ~n + ~k i ⊗ R ~k , (2)where | ~n i is the input state, U denotes unitary trans-formation, R in l.h.s. denotes blank state andthe initial ancillary state, R ~k is the ancillary statewhich can be realized also by symmetric state | ~k i ,the whole normalization factor takes the form, η = p ( M − N )!( N + d − / ( M + d − P k j = M − N . For N identical purestates | ϕ i , the input takes the form, | ϕ i ⊗ N = √ N ! N X ~n Y j x n j j p n j ! | ~n i , (3)we can apply Eq. (2), trace out the ancillary statesand thus obtain the output density operator. Similarlyas for qubit case in theory [8] and in experiment [19],this cloning machine might naturally be realized by lightemission from multilevel atomic system [9].Typically, two different figures of merit are applied forthe universal cloning machines. One is the global fi-delity between the whole output density operator ρ out and the ideal output | ϕ i ⊗ M with perfect M copies, F M = ⊗ M h ϕ | ρ out | ϕ i ⊗ M . The other is the single-copyfidelity defined between an individual output density op-erator and a single input pure state, F = h ϕ | ρ out | ϕ i ,where each individual output density operator ρ out is thesame for all M copies. It is shown that those two fidelitiesfor Werner cloning machine and for cloning machine in(2) are the same. Yet the similarity of the output densityoperators is necessary. Next, explicitly we shall show theoutput state of M copies from the two cloning machinesare the same.First, we can find that the symmetric state | ~m i of M qudits can be divided into two parts with N qudits and M − N qudits, respectively, | ~m i = 1 q C NM M − N X ~k Y j s m j !( m j − k j )! k j ! | ~m − ~k i| ~k i . (4)The symmetric projector s M can thus be reformulatedby this splitting, then with the help of the expansion FIG. 1: Generally, the input of the cloning machines are iden-tical pure input states, blank states and initial ancilla, thecloning is realized by unitary transformation. (3), substituting σ ⊗ N into (1), by some calculations, theoutput density operator of Werner cloning machine takesthe form, ρ out = N ! η M X ~m,~m ′ | ~m ih ~m ′ |× M − N X ~k Y j x m j − k j j x ∗ ( m ′ j − k j ) q m j ! m ′ j !( m j − k j )!( m ′ j − k j )! k j ! , (5)where we have already used h ~l | ~l ′ i = δ ~l~l ′ .For the second cloning machine (2), substituting theresult of (3) into (2) and tracing out the ancillary states,simply the output state is written as ρ ′ out = N ! η N X ~n,~n ′ M − N X ~k | ~n + ~k ih ~n ′ + ~k |× Y j x n j j x ∗ ( n ′ j ) j q ( n j + k j )!( n ′ j + k j )! n j ! n ′ j ! k j ! . (6)Considering we have the constraint P k j = M − N , ap-parently, the output states from two universal quantumcloning machines are the same, ρ out = ρ ′ out . Unified universal quantum cloning machine .—Stimulated by the fact that the two existing universalcloning machines are the same, we may try further toexplore the possibility to unify the two cloning machinestogether. For N identical pure states, we propose the N → M universal cloning machine as the following, | ϕ i out = λ (cid:16) s M ⊗ I ⊗ ( M − N ) (cid:17) | ϕ i ⊗ N | Φ + i ⊗ ( M − N ) , (7)where | Φ + i ≡ √ d P j | jj i is a maximally entangled statein H ⊗ H , λ is the normalization factor. Here operator s M acts on the N identical pure input states and half ofthe M − N maximally entangled states, the left half ofthe maximally entangled states are the ancillary states,see Fig.2.Next we will show that this cloning transformation isthe same as (2). We consider a property of the symmetric FIG. 2: The unified cloning machine is constituted by identi-cal pure input states and the prepared maximally entangledstates, the cloning is always realized by symmetric projection. projector, s M = s M (cid:0) I ⊗ N ⊗ s M − N (cid:1) , this is due to Eq.(4)and also s M − N is the identity operator on H ⊗ ( M − N )+ .By symmetric projection, M − N maximally entangledstates can be mapped as a maximally entangled state insymmetric subspace H ⊗ ( M − N )+ ⊗ H ⊗ ( M − N )+ , (cid:16) s M − N ⊗ I ⊗ ( M − N ) (cid:17) | Φ + i ⊗ ( M − N ) = M − N X ~k | ~k i| ~k i , (8)where an unimportant whole factor is omitted, and someother unimportant whole factors will also be omittedlater without specification. Note that the entanglementcutting is unchanged here. Since quantum mechanics islinear, so we just consider the input state be a symmetricstate for (7), with the help of the result in (8), we canfind the unified cloning machine (7) can be rewritten asthe form (cid:16) s M ⊗ I ⊗ ( M − N ) (cid:17) | ~n i| Φ + i ⊗ ( M − N ) = (cid:16) s M ⊗ I ⊗ ( M − N ) (cid:17) | ~n i M − N X ~k | ~k i| ~k i = M − N X ~k vuutY j ( n j + k j )! n j ! k j ! | ~n + ~k i| ~k i , (9)where the splitting relation (4) is used in the last equa-tion. Thus considering that the last M − N qudits areancillary states, the universal cloning transformation (2)is re-obtained by the unified cloning transformation (7).Also (7) can be considered to be an equivalent form ofWerner cloning machine since by taking trace over thelast M − N qudits where each maximally entangled statewill provide an identity on H , we will re-obtain Wernercloning machine (1).We remark that the unified cloning machine (7) canbe easily understood, which is a property inherited fromWerner cloning machine, and it has also the explicittransformations as those in, such as Refs.[2, 4, 6, 7]. General fidelities .—The merit of the cloning machineis generally quantified by fidelity between input andoutput, the global fidelity and single-copy fidelity are presently known. As we mentioned, it is also neces-sary to have all possible arbitrary-copy fidelities to offera full description of the merit of universal cloning ma-chine. With output density operator ρ out available, wecan find the reduced density operators of L qudits, ρ outL ,where 1 ≤ L ≤ M . The general fidelities are definedas F L ≡ ⊗ L h ϕ | ρ outL | ϕ i ⊗ L . We remark that all reduceddensity operators of L qudits at different positions arethe same which is ensured by the fact that the outputis in symmetric subspace H ⊗ M + . By straightforward buttedious calculations, we find, F L = ( d + N − M − N )!( M − L )!( d + M − M ! N ! × X m ( M − m + d − m !) ( m − L )!( m − N )!( d − M − m )! , (10)where m is one entry of the vector ~m , here we have con-sidered the property that the output state ρ out is covari-ant for the cloning transformation, i.e., ρ out is changedas u ⊗ M ρ out u †⊗ M when | ϕ i is changed as u | ϕ i [5].For L = 1 , L = M , we recover the knownresults [5, 13]. F M = d [ N ] /d [ M ], and F =( N ( d + M ) + M − N ) / ( d + N ) M .
For special case N = 1, the general fidelity (10) can besimplified as, F L ( N = 1) = L ! d ![ L ( d + M ) + M − L ]( d + L )! M . (11)Here we have used the identity:( M − N )!( N + d − M + d − N ! × M − N X m =0 (( N + m )!) ( M − N − m + d − M m ( M − N − m )!( d − N ( d + M ) + M − N ( d + N ) M The proof of this identity is mainly based on a permuta-tion and combination equation [28].
Extension of the unified cloning machine to asymmet-ric case and examples .—As an example, let us considerthe 1 to 2 cloning machine for qudit and qubit. Thesymmetric projector takes the form s = P | jj ih jj | + P j = l ( | jl i + | lj i )( h jl | + h lj | ). Up to a whole factor, theunified cloning machine can be written as,( s ⊗ I ) | l i | Φ + i a = | ll i | l i a + 12 X j = i ( | lj i + | jl i ) | j i a , (12)where states with subindex a is the ancilla. Really thisis the optimal universal cloning machine presented inRef.[6]. For qubit case, we have the well known Buˇzek-Hillery cloning machine, | i → r | i| i a + r
16 ( | i + | i ) | i a , | i → r | i| i a + r
16 ( | i + | i ) | i a . As we already know, besides the case of symmetricoutput, we can adjust the qualities of the individual out-put states in an imbalanced way. This is realized bythe asymmetric cloning machine [14]. For 1 to 2 unifiedcloning machine, where projection s is used, we knowthat, s can be written as a summation of identity anda permutation, s = (cid:0) I ⊗ + P (cid:1) , where P is the permu-tation (SWAP) operator, P| jl i = | lj i . We can then con-sider to adjust the weights of identity and permutation inan imbalanced way. Naturally, we can change symmetricprojector to asymmetric case as, s → α I ⊗ + β P , where α and β are weights for adjusting. The correspondingasymmetric unified cloning machine is now changing as, | ϕ i → α | ϕ i | Φ + i a + β | ϕ i | Φ + i a , (13)note the orders of the subindices in these two terms aredifferent, also those two terms are not orthogonal. Theproblem now is whether this cloning procession is opti-mal. We know that, | ϕ i | Φ + i a = d P ( U jl | ϕ i ) | Φ jl i a ,where U jl are generalized Pauli matrices and identity, | Φ jl i a are orthonormal maximally entangled states with | Φ i = | Φ + i . Now exactly, we find that (13) is the opti-mal asymmetric cloning proposed by Cerf [14, 15].So far only limited cases of the asymmetric cloningmachine have been presented [14–16]. The general asym-metric cloning is still absent possibly because that theformulae are too complicated to be extended. Here sim-ilar as for the case of 1 to 2, the unified cloning machinecan be adjusted to the general asymmetric cloning ma-chine and the related entanglement sharing inequalities[27]. The method is to plug into a weight for each essen-tial permutation to modify the symmetric operator s M in (7), the problem is like to put N balls into M boxeswith a weight for each choice. Thus we offer a simplerealization of the asymmetric cloning. When all weightsare equal, it reduces to the symmetric case. Conclusions .—We present a unified optimal universalcloning machine. The cloning procession is equivalentwith Werner cloning machine [5] and the one proposedby Fan et al. [7] and can be easily adjusted to asymmet-ric cloning machines [14–16, 27] and to the general case.This simple cloning machine is always realized by a sym-metric projection and initially prepared maximally en-tangled states and thus should reduce the difficulties forimplementation. Also the general fidelities are obtained.Our result offers a new platform for other cloning tasksfor cases like phase-covariant and state-dependent.This work is supported by NSFC (10974247, 11047174),“973” program (2010CB922904) and NFFTBS(J1030310). [1] W.K. Wootters and W. H. ZureK, Nature (London) ,802(1982).[2] V. Buˇzek and M. Hillery, Phys. Rev. A , 1844 (1996).[3] L.M.Duan and G.C.Guo, Phys.Rev.Lett. ,4999(1998).[4] N. Gisin and S. Massar, Phy. Rev. Lett. , 2153 (1997).[5] R. F. Werner, Phy. Rev. A , 1827 (1998).[6] V.Buˇzek and M.Hillery, Phys.Rev.Lett. , 5003 (1998).[7] H. Fan, K. Matsumoto, and M. Wadati, Phys. Rev. A , 064301 (2001).[8] C. Simon, G. Weihs, and A. Zeilinger, Phys. Rev. Lett. , 2993 (2000).[9] H. Fan, G. Weihs, K. Matsumoto, and H. Imai, Phys.Rev. A , 024307 (2002).[10] D. Bruß, D. DiVincenzo, A. Ekert, C. A. Fuchs, C.Macchiavello, and J. A. Smolin, Phys. Rev. A , 2368(1998).[11] N. Gisin and S. Massar, Phys. Rev. Lett. , 2153 (1997).[12] D. Bruß, A. Ekert, and C. Macchiavello, Phys. Rev. Lett. , 2598 (1998).[13] M. Keyl and R. F. Werner, J. Math. Phys. , 3283(1999).[14] N. J. Cerf, Phys. Rev. Lett. , 4497 (2000).[15] N. J. Cerf, J. Mod. Opt. , 187 (2000).[16] J. Fiurasek, R. Filip, J. J. Cerf, Quant. Inform. Comp. , 583 (2005).[17] V. Scarani, S. Iblisdir, N. Gisin, and A. Acin, Rev. Mod.Phys. , 1225 (2005).[18] H. Fan, K. Matsumoto, X. B. Wang, and M. Wadati,Phys. Rev. A , 012304 (2002).[19] A. Lamas-Linares, C. Simon, J. C. Howell, and D.Bouwmeester, Science , 712 (2002).[20] E. Nagali et al ., Nature Photonics , 720 (2009).[21] F.Sciarrino, F.De Martini, Phys. Rev. A , 062313(2005).[22] C. Vitelli et al. Phys. Rev. Lett. , 113602 (2010).[23] J. F. Du et al . Phys. Rev. Lett. , 040505 (2005)[24] H. Chen, X. Zhou, D. Suter, and J. F. Du, Phys. Rev. A , 012317 (2007).[25] X. Y. Pan, G. Q. Liu, L. L. Yang, and H. Fan,arXiv:1009.2618.[26] H. Fan, K. Matsumoto, X. B. Wang, and H. Imai, J.Phys. A , 7415 (2002).[27] A. Kay, D. Kaszlikowski, and R. Ramanathan, Phys.Rev. Lett. , 050501 (2009).[28] The equation can be written as, M − N X m =0 (cid:18) ( N + m )! N ! m ! (cid:19) (cid:18) ( M − N − m + d − M − N − m )!( d − (cid:19) ( N + m )= N M − N X m =0 C NN + m C d − M − N − m + d − ++( N + 1) M − N X m =0 C N +1 N + m C d − M − N − m + d − = N · C M − NM + d − + ( N + 1) C M − N −1
16 ( | i + | i ) | i a . As we already know, besides the case of symmetricoutput, we can adjust the qualities of the individual out-put states in an imbalanced way. This is realized bythe asymmetric cloning machine [14]. For 1 to 2 unifiedcloning machine, where projection s is used, we knowthat, s can be written as a summation of identity anda permutation, s = (cid:0) I ⊗ + P (cid:1) , where P is the permu-tation (SWAP) operator, P| jl i = | lj i . We can then con-sider to adjust the weights of identity and permutation inan imbalanced way. Naturally, we can change symmetricprojector to asymmetric case as, s → α I ⊗ + β P , where α and β are weights for adjusting. The correspondingasymmetric unified cloning machine is now changing as, | ϕ i → α | ϕ i | Φ + i a + β | ϕ i | Φ + i a , (13)note the orders of the subindices in these two terms aredifferent, also those two terms are not orthogonal. Theproblem now is whether this cloning procession is opti-mal. We know that, | ϕ i | Φ + i a = d P ( U jl | ϕ i ) | Φ jl i a ,where U jl are generalized Pauli matrices and identity, | Φ jl i a are orthonormal maximally entangled states with | Φ i = | Φ + i . Now exactly, we find that (13) is the opti-mal asymmetric cloning proposed by Cerf [14, 15].So far only limited cases of the asymmetric cloningmachine have been presented [14–16]. The general asym-metric cloning is still absent possibly because that theformulae are too complicated to be extended. Here sim-ilar as for the case of 1 to 2, the unified cloning machinecan be adjusted to the general asymmetric cloning ma-chine and the related entanglement sharing inequalities[27]. The method is to plug into a weight for each essen-tial permutation to modify the symmetric operator s M in (7), the problem is like to put N balls into M boxeswith a weight for each choice. Thus we offer a simplerealization of the asymmetric cloning. When all weightsare equal, it reduces to the symmetric case. Conclusions .—We present a unified optimal universalcloning machine. The cloning procession is equivalentwith Werner cloning machine [5] and the one proposedby Fan et al. [7] and can be easily adjusted to asymmet-ric cloning machines [14–16, 27] and to the general case.This simple cloning machine is always realized by a sym-metric projection and initially prepared maximally en-tangled states and thus should reduce the difficulties forimplementation. Also the general fidelities are obtained.Our result offers a new platform for other cloning tasksfor cases like phase-covariant and state-dependent.This work is supported by NSFC (10974247, 11047174),“973” program (2010CB922904) and NFFTBS(J1030310). [1] W.K. Wootters and W. H. ZureK, Nature (London) ,802(1982).[2] V. Buˇzek and M. Hillery, Phys. Rev. A , 1844 (1996).[3] L.M.Duan and G.C.Guo, Phys.Rev.Lett. ,4999(1998).[4] N. Gisin and S. Massar, Phy. Rev. Lett. , 2153 (1997).[5] R. F. Werner, Phy. Rev. A , 1827 (1998).[6] V.Buˇzek and M.Hillery, Phys.Rev.Lett. , 5003 (1998).[7] H. Fan, K. Matsumoto, and M. Wadati, Phys. Rev. A , 064301 (2001).[8] C. Simon, G. Weihs, and A. Zeilinger, Phys. Rev. Lett. , 2993 (2000).[9] H. Fan, G. Weihs, K. Matsumoto, and H. Imai, Phys.Rev. A , 024307 (2002).[10] D. Bruß, D. DiVincenzo, A. Ekert, C. A. Fuchs, C.Macchiavello, and J. A. Smolin, Phys. Rev. A , 2368(1998).[11] N. Gisin and S. Massar, Phys. Rev. Lett. , 2153 (1997).[12] D. Bruß, A. Ekert, and C. Macchiavello, Phys. Rev. Lett. , 2598 (1998).[13] M. Keyl and R. F. Werner, J. Math. Phys. , 3283(1999).[14] N. J. Cerf, Phys. Rev. Lett. , 4497 (2000).[15] N. J. Cerf, J. Mod. Opt. , 187 (2000).[16] J. Fiurasek, R. Filip, J. J. Cerf, Quant. Inform. Comp. , 583 (2005).[17] V. Scarani, S. Iblisdir, N. Gisin, and A. Acin, Rev. Mod.Phys. , 1225 (2005).[18] H. Fan, K. Matsumoto, X. B. Wang, and M. Wadati,Phys. Rev. A , 012304 (2002).[19] A. Lamas-Linares, C. Simon, J. C. Howell, and D.Bouwmeester, Science , 712 (2002).[20] E. Nagali et al ., Nature Photonics , 720 (2009).[21] F.Sciarrino, F.De Martini, Phys. Rev. A , 062313(2005).[22] C. Vitelli et al. Phys. Rev. Lett. , 113602 (2010).[23] J. F. Du et al . Phys. Rev. Lett. , 040505 (2005)[24] H. Chen, X. Zhou, D. Suter, and J. F. Du, Phys. Rev. A , 012317 (2007).[25] X. Y. Pan, G. Q. Liu, L. L. Yang, and H. Fan,arXiv:1009.2618.[26] H. Fan, K. Matsumoto, X. B. Wang, and H. Imai, J.Phys. A , 7415 (2002).[27] A. Kay, D. Kaszlikowski, and R. Ramanathan, Phys.Rev. Lett. , 050501 (2009).[28] The equation can be written as, M − N X m =0 (cid:18) ( N + m )! N ! m ! (cid:19) (cid:18) ( M − N − m + d − M − N − m )!( d − (cid:19) ( N + m )= N M − N X m =0 C NN + m C d − M − N − m + d − ++( N + 1) M − N X m =0 C N +1 N + m C d − M − N − m + d − = N · C M − NM + d − + ( N + 1) C M − N −1 M + d −1