Optimal Non-Universally Covariant Cloning G. M. D’Ariano, P. Lo Presti Theoretical Quantum Optics Group Universit`a degli Studi di Pavia and INFM Unit`a di Pavia via A. Bassi 6, I-27100 Pavia, Italy (February 1, 2008) describe a quantum cloning transformation and its rela- We consider non-universal cloning maps, namely cloning tion to CP-maps. Sec. III is devoted to the description transformationswhicharecovariantunderapropersubgroup of CP-maps in terms of positive operators, while in Sec. GoftheuniversalunitarygroupU(d),wheredisthedimen- IV we treat the case of covariant CP-maps, giving their sion of the Hilbert space H of the system to be cloned. We 1 parametrization with suitable covariant positive opera- 0 give a general method for optimizing cloning for any cost- tors. In Sec. V we use the previously explained tech- 0 function. Examples of applications are given for the phase- niquestodealwithcloningoptimization,focusingonthe 2 covariant cloning (cloning of equatorial qubits) and for the covariantcase. Weyl-Heisenberggroup (cloning of “continuous variables”). n a J 9 I. INTRODUCTION II. CLONING TRANSFORMATIONS 1 The impossibility of perfectly cloning an unknown in- In a quantum cloning transformation, the input state 1 ρ ( ) is processed in order to produce N output putstateisatypicalquantumfeature[1],nonetheless,in ∈ L H v clones(throughoutthepaper ( )willdenotethevector 0 the laws of quantum mechanics there’s enough room ei- space of linear bounded operLatoHrs on the Hilbert space ther to systematically produce approximate copies [2] or 0 ). This requires a “spreading” of ρ into the joint state 1 to make perfect copies of orthogonalstates [3] or of non- Hρ′ ( ⊗N)ofN identicalquantumsystems. Themost 1 orthogonal ones with a non-unit probablility [4]. These ∈L H general setup for such purpose is the following. Initially, 0 possibilities have been studied in several works [5–7]. 1 Recently, quantum cloning has entered the realm of ρisencodedinaquantumsystemS1,whileN−1equiva- 0 lentsystems S , i=2...N, are preparedin a fixedstate experimental physics [8] [9]. Moreover it has became in- i / ω . An auxiliary system E is provided in a state h teresting from a pratical point of view, since it can be | i(N−1) p used to speed-up some quantum computations [10] or to |ei, in order to make the whole system isolated. A uni- - perform some quantum measurements [11,12]. All these tarytransformationU actsontheoverallstateproducing t n tasks require a spreading of the quantum information the output a u contained in a system into a larger system, and quan- Λ=Uρ (ω ω )(N−1) e e U† . (1) tum cloning is a way to achieve such a spreading. ⊗ | ih | ⊗| ih | q : In this paper we will see how any “spreading” cor- By taking the partialtrace of Λ on the auxiliarysystem, v responds to a particular completely-positive (CP) map. wegetthejointstateρ′oftheN outputsystemsS . This i i X By exploiting the correspondencebetween CP-maps and state will eventually support the clones. Upon calculat- r positive operators on the tensor product of the output ing the trace with respect to a chosen basis j E for a and input spaces [13], we can parametrize all the possi- one has {| i } E H ble spreading transformation. Then we focus on covari- antCP-map,showingthatquantumcloningisaparticu- dimHE dimHE . ρ′ = j Λj = A ρA† = (ρ), (2) lar case of permutation-covariance. By means of Schur’s Eh | | iE j j E lemmaswecompletelycharacterizethepositiveoperators Xj=1 Xj=1 corresponding to quantum cloning transformations. By where A = j U ω e . the same technique, we characterize G-covariantcloning j Eh | | i(N−1)| iE The map ρ (ρ) in Eq. (2) is a completely- transformation, where G is any single-copy covariance → E positive(CP)andtrace-preservinglinearmapfrom ( ) group. to ( ⊗N). Trace preserving CP-maps generallyLdHe- The parametrizationof CP-maps, and in particular of L H scribe the evolution of open quantum systems. To un- cloning and covariant cloning, stands at the base of any derstandthegeneralfeaturesofquantumcloningandfor further optimization. In fact, quantum cloning can be sakeofoptimization,it is convenientto treatthese maps used to perform some tasks on the copies, and depend- at an abstract level: a realization theorem guarantees ing on what these copies will be used for, one defines that any CP-map can be achieved as a unitary transfor- a “goodness” criterion for the cloning process and opti- mationonanextendedHilbertspace[16,17],similarlyto mizes accordingly. Eq. (1). CP-maps will be shortly reviewed in the next Thepaperisorganizedasfollows. InSec. II,webriefly session. 1 III. CP-MAPS AND POSITIVE OPERATORS where there are many different choices of the vectors A , which are not necessarily eigenvectors of R , and i E | ii A linear map : ( ) ( ) is completely-positive generally are not normalized. if its trivial exteEnsioLnH → L tKo ( ′) is positive, Substituting this relation in Eq. (7) and remembering H for any H′ (IH′ denotiEng⊗thIe′triviLalHma⊗pHon L(H′)). that Ai ∈L(H,K), we find Here we recall a convenient notation [14]. Fixing two osprtehcotinvoerlym,aalnbyasviesc{to|iri1}Ψand {|j1i2} fo2r Hca1nabnedwHr2ittreen- E(ρ)=Xi TrH(cid:2)11⊗ρT |AiiihhAi|(cid:3)=Xi AiρA†i , (9) | ii ∈ H ⊗ H as thus recovering that any CP-map admits different . Ψ = c i j = C , (3) Kraus’s decompositions [16], depending on the choice of ij 1 2 | ii Xij | i | i | ii the vectors |Aiii in Eq. (8). Clearly, Eq. (8) holds for any positive operator R on where C = c i j ( , ) is a linear . The map defined by R through Eq. (7) is ij ij| i12h | ∈ L H2 H1 K ⊗ H bounded operaPtor from 2 to 1. completely-positive,sinceitcanbeexpressedintheform H H The following relations can be easily verified ofEq. (9)whichtriviallygivesaCP-map. Thenthe cor- respondence from CP-maps to operators is also “onto”. A B C = ACBT , (4) Concluding, Eq. (7) defines a one-to-one correspon- ⊗ | ii | ii TrH2 |AiihhB| =AB† ∈L(H1). (5) dencebetweenCP-mapsfromL(H)toL(K)andpositive (cid:2) (cid:3) operators on . By exploiting this correspondence, For every CP-map : ( ) ( ) we define the properties ofK⊗caHn be translated into properties of R . E L H → L K E positive operator R in ( ) E E For example, the trace-preserving condition for L K⊗H E . RE =E ⊗I |11iihh11| , (6) TrK[ (ρ)]=1=TrH ρT TrK[RE] , (cid:0) (cid:1) E where denotes the identical map over the extention (cid:2) (cid:3) I for all ρ ( ) such that Tr[ρ]=1, becomes space , and for the vector 11 we used the ∈L H H | ii ∈ H⊗H notation (3) for Ψ = 11 the identity matrix with respect TrK[RE]=11 ( ). (10) to a fixed basis on . The action of on ρ ( ) can ∈L H H E ∈ L H be expressed as In the following, it will be useful to consider the dual map ∨ofaCP-map ,namelythetransformationinthe E(ρ)=TrH 11⊗ρT RE , (7) HeiseEnberg picture veErsus the Schr¨oedinger picture map (cid:2) (cid:3) ρ (ρ). The dual map ∨ is defined by the identity wherethetranspositionρ ρT isperformedwithrespect →E E → to the same fixed basis. In fact, substituting Eq. (6) in Tr ρ ∨(O) =Tr (ρ)O , (11) Eq. (7), one has E E (cid:2) (cid:3) (cid:2) (cid:3) TrH 11⊗ρT RE =TrH 11⊗ρT E ⊗I |11iihh11| . wofhtihchemopuesrtabtoervRalidofnoerahlalsoperatorsO∈L(K). Interms (cid:2) (cid:3) (cid:2) (cid:0) (cid:1)(cid:3) E Then, it is possible to take the factor 11 ρT inside the CP-map , since they act independen⊗tly ondifferent E∨(O)=TrK O⊗11RETH], (12) E⊗I (cid:2) spaces. By applying Eq. (4), one obtains where T denotes partial transposition on the Hilbert H TrH 11 ρT RE =TrH ρ 11 , space H only [15]. ⊗ E ⊗I | iihh | Inthenextsession,thecorrespondence R willbe (cid:2) (cid:3) (cid:2) (cid:0) (cid:1)(cid:3) E ↔ E and thus, commuting the partial trace with , and appliedto the covarianceconditionfor a CP-map,which E ⊗I using Eq. (5), one finally gets Eq. (7). turns out to be the key idea to deal with cloning and The operator R is the only one for which Eq. (7) covariantcloning. E holds true. In fact, suppose R and R′ give the same E CP-map by means of Eq. (7), then E IV. COVARIANT CP-MAPS TrH 11 ρT (RE R′) =0 ( ), ρ ( ). ⊗ − ∈L K ∀ ∈L H (cid:2) (cid:3) Let : ( ) ( ) be a CP-map, and let G be a Since an operator O ( ) is null if v Ov =0 E L H → L K L(K) for all |vi ∈ H∈, LitHfol⊗lowKs that RE h=|R|′.i Thu∈s, grersopuepctwivitehlyu.nitiasrGyr-ceopvraesreiannttatwioitnhsrUesapnedctVtooUnHanadnVdKif the correspondence from CP-maps to positive operators E is “into”. U ρU† =V (ρ)V† , (13) Since RE is positive, it can be written as E g g gE g (cid:0) (cid:1) for any ρ ( ) and g G. R = A A , (8) ∈L H ∈ E | iiihh i| By means of Eq. (7), the covariance condition becomes Xi 2 (ρ)=TrH 11 ρT RE = and finally E ⊗ ≡TrH(cid:2)(cid:2)11⊗ρT Vg†⊗(cid:3) UgT RE Vg⊗Ug∗(cid:3). (14) R= cij11ij . (21) Fromthe uniqueness ofthe operatorassociatedto a CP- Xij map, we conclude that is G-covariant if and only if In order to have a positive R, the matrix c must be E ij RE =Vg†⊗UgT RE Vg⊗Ug∗ , ∀g ∈G, (15) positive, since taking |ψii=PiPdl=im1Miψil|i,li one has dim i or equivalently ψ Rψ = M ψ∗ c ψ . hh | | ii li ij lj R , V U∗ =0, g G. (16) Xij Xl=1 E g ⊗ g ∀ ∈ (cid:2) (cid:3) Recalling that c = 0 if i ≁ j, and reordering the in- Thus, G-covariance of a CP-map is equivalent to ij dices of the representations by grouping the equivalent G-invariance of the corresponding poEsitive operator RE. ones, the matrix c assumes a block diagonal form, dif- ij ferent blocks corresponding to inequivalent representa- tions, each block including all representationsequivalent Group invariant operators to the same one. In this way, each block has dimension equaltothemultiplicityoftherepresentation. Positivity The G-representation W = diag(V U∗) on , of R implies positivity of each block of matrix c . This definedas Wg =Vg⊗Ug∗, is generallyr⊗educible,i.Ke⊗. tHhe structure of cij is reflected on R by means of Eqi.j(21). space can be decomposed into a direct sum of minimal invariant subspaces i M V. OPTIMAL COVARIANT CLONING = , (17) i K⊗H M Mi=1 A cloning map is just a CP-map from ( ) to ( ⊗N) with the output copies invariaCntunderLthHe per- each Mi supporting a unitary irreducible representation mLuHtations of the N output spaces. This is equivalent to (UIR) of the group. Given this decomposition one can a particular covariance of the CP-map for the group look at any operator O on K⊗H as a set of operators ofpermutationsS ,namelyitcorresponCdstotheinvari- Oi in ( , ), so that O = Oi. N j L Mj Mi ij j anceofthepositiveoperatorR undertherepresentation Due to irreducibility of the suPbspaces Mi, Wg will be W = diag(V I), where V isCthe representation of S decomposed as follows ⊗ N permuting the N identical output spaces, and I (corre- (Wg)ij = δijTgj , sponding to U in Eq. (13)) is the SN-trivial representa- tion on the input space. One has where Tj is the UIR supported by . Two UIR Ti and Tj are equivalent, i j, if theyMajre connected by C(ρ)=VπC(ρ)Vπ† , ∀π ∈SN . (22) similarity, i. e. through a∼n isomorfism Ii ( , ) j ∈ L Mj Mi Notice that permutation covariance does not imply that such that Tj =(Ii)−1TiIi. the output state has support in the symmetric subspace j j The invariance equation (15) becomes of the output space ⊗N. H As explained in the previous section, Eq. (22) deter- TiRiTj =Ri g G, g j g j ∀ ∈ minesapeculiarblockstructurefortheoperatorRC asso- ciatedtothemap . Suchastructureisstrictlyrelatedto so that, by Schur’s lemmas (see, for example, Ref. [18]), the decompositionCof ⊗(N+1) into invariant subspaces one finally has H forVπ 11. AnypossibleclonerisdescribedbyanRC with ⊗ Ri =c Ii , (18) that structure and satisfying the trace-preserving condi- j ij j tion of Eq. (10). In this way, one classifies all possible where if i ≁ j then c = 0, and, if i j, c can be cloningmapsthroughthedecompositionintoirreducibles ij ij different from zero. ∼ of the SN-representation V on ⊗N. H Since equivalent representations are related by simi- Inadditiontopermutationinvariance,inthispaperwe larity, in any invariant subspace one can choose the willconsidercovarianceunderagroupoftransformations i basis {|i,li, l =1...dimMi} so tMhat for i∼j Gfol,lowwitinhgreidpernesteitnytationT on H. This corresponds to the i,l Ti i,m = j,l Tj j,m , (19) h | | i h | | i (T ρT†)=T⊗N (ρ)T†⊗N . (23) C g g g C g hence One can choose a cost function C(R ) (related to the C . Ii = i,l j,l =11i , (20) followingusageoftheclones). Covariantcloningissuited j Xl | ih | j to invariant cost-functions of the form 3 C(R ) C (ρ ,R )=C (T ρ T†,R ), (24) C ≡ 0 0 C 0 g 0 g C M1 |0001i -1 T M5 where ρ0 is the seed of the covariant family of states on M2 |0000i 0 T M6 which we are interested in having the minimum C0. M3 |1001i+|0101i+|0011i 0 T M7 The best cloner is found by minimizing C(R ) vs |1001i−|0101i, RC, with the constraints of positivity RC 0, tCrace- M4 12|1001i+ 21|0101i−|0011i 0 D M8 preserving (Eq. (10)), and covariance unde≥r permuta- M5 |1110i 3 T M1 tions and G (Eqs. (22) and (23)). M6 |1111i 2 T M2 M7 |0110i+|1010i+|1100i 2 T M3 |0110i−|1010i, VI. EXAMPLES M8 12|0110i+ 21|1010i−|1100i 2 D M4 M9 |1000i+|0100i+|0010i 1 T M10 Phase covariant qubit cloning M10 |0111i+|1011i+|1101i 1 T M9 |1000i−|0100i, M11 1|1000i+ 1|0100i−|0010i 1 D M12 2 2 Here, we consider the problem of cloning a qubit in a |0111i−|1011i, U(1)-covariant fashion, where the group representation M12 1|0111i+ 1|1011i−|1101i 1 D M11 2 2 is given by TABLE I. H⊗3+1 decomposition into U(1) − S3 irre- i ducibles. U(1) acts on each subspace as a phase shift einφ, Tφ =exp φ(11 σz) . (25) where n∈ Z (column III) labels inequivalent representation. (cid:20)2 − (cid:21) S3 acts trivially (T) on one-dimensional subspaces, whereas Since the cloning to two copies is already given in Ref. on bidimensional ones it acts as the defining representation [19], whereas the general case for N copies is very com- (D).Spin flippingconnects subspaces (column V). plicated, here for simplicity we will consider the case of N =3 copies. We want to achieve the maximum fidelity betweeninputandclones,whentheinputisanequatorial qubit LookingatTableI,onecanseethatinthisexamplethe matrix c defined in Sec. IV has the following positive ij 1 1 diagonal blocks: ψ =T [0 + 1 ]= [0 +eiφ 1 ]. (26) φ φ | i √2 | i | i √2 | i | i 1 , 2,3 , 4 , 5 , 6,7 , 8 , 9,10 , 11,12 . { } { } { } { } { } { } { } { } In other terms, we want to maximize the fidelity To ensure spin flipping covariance, the elements of c ij F =Tr (ψ ψ ) ψ ψ F = connectedbyaflipmustbe equal,forexamplec =c . φ φ φ φ φ 0 23 67 C | ih | | ih | ≡ =Tr(cid:2)11⊗2⊗|ψ0ihψ0|⊗(|ψ0(cid:3)ihψ0|)T RC . (27) neAedttthhee epnadra,mtoetfielrlsthae, bbl,occ,ksdo,fec,ijfi,ngthe rRig+h,tvway,Rw3e, (cid:2) (cid:3) where d e, f g, and c v . Table I∈I explains∈how Since the equator is invariantevenfor spin flipping, here ≥ ≥ ≥ || || to employ them. we will require the additional covariance with respect to the group Z2, with representation 11,σx . { } In order to satisfy all the covariancerequirements, R C Blocks Content mustbeinvariantforpermutations,phaseshift,andspin {1}, {5} a flip, i. e. for products of any of the following unitary {4}, {8} b operators {2,3}, {6,7} c11+v·σ Tφ⊗3⊗Tφ∗ , Vπ ⊗11, σx⊗3⊗σx∗ . {{191,,1102}} fd1111++egσσxx TheHilbertspace ⊗3+1 canbedecomposedintosub- TABLEII. Content oftheblocksof thematrix cij,chosen spaces whichare irreHducible with respect to the joint ac- inordertohaveRC describingthemostgeneralCP-mapfrom tion of U(1) and S , as shown in Table I. L(H) toL(H⊗3) which iscovariant with respect topermuta- 3 tions, phase shift, and spin flip. Space Unnormalized Basis U(1) S3 Flipped 4 1 The parameters must satisfy another constraint given RC = P12 113 111 (11 11 )23 P12 113 , (32) by the trace-preserving condition defined in Eq. (10). 2 ⊗ ⊗ | iihh | ⊗ Within this parametrizationit reads whereP =V 0 0 11V†,andV isthe50%beamsplitter | ih |⊗ a+2b+2c+d+2f =1. (28) unitary transformation V =exp[π4(a†1a2−a1a†2)]. A simple calculation shows that Substituting this equation into the equatorialfidelity F 0 2 defined in Eq. (27), one has P = d2α α α ⊗2 , (33) π Z | ih | 1 1 √3 F0 = 2 + 3(e−g)+ 3 vx . (29) where α =D(α)0 ,andD(α)=eαa†−α¯aisthedisplace- | i | i ment operator generating the Weyl-Heisenberg (WH) This quantity can be easily maximized by hand, taking group. By means of Eq. (33), the invariance of R C into account the constraint given by Eq. (28) and the defined in Eq. (32) with respect to permutations and properties of the parameters. The maximum fidelity is displacements can be easily verified. F = 5 (see note [22]) and is achieved for d = e = 1 Using the dual cloning map as in Eq. (31), we should 6 and all the other parameters equal to zero. The optimal check that phasecovariantcloningisthusdescribedbytheoperator 1 ∨(E0 Eπ/2)= α α , α=x+iy , (34) Ropt = Φ Φ , (30) C x⊗ y π| ih | C | iihh | where where Evφ = |viφφhv|, and Xφ|viφ = v|viφ. In fact, the last term of Eq. (34) is the well-known optimal Φ = 1 [1000 + 0100 + 0010 + POVMfor the joint measurement of conjugated quadra- | ii √3 | i | i | i tures, whereas Eφ is the POVM of the φ-quadrature + 0111 + 1011 + 1101 ]. v | i | i | i measurement. Hence identity (34) guarantees that the cloning achieves the optimal joint measurement of the The Kraus’s decomposition of the optimal cloner is (ρ)= BρB†, where twoconjugatedquadratureviacommutingmeasurements C on clones. B = 1 [100 0 + 010 0 + 001 0 + Noticing that √3 | ih | | ih | | ih | + 011 1 + 101 1 + 110 1]. | ih | | ih | | ih | E0 Eπ/2 =D(α)⊗2E0 Eπ/2D(α)⊗2† , (35) x⊗ y 0 ⊗ 0 and exploiting the WH covariance,Eq. (34) reduces to Cloning of continuous variables 1 ∨(E0 Eπ/2)= 0 0 . (36) C 0 ⊗ 0 π| ih | The parametrizationofCP-mapsgiveninSec. III and its specialization to the covariant case are useful tools Substituting Eq. (31) into this last equation, and taking for engineering measurements. The idea is to “spread” matrix elements i ... j , one finally must check that h | | i a quantum state on a larger system with a CP-map , E 1 aTnhdetchoennnetoctpioenrfobremtwaeemnetahseurPeOmVenMtonMtheosnprtehaedlsatragteer. 0h0|π2h0|hi|RC|0i0|0iπ2|ji= πδi0δj0 . (37) i space and the resulting one M∨ {on } is given by KMi∨ =E∨(Mi)=. TrK(cid:2)M{i⊗i }11RETHH(cid:3), (31) [S2i1n]c)e,oVn|e0ih0a|0sitπ2ha=tPq|0π2i|π211|i0iia0n=dqV|0π2i||00ii|0=i.|0Tih|0uis(Esqee. (R3e7f). where ∨ isthedualmapof ,andthesymbolT stands holds,andthecloningreallyachievesthewantedPOVM. H E E for transposition with respect to only (see Sec. III). H In Ref. [11], the cloning map for continuous variables of Ref. [20]is usedto achievethe optimal POVMfor the Universal cloning jointmeasurementoftwoconjugatedquadraturesX and 0 Xπ2 ofanoscillatormodea(whereXφ = 12[a†eiφ+ae−iφ]) Clearly, the universal covariant cloning of Werner [6] by measuring them separately on the two clones. Here, is a special case of covariant cloning for the covariance we will briefly show how our general method works on group U(d), d = dim , of all unitary operators on . H H this problem. Here,forsakeofcomparisontoRef.[6],weconsidermore Denote by 3 the input space and by 1, 2 the two generally the cloning from M to N > M copies. Hence H H H output spaces of the oscillator modes a3, a1, a2 respec- the cloning is a CP-map from ( ⊗M) to ( ⊗N) C L H L H tively. The cloning is described by such that for any U U(d) and σ ( ) ∈ ∈L H 5 (U⊗Mσ⊗MU†⊗M)=U⊗N (σ⊗M)U†⊗N . (38) [14] See, for example, G. M. D’Ariano, P. Lo Presti, and M. C C F. SacchiPhys. Lett. A 272, 32 (2000). The cost-function for optimization is the (negative) fi- [15] Remember that transposition is not a CP-map, hence delity between clones and input partial transposition is not a positive map. Thus the partial transposition of a positive operator can be non- C(RC)= F = Tr σ⊗N (σ⊗M) , (39) positive. − − C (cid:2) (cid:3) [16] K. Kraus, States, Effects, and Operations (Springer- whereσ ispure. Owingtocovariance,thefidelityF does Verlag, Berlin, 1983). not depend on σ, since any pure state lies in the U(d) [17] M. Ozawa, J. Math. Phys. 25, 79 (1984). orbit of any other pure state. [18] H. F. Jones, Groups, representations, and Physics (In- The optimal cloning map of Ref. [6] is given by stitute of Physics Publishing, Bristol and Philadelphia, 1990). d(M) [19] D. Bruß, M. Cinchetti, G. M. D’Ariano, and C. Macchi- C(ρ)= d(N)SN(ρ⊗11⊗(N−M))SN , (40) avello, Phys. Rev.A 62 012302 (2000). [20] N. J. Cerf, A. Ipe, and X. Rottenberg, Phys. Rev. Lett. where ρ ( ⊗M), S is the projector on the sym- 85, 1754 (2000). N metric su∈bspLacHe ⊗N, and d(N) = dim( ⊗N). In our [21] G. M. D’Ariano, and M. F. Sacchi, Mod. Phys. Lett. B H+ H+ 11, 1263 (1997). framework, one has [22] Wenoticethatthisfidelityvalueislargerthanthebound d(M) given in Ref. [19]. The correctness of the present value RC = d(N) S˜11H⊗(N−M)(|11iihh11|)H⊗(M+M)S˜, (41) andtheoptimalityofthemapcanbecheckedaposteriori. whereS˜=SN 11⊗M. ItcanbeeasilyverifiedthatRC is ⊗ both covariant and permutation invariant as it must be. ACKNOWLEDGEMENTS ThisworkhasbeensupportedbytheItalianMinistero dell’Universit`a e della Ricerca Scientifica e Tecnologica (MURST) under the co-sponsored project 1999 Quan- tum Information Transmission And Processing: Quan- tum Teleportation And Error Correction. 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