Long quantum channels for high-quality entanglement transfer
Leonardo Banchi, Tony John George Apollaro, Alessandro Cuccoli, Ruggero Vaia, Paola Verrucchi
aa r X i v : . [ qu a n t - ph ] M a y Long quantum channels for high-quality entanglement transfer
L. Banchi,
1, 2
T. J. G. Apollaro, A. Cuccoli,
1, 2
R. Vaia, and P. Verrucchi
3, 1, 2 Dipartimento di Fisica, Universit`a di Firenze, Via G. Sansone 1, I-50019 Sesto Fiorentino (FI), Italy INFN Sezione di Firenze, via G.Sansone 1, I-50019 Sesto Fiorentino (FI), Italy Istituto dei Sistemi Complessi, Consiglio Nazionale delle Ricerche,via Madonna del Piano 10, I-50019 Sesto Fiorentino (FI), Italy (Dated: June 10, 2018)High-quality quantum-state and entanglement transfer can be achieved in an unmodulated spinbus operating in the ballistic regime, which occurs when the endpoint qubits A and B are coupledto the chain by an exchange interaction j comparable with the intrachain exchange. Indeed, thetransition amplitude characterizing the transfer quality exhibits a maximum for a finite optimalvalue j opt0 ( N ), where N is the channel length. We show that j opt0 ( N ) scales as N − / for large N and that it ensures a high-quality entanglement transfer even in the limit of arbitrarily longchannels, almost independently of the channel initialization. For instance, the average quantum-state transmission fidelity exceeds 90 % for any chain length. We emphasize that, taking the reversepoint of view, should j be experimentally constrained, high-quality transfer can still be obtainedby adjusting the channel length to its optimal value. I. INTRODUCTION
Quantum state transfer between distant qubits (say,A and B) is a fundamental tool for processing quantuminformation. The task of covering relatively large dis-tances between the elements of a quantum computer,much larger than the qubit interaction range, can beachieved by means of a suitable communication channelconnecting the qubits A and B.Spin chains are among the most studied channel pro-totypes [1]. In particular, the
XY S = 1 / N / ,allows us to devise an optimal value of the endpoint in- teractions for any N , and vice versa. Remarkably, thecorresponding transmission quality, as witnessed by thestate- and the entanglement fidelity, does not decrease tozero when the channel becomes very long, but remainssurprisingly high.We consider the setup illustrated in Fig. 1: the channelconnecting the qubits A and B is a one-dimensional arrayof N localized S = 1 / XX Heisenberg type and a possible external magneticfield applied along the z direction. This gives the totalHamiltonian the following structure H = − N − X i =1 (cid:0) S xi S xi +1 + S yi S yi +1 (cid:1) − h N X i =1 S zi (1) − j X i =0 ,N (cid:0) S xi S xi +1 + S yi S yi +1 (cid:1) − h (cid:0) S z + S zN +1 (cid:1) , where the qubits A and B sit at the endpoint sites 0and N +1 of a one-dimensional discrete lattice on whosesites 1 , , ..., N the spin chain is set. The exchange inter-action (chosen as energy unit) and the magnetic field h are homogeneous along the chain, and an overall mirrorsymmetry is assumed, implying the endpoint coupling j and field h to be the same for both ends. The N spinsconstituting the XX channel are collectively indicatedby Γ. We will focus our attention on how the state of thequbit B evolves under the influence of the chain Γ, anddepending on the initial state of the qubit A; the latteris possibly entangled with an ancillary qubit C. The re-sults of the analysis are used for gathering insights on thequantum-information transmission through the chain, soas to characterize the dynamical evolution of the overallsystem and to maximize the quality of the quantum-statetransfer.Even though the overall scheme could also be used forrealizing tasks other than quantum information transfer,via the dynamical correlations that the chain induces be-tween A and B [20], our approach is specifically tailoredfor studying transfer processes along the chain: the qubit FIG. 1. The endpoints of a quantum channel Γ are coupledto the qubits A and B, via a tunable interaction j ; A can beentangled with an external qubit C. B or the qubit-pair BC are considered as target system,depending on whether the quantum-state of the qubit Aor that of the qubit-pair AC are to be transferred, re-spectively.In Section II the formalism used to study the dynami-cal evolution of the composite system is introduced andwe derive the corresponding time-dependent expressionsfor the quantities used for estimating the quality of thequantum-state and entanglement transfer processes. InSection III the proposed formalism is applied to the XX model described by Eq. (1). In Section IV we put forwardan analytical framework in order to improve the under-standing of the conditions inducing an optimal ballisticdynamics. The resulting high-quality transfer processesalong the spin chain are analyzed in Section V. Conclu-sions are drawn in Section VI, where comments aboutpossible implementation of the procedure are also putforward. Relevant details of calculations are reported inthe Appendices. II. DYNAMICS AND TRANSFER QUALITYA. Dynamics
The density matrix of a qubit B can be written as ρ = P µ b µ ζ µ where µ = 1 , ..., b µ are complex num-bers, and { ζ µ } is an orthonormal basis in the Hilbertspace of 2 × ζ † µ ζ ν ] = δ µν . In some cases it is usefulto choose Hermitian ζ µ , for instance the identity and thethree Pauli matrices. However, a more direct connectionwith the computational basis {| i ≡ |↑i , | i ≡ |↓i} forthe pure states of B is established by choosing ζ µ = | i ih j | and setting µ = 1 +2 i + j , with i, j = 0 ,
1. In what fol-lows we will use this description, and hence use indices µ, ν, λ running from 1 to 4, understanding summationover any repeated index, e.g., ρ = b µ ζ µ . Hermitic-ity of ρ implies b and b to be real and b = b ∗ , whileTr ρ = 1 means b + b = 1, so that only three real pa-rameters are independent. The positivity of ρ requiresthat | b | ≤ b b .The dynamics of B is described by its time-dependent density operator ρ ( t ); this can always be expressed asa linear function of a suitable input density matrix ρ in ,which may be the initial one of qubit B itself or that ofany other qubit playing a role in its evolution: ρ ( t ) = E t ρ in , (2)where the quantum operation E t is a trace-preserving,completely positive, convex-linear map from input den-sity operators ρ in to density operators of the qubit attime t (see, e.g., Ref. 21). The effect of the linear mapcan be represented in terms of a 4 × T µν ( t ) = Tr (cid:2) ζ † µ E t ζ ν (cid:3) . (3)In fact, by writing ρ ( t ) = b µ ( t ) ζ µ and ρ in = a µ ζ µ , onegets b µ ( t ) = T µν ( t ) a ν . (4)If B is part of a larger system S ruled by a total Hamil-tonian H and prepared in the initial state ρ tot , it is also b µ ( t ) = (cid:10) ζ † µ ( t ) (cid:11) , (5)where h · i ≡ Tr[ · ρ tot ] and ζ µ ( t ) = e ı H t S r B ⊗ ζ µ e − ı H t , ζ µ acting on the Hilbert space of the qubit B. The matrixelements T µν ( t ) explicitly follow from Eq. (4) and (5).Here we will essentially focus on the time evolutionof the qubit B and take the input density operator inEq. (2) to represent the initial state of the qubit A, asthis is the most natural setup for studying quantum stateand entanglement transmission processes from A to B.Referring to the specific setup described in Fig. 1 weprepare the overall system A ∪ Γ ∪ B in the initial state ρ tot = ρ A ⊗ ρ Γ ⊗ ρ B and let it evolve into e − ı H t ρ tot e ı H t ,where H is the total Hamiltonian (1). Notice that ρ tot has a fully separable structure: this is not a necessarycondition for the explicit determination of the dynamicalmatrix (a structure such as ρ A ⊗ ρ Γ B would also work),but it is quite meaningful in our scheme, as the qubit Bis assumed not to interact with the chain during the ini-tialization process, which makes quite artificial its beingentangled with the chain before the dynamics starts.The density matrix of B evolves according to ρ B ( t ) =Tr A ∪ Γ (cid:2) e − ı H t ρ tot e ı H t (cid:3) , from which T µν ( t ) = Tr B h ζ † µ Tr A ∪ Γ (cid:2) e − ı H t ζ ν ⊗ ρ Γ ⊗ ρ B e ı H t (cid:3)i = (cid:10) A ⊗ Γ ⊗ ζ † µ ( t ) (cid:11) ν (6)where h · i ν ≡ Tr[ · ζ ν ⊗ ρ Γ ⊗ ρ B ]. Notice that the initialstate of the qubit A does not enter the expression of T µν ( t ), setting this procedure in the framework of generaltomographic approaches.Let us now consider the time evolution of the quan-tum state describing the qubit-pair C ∪ B when the to-tal system is initially prepared in the state ρ tot = ρ CA ⊗ ρ Γ ⊗ ρ B . By a trivial generalization of the descrip-tion introduced at the beginning of this section, we write ρ CA = g CA µν ζ µ ⊗ ζ ν , and find ρ CB ( t ) = h g CA µλ T νλ ( t ) i ζ µ ⊗ ζ ν ≡ g CB µν ( t ) ζ µ ⊗ ζ ν . (7)This equation shows that T µν ( t ) is the only ingredientneeded not only for deriving the dynamics of the qubitB, but also of the qubit pair C ∪ B, providing the pairC ∪ A was prepared separately from the rest of the system,and C is non-interacting [22]. In particular, if C andA are initially prepared in one of the Bell states, say( | i + | i ) / √
2, it is g CA µν = δ µν (similar conditionshold for the other Bell states), i.e., ρ CABell = 12 ζ µ ⊗ ζ µ , (8)and hence ρ CBBell ( t ) = T νµ ( t ) ζ µ ⊗ ζ ν . (9)The trace-preserving property of the dynamical map,Tr (cid:2) E t ζ µ (cid:3) = Tr ζ µ , together with the specific choice ofthe basis { ζ µ } , entails T + T = T + T = 0 and T + T = T + T = 1; the Hermiticity of ρ B furtherimplies that T µ a µ and T µ a µ are real and T µ a µ = T ∗ µ a ∗ µ .Moreover, from Eq. (6) it clearly follows that symmetryproperties of the total Hamiltonian and of the initial state ρ Γ ⊗ ρ B may result in constraints for the matrix elements T µν . In particular, when both are symmetric under ro-tations around the z -axis, it is T µ = T µ = T δ µ , T µ = T µ = T δ µ , (10)and only three matrix elements, say T , T , and T ,need to be determined. B. Quality of transfer processes
In order to study the quality of the transfer processesmediated by the spin chain, we specifically consider theentanglement transfer from C ∪ A to C ∪ B, when the for-mer spin pair is initially prepared in a pure state. In thiscase, the entanglement fidelity, which is a proper mea-sure of the quality of entanglement transmission, reads F ent ( t ) ≡ Tr (cid:2) ρ CA † ρ CB ( t ) (cid:3) , which is, via Eq. (7), F ent ( t ) = T µν ( t ) (cid:0) g CA λµ (cid:1) ∗ g CA λν . (11)The entanglement fidelity measures how close the state ofC ∪ B at time t is to the initial state of C ∪ A. In particular,if C and A are initially prepared in a Bell state, Eq. (8),it is F Bellent ( t ) = T µµ ( t ) , (12)while the entanglement of the pair C ∪ B is measured by C Bell CB ( t ) = C (cid:2) T νµ ( t ) ζ µ ⊗ ζ ν (cid:3) , (13) where C [ ρ ] is the concurrence [23] between two qubitsin the state ρ . Note that Eqs. (9), (12), and (13) areindependent of which Bell state is chosen, since differentBell states are connected to ( | i + | i ) / √ F AB ( t ) ≡ Tr (cid:2) ρ A † ρ B ( t ) (cid:3) provided that ρ A is a pure state, i.e., by Eq. (4), F AB ( t ) = b µ ( t ) a ∗ µ = T µν ( t ) a ∗ µ a ν . (14)If A is initially prepared in the state | ψ θϕ i = cos θ | i + e ıϕ sin θ | i , meaning a = cos θ , a = sin θ , and a = a ∗ = e ıϕ sin θ cos θ , Eq. (14) can be averaged over allpossible initial pure states by integrating over the Blochsphere, resulting in F AB ( t ) = + T µµ ( t ) (cid:3) , (15)which can be compared with Eq. (12) to obtain the rela-tion [24, 25] F AB ( t ) = + F Bellent ( t ) . (16)It is worth noticing that a high average fidelity couldstill allow for states that are poorly (or even not at all)transferred, while the ultimate goal is the transmissionof any state.When the setup is such that Eqs. (10) hold, fromEqs. (12) and (13) it follows that F Bellent ( t ) = (cid:0) T + T + 2 Re T (cid:1) , (17) C Bell CB ( t ) = max (cid:8) , | T | − p T T (cid:9) . (18)Moreover, if A is prepared in the pure state | ψ θϕ i ,Eq. (14) explicitly reads F AB ( t ) = T cos θ + T sin θ + (cid:2) T ) − ( T + T ) (cid:3) sin θ cos θ , (19)which shows that f ≡ T and f ≡ T are the trans-mission fidelities of the states | i = | ψ ϕ i and | i = | ψ πϕ i ,respectively, while f = (1+ Re T ) represents the trans-mission fidelity of the states (cid:0) | i + e ıϕ | i (cid:1) / √ | ψ π ϕ i .The fidelity (19) only depends on θ ∈ [0 , π ] and its ex-trema can be easily determined. To this purpose rewriteEq. (19) as F AB ( t ) = f + ( f − f ) cos θ + ( f + f − f ) cos θ ;(20)it follows that F AB ( t ) takes the values f and f at theextrema of the range cos θ ∈ [ − ,
1] and can have a mini-mum f m = f −
18 ( f − f ) f + f − f (21)in between provided that c m = f − f f + f − f ) ∈ ( − , F AB ( t ) are: F min AB ( t ) = (cid:26) min { f , f } if | c m | ≥ f m if | c m | < , (22) F max AB ( t ) = max { f , f } . (23)One can see that if f = f the best transmitted states are | i and | i which, on the other hand, are the best andthe worst transmitted ones (or viceversa) if | c m | ≥
1. Thispeculiar role of the computational states is obviously aconsequence of the assumed symmetry.
III. THE XX MODELA. Dynamical evolution
In this section we specifically consider the Hamilto-nian (1). The system A ∪ Γ ∪ B is prepared in the state ρ tot = ρ A ⊗ ρ Γ ⊗ ρ B where ρ Γ is any state invariant underrotations around the z -axis, and ρ B = b ζ + b ζ : thischoice fulfills the requisite of U (1) symmetry of ρ Γ ⊗ ρ B leading to Eq. (10). Referring to the usual Jordan-Wigner transformation, we cast Eq. (1) in the fermionicquadratic form H = X i,j c † i Ω ij c j = X n ω n c † n c n , (24)where { c i , c † i } are fermionic operators whose nearest-neighbor interaction is described by a ( N +2) × ( N +2)tridiagonal mirror-symmetric matrixΩ = − h j j h
11 2 h
1. . . . . . . . .1 2 h
11 2 h j j h ; (25)an orthogonal transformation O = {O ni } diagonalizes H (see Appendix A) in terms of Fermi operators c n = P i O ni c i and c † n which annihilate/create excita-tions of energy ω n [26, 27]. The trivial time-evolutionof the c n ’s entails a time-dependent transformation c i ( t ) = N +1 X i =0 U ij ( t ) c j , (26)where U ij ( t ) = X n O ni O nj e − ıω n t . (27)Reminding that ζ , = ( ± σ z ), ζ = ζ † = σ + , andthat ζ µ in Eq. (6) acts on the qubit B, and provided that Eqs. (10) hold, we find T ( t ) = | u ( t ) | + v ( t ) T ( t ) = 1 − v ( t ) T ( t ) = − p h σ zN +1 i u ( t ) e ıα ( t ) (28)where u ( t ) = | U N +1 , ( t ) | , (29) α ( t ) = arg[ U N +1 , ( t )] (30) v ( t ) = | U N +1 ,N +1 ( t ) | h σ zN +1 i + 12 + C N +1 ( t ) , (31) C i ( t ) = N X j,j ′ =1 U ∗ ij ( t ) U ij ′ ( t ) Tr (cid:2) ρ Γ c † j c j ′ (cid:3) , (32)with p = Tr[ P ρ Γ ], and P = exp (cid:0) ıπ P Ni =1 c † i c i (cid:1) ≡ Q Ni =1 ( − σ zi ) is the chain parity operator, which is a con-stant of motion. In particular, when ρ Γ is the groundstate of the chain, p = sign (cid:2) det ˜Ω (cid:3) [27], where ˜Ω is ob-tained from Ω by referring only to the chain, i.e., delet-ing the first and last row and column. In our case, p = ( − [( N cos − h ) /π ] , where [ · ] denotes the integer part,meaning that p equals ( −
1) to the power of the numberof negative energy fermionic eigenmodes of Γ. For h = 0it reduces to p = ( − [ N/ .Once the above expressions are evaluated, the dynam-ics of B directly follows from the initial state of A viaEq. (4). Similarly, the dynamics of the qubit pair C ∪ Bfollows from the initial state of the qubit pair C ∪ A, viaEq. (7). Moreover, the time evolution of the magnetiza-tion along the chain can be straightforwardly obtainedfrom the formalism described above: (cid:10) σ zi ( t ) (cid:11) = (cid:12)(cid:12) U i ( t ) (cid:12)(cid:12) h σ z i + (cid:12)(cid:12) U i,N +1 ( t ) (cid:12)(cid:12) h σ zN +1 i + G i ( t ) , (33)where G i ( t ) = 2 C i ( t ) + | U N +1 , | + | U N +1 ,N +1 | − h = 0 and N even, G i ( t ) = 0, meaning that h σ zi ( t ) i aresolely determined by two traveling excitations, startingfrom the edges of the chain. These excitations, becauseof the single-particle nature of the Hamiltonian, do notscatter, which has great relevance as far as the transportproperties of the chain are concerned, as discussed inSection IV. On the other hand, when the chain is initial-ized in the fully polarized state N Nj =1 |↓i the contribution C i ( t ) vanishes and G i ( t ) has the only effect of redefiningthe range of the magnetization during the dynamics. B. Fidelities and concurrence
We again consider the system ruled by the Hamilto-nian (1) in the setup described above, so that Eqs. (10)hold. Using Eq. (28) we find the entanglement fidelityand the average transmission fidelity of pure states F Bellent ( t ) = + u ( t ) − p cos α h σ zN +1 i u ( t ) , (34) F pure AB ( t ) = + u ( t ) − p cos α h σ zN +1 i u ( t ) , (35)as well as the concurrence, C Bell CB ( t ) = max (cid:8) , C (cid:9) , (36)where C = (cid:12)(cid:12) p h σ zN +1 i (cid:12)(cid:12) u ( t ) − p v ( t )[1 − u ( t ) − v ( t )] . (37)From the above formulas it appears that the choice ofthe initial state ρ Γ ⊗ ρ B plays an important role [28]: inparticular, in order to get the largest concurrence it mustbe p h σ zN +1 i = ± , (38)meaning that ρ Γ is an eigenstate of P and the qubit Bis initially in a polarized state, ρ B = ζ or ρ B = ζ ; as forthe initial state of the channel, the choices range, forexample, from its ground state to a fully polarized state.Such limitation in the choice of the initial state might beovercome by applying a two-qubit encoding and decodingon states ρ A and ρ B , respectively [19, 29]. Similarly, forthe transmission fidelity the condition − p h σ zN +1 i cos α = 1 (39)must hold. Notice that the l.h.s. of Eq. (39) follows fromthe rotation around the z -axis undergone by the stateduring the transmission and can be treated by choosinga proper magnetic field [2] or the parity of N , as well asby applying a counter-rotation on the qubit B [28].The above analysis shows that, once condition (39) isfulfilled the quality of the state and entanglement transfermainly depends on u ( t ) and increases with it; the residualdependence on v ( t ) suggests that the latter quantity needbeing minimized. In Ref. [28] it has been shown that v ( t )is exactly zero provided that both Γ and B are initializedin the fully polarized state. On the other hand, since U ij ( t ) entering Eqs. (29) and (31) is a unitary matrix, ifa certain time t ∗ exists such that u ( t ∗ ) is very close tounity, then v ( t ∗ ) must be close to zero, no matter theinitialization. This situation is related with the optimaldynamics studied in Ref. [6]: such relation stands on theanalytical ground developed in the next Section. IV. OPTIMAL DYNAMICS
We now have the tools for determining the conditionsfor a dynamical evolution that corresponds to the bestquality of the transmission processes. In Appendix Athe algebraic problem of diagonalizing the XX Hamilto-nian (24) in the case of nonuniform mirror-symmetricendpoint interactions, Eq. (25) is analytically solved.The eigenvalues of Ω can be written as ω k = − h − cos k , (40) in terms of the pseudo-wavevector k , which takes N +2discrete values k n in the interval (0 , π ): from Eqs. (A15)and (A16) it follows that these values obey k n = π n + ϕ k n N +3 , ( n = 1 , ..., N +2) , (41)with ϕ k = 2 k − − (cid:16) cot k ∆ (cid:17) ∈ ( − π, π ) , (42)∆ = j − j , (43)where we have set h = h . From the above equationsit follows that the k ’s correspond to the equispaced val-ues πn/ ( N +3), slightly shifted towards π/ π/ ( N +3), so that their order is pre-served: therefore k can be used as an alternative indexfor n , understanding that it takes the values k n , as donein Eq. (40). According to the conclusions of the previoussection, we focus on the transition amplitude, Eqs. (27)and (29)-(30), which explicitly reads: U N +1 , ( t ) ≡ u ( t ) e ıα ( t ) = − X n ρ ( k n ) e ı ( πn − ω kn t ) , (44)where, after Eq. (A21), it is ρ ( k ) = 1 N +3 ∆(1+∆)∆ + cot k , (45)and mirror symmetry is exploited according to Eq. (A3):the transition amplitude above is a superposition of phasefactors with normalized weights, P n ρ ( k n ) = 1, entailing | u ( t ) | ≤ ρ ( k ) is peaked at k = k = π/ j the narrower ρ ( k ).As u ( t ) essentially measures the state-transfer qual-ity, the condition for maximizing it at some time t ∗ , i.e. u ( t ∗ ) ≃
1, is that all phases πn − ω k n t ∗ almost equal eachother. Assume for a moment that the k ’s be equispacedvalues, as in (A8), and that the dispersion relation belinear, ω k = vk : then Eq. (44) would read u ( t ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X n ρ ( k n ) e ıπn (1 − t/t ∗ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (46)with t ∗ =( N +3) /v , so that u ( t ∗ ) = P n ρ ( k n ) = 1, i.e., allmodes give a coherent contribution and entail perfecttransfer. On the other hand, in our case ω k is nonlin-ear in k , and the k n are not equally spaced due to thephase shifts (42) entering Eq. (41), so generally the dif-ferent modes undergo dispersion and lose coherence. A. Transfer regimes
The dependence of ∆ upon j reveals the possibility ofidentifying different dynamical regimes, characterized bya qualitatively different distribution ρ ( k ), Eq. (45), andhence, as for the transfer processes, a different behaviorof the transition amplitude u ( t ). For extremely small j the distribution ρ ( k ) can be so thin that (for even N )only two opposite small eigenvalues come into play, saydiffering by δω , and perfect transmission will be attainedat a large time t = π ( δω ) − (for odd N there is a thirdvanishing eigenvalue at k = π/ δω do matter). This is the Rabi-like regime alsomentioned in the Introduction.A different regime is observed when j is increased:a few more eigenvalues come into play and it may oc-cur, in a seemingly random way, that their spacings be(almost) commensurate with each other, i.e., they can beapproximated as fractions with the same denominator K ,yielding phase coherence at t K = πK . By recording themaximum of u ( t ) over a fixed large time interval T , as j is varied (see Ref. [3]), a rapid and chaotic variation isobserved. This regime is clearly useless for the purposeof quantum communication.As j further increases, the ballistic regime eventuallymanifests itself: ρ ( k ) involves so many modes that com-mensurability is practically impossible, and a more regu-lar behavior with short transmission time t ∗ ∼ N sets in.The ballistic regime is characterized by relatively largevalues of u ( t ∗ , ∆) which is the quantity plotted in Fig. 2,reporting numerical results for increasing chain lengths.It appears that each curve shows a maximum for a par-ticular optimal value of ∆ = ∆ opt ( N ) or, equivalently, of j = j opt0 ( N ): such maxima are remarkably stable for veryhigh N and yield very high transmission quality. In Ta-ble I we report some of the optimal values ∆ opt ( N ) and j opt0 ( N ) for a wide interval of chain lengths. This last‘ballistic-transfer’ regime is the one we are interested in,since it has two strong advantages: first, the transmis-sion time t ∗ ∼ N is the shortest attainable, and second,the maximum value u ( t ∗ , ∆ opt ) of u ( t ∗ , ∆) is such thatone can achieve very good state transfer, e.g., the corre-sponding transmission fidelity is far beyond the classicalthreshold, even for very long chains.The above analysis gives a physical interpretation ofwhat is observed in Fig. 3 of Ref. [3], where the Rabi-like, intermediate and ballistic regimes emerge.A qualitative picture of the ballistic regime can beobtained by viewing the transition amplitude (44) asa wavepacket with N +2 components. It can be evalu-ated by progressively adding the contributions from sym-metric eigenvalues, i.e., for odd N summing between( N +1) / ∓ ℓ , for ℓ = 0 , , ..., ( N +1) /
2. This yields thepartial sum u ℓ ( t ∗ ) shown in Fig. 3, together with thecorresponding frequency and density. One can see thatthe amplitude increases only over the modes of the linear-frequency zone, i.e. where frequencies are equally spaced,indicating that only those wavepacket components whose FIG. 2. (color online) Value of u ( t ∗ , ∆) as a function of ∆,for different wire lengths N . t ∗ is obtained numerically bymaximizing Eq. (44) around t ≃ N +3.FIG. 3. (color online) Partial sum of the amplitude u ℓ ( t ∗ ) vs ℓ for N + 2 = 51 and j = 0 .
58, together with the correspondingfrequency and density. frequency lies in such zone play a role in the transmissionprocess.
B. Ballistic regime and optimal values
From the above reasoning, since the modes contribut-ing to the amplitude lie in a range of size ∆ around k ,in order to get high-quality transfer processes it is neces-sary that the corresponding frequencies be almost equallyspaced, meaning that ω k n is approximately linear in n .Actually, ω k has an inflection point in k : its nonlinearityis of the third order in k − k and the modes close to k satisfy the required condition. However, from the phase-shifts (42) a further cubic term arises, which depends on∆. As ∆ varies with j , the latter can be chosen so as toeliminate the cubic terms, yielding a wide interval withalmost constant frequency spacing. The latter can beexpressed just as the derivative of ω k n with respect to n , ∂ n ω k n = sin k ∂ n k . The last term is evaluated from N + 2 25 51 101 251 501 1001 2501 5001 10001 25001 50001 100001 250001 500001∆ opt j opt0 u ( t ∗ , ∆ opt ) 0.972 0.953 0.936 0.916 0.902 0.891 0.880 0.873 0.868 0.862 0.859 0.857 0.854 0.853TABLE I. Optimal values ∆ opt and the corresponding j opt0 and u ( t ∗ , ∆ opt ) (see text for details), for different N . Eqs. (41) and (42), ∂ n k = π + ϕ ′ k ∂ n kN +3 = πN +3 − ϕ ′ k , (47) ϕ ′ k = − − ∆∆ + 2(1 − ∆ ) cos k ∆[∆ + (1 − ∆ ) cos k ] , (48)so that ∂ n ω k n = π sin kN +3 − ϕ ′ k = πt ∗ h (cid:16) − ∆ t ∗ ∆ − (cid:17) cos k + O (cos k ) i , (49)where t ∗ = N +3 + 2 (1 − ∆) / ∆ is the arrival time. It fol-lows that one can minimize the nonlinearity of ω k n bysetting the width to the value ∆ satisfying∆ = h t ∗ (1 − ∆ ) i / −→ N ≫ / N − / , (50)and j ≃ / N − / for large N . Therefore the mainmechanism that produces an optimal ballistic transmis-sion is that of varying the endpoint exchange parameterto the value j that ‘linearizes’ the dispersion relation.Actually, if the corresponding ∆ = ∆( j ) is such that ρ ( k ) exceeds the region of linearity, further gain arises bylowering j so as to tighten the relevant modes towards k . However, at the same time ω k n becomes less linearand the trade-off between these two effects explains whya maximum is observed. This is well apparent in Fig 4,where for different values of ∆ the shapes of ∂ n ω k can becompared with the excitation density ρ ( k ): for ∆ = ∆ the density still has important wings in the nonlinearzone, so the optimal value ∆ opt turns out to be smaller.The dynamics in the ballistic regime is best illustratedby the time evolution of the magnetization Eq. (33) alongthe chain, plotted in Fig. 5 when the initial state is |↑i ⊗ |↓↓ · · · ↓i ⊗ |↑i . The initial magnetizations at theendpoints generate two traveling wavepackets: for non-optimal couplings ( j = 1, upper panel) they change theirshape and quickly straggle along the chain; for optimalcoupling ( j = j opt0 , lower panel) they travel with minimaldispersion. This confirms that the coherence is best pre-served when the optimal ballistic dynamics is induced:In the next session we show that to such dynamics do infact correspond high values of the quality estimators forthe state and entanglement transfer. FIG. 4. (color online) ‘Group velocity’ v k ≡ [( N +3) /π ] ∂ n ω k n and ρ ( k ) vs k for different values of ∆. The thicker curvescorrespond to ∆ = 0 . k , and to ∆ opt ≃ . V. INFORMATION TRANSMISSIONEXPLOITING OPTIMAL DYNAMICS
The requirement Eq. (39), means that the state isnot rotated by the dynamics when it arrives on site B,though during the evolution it may undergo a rotationaround the z axis. In Ref. [20] it has been shown that α = − π ( N +1) at the transmission time t ∗ . Therefore,also without applying a counter-rotation on qubit B [28],condition (39) can be fulfilled by choosing N = 4 M ± ± is given by (38) and thus depends onthe initial state of the chain. In the following we assumethat conditions (38) and (39) are always satisfied.Let us consider for the moment that Γ and B are ini-tially in the fully polarized state |↓↓ · · · ↓i ⊗ |↓i . In thatcase v ( t ) ≡ u ( t ). The best attainableinformation transfer quality corresponds therefore to themaximum amplitude u opt ≡ u ( t ∗ , ∆ opt ). In Fig. 6 and inTable I we report these values together with the corre-sponding optimal ∆ opt as a function of the chain length N in a logarithmic scale; the inset shows that ∆ opt obeysthe same power-law behavior predicted in Eq. (50) for∆ . Fig. 6 also shows that for larger and larger N themaximal amplitude u opt does not decrease towards zero,but it rather tends to a constant value of about 0.85,which is surprisingly high, as, e.g., it corresponds to an FIG. 5. (color online) Dynamics of the magnetization σ zi ( t )at time t and site i when a) j = 1 and b) j = j opt0 . The initialstate of the whole system is |↑i⊗|↓↓ · · · ↓i⊗|↑i and the lengthof the chain is N + 2 = 250. average fidelity F AB ( t ∗ ) & .
9. This is indeed true: Weshow in Appendix B that in the limit of N → ∞ the op-timized amplitude tends to u opt = 0 . j opt0 ≃ . N − / . (51)In the optimal ballistic case the channel initialization isnot crucial, as different initial states satisfying (38) giverise to almost the same dynamics as discussed at the endof subsection III B. In fact, the term C N +1 ( t ) enteringEq. (31) essentially embodies the effect of channel initial-ization and it is expected to be small at t = t ∗ . This isapparent in Fig. 7, where for j = j opt0 , C N +1 ( t ∗ ) stayswell below 0.1 for N as long as 1000.The transmitted entanglement, as measured by theconcurrence (36), is shown in Fig. 8 as a function of j and t , with the channel initially prepared in its groundstate. As expected, the peak of the transmitted con-currence is observed for j = j opt0 ; away from j opt0 thequality of transmission falls down because u ( t ∗ ) decreases FIG. 6. (color online) Behavior of the maximum attainableamplitude u opt and (inset) of the corresponding optimal valueof ∆ opt vs logarithm of the chain length N . The horizontaldashed line is the infinite N limit of u opt . C N + ( t ∗ )
10 100 1000 Nj = j opt0 GSNe´el j = 1GSNe´el FIG. 7. (color online) C N +1 ( t ∗ ) for different initial states ofthe chain (ground state, anti-ferromagnetic Ne´el state, andseries of singlets [28]) when j = j opt0 and j = 1. The resultsfor a series of singlets are numerically indistinguishable fromthose with the Ne´el state.FIG. 8. (color online) Evolution of the concurrence C BellCB vs j t . The length of the chain is N + 2 = 250. F m i n A B ( t )
200 220 240 260 280 300 320 tj = j opt0 j = 1 j = 0 . j = 0 . j = 0 . j = 0 . FIG. 9. (color online) Minimum fidelity vs time for differ-ent values of j . The length of the chain is N + 2 = 251 and j opt0 = 0 . and, accordingly, v ( t ∗ ) is allowed to increase. In fact, inthe non-optimal ballistic case the quality of entanglementtransfer does depend on the the initial state of the chan-nel [28, 30]; for instance, when j = 1 and the chain isinitially in its ground state, the contribution of the over-lap terms Tr[ ρ Γ c † j c j ′ ] in Eq. (32) is not quenched by thedynamical prefactors, and higher values of C N +1 ( t ∗ ) (seeFig. 7) inhibit the transmission of entanglement even if u ( t ∗ ) = 0.The effect of the optimization of j is clearly evidentin the time behavior of the minimum fidelity, Eq. (22),reported in Fig. 9. The peak of F min AB ( t ) for j opt0 occursat the arrival time N +3 + s with a time delay s thatagrees with the asymptotic value s ≃ . N / derived inAppendix B. The ‘reading time’, i.e., the time intervalduring which the qubit B keeps being in the transferredquantum state, is t R ≃ ∆ − , as the same figure alsoshows; note that, in the optimal case, t R increases with N according to the asymptotic behavior t R ≃ . N / . VI. CONCLUSIONS
In this paper we have shown that high-quality quan-tum state and entanglement transfer between two qubitsA and B is obtained through a uniform XX channel ofarbitrary length N by a proper choice of the interaction j between the channel and the qubits. The value of suchinteraction is found to control the transfer regime of thechannel, which varies, as j increases, from the Rabi-likeone, characterized by very long transmission time, to anintermediate regime, which turns useless for the purposeof quantum communication, and finally becomes ballisticfor j of the order of the intrachannel interaction.In order to get coherent transfer in the ballisticregime, it is desirable that the k -density of the travel-ing wavepacket generated by Alice’s initialized qubit Abe narrow and concentrated in the linear zone of the dis- persion relation, i.e., with equispaced frequencies. As theparameter j controls both the width of the k -density andthe spacings of the frequencies entering the dynamics, onecan therefore improve the transmission quality up to abest trade-off arising for an optimal value j opt ( N ) whichfor large N behaves as j opt ( N ) ≃ . N − / . Remark-ably, we have found that for such a choice the quantum-state-transfer quality indicators are very high and, in-deed, have a lower bound for N → ∞ that still allows toefficiently perform quantum-information tasks: e.g., theaverage fidelity of state transmission is larger than 90 %.The ballistic regime ensures fast transmission on a timescale of the order of N , at variance with the Rabi-likeregime, and in the optimal case the reading time increasesas N / . It is also to be noted that, if experimental set-tings constrain to a given value j exp0 , yet one can opti-mize the chain length in such a way that j exp0 = j opt0 ( N ).The only requirement on the initial state of the receivingqubit B and the spin bus is to possess U (1) symmetry, acondition that can be fulfilled by several configurationsconcerning the spin bus, ranging from the fully polarizedstate to the highly-entangled ground state. If a largemagnetic field can be switched on during the initializa-tion procedure (in order to fully polarize the channel),and switched off as soon as the transmission starts, then,from our analytical treatment it emerges that tempera-ture is not a major issue as far as the dynamical evolutionof the channel is concerned, though low temperatures areobviously necessary to protect the qubits from phase andamplitude damping due to the solid-state environment.To judge if the proposed scheme identifies a reasonableexperimental framework, let us estimate the magnitudeof the involved physical quantities. Consider a solid-state implementation with lattice spacing of about 10 ˚Aand intrachain exchange J ≃ K. A quantum state willthen be transferred with fidelity 90 % along a channel oflength 1 cm ( N ≃ ) using j ≃ . N − / J ≃ . t = N ~ / ( k B J ) ≃ . µ s and read-ing time t R ≃ . N / ~ / ( k B J ) ≃ .
03 ns.
ACKNOWLEDGMENTS
We acknowledge the financial support of the ItalianMinistry of University in the framework of the 2008 PRINprogram (contract N. 2008PARRTS 003). LB and PVgratefully thank Dr. A. Bayat and Prof. S. Bose for use-ful discussions, and TJGA thanks ISC-CNR for the kindhospitality.0
Appendix A: Quasi-uniform tridiagonal matrices
The matrix Ω, Eq. (25), can be written as Ω = − h − M where M ( x, y ) = x yy yy x (A1)is a square tridiagonal matrix of dimension M = N + 2,and x = 2( h − h ) and y = j . This real symmetricmatrix is diagonalized by an orthogonal matrix O ( x, y ), M X i,j =1 O ki M ij O k ′ j = λ k δ kk ′ , (A2)and it is known that ( i ) if y = 0 the eigenvalues arenondegenerate [31], ( ii ) the eigenvectors correspondingto the eigenvalues ordered in descending order are alter-nately symmetric and skew symmetric [32], i.e., O ki = ± O k,M +1 − i . (A3)The eigenvalues are the roots of the associated charac-teristic polynomial χ M ( λ ; x, y ) ≡ det[ λ − M ( x, y )] . (A4)In the fully uniform case the characteristic polynomialis η M ( λ ) ≡ χ M ( λ ; 0 ,
1) and one easily obtains the recur-sion relation η M = λ η M − − η M − , (A5)that can be solved in terms of Chebyshev polynomials ofthe second kind, η M = sin( M +1) k sin k , (A6)where λ ≡ k , (A7)so the eigenvalues of M (0 ,
1) correspond to M discretevalues of k , k = π nM +1 [ n = 1 , . . . , M ] ; (A8)the corresponding eigenvectors are O ki (0 ,
1) = q M +1 sin ki . (A9) The general determinant (A4) can be expressed interms of the η M ’s by expanding it in the first and thenin the last column, χ M = ( λ − xλ + x ) η M − − y ( λ − x ) η M − + y η M − , (A10)and using Eq. (A5) one can eliminate the explicit appear-ances of λ , χ M = η M − x η M − + x η M − +(1 − y ) (cid:2) η M − − x η M − +(1 − y ) η M − (cid:3) . (A11)By rewriting Eq. (A6) as sin k η M = Im (cid:2) e ı ( M +1) k (cid:3) anddefining z ≡ − y , z k ≡ z e − ık , x k ≡ x e − ık , (A12) u k ≡ − x k + z k = e − ık (cid:8) [(2 − y ) cos k − x ] + ı y sin k (cid:9) , (A13)Eq. (A11) takes the formsin k χ M ( k ) = Im (cid:8) e ı ( M +1) k u k (cid:9) . (A14)The secular equation Im (cid:8) e ı ( M +1) k u k (cid:9) = 0 entails thatwhen k corresponds to an eigenvalue the quantity inbraces is real and equal to either ±| u k | ; by Eq. (A13) itturns into sin (cid:2) ( M +1) k − ϕ k (cid:3) = 0, with the phase shift ϕ k = 2 k − − y sin k (2 − y ) cos k − x , (A15)so the M eigenvalues correspond to k n = π n + ϕ k n M +1 , ( n = 1 , . . . , M ) . (A16)We are interested in the squared components of thefirst column of the diagonalizing matrix O ki , which canbe expressed as [31] O k = ξ M − ( λ k ) ∂ λ χ M ( λ k ) = − k ξ M − ( k ) ∂ k χ M ( k ) , (A17)where k assumes the values (A16) and ξ M − ( λ ; x, y ) is thecharacteristic polynomial associated to the first minormatrix M that, expanded in the last column and usingEq. (A5), reads ξ M − ≡ det[ λ − M ( x, y )]= ( λ − x ) η M − − y η M − = η M − − x η M − + (1 − y ) η M − . (A18)Then the numerator of Eq. (A17) issin k ξ M − = Im (cid:8) e ıMk u k (cid:9) , (A19)while from Eq. (A14) one hassin k ∂ k χ M ( k ) = ( M +1) Re (cid:8) e ı ( M +1) k u k (cid:9) +2 Im (cid:8) e ı ( M +1) k u k u ′ k (cid:9) , (A20)1where the argument of Re is real indeed; retaining onlythe dominant term for M ≫ O k = 2 M +1 y sin k [(2 − y ) cos k − x ] + y sin k . (A21)For x = 0 the above expression is in agreement withRef. [3]. In the most common case x < − y the maxi-mum k of O k is located atcos k = x − y , (A22)so that switching x on the maximum shifts from π/ k = π − sin − x − y ; (A23)the ‘eigenvalue’ corresponding to the maximum is λ = 2 cos k = 2 x − y (A24)so that for y ∼
1, the maximum shifts the ‘energy’ lin-early with x . Expanding O k around the maximum, theleading behavior is found to be a Lorentzian, O k ≃ M +1 y y + [(2 − y ) − x ]( k − k ) , (A25)whose width (HWHM) is given by∆ ≃ y p (2 − y ) − x . (A26)When x and y are small, k ≃ ( π − x ) / ≃ y /
2, so x rules the position of the peak, while y determines itswidth. Appendix B: large- N limit of the amplitude The transition amplitude, Eq. (44), in the case of odd N = 2 M − u ( t ) = ∆(1 + ∆) N + 3 M X m = − M e ı ( πm − t sin q m ) ∆ + tan q m , (B1)where the summation has been made symmetric throughthe change of variable q = π/ − k . The shift equa-tion (41) turns into π m = ( N +3) q m + πϕ q m , (B2)with πϕ q = 2 (cid:16) tan − tan q ∆ − q (cid:17) . (B3) In the limit N → ∞ one can write the sum as an integralsetting1 N +3 X m −→ Z dqπ (cid:16) πϕ ′ q N +3 (cid:17) −→ Z dqπ . (B4)As we deal within the region of the optimal value of∆ ∼ N − / →
0, we have u ∞ ( t ) = lim N →∞ ∆ Z π − π dqπ e ı [( N +3) q + πϕ q − t sin q ] ∆ + tan q . (B5)Writing the arrival time as t = N +3 + s , where s is thearrival delay, one has then u ∞ ( t ) = lim t →∞ ∆ Z π − π dqπ e ı [ t ( q − sin q ) − sq + πϕ q ] ∆ + tan q (B6)The relevant q ’s are of the order of ∆ ∼ N − / →
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