A deeper insight into quantum state transfer from an information flux viewpoint
aa r X i v : . [ qu a n t - ph ] S e p December 19, 2018 12:16 WSPC/INSTRUCTION FILE InformationFlux
International Journal of Quantum Informationc (cid:13)
World Scientific Publishing Company
A DEEPER INSIGHT INTO QUANTUM STATE TRANSFERFROM AN INFORMATION FLUX VIEWPOINT
C. DI FRANCO, M. PATERNOSTRO
School of Mathematics and Physics, Queen’s University, Belfast BT7 1NN, United [email protected]
G. M. PALMA
NEST-CNR (INFM) & Dipartimento di Scienze Fisiche ed Astronomiche, Universita’ deglistudi di Palermo, via Archirafi 36, Palermo, 90123, Italy
We use the recently introduced concept of information flux in a many-body register inorder to give an alternative viewpoint on quantum state transfer in linear chains of manyspins.
Keywords : Quantum many-body systems; Quantum state transfer; Information flux
1. Introduction
The control over multipartite devices, used for the purposes of quantum communica-tion and quantum computation, can be reduced by means of specific form of built-inand permanent intra-register couplings. This possibility has recently been the centerof a strong interest of the quantum information processing (QIP) community.1 , , control-limited . The efforts are inthis case directed towards the determination of the exact distribution of couplingstrengths for a given inter-qubit interaction model in a control-limited setting. Herewe discuss a recently proposed approach that sheds new light onto the achievementof this task and the design of efficient QIP protocols.In Ref. 4 we have introduced the concept of information flux in a quantummechanical system. This can be seen in terms of the influences that the dynamics ofa selected element of a multipartite register experience due to interaction channels with any other parties. We showed that information can be effectively processed byarranging the network of interactions in a way so as to privilege or repress specificinteraction channels. The realization of an optimal QIP task is therefore translatedinto the maximization of the information flux associated with such channels. Here ecember 19, 2018 12:16 WSPC/INSTRUCTION FILE InformationFlux C. Di Franco, M. Paternostro, and G. M. Palma
Fig. 1. Scheme of the processes we consider. A computation or communication step is interpretedas a black box (whose operation depends on the coupling scheme within a multipartite registerof qubits) with movable input and a detection terminal. Through the study of information-fluxdynamics, we can design the best coupling scheme for a chosen QIP operation. we give a deeper insight into the mechanism behind this tool and, therefore, a clearand intuitive picture of it by providing an explicit example.This paper is organized as follows. In Sec. 2 we resume the concept of informationflux. In Sec. 3 we address the case of quantum state transfer in spin chains anddescribe a graphical method as an additional tool to perform such an analysis.Finally, we summarize our results in Sec. 4.
2. Concept
Let us consider a register of N interacting qubits coupled via the Hamiltonianˆ H { g } ( t ) whose structure we do not need to specify here. Our assumption is thatˆ H { g } ( t ) depends on a set of parameters g j (which could stand for the couplingstrengths between the elements of the register) and a generalized time parameter t . We adopt the notation according to which ˆΣ j = ⊗ j − k =1 ˆ I k ⊗ ˆ σ Σ j ⊗ Nl = j +1 ˆ I l (Σ = X, Y, Z ) is the operator that applies the ˆ σ Σ Pauli-matrix only to the j -th qubitof the register. Here ˆ I j is the 2 × j . In the remainderof the paper, we work in the Heisenberg picture where time-evolved operators areindicated as ˆ˜ O j ( t ) = ˆ U † ( t ) ˆ O j ˆ U ( t ) with ˆ U ( t ) = exp[ − ( i/ ~ ) R ˆ H { g } ( t ′ ) dt ′ ]. We saythat there is information extractable from qubit j at time t whenever there is atleast one ˆΣ for which h ˆ˜Σ j ( t ) i 6 = 0. Here, the expectation value is calculated overthe initial state of the register | Ψ i ..N .We adopt, hereafter, the following schematic description of a computation orcommunication process: we suppose to have access to a selected qubit of a multi-partite register and we consider it as the input terminal of the black box given bythe rest of the elements and their mutual coupling. We then use a detection stagewhich can be attached to a suitable output port, connected to one of the qubitsin the black box (a sketch is given in Fig. 1). In this picture, the initial state ofa quantum system is described by the state vector | Ψ i ..N = | φ i | ψ i ..N . Thisis the case in which the first qubit is initialized in a generic input state (and isecember 19, 2018 12:16 WSPC/INSTRUCTION FILE InformationFlux A deeper insight into quantum state transfer from an information flux viewpoint separable with respect to the rest of the register) and | ψ i ..N represents the initialstate of the other qubits that, in general, can be mutually entangled. We assumethis state to be known and independent of the input state. The assumption of aknown initial state of the register { , .., N } corresponds to the situation assumed inmany control-limited QIP protocols. We can thus interpret a quantum process asthe flux of appropriately processed information from the input qubit to the remain-ing components of the register. Such a flux is witnessed by any explicit dependenceof the dynamics of the i -th qubit on the operators associated with the input one.Therefore, in order to find if qubit i has developed any extractable information attime t , as a result of an information flux from the input qubit, we need to study thedependence of h Ψ | ˆ˜Σ i ( t ) | Ψ i ’s on at least one of h Ψ | ˆΣ ′ | Ψ i ’s (Σ , Σ ′ = X, Y, Z ).We refer to Ref. 4 for mathematical details and for the formal definition of theinformation flux. Here we would like just to specify that this approach consists inthe decomposition of each ˆ˜Σ i ( t ) over the operator-basis built out of all the possibletensorial products of single-qubit operators acting on the elements of the system { , .., N } . The information flux can be evaluated from the expectation value (over | ψ i ..N ) of the participants to this decomposition which include operators of thefirst qubit.The control over the set { g } can be fully utilized in the preparation of a multi-partite device in the most appropriate configuration of couplings and initial state,for a given QIP task. The potential of this approach is better illustrated in Refs. 4, 5.In what follows, in order to show a practical application of this method, we willaddress explicitly the case of quantum state transfer.
3. Quantum State Transfer
Quantum state transfer in spin chains is a scenario where the information flux ap-proach is particularly useful. For short-distance communication, the idea of usingspin chains as quantum wires has been put forward by Bose.2 With an isotropicHeisenberg interaction and a local magnetic field, a transmission fidelity that ex-ceeds the maximum value achievable classically can be obtained for a chain up to ∼
80 qubits. Later, Christandl et al. showed that, by engineering the strength ofthe couplings in the chain, a unit fidelity can be reached in for end-to-end transferin a linear chain, among topologically more complicated situations.6We will first analyze an open three-qubit chain, whose Hamiltonian reads ˆ H = P i =1 J ( ˆ X i ˆ X i +1 + ˆ Y i ˆ Y i +1 ) with J the coupling strength of the pairwise interactionbetween the qubits. This is an instance of the linear model considered in Ref. 6 andthus perfect state transfer from the first to the third qubit is obtained after a time t ∗ = π/ (2 √ J ) [we set ~ = 1 throughout the paper], if the initial state of spin 2and 3 is | ψ i , = | i , . If we want to determine the information flux from thefirst to the last qubit, we should evaluate the evolution of ˆ X and ˆ Y ( ˆ Z can beecember 19, 2018 12:16 WSPC/INSTRUCTION FILE InformationFlux C. Di Franco, M. Paternostro, and G. M. Palma seen as − i ˆ X ˆ Y ) in the Heisenberg picture, at time t . It is easy to verify thatˆ˜ X ( t ) = α ( t ) ˆ X ˆ Z ˆ Z + α ( t ) ˆ Y ˆ Z + α ( t ) ˆ X , ˆ˜ Y ( t ) = β ( t ) ˆ Y ˆ Z ˆ Z + β ( t ) ˆ X ˆ Z + β ( t ) ˆ Y . (1)with α ( t ) = β ( t ) = − sin ( √ Jt ), α ( t ) = − β ( t ) = (1 / √
2) sin(2 √ Jt ),and α ( t ) = β ( t ) = cos ( √ Jt ). The only term in the decomposition of theevolved operator ˆ˜ X ( t ) ( ˆ˜ Y (t)) in which ˆ X ( ˆ Y ) is present is α ( t ) ˆ X ˆ Z ˆ Z ( β ( t ) ˆ Y ˆ Z ˆ Z ). The information flux from ˆ X to ˆ X ( ˆ Y to ˆ Y ) is therefore I XX ( t ) = α ( t ) , h | ˆ Z ˆ Z | i , = α ( t ) [ I Y Y ( t ) = β ( t )]. For t = t ∗ , we have I XX ( t ) = I Y Y ( t ) = −
1. A perfect state transfer should correspond to an infor-mation flux between homonymous operators equal to 1. The minus sign we foundcan be explained by remembering that, in Christandl’s protocol, we need to applya single-qubit rotation ˆ R ( N ) = e iπ ( N − ! (2)on the last qubit, after the action of the Hamiltonian ˆ H , in order to compensate anadditional phase factor arising from the evolution. The inclusion of this gate intoour evolution just corresponds to change ˆ X in − ˆ X and ˆ Y in − ˆ Y , keeping ˆ Z unchanged. We thus obtain information flux from ˆ X to ˆ X and from ˆ Y to ˆ Y (andobviously from ˆ Z to ˆ Z ) equal to +1, at time t = t ∗ .In general, a decomposition over the operator-basis can be demanding, especiallyfor a large number of qubits. Indeed, the dimension of this basis is 4 N . In some cases,in virtue of the symmetries of the interaction model (as in the example analyzedabove), the evolution of ˆΣ i involves only few elements of this basis. A simple methodto estimate which terms are included in this evolution is presented here.For a time-independent Hamiltonian, it is ˆ˜Σ i ( t ) = e i ~ ˆ H t ˆΣ i e − i ~ ˆ H t and by meansof the operator expansion formula, we haveˆ˜Σ i ( t ) = ˆΣ i + i ~ t [ ˆ H , ˆΣ i ] + 12! ( i ~ t ) [ ˆ H , [ ˆ H , ˆΣ i ]] + .. (3)If the Hamiltonian ˆ H is expressed in terms of operators ˆΣ ′ j ’s all the commutatorscan be easily represented in a graph. Suppose, for instance, that we want to analyzea 5-qubit chain, whose Hamiltonian reads ˆ H = P i =1 J i ( ˆ X i ˆ X i +1 + ˆ Y i ˆ Y i +1 ), focusingon the evolution of ˆ X . The first commutators are[ ˆ H , ˆ X ] = − i J ˆ Y ˆ Z , [ ˆ H , ˆ Y ˆ Z ] = 2 i J ˆ X ˆ Z ˆ Z + 2 i J ˆ X ,.. (4)The only operators involved in this iterative sequence are ˆ X , ˆ Y ˆ Z , ˆ X ˆ Z ˆ Z ,ˆ Y ˆ Z ˆ Z ˆ Z , and ˆ X ˆ Z ˆ Z ˆ Z ˆ Z . Therefore it is possible to write the evolved oper-ecember 19, 2018 12:16 WSPC/INSTRUCTION FILE InformationFlux A deeper insight into quantum state transfer from an information flux viewpoint H = P i =1 J i ( ˆ X i ˆ X i +1 + ˆ Y i ˆ Y i +1 ). ators ˆ˜ X ( t ) as ˆ˜ X ( t ) = γ ( t ) ˆ X ˆ Z ˆ Z ˆ Z ˆ Z + γ ( t ) ˆ Y ˆ Z ˆ Z ˆ Z ++ γ ( t ) ˆ X ˆ Z ˆ Z + γ ( t ) ˆ Y ˆ Z + γ ( t ) ˆ X . (5)In those cases where the parameters γ j ( t ) cannot be analytically evaluated due tothe complications in the evolution, it is possible to approximate them by means ofrecurrence formulas. In our case, we have γ j ( t ) ∼ P Ml =1 [(2 t ) l /l !] γ ( l ) j , where M is aproper cut-off and γ ( l )1 = − J γ ( l − , γ ( l )2 = J γ ( l − + J γ ( l − ,γ ( l )3 = − J γ ( l − − J γ ( l − , γ ( l )4 = J γ ( l − + J γ ( l − ,γ ( l )5 = − J γ ( l − (6)with γ (0) j = 0 (1) for j = 5 ( j = 5). This analysis can be summarized in theoriented graph in Fig. 2. Each node corresponds to an operator involved in thedecomposition, while an edge connect a node to the operator resulting from itscommutator with the Hamiltonian. We show the corresponding coefficient and anoutgoing (ingoing) edge corresponds to a + (-) sign. The factor 2 i has been omittedfor the sake of simplicity. The recurrence formulas can be easily derived from thisgraph. For J k = J p k (5 − k ), the spin chain becomes the one in Ref. 6 and therecurrence formulas give γ ( t ) = sin (2 Jt ), γ ( t ) = − Jt ) sin (2 Jt ), γ ( t ) = − p (3 /
8) sin (4 Jt ), γ ( t ) = 2 cos (2 t ) sin(2 Jt ), and γ ( t ) = cos (2 Jt ). At t ∗ = π/ (4 J ) we have γ ( t ∗ ) = 1 and γ ( t ∗ ) = γ ( t ∗ ) = γ ( t ∗ ) = γ ( t ∗ ) = 0, whichcorresponds to perfect state transfer.Of course, not all the spin chains are associated to a graph with a linear struc-ture. For example, if we analyze the Hamiltonian ˆ H = P i =1 J ( ˆ X i ˆ X i +1 + ˆ Y i ˆ Y i +1 +ˆ Z i ˆ Z i +1 ), we obtain the graph in Fig. 3 (a) . In this case, the decomposition ofˆ˜ X involves two elements in which ˆ X is present. Therefore, the information fluxfrom ˆ X to ˆ X (this time again we consider the initial state | ψ i = | i ) is I XX ( t ) = δ ( t ) h | ˆ I ˆ I | i + δ ( t ) h | ˆ Z ˆ Z | i = δ ( t ) + δ ( t ), where δ ( t )and δ ( t ) are the coefficients of ˆ X and ˆ X ˆ Z ˆ Z , respectively. A plot of I XX ( t ),evaluated by means of recurrence formulas, is presented in Fig. 3 (b) . In this case I XX ( t ) never reaches 1 (it is well known that this Hamiltonian does not allow per-fect state transfer). To highlight the fact that the information flux can be used as aproper figure of merit in quantum state transfer processes, we have also plotted thestate fidelity F in the worst case (i.e., the transmission of the initial state | i ) inFig. 3 (c) : The maxima in the two plots appear for the same values of scaled time Jt , showing the useful correspondence between information flux and state fidelity.ecember 19, 2018 12:16 WSPC/INSTRUCTION FILE InformationFlux C. Di Franco, M. Paternostro, and G. M. Palma
Fig. 3. (a) : Oriented graph for a 3-qubit chain with Heisenberg interaction. The coefficient J above all the edges has been omitted for the sake of simplicity. (b) : Evaluation of I XX ( t ) bymeans of recurrance formulas. (c) : State fidelity F in the worst transmission case.
4. Remarks
We have shown how the information flux approach helps in the analysis of many-body systems, particularly in the case of spin chains. We have presented a graphicalmethod that is useful to derive the recurrence formulas to evaluate the informationflux, when it is not possible to obtain it analytically. The analogy between theinformation flux in the model of Ref. 6 and in a transverse Ising model with localmagnetic fields has paved the way to the idea presented in a recent paper.7 Thelinear structure of the graphs associated with the two problems allowed us to obtaina coupling-strength pattern that guarantees a perfect state transfer also in a spin-non-preserving chain.
Acknowledgments
We thank M. S. Kim for discussions. We acknowledge financial support from UKEPSRC and QIP IRC. G.M.P. acknowledges support under PRIN 2006 “Quantumnoise in mesoscopic systems” and under CORI 2006. M.P. is supported by TheLeverhulme Trust (Grant No. ECF/40157).
References
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