Quantum states experimentally achieving high-fidelity transmission over a spin chain
Zhao-Ming Wang, Lian-Ao Wu, C. Allen Bishop, Yong-Jian Gu, Mark S. Byrd
aa r X i v : . [ qu a n t - ph ] O c t Quantum states experimentally achieving high-fidelity transmission over a spin chain
Zhao-Ming Wang,
1, 2
Lian-Ao Wu, C. Allen Bishop, ∗ Yong-Jian Gu, and Mark S. Byrd Department of Physics, Ocean University of China, Qingdao, 266100, China Department of Theoretical Physics and History of Science,The Basque Country University(EHU/UPV) and IKERBASQUE,Basque Foundation for Science, 48011 Bilbao, Spain Center for Quantum Information Science, Computational Sciences and Engineering Division,Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-6418, USA Department of Physics, Southern Illinois University, Carbondale, Illinois 62901-4401, USA
A uniformly coupled double quantum Hamiltonian for a spin chain has recently been implementedexperimentally. We propose a method for the determination of initial quantum states that willprovide perfect or near-perfect state transmission for an arbitrary Hamiltonian including this one.By calculating the eigenvalues and eigenvectors of a unitary operator obtained from the free evolutionplus an exchange operator, we find that the double quantum Hamiltonian spin chain will supporta three-spin initial encoding that will transfer along the chain with remarkably high fidelity. Thefidelity is also found to decrease very slowly with increasing chain length. In addition, we are ableto explain previous results showing exceptional transfer using this method.
PACS numbers: 03.67.Hk,75.10.Pq
Introduction.—
Quantum information processing(QIP) often requires the transfer of known or unknownquantum states from one subspace to another withinan information processing device. In recent years, thequantum spin chain has become a prime candidatefor quantum communication purposes such as these[1–3]. In the simplest configuration, where the nearestneighbor couplings are considered to be equal, perfectstate transmission is typically not possible between twosingle spin processors within a linear chain. In otherwords, there is typically a non-vanishing probability thatthe initial excitation amplitude can be found outsidethe receiving spin location [4] at any given time. Inprinciple, however, perfect state transfer (PST) can berealized by properly engineering the couplings betweenneighboring sites [5]. High fidelity state transmissionscan also be obtained using weakly coupled externalqubits [6, 7], modifying only one or two couplings [8, 9],or by encoding the states using multiple spins [10–14]. In[13, 14], a class of states were found to transfer very wellacross long XY coupled spin chains. The existence ofPST has also been established for a variety of interactingmedia, including, but not limited to, the spin chainmodel [15]. Recently, exact state swap through a spinring has been investigated. It was shown that there is astraightforward approach to calculating the probabilityof the occurrence of an exact state swap [16].The schemes developed in Ref. [15] prompted the fol-lowing question. Given an arbitrary spin chain Hamil-tonian, can we find initial states which can be used toenable high-fidelity state transmission? In this letter, weanswer this question and show that for a uniformly cou-pled chain, there exists a particular state which reliablytransfers quantum information over large distances. Weuse a multi-spin encoding scheme and find the existence of a three-spin encoding which can provide reliable statetransmission. In this case, the encoding and decodingprocesses can also be realized easily [13]. This reportis therefore important from an experimental perspectivedue to the ease of implementation which is typically fa-vorable.
The method for identifying high-fidelity states.—
Con-sider a spin chain consisting of N sites which evolvesaccording to some Hamiltonian H in a single excitationsubspace. Suppose for the moment the initial state of oursystem is | Ψ(0) i = | i = | i A ⊗ | ... i ⊗ | i B , whereA and B denote separate processors. After the systemevolves, the state at time t will be | Ψ( t ) i = U ( t ) | i = exp( − iHt ) | Ψ(0) i , (1)where ~ is taken to be 1 throughout. Suppose that atsome time τ PST occurs, then | Ψ( τ ) i = U ( τ ) | i = | N i , (2)where | N i = | i A ⊗ | ... i⊗ | i B . We can use a permu-tation operator P AB to swap all states in A and B, thenthe quantum information can be transferred from A toB. The permutation operator can be expressed as: P AB = X αβ ( | β A i h α A | ⊗ | α B i h β B | ) , (3)where α, β = 1 , , ..., k represent the standard basisfor the k qubits located in processors A and B. Clearly P † AB = P AB and P AB = 1. (cid:12)(cid:12) α ( β ) A ( B ) (cid:11) refers to a state | α ( β ) i in processor A (B). From Eq. (2) U ( τ ) | i = P AB | i , (4)Then P AB U ( τ ) | i = W ( τ ) | i = | i . (5)We introduce the unitary operator W ( τ ) = P AB U ( τ ).From Eq. (5), if the state | i is an eigenvector of theoperator W at time τ , PST occurs. The eigenvectors of W reveal information about the possibilities of a specificstate transmission. The problem of solving Schr¨odinger’sequation now becomes a standard eigen-problem of theoperator W .Since W ( τ ) is a unitary operator it has a complete setof orthonormal eigenvectors {| Ψ m (0) i} τ corresponding toeigenvalues { E m } τ , W ( τ ) | Ψ m (0) i = E m | Ψ m (0) i . (6)This can also be written as U ( τ ) | Ψ m (0) i = E m P † AB | Ψ m (0) i , (7)where U ( τ ) | Ψ m (0) i is the wave function | Ψ m ( τ ) i of thesystem which was initially prepared in the eigenstate | Ψ m (0) i . If | Ψ m (0) i is a product state | Ψ m (0) i = | A i ⊗ | C i , (8)with | A i describing the state of processor A and | C i de-scribing the rest of the system, we can then obtain | Ψ m ( τ ) i = E m P † AB | A i ⊗ | C i = E m | B i ⊗ | C ′ i . (9)For the single excitation subspace, if one of the eigen-vectors | Ψ m (0) i = | i at time τ , PST occurs. If theeigenvectors are degenerate, an arbitrary linear superpo-sition of these degenerate states is also suitable for PST.Suppose there are L degenerate eigenvectors | Ψ l (0) i ( l =1 , , ...L ), which have common eigenvalues E L . The state | Ψ(0) i = L X l =1 C l | Ψ l (0) i . (10)is an eigenvector of W ( τ ), where C l is an arbitrary num-ber. Our analysis describes a method for finding a statewhich can realize PST. (Note that these states are not allunique.) For a given Hamiltonian, if we initially preparethe state | Ψ(0) i as an eigenvector of the operator W ( τ ),then after time τ PST occurs. (a)(b) P FIG. 1: Schematic of our quantum transmission protocol: (a)single-site encoding (b) three-site encoding.
For state transmission from one end spin 1 to an-other end spin N , the exchange operator is given by P = | i h N | + | N i h | + P which is shown in Fig. 1, where P = P j | j i h j | ( j = 1 , N ). We now considered aparticular Hamiltonian, but emphasize that our methodcan be used for any Hamiltonian not only this example. An experimentally implementable Hamiltonian. – Nowconsider the recently implemented Hamiltonian called adouble quantum (DQ) Hamiltonian [17]: H = − N − X i =1 J i,i +1 ( X i X i +1 − Y i Y i +1 ) . (11)where J i,i +1 denotes the coupling between sites i and i +1. This nearest neighbor coupled one-dimensional spinchain can be experimentally implemented using solid-state nuclear magnetic resonance [17–19] in F spins ina crystal of fluorapatite ((FAp-Ca (POa) )F) [18, 19].The system described by Eq. (11) will exhibit free evo-lution such that the evolution operator at time τ willbe U ( τ ) =exp[ − iτ H ]. We can diagonalize the Hamil-tonian H such that H d = W † HW in the single exci-tation subspace. The evolution operator can thereforebe expressed by U ( τ ) = W exp[ − iτ H d ] W and the N eigenvectors of W ( τ ) can be obtained as a function of τ . Furthermore, we consider a natural configuration fora DQ Hamiltonian with open ends. The z -componentof the total for the staggered spins is a conserved quan-tity, [( P i ∈ odd Z i − P i ∈ even Z i ) , H ] = 0. For simplicity,we will only consider the single excitation subspace of thefull Hilbert space. In this case the total number of flippedspins is one. The basis for this subspace will be denotedas | j i which indicates that, after flipping, the even (odd)site spins all of the spins reside in the | i ( | i ) state exceptfor the spin at site j which is in the | i ( | i )state. Forexample, in a N = 5 site chain, the single excitation sub-space will be spanned by | i = | i , | i = | i ,...,etc. If we flip the even numbered states we find that thetotal up spin is actually one. We will use this descriptionthroughout this paper. Example I: nonuniform couplings—
We will considerseveral different coupling configurations with the poten-tial for high-fidelity state transmission and the best re-sults will be provided at the end of our analysis. Firstas an example, we consider two pre-engineered couplings:(1) weak couplings at both ends, where J , = J N − ,N = J and J i,i +1 = J elsewhere. (2) couplings termed PST,where J i,i +1 = p i ( N − i ). It is already known that highfidelity ( F max ≈
1) state transmission for the first con-figuration [6, 7] and perfect fidelity ( F max = 1) for thesecond configuration can be gained in a spin system [5].Here we will use these two kinds couplings to show theapplicability of our methods.For a five-spin system with weak couplings at bothends, we take J , = J , = 0 . J . J equals -1 elsewhere.The eigenvalues and eigenvectors of the operator W ( τ ) atan arbitrary time τ can be obtained numerically. We willconsider those which span the single-excitation subspace.In Table I, we plot the results for τ = 31. The first col- | i | i | i | i | i W ( τ ) at τ = 31 in a spin chain wherethe weak coupling conditions J , = J , = 0 . J are satisfied.We take N =5, J = − umn labels the complex eigenvalues while the remainingcolumns are associated with the amplitudes of the statesat the top of each column. The coefficients in columns2-6 could be complex numbers but our results show thatthe imaginary part always equals zero, so we take themto be real throughout. The same meaning holds for Ta-bles I, II, IV, and V. For example, at the line labeledwith a 4 the eigenvalue is (1.000,0) and the eigenvectoris 0 . | i − . | i − . | i . The state | i closelyapproximates this eigenvector and it can be written asthe aforementioned product state. If we use the state | i as the initial state of the whole system, then at time τ = 31, the system will closely approximately the state | N i .We use the fidelity between the received state and theideally transfered state, F = p h Φ(0) | ρ ( t ) | Φ(0) i as ameasure of the quality of the transfer. Here | Φ(0) i isa state at the receiving end which has the same form asthe state initially prepared by the sender. ρ ( t ) is the re-duced density matrix of the receiver’s spin at time t andis obtained by tracing over all but the receiver’s sites. InFig. 2 (a) we plot the fidelity versus time t for the weaklycoupled chain. The initial state is | i has the maximumfidelity, F = 1 at time t = 31. | i | i | i | i W ( τ ) at τ = 3 .
14 in a N =4 site spinchain with couplings given by J i,i +1 = p i ( N − i ). Next we discuss PST. For the simple case of N = 4,the results at time τ = 3 .
14 are listed in Table II. None ofthe eigenvectors can be written in the form of a productstate | Ψ m (0) i = | A i ⊗ | C i , but the eigenvalues of 1, 3, 4are roughly degenerate. Consider the superposition √ | Ψ (0) i + r | Ψ (0) i + r | Ψ (0) i = | i , (12) where | Ψ (0) i = r
12 ( | i − | i ) , | Ψ (0) i = r
38 ( | i + | i ) − r
18 ( | i + | i ) , (13) | Ψ (0) i = r
18 ( | i + | i ) + r
38 ( | i + | i ) . The state | i at site 1 can be transferred exactly to site4 at time τ = 3 .
14. In Fig. 2(b) we plot the time evo-lution of the fidelity when transferring a state | i fromsite 1 to 4. We also see that at time τ = 3 .
14 the fi-delity is nearly 1. These examples illustrate the validityand practicality of our method while providing a generalmethod to obtain the results. F t(a) F t(b) FIG. 2: The fidelity as a function of time for (a) N = 5, weakcoupling conditions for J , = J , = 0 . J (b) N = 4, channelcoupling conditions J i,i +1 = p i ( N − i ). See text for moredetails. Example II: uniform couplings. —
Consider the mostnatural configuration for a spin chain; a uniformly cou-pled spin chain. We take the ferromagnetic coupling J i,i +1 = J = −
1. Note that PST is typically unattain-able in these systems using single-spin encodings [5, 10].We will consider both single spin encodings as well asmulti-spin encodings. For single-spin encodings, our cal-culations confirm that PST cannot occur in this modelas is known [16]. In table III we list the the maximal p m and the corresponding τ for a N = 7 uniform chain fordifferent eigenvectors. Here p m is defined as the overlapbetween the eigenvector | Ψ m (0) i and the initial state | i ,i.e., p m = |h Ψ m (0) | i| . | Ψ m i | Ψ i | Ψ i | Ψ i | Ψ i | Ψ i | Ψ i | Ψ i τ p m p m and corresponding values of τ . N =7, the maximum values are found in a time interval [5 , Now we will examine multi-spin encoding schemes.As an example, we first consider a three-spin encoding.Specifically, suppose we wish to transfer a state of theform | Ψ(0) i = ( α | i + β | i + γ | i ) A ⊗ | ... i ⊗| i B . As shown in Fig. 1(b), we intend to transfer thestate of the first three spins to the opposite end. Theresults for a N = 6 site chain are given in Table IV fortime τ = 4 .
0. The eigenvalues of 1 and 2 are roughlydegenerate and the approximate relation | Ψ (0) i + | Ψ (0) i = − | i + | i (14)can be written in the form of a product state ( − | i + | i ) A ⊗| ... i . The state ( − | i + | i ) / √ N = 7 and find that at time τ = 28 . | i | i | i | i | i | i W ( τ ) at τ = 4 . N =6. | i | i | i | i | i W ( τ ) at τ = 47 . N =5.
10 20 30 40 50 60 70 800.40.50.60.70.80.91.0 (a) F m a x N
10 20 30 40 50 60 70 80051015202530354045 T m a x N (b) FIG. 3: (Color online.) Length dependence of the maximumfidelity achievable F max and the associated arrival times T max for the state (a) ( − | i + | i ) / √ | i−| i ) / √ , Table V lists the results corresponding to a N = 5chain for the two-spin encoding. At τ = 47 .
2, the relation | Ψ (0) i + | Ψ (0) i = | i − | i approximately holds whichcan also be written in the form of the product state inEq. (8).We have found some states realizing high-fidelity statetransmission, for small N . Now we will check to see if the state ( − | i + | i ) / √ − | i + | i ) / √ N . In Fig. 3, we plot the maximumfidelity F max and the associated arrival time T max asa function of chain length N . The analytic expressionwith eigenvalues E m = − J cos[ πm/ ( N + 1)] and eigen-vectors | Ψ m (0) i = p / ( N + 1) P j sin( q m j ) | j i are used.For practical implementation of our protocol, the maxi-mum fidelity is found in the time [0,50]. For the two-spinencoding, the high fidelity associated with short chainlengths cannot be acheived with increasing chain length. F max quickly decreases with increasing N . However, thisrobustness can be observed even for long chains using thethree-spin encoding. The fidelity is exceptionally largefor a relatively long chain. Therefore, using this state, ahigh-fidelity state transfer can be gained. F max = 0 . N = 6 at t = 4 . F max = 1 .
00 for N = 7 at t = 28 . T max typi-cally increases with increasing chain length N except forsome deviation with small values of N . We also find thatthe T max associated with the three-spin encoding is a lit-tle longer than in the two-spin encoding case for N >
Conclusions.–
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