The orthogonality speed of two-qubit state interacts locally with spin chain in the presence of Dzyaloshinsky-Moriya interaction
TThe orthogonality speed of two-qubit state interacts locally with spin chainin the presence of Dzyaloshinsky-Moriya interaction
D. A. M. Abo-Kahla a,b , M. Y. Abd-Rabbou c , and N. Metwally d,ea Department of Mathematics, Faculty of Science, Taibah University, KSA. b Math. Dep. , Faculty of Education, Ain Shams University, Cairo, Egypt. c Math. Dep., Faculty of Science, Al-Azhar University, Nasr City 11884, Cairo. d Department of Mathematics, College of Science, University of Bahrain, Bahrain. e Department of Mathematics, Aswan University, Aswan, Sahari 81528, Egypt.
Abstract
The orthogonality time is examined for different initial states settings interactinglocally with different types of spin interaction: XX , Ising and anisotropic models.It is shown that, the number of orthogonality increases, and consequently the timeof orthogonality decreases as the environment qubits increase. The shortest time oforthogonality is displayed for the XX chain model, while the largest time is shownfor the Ising model. The external field increases the numbers of orthogonality, whileDzyaloshinsky-Moriya interaction decreases the time of orthogonality. The initial statesettings together with the external field has a significant effect on decreasing/increasingthe time of orthogonality. Keywords : Orthogonality, Dzyaloshinsky-Moriya interaction, Ising Model, quantum speed,
Quantum speed is a significant tool for developing the quantum computer. Its importance lieswith the physical limits levied by the quantum mechanical laws on the speed of the quantuminformation processing and transfer of this information between users [1, 3]. Quantum speedlimit is based on the Heisenberg uncertainty principle associated with the time and energy, bywhich the minimum time that an initial state needs to evolve to its final state is determined.This determination is crucially important for quantum technology [5], such as, quantumcommunication [6, 7], quantum computation [3, 8], and metrology [9]. In other words, theminimum time prerequisite for a quantum system to pass from one orthogonal state (node)to another is called the speed of orthogonality [4]. It was investigated for a single two-levelatom (single qubit) interacting with a rectangular pulse [10], and interacting quantized field[11, 12].On the other hand, the exchange interaction of the antisymmetric Dzyaloshinsky–Moria(DM) was introduced by Dzyaloshinsky [ ? ]. Several studies have discussed the qubit systemin the Heisenberg spin chain. For example, the quantum speed limit of a quantum systemconsist a central spin coupled to a Heisenberg spin-1/2 XY chain has been studied [14]. For afinite XY spin chain coupled to the external magnetic field the quantum speed limit has beendiscussed [15]. Also, quantum speed has been studied for a central qubit interacting withthe isotropic Lipkin-Meshkov-Glick environment [16]. The effect of dynamical dissociate on1 a r X i v : . [ qu a n t - ph ] F e b he speed-up role by applying the strong ”bang-bang” (BB) pulses control on the centralspin system has been examined [17]. addition, the criticality of quantum speed limit fora coupled system of a two-qubit system in the Heisenberg spin chain interacting with theantisymmetric Dzyaloshinsky-Moria (DM) has been explored [18].Our motivation in this work is to investigate the behavior of the quantum speed orthog-onality of a two qubit system coupled to a Heisenberg XY spin chain in the presence ofan external magnetic field and DM interaction. The initial state is prepared in generalizedWerner state. So, we have organized this paper as a follows: In Sec. 2, we present theanalytical solution of the physical model, and we obtain the final density operator. In Sec.3, we discuss the quantum speed orthogonality in Werner state. In Sec. 4, we discuss thequantum speed orthogonality in a maximum entangled state. The final section is devoted topresent our conclusion. Let us consider a two qubit system that is coupled to a Heisenberg XY chain in the presenceof DM interaction in an external field. The total Hamiltonian of this system is H T = H E + H I ,with; H E = − N (cid:88) l (cid:26) γ σ xl σ xl +1 + 1 − γ σ yl σ yl +1 + λσ zl + D ( σ xl σ yl +1 − σ yl σ xl +1 ) (cid:27) , (1)and H I = − g σ zA + σ zB ) N (cid:88) l σ zl . (2)where, H E and H I are representing the Hamiltonian of the environment, and the interactionbetween the two qubit system and the spin–chain environment. l is the site of the chain, N is the total number of sites, and σ il , i = x, y, z are the Pauli matrices. The factor γ (0 < γ < xy plane. D ( − ≤ D ≤ g describes the coupling between thesystem and surrounding environment chain. By using the Jordan–Wigner transformation,and replacing σ xj = σ + j + σ − j , σ yj = i (cid:0) σ − j − σ + j (cid:1) , and σ zl = 1 − n l . The energy Hamiltonian H λ ν E may be rewritten as [19]: H λ ν E = − N (cid:88) l (cid:0) C + l +1 C l + C + l C l +1 (cid:1) + γ (cid:0) C + l C + l +1 + C l +1 C l (cid:1) +2 iD (cid:0) C + l +1 C l − C + l C l +1 (cid:1) − λ ν C + l C l + λ ν , (3)where σ + l = (cid:81) l − j =1 (1 − n j ) C l , σ − l = (cid:81) l − j =1 (1 − n j ) C + l , n j = C + j C j , and C + l ( C l ) is fermioniccreation (annihilation) operator. However, λ ν are the eigenvalues ( λ = λ + g, λ = λ = λ , λ = λ − g, )The energy environmental Hamiltonian H λ ν E can be written under the Fourier series as:2 λ ν E = − (cid:88) k (cid:26) λ ν − cos K ) C + k C k + iγ sin K (cid:0) C + − k C + k + C − k C k (cid:1) (4)+4 D sin KC + k C k − λ ν (cid:27) , (5)where C l = 1 √ N (cid:88) k e iKl C k , C + l = 1 √ N (cid:88) k e − iKl C + k , e iKN = 1 , N (cid:88) k e i ( K ∓ R ) l = δ K, ± R , K = 2 πkN ,C N +1 = C , K = − M, ......, − , , , ....., M, and M = N − . (6)The final diagonalized Hamiltonian H λ ν E by Bogoliubov transformations is expressed by; H λ ν E = (cid:88) k Ω λ ν k (cid:18) b + k b k − (cid:19) (7)where,Ω λ ν k = 2 (cid:112) ( λ ν − cos K ) + γ sin K + 4 D sin K , and b + k ( b k ) is the creation (annihila-tion) operator. The relation between b + k ( b k ) and C + l ( C l ) is given by; C k = cos θ λ ν k b k,λ ν + i sin θ λ ν k b + − k,λ ν , C + k = cos θ λ ν k b + k,λ ν − i sin θ λ ν k b − k,λ ν , and θ λ ν k = tan − ( γ sin Kλ ν − cos K ) , ν = 1 , , , , (8)where we choose θ λ ν k such that the coefficients of all ”anomalous”, b k b − k and b + k b k , in theenergy Hamiltonian H λ ν E vanish. The initial density state of the total system can be expressedas ρ (0) = ρ AB (0) ⊗ | G (cid:105) (cid:104) G | , where ρ AB (0) = (cid:88) α,β c αβ | χ α (cid:105) (cid:104) χ β | , α, β = 1 , , , | G (cid:105) is the ground state of the energy Hamiltonian H λ ν E , which is defined as; | G (cid:105) λ = M (cid:89) k =1 cos θ λk | (cid:105) k | (cid:105) − k + i sin θ λk | (cid:105) k | (cid:105) − k . (10)Meanwhile, ρ AB ( t ) = T r [ U t ρ AB (0) ⊗ | G (cid:105) (cid:104) G | U † t ] (11)= (cid:88) α,β c αβ S αβ | χ α (cid:105) (cid:104) χ β | ,S αβ = (cid:104) G | exp( iH λ β T t ) exp( − iH λ α T t ) | G (cid:105) ,U t = exp( − iH λ ν T t ) . (12)3ence, S αβ = (cid:89) k> (cid:26) cos( η λ α k − η λ β k ) (cid:18) cos η λ α k cos η λ β k e it (Ω λαk − Ω λβk ) + sin η λ α k sin η λ β k e − it (Ω λαk − Ω λβk ) (cid:19) − sin( η λ α k − η λ β k ) (cid:18) cos η λ α k sin η λ β k e it (Ω λαk +Ω λβk ) − sin η λ α k cos η λ β k e − it (Ω λαk +Ω λβk ) (cid:19)(cid:27) . (13)where, η λ j k = θ λjk − θ λk , is the difference angle between the normal mode dressed by thesystem-environment interaction and the purely environment. Let us assume that, the initial system is initially prepared in a generic pure state [20]. Bymeans of the Bloch vectors and the cross dyadic, this pure state can be written as, ρ p (0) = 14 (cid:0) I + p ( σ x − τ x ) − σ x τ x − q ( σ y τ y + σ z τ z ) (cid:1) (14)with q = (cid:112) − p . The time evolution of this initial state is given by, ρ PAB ( t ) = 14 (cid:26) (1 − q ) (cid:0) | (cid:105) (cid:104) | + | (cid:105) (cid:104) | (cid:1) + (1 + q ) (cid:0) | (cid:105) (cid:104) | + | (cid:105) (cid:104) | (cid:1) (15)+ ( q − S ( t ) | (cid:105) (cid:104) | − (1 + q ) | (cid:105) (cid:104) | + H.C. ) (cid:27) , where q = (cid:112) − p . The eigenvectors of the initial pure state is obtained as, µ (0) = (cid:26) √ , , , √ (cid:27) , µ (0) = (cid:26) , √ , √ , (cid:27) ,µ (0) = − (cid:113) | p | + 2 , | p | (cid:16) q − (cid:112) p + q (cid:17) √ p (cid:113) | p | + X1 , | p | (cid:16)(cid:112) p + q − q (cid:17) √ p (cid:113) | p | + X1 , (cid:113) | p | + 2 ,µ (0) = − (cid:113) | p | + 2 , | p | (cid:16)(cid:112) p + q + q (cid:17) √ p (cid:113) | p | + X1 , − | p | (cid:16)(cid:112) p + q + q (cid:17) √ p (cid:113) | p | + X1 , (cid:113) | p | + 2 ; (16)where, X = | q (cid:16)(cid:112) p + q + q (cid:17) + p | . The eigenvectors of the final state is given by: ψ ( t ) = (cid:26) , √ , √ , (cid:27) , ψ ( t ) = (cid:26) , − √ , √ , (cid:27) ψ ( t ) = − | S (1 , t | S (4 , (cid:113) | S (1 , S (4 , | + 1 , , , (cid:113) | S (1 , S (4 , | + 1 ,ψ ( t ) = | S (1 , t | S (4 , (cid:113) | S (1 , S (4 , | + 1 , , , (cid:113) | S (1 , S (4 , | + 1 . (17)4 Dynamics of orthogonality
The aim of this section is shedding the light on the orthogonality behavior of the final stateof the qubits system, where it is assumed that the two qubit system is initial prepared ina maximum entangled state (MES) or partial entangled state (PES). In this context, it isimportant do define the concept of the orthogonality. Now, consider that a system is initiallyprepared in the state ψ (0), this state is evolved under the unitary operator U ( t ) = e − i H t ,where H is the Hamiltonian which describes the system. Then, the final state (cid:12)(cid:12) ψ ( t ) (cid:11) = U ( t ) (cid:12)(cid:12) ψ (0) (cid:11) . The orthogonality is given by [1, 10, 12]: S or = (cid:104) ψ ( t ) | ψ (0) (cid:105) . (18)In our treatment, we shall investigate the orthogonality of the eigenvectors of the initial andthe final density operators. In our numerical calculations, it is enough to consider only twocomponents of the eigenvectors to examine the effect of the parameters of the field and theinitial state settings. The system is initially prepared in a partial pure state ( a ) S or t ( b ) S or t Figure 1: The orthogonality speed of the qubit system where the two qubit system interactswith environment consists of N sites, 7 ,
27 for (a),(b), respectively, where D = 0 ,g=0.1 λ = 0,p=0.5.In Fig.(1), we discuss the speed of orthogonality of a system prepared in a pure stateConsisted of two qubits. This qubit system interacts locally with environment consists of N sites of 7 and 27 qubits. Moreover, we consider that the interaction system is preparedin the XX chain model, i.e., γ = 0, in anisotropic case,0 < γ <
1, and the Ising modelwith γ = 1. It is clear that, the speed of orthogonality S or depends on the sites number,where as one increases the site number N , the number of oscillations increasing and theorthogonality is displayed at short time. This means that the possibility of transmittingthe information increases as N increases. On the other hand, the type of the interaction5odel plays an important role on decreasing the time of orthogonality. However, as it isdisplayed from Fig.(1a) the shortest orthogonal time is depicted for the XX - chain model( γ = 0), while the largest one is displayed for Ising model. These results are coincide withthe definition of the orthogonal time, which is defined T ≥ ¯ h E , where E is the energy, where E γ =0 > E γ =0 . > E γ =1 . However, this results will be changed depending on the site numbers N . Therefore as N increase E γ =1 will be the largest and consequently the orthogonal timewill be the smallest. ( a ) S or t ( b ) S or t Figure 2: The orthogonality speed S or , where the initial state is prepared with p = 0 . N = 7, D = 0 , g = 0 . λ = 0 .
2, (b) λ = 0 . λ ; the strength of the external filed, we consider that thenumber of sites N = 7 and the initial pure state is prepared with p = 0 .
5. Fig.(2 displaysthe behavior of S or for the three different types of the interaction, namely, the XX chainmode, Ising model, and the isotropic model at large strength where we set λ = 0 . , . S or t Figure 3: The effect of the coupling constant same as Fig(1a) but we set g = 0 . g = 0 .
3. It is clear that, the number oforthogonality increases as the coupling constant increases. These result appears clearly bycomparing Fig.(1a) and Fig.(3), where the first orthogonality is depicted for the XX chainmodel at short time compared with that displayed in Fig.(2a). Although the number oforthogonality that depicted for Ising and anisotropic models are the same, but the time oforthogonality that depicted for the anisotropic interaction is smaller than that shown for theIsing model. ( a ) S or t ( b ) S or t Figure 4: The effect of the DM interaction on orthogonality speed S or , where the initial stateis prepared with p = 0 . N = 13 , g = 0 . λ = 0 .
1, (a) DM = 0 and (b) DM = 0 . DM interaction on the behavior of the orthog-onality speed, where we consider that the initial system is initially prepared in a partialentangled pure state with p = 0 .
5. In this context, we would like to mention that, theimpact effect of DM appears only at large numbers of the environment’s qubits and theexistence of the external field. As it is displayed from Fig.(4a), due to the large numbers ofthe initial environment’s qubit, i,e. we set N = 13, the number of orthogonality increasesand consequently the time of orthogonality decreases. In Fig.(4b), we increase the strengthof the Dzyaloshinsky-Moriya interaction, where DM = 0 .
3. It is clear that, the behavior of S or is similar to that displayed in Fig.(4a), but with larger numbers of orthogonality andshorter time of orthogonality. Moreover, by switching the DM interaction, the effectivenessof the interactions types will change. However, the impact of Ising model on the orthogo-nality time is the shortest one, while the longest one is depicted for the XX chain model.Therefore, to decrease the orthogonality time, and consequently, the computational speed,one has to switch on the Dzyaloshinsky-Moriya interaction in the presences of any strengthof the external field with large number of environment’s qubits.Fig.(5), shows the behavior of the orthogonality S or as contour plot, where it is assumedthat the initial environment consists of small number of qubits, N = 3 ,
9. It is clear that, atsmall number ( N = 3, Fig. (5a)), the first orthogonality for the three types of interactionsappears at t (cid:39)
30. However as γ increases, namely the interaction turns into anisotropic(0 < γ < a ) ( b )Figure 5: The behavior of S or ( γ ) for a system is initially prepared in a PES ( p = 0 .
5) with , D = 0 .
5, g=0.05 λ = 1, where (a) N = 3, (b) N = 9.orthogonal time is displayed for the Ising model, namely at γ = 1. As one increases the qubitsof the environment ( N = 9, Fig. (5b)), the number of orthogonality increases. Moreover,the time delay of orthogonality is displayed as γ increases. The initial system is initial prepared in a MES
Finally, we assume that the system is initially prepared in a maximum entangled state ofBell type. | φ + (cid:105) = √ (cid:0) | (cid:105) + | (cid:105) (cid:1) . The eigenvectors for this state is given by; µ (0) = { , , , } , µ (0) = { , , , } ,µ (0) = (cid:26) √ , , , √ (cid:27) , µ (0) = (cid:26) − √ , , , √ (cid:27) (19)The time evolution of this state is given by, ρ mAB ( t ) = 12 (cid:26) | (cid:105) (cid:104) | + | (cid:105) (cid:104) | + S ( t ) | (cid:105) (cid:104) | + H.C. (cid:27) . The eigenvectors of the final state are defined as, ψ ( t ) = { , , , } , ψ ( t ) = { , , , } ψ ( t ) = (cid:40)(cid:115) S ( t )2 S ( t ) , , , √ (cid:41) , ψ ( t ) = (cid:40) − (cid:115) S ( t )2 S ( t ) , , , √ (cid:41) (20)The effect of the interaction’s parameters on the behavior of the orthogonality speed fora system is initially prepared in a maximum entangled state, is similar to that displayedfor seperabpe/partial entangled states. However, it is important to clarify our results byexamining the behavior of S or , where we assume that the system is initially prepared in thestate ρ (0) = (cid:12)(cid:12) φ + (cid:11)(cid:10) φ + (cid:12)(cid:12) , φ + = √ ( (cid:12)(cid:12) (cid:11) + (cid:12)(cid:12) (cid:11) ). In Fig.(6a), we assume that the environmentconsists of 3 qubits and the same parameters are fixed as in Fig. (5)). The behavior of S or
8s similar to that displayed in Fig. (5a)), where the initial system is initially prepared in apartial entangled state with p = 0 .
5. However, the numbers of orthogonality that displayedin Fig.(6a) is larger than that displayed in Fig. (5a)), namely the time of orthogonality issmaller, where the first orthogonality is displayed at t (cid:39)
15. In Fig.(6b), we increase thequbits of the environment, ( N = 9). The behavior of S or shows the number of orthogonalityincreases, and consequently the time of orthogonality decreases. Moreover, the delay oforthogonality time increases as γ increases.From Fig.(5) and (6) , one may conclude that preparing the initial two-qubit system in amaximum entangled state, (MES) increases the number of orthogonality and as a result thespeed of transporting information and the computations increases. γ ( a ) γ ( b )Figure 6: The behavior of S or ( γ ) for a system is initially prepared in a MES with , D = 0 . λ = 1, where (a) N = 3, (b) N = 9.Fig.(7) shows the behavior of the orthogonality S or ( DM ) as a contour plot, where weconsider that the environment system consists of 7 qubits. The small numbers of the or-thogonality are displayed for the XX chain model, namely as one increases γ , the number oforthogonality decreases, and consequently the time of orthogonality increases. However, aswe have discussed above, these results will be changed if the initial environment’s qubits arelarge. On the other hand, the largest tie of orthogonality is displayed for the Ising interactionmodel.In Fig.(8), we investigate the behavior of S or ( DM ) for the different interaction types,where we assume that the number of the environment’s qubits , N = 7. It is clear that,the largest number orthogonality are displayed at γ = 0, namely the interaction is describedby the XX chain model. However the smallest numbers are predicted for the Ising model,where γ = 1. Moreover, the shortest time of orthogonality and consequently the speed oftransforming the information is shown for the XX chain model. In this manuscript, we discuss the orthogonality speed of a two qubit system interactinglocally with different types of spin chain; XX , Ising and the anistropic, in the presence of9 M ( a ) DM ( b ) DM ( c )Figure 7: The behavior of S or ( DM ) for a system is initially prepared in a MES with ,g=0.1 λ = 1, and N = 7, where (a) γ = 0, (b) γ = 0 . γ = 1. S or t Figure 8: The effect of the DM interaction for MES same as Fig.(2a) but we set g = 0 . XX spin chain, while the largesttime is depicted when the Ising interaction model is applied. The effect of the external fieldis examined in the presence of all the three different types of spin interaction. It is shownthat in the absences of the external field, the numbers of orthogonality are small during aperiod of interaction time. However, as one increases the strength of the external field theorthogonality’s numbers increase. Moreover, the difference between the orthogonality timethat displayed for the three different spin interactions, may be minimized by increasing thestrength of the external field. Furthermore, the behavior of the orthogonality is examinedwhen the interaction of Dzyaloshinsky-Moriya (DM) is switched on. As it is shown from10he listed figures, the numbers of orthogonality increase as one increases the strength of DMinteraction. However, this effect is clearly displayed if one increases the numbers of the sites’qubits of the environment.Finally, the effect of the initial state settings on the orthogonality speed is investigated,where we consider that the initial system either prepared in a maximum, partial or separablestate. The interaction’s Hamilations parameters have the same effect for all the initial statesettings. Moreover, for small values of the external field strength, the maximum entangledstate is robust against the decoherence induced by the interaction of the qubit system withthe qubit’s environment. Therefore, the orthogonality speed is larger than that displayedfor the separable/partial entangled states. However, as the strength of the external fieldincreases, the possibility that the maximum entangled state losses its coherence increases. In conclusion: the orthogonality time is examined for different initial state settings inter-acts locally with different types of spin interaction models. It is shown that, the shortest timeof orthogonality is displayed for the XX chain model, while the largest time is shown for theIsing model. The external field increases the numbers of orthogonality, while Dzyaloshinsky-Moriya interaction decreases the time of orthogonality. The initial state settings togetherwith the external field have a significant effect on decreasing/increasing the time of orthog-onality. References [1] Margolus N., Levitin Lev B, 1998 The maximum speed of dynamical evolution
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