Transfer from Arbitrary Pure State to Target Mixed State for Quantum Systems

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1 reprints o the 8th FAC World Congress ilano (taly August 8 - epteber, ranser ro Arbitrary ure tate to arget ixed tate or Quantu ystes J. WEN,. CONG Departent o Autoation, University o cience and echnology o China, eei, Anhui,.R.China, (e-ail:scong@ustc.edu.cn Abstract: Considering two-level quantu systes, this paper proposes a control strategy that drives an arbitrary pure state to target ixed state via three-step controls. he irst step is the preparation o eigenstate: the state o syste controlled is driven to eigenstate ro an arbitrary pure state. he second one is to drive the eigenstate prepared to the ixed state in which not all o the o-diagonal eleents in the density atrices are eros. n this step, one needs to use an auxiliary syste and achieves the transer through the interaction between controlled syste and auxiliary syste. the target state is the ixed state in which all o the o-diagonal eleents in the density atrices are eros, then one needs to peror the third step: to drive the ixed state obtained in the second step to the target ixed state. Nuerical siulation experients and results analysis are given. he control strategy proposed in this paper is useul in the application o quantu coputation. Keywords: quantu systes, pure state, ixed state, interaction, yapunov unction, density atrix. NRODUCON n quantu coputation, the restriction to unitary gates and pure state is unnecessary, and or the odel o quantu circuits in which the state is ixed state, it can deal orally with several central issues: easureents in the iddle o the coputation, decoherence and noise, using probabilistic subroutines and so orth while it's diicult or ipossible or pure state (Aharonov et al., 998. Generaliations which are obtained ro the prograing approach or a odel o quantu coputation based on ixed state are useul to analyse the propagation o errors in a quantu coputation involving ixed state (Zuliani, 7. ixed state plays an iportant role in quantu coputation, so it is worth to study the preparation o ixed state. n general, there are two situations which can produce ixed state. One is when a quantu syste has interaction with environent, the dissipative phenoenon occurs due to the existence o the environent. he evolution o density atrix is not unitary any ore in such a case. Another situation is a large nuber o particles which are in dierent pure states are statistical incoherent ixture (Zhang and Cong, 9. We only consider the irst situation in this paper. When the initial state is ixed state, it is easy to get the target ixed state and the control strategy has been given (Zhang and Cong, 9. owever, pure state is the state which can be prepared easilier. When the initial state is pure state, or the irst situation, the act is that noise taes pure state only into a liited range o ixed state, rather than to the whole space o ixed state (enderson et al.,. hus it is necessary to see an eective control strategy to drive arbitrary pure state to the target ixed state. Bans, esin and ussind (B suggested a iouville equation or the density atrix ρ, its generic or is i ρ = [, ρ] i hn( QQn ρ ρqqn Qn ρq ( t n, where Qn is any eritian operator and hn is c-nuber eritian atrix. A suicient condition which can ensure the positivity o ρ is that the atrix h is positive. ρ is a eritian atrix or quantu systes, so its eigenvalues are non-negative and tr ( ρ =. Nevertheless, ( can not preserve the value o tr ( ρ. When ρ represents a pure state, the value o tr ( ρ is, and the value is less than when ρ represents a ixed state, ro which one can see that pure state can indeed evolve to ixed state(reni, 996. his paper proposes a control strategy that drives an arbitrary pure state to target ixed state or two-level systes via three-step controls. he irst step is the preparation o eigenstate: drives an arbitrary pure state to eigenstate through the control o pure state quantu systes. he second step is to drive the eigenstate prepared to the ixed state in which not all o the o-diagonal eleents in the density atrices are eros. n this step, we use an auxiliary syste that is put in interaction with the controlled syste, and achieve the transer through the interaction o two systes. the target state is the ixed state in which all o the o-diagonal eleents in the density atrices are eros, then one needs to drive the ixed state obtained in the second step to the target state through the control o ixed state quantu systes.. RANFER FRO AN ARBRARY URE AE O AN EGENAE Considering a closed-loop quantu syste, its atheatical odel is described as the ollowing chrödinger equation i =, = ( c Copyright by the nternational Federation o Autoatic Control (FAC 4638

2 reprints o the 8th FAC World Congress ilano (taly August 8 - epteber, where is the inner ailton, t is the interaction ( ( ( ailton and t = u t. Both and are c = linear eritian operators. u (t is a real-valued control unction, and the lanc constant is set =. First, the ollowing unitary transoration is perored = U (3 iλt iλt iλnt where U = diag( e, e,, e is a unitary atrix and λ is the eigenvalue o. ubstituting (3 into ( one i has i = ( (4 u Λ = = diag ( λ, λ,, λn c where Λ, = U U and = U U. ( and (4 describe the sae syste(boscain et al.,, so we design a control law based on the syste odel described by (4. he design o control ethods is plentiul, in which the yapunov ethod is adapted to the tie-varying syste and nonlinear syste, and the control can ensure the stability o controlled syste. o we use the yapunov ethod to design all controls in this paper. he idea o yapunov control is to choose a suitable yapunov unction V and then try to ind a control so that ensures that V is onotonically decreasing along any dynaical evolution o control syste. ere, the yapunov unction selected V is taen as (Zhang and Cong, 9 V = (5 where is the actual state o the syste, and is the target state and = cn n. c n is the probability o = n. n t can be calculated that the irst-order tie derivative o V is given by V = = ( ( Λ = u ( t Due to = cn n and n = λn n, we can get n ( ( Λ =. o the expression o V in (6 can be written as V = u ( (7 = Observing (7, or the sae o ensuring V, we choose the control law as u u = K (8 ( (6 where K >. V decreases gradually under the action o u, and reaches ero when =. At the oent, V =, u =, the syste state would be stabilied at. Just choosing to be an eigenstate o the syste, arbitrary pure state can be driven to through using the control law (8, and the state reains in the eigenstate. 3. RANFER FRO AN EGENAE O XED AE BY NERACON CONRO 3. Establishent o atheatical odel n quantu control, the eect o control to the dynaics o controlled syste is the sae as to peror a suitable unitary transoration (Cong, 6. owever, the unitary transoration can not change the purity o state; it is ipossible to drive pure state to ixed state or single particle by using a unitary transoration. hus we need an initially uncorrelated auxiliary syste that is put in interaction with the controlled syste, and the transer o states is achieved through the interaction o the two systes(roano and D Alessandro, 6. We have driven an arbitrary pure state to the eigenstate o syste through the control used in section. An eigenstate can be expressed as ρ = with density atrix. he initial state o auxiliary syste is ixed state ρ = ρ ( u, and the ability to odiy the state o controlled syste depends on the correlations between and, which is generated by the interaction. he dynaics o is given by ( t u r ρ ρ, = (9 with t [ ρ ( ρ ( u ] ρ = γ ( where denotes the coposite syste, and γ t is an operator. is a closed syste, then γ t is given by t i t it [ ] = e e γ ( According to (9 and in consideration o ( and (, when syste and copose a closed coposite syste, the dynaics o is given by ( ( t u = r U ρ ρ ( u U ρ, ( i t where U e = is a unitary propagator, is the total ailton and given by = (3 where and are the ree ailton o and, respectively. is the interaction ailton o and. When syste and are both two-level, is given by considering a particular dynaical odel(roano, 7: = ω σ, = ω σ, = gσ σ (4 x x 4639

3 reprints o the 8th FAC World Congress ilano (taly August 8 - epteber, ω where ω and are the eigen-requencies o and, and g is the coupling constant. σ and σ x are the auli atrices in syste, σ and σ x are the auli atrices in. We deine the purity o a state ρ is the von Neuann distance o the state ρ ro the axially ixed state /: ( ρ π = r ρ = r (5 the state ρ is represented by a Bloch vector ( s( t σ ρ = (6 hen the tie-dependent purity π ( t is given by ( t π = s (7 o the tas one transers a state ro pure initial state to ixed state becoes a decreasing process o π ( t. 3. Design o the control law Considering the atheatical odel o syste in (, the total ailton o contains two parts: one is local ailton = coposited o the ree ailton and, and another is the interaction ailton. We still use the yapunov ethod to design the control law u in this section. When is a closed syste, ollows the iouville equation (with = ρ ( t = i[ u( t, ρ ( t ] (8 where =, and the dynaics o is ( t = U ( t ρ ρ ( u U ( t ρ (9 i where t U = e is a unitary propagator. he initial state o and is the eigenstate and ixed state, respectively, and and is initially uncorrelated. ρ and ρ represents the inal state o and, respectively, so the inal state o is ρ where = ρ ρ ( ρ is a ixed state. he ilbert chidt distance ρ ρ is chosen as the yapunov unction (Kuang and Cong, 8 V ( ρ ( t, ρ ρ ( t ρ = r ρ ( t [( ] ρ = ( he irst-order tie derivative o V is ( ρ ( t ρ ρ ( t V = r ( ubstituting (8 into (, one has V = r = r = r = r ( ρ ρ ( i[ u( t, ρ ( t ] ( ρ ( t ρ ( i[, ρ ( t ] i[ u( t, ρ ( t ] ( iρ [, ρ ( t ] iρ [ u( t, ρ ( t ] i, ρ ρ t u t r ρ [ i, ρ t ] ( ([ ] ( ( ( ( (3 when [, ρ ] = is satisied, the irst ite in the right side o (3 is ero. n order or V to be nonnegative, the second ite in the right side o (3 should be nonnegative. o assuing >, we choose the ollowing control ( r ρ [ i, ρ ] u = (4 and (3 becoes ( t, > V = u (5 he yapunov unction V is onotonically decreasing along any dynaical evolution because V. On the basis o the yapunov stability theore, the closed quantu syste is stable under the action o the control law (4, which drives the syste state ρ ( t to the inal state ρ, at the sae tie the inal state in evolves to, r ρ ( ρ r ( ρ ρ = ρ = (6 As long as the inal state o satisies, ρ =, the control (4 can drive an eigenstate to the ixed state in which not all o the o-diagonal eleents in the density atrices are eros or syste through the interaction between controlled syste and auxiliary syste. [ ] 4.CONRO DEGN FOR A XED AE QUANU YE he state o controlled syste has been driven ro an arbitrary pure state to the ixed state in which not all o the o-diagonal eleents in the density atrices are eros via the controls in section and section 3. When the target state is the ixed state in which all o the o-diagonal eleents in density atrices are eros, it is necessary to peror the urther control to the ixed state which is obtained in section 3, so that the state o syste can be driven to the target ixed state. his section will give the speciic design procedure o controller. he density atrix iouville equation ρ o controlled syste ollows i ρ ( t = [ ( t, ρ ( t ], ( t = ( (7 t t = where is the inner ailton, is the interaction ailton generated by the interaction o external controls and the syste, is the external control. he collection o bounded linear operators in ilbert space ors own ilbert space which is called iouville space. Every bounded linear operator A in ilbert space 464

4 reprints o the 8th FAC World Congress ilano (taly August 8 - epteber, corresponds to a vector A in iouville space. Using ρ to express the density atrix ρ in iouville space, (7 can be written as(ohtsui and Fujiura, 989 i ρ ( t = ρ ( t, ( t = (8 t = he control law is designed based on yapunov stability theore in this section and the yapunov unction selected is the ollowing equation(zhang and Cong, 9 V ( ρ = ρ ρ ρ ρ (9 where ρ is actual state o, and ρ is target ixed state. is a positive deinite syetric atrix and satisies = (3 According to (8 and (3, the irst-order tie derivative o V ( ρ is given by V ( ρ = Re( ρ ρ ρ = ( ρ ρ ρ ( t ( ρ ρ ρ = = ( t ( ρ ρ ρ = (3 n order to ensure V, we choose the control law as = ( ρ ρ ρ, ( =,,, (3 r and (3 becoes [ ( ρ ρ ρ ] V ( ρ = r = ( = (.866,,. 5 to the target ixed state inal = (,,.7 (33 n such a way, one can realie to drive the state o to the ixed state in which all o the o-diagonal eleents in density atrices are eros under the action o control law (3. Note that the purity o initial state is equal to the purity o inal state when control law (3 is adopted. t eans that the inal state in section is not arbitrary when it is necessary to peror the third step. he purity o inal state selected in section should be equal to the purity o target state. 5.NUERCA UAON EXEREN AND REU ANAY his section will illustrate the eectiveness o the proposed control strategy through a speciic nuerical siulation. he siulation experient is to drive the state o controlled syste ro the initial superposition state, and the states o systes are all expressed in Bloch vector in siulation. he irst step: Drive controlled syste ro superposition state ( = (.866,,. 5 to the eigenstate ( = (,,. We can obtain the needed control law according to (8: u = K (. K is set to be in the experient. he control syste siulation results are shown in Fig., in which Fig.(a shows the evolution trajectory o syste and Fig.(b shows the value o control. Fro Fig. (a one can see that the state o is driven ro the superposition state ( = (.866,,.5 to the eigenstate ( = (,, under the action o control designed. As can be seen ro Fig.(b, the value o control stays at ero once the eigenstate is reached and the state can reain in the reached eigenstate stably and continuously. o the state o reaches the eigenstate at the end o control. Fig. (a rajectory o syste Fig. (b Value o Control u Fig.. iulation results o the irst-step control or syste he second step: Drive the eigenstate ( = (,, ixed state = (.7,, to the in which not all o the o-diagonal eleents in the density atrices are eros. he initial state o the auxiliary syste is set as ( = (.7,,, and its inal state is = (,,. he eigen-requencies ω and ω are both set as. = diag(, and = diag(, is the ailton o and respectively,the ailton o are = δ δ and = δ x δ x. he control law in this step is designed by (4: u( t = r( ρ [ i, ρ ], in which is selected as in the experient. he second step control siulation results are shown in Fig., in which Fig.(a shows the evolution trajectory o, Fig. (b shows the purity curve o, Fig.(c shows the value o control and Fig.(d shows the projection o the evolution trajectory in the tie period 5- in the X-Y plane. t can be seen ro Fig.(a that the evolution trajectory enter the inside ro 464

5 reprints o the 8th FAC World Congress ilano (taly August 8 - epteber, North ole o the Bloch ball under the control and aes circular otion around the Z axis ater the interaction. According to the deinition o purity, the purity is when it is pure state and the purity is less than when it is ixed state. Fig. (b shows the purity o decreases ro and reains in.7, so the controlled syste is driven to the ixed state ro the eigenstate. Fro Fig.(c one can see that the value o the control tends to ero when tie is about 5 and it eans the end o the interaction o and. According to Fig.(a and Fig.(d, it can be concluded that is driven to the ixed state in which not all o the o-diagonal eleents in the density atrices are eros ro the eigenstate under the control. n order to drive the controlled syste to the target ixed state in which all o the o-diagonal eleents are eros, it is necessary to peror the control o ixed state quantu syste. his purpose can be achieved by using the control law (3. he initial state o the third step is = (.7,, which is the inal state o the second step and the inal state is the target ixed state inal = (,,. 7. he purities o and inal are all equal to.7 and satisy the liiting condition o control law (3. he ailton o controlled syste is = diag(,, = and the paraeter in the control law is = diag(6.5,,,. he third step control siulation results are shown in Fig.3, in which Fig.3(a shows the evolution trajectory, Fig.3(b shows the purity curve, Fig.3(c shows the value o control and Fig.3(d shows the curves o coordinates x and y in Bloch vector which represents the state o syste. (a rajectory o syste (b urity o syste (c Value o Control u (d rojection o trajectory or 5 t in X-Y plane Fig.. iulation o the second-step control or syste One can see that syste is driven to the target ixed state under the control law ro Fig.3(a. t can be seen ro Fig.3(b that the purity o state has been ept at.7 in the control process, thus the control law (3 don't change the purity. he control law (3 is related to the choice o paraeter, and selecting dierent paraeters will get dierent curves and eects o the controls, so it s iportant to select a suitable paraeter in control. is selected as = diag(6.5,,, in our experient, and ro Fig.3(c one can see that the control value in tends to ero inally which indicates that is chosen suitably. When coordinates x and y in Bloch vector are eros, all o the o-diagonal eleents in the density atrices are eros. t can be seen ro Fig.3(d that: the value o coordinates x and y tend to ero, so the inal state is the ixed state in which all o the o-diagonal eleents in the density atrices are eros. hus the state o is driven to target ixed state ro the initial superposition state via three-step controls. When the initial state o the controlled syste is an eigenstate, the irst step control is not necessary and the second step control can be perored directly. When the target state is the ixed state in which not all o the o-diagonal eleents in the density atrices are eros, the third step control is not necessary and the transer can be achieved through the irst two steps. n addition, can be driven to the target ixed state in which not all o the o-diagonal eleents are eros only when the initial state o is the sae as the inal state o in the second step. 6. CONCUON 464

6 reprints o the 8th FAC World Congress ilano (taly August 8 - epteber, his paper studied the transer ro an arbitrary pure state to the target ixed state or two-level quantu systes. When the initial state is superposition state, the state is driven to the eigenstate through the control o pure state quantu syste irst. he eect o control to the dynaics o syste controlled is the sae as to peror a suitable unitary transoration. Unitary transoration do not change the purity o the state, so it is ipossible to drive pure state to ixed state or single particle. hus the auxiliary syste is necessary and coponents the coposite syste with controlled syste. he control law is designed to control the interaction by yapunov ethod. he state o is driven to the inal state, at the sae tie the state o syste evolves to the target ixed state ro the eigenstate through the interaction o and. the target state is the ixed state in which all o the o-diagonal eleents are eros, one needs to drive the ixed state to the target ixed state through the control o ixed state quantu systes. hus we achieve the control purposes via three-step controls at ost. (arajectory o syste ACKNOWEDGEEN his wor was supported in part by the National Key Basic Research rogra under Grants No. 9CB996, the National cience Foundation o China under Grant No he Doctoral Fund o inistry o Education o China under Grant No (b urity o syste (c Value o Control u (d Value o coordinates x and y in Bloch vector Fig. 3. iulation o the third-step control or syste REFERENCE AARONOV, D., KAEV, A., and NAN, N. (998. Quantu Circuits with ixed tates. n OC 98: roceedings o the thirtieth annual AC syposiu on heory o coputing, -3. BOCAN, U., CABRON,. and GAUER, J.. (. On the K proble or a three-level quantu syste. Journal o Dynaical and Control ystes, 8(4, CONG,. (6. ntroduction to quantu echanical syste control.cience ress,beijing. ENDERON,., NDEN, N. and OECU.. (. Are All Noisy Quantu tates Obtained ro ure Ones. hysical Review etters, 87(3, 379. KUANG,., CONG,. (8. yapunov control ethods o closed quantu systes. Autoatica, 44(, OUK, Y. and FUJURA, Y. (989. Bath-induced vibronic coherence transer eects on etosecond tieresolved resonant light scattering spectra ro olecules. Journal o Cheical hysics, 9(7, REZNK, B. (996. Unitary Evolution Between ure and ixed tates. hysical Review etters, 76(8, ROANO, R. (7. Resonant puriication o ixed states or closed and open quantu systes. hysical Review A, 75(, 43. ROANO, R., D AEANDRO, D. (6. ncoherent control and entangleent or two diensional coupled systes. hysical Review A, 73(, 33. ZANG, Y. Y. (9. Optial Control o Quantu yste Based on yapunov tability heore. University o cience and echnology o China. ZUAN,. (7. Quantu rograing With ixed tates. Electronic Notes in heoretical Coputer cience, 7,