Quantum Lyapunov Control Based on the Average Value of an Imaginary Mechanical Quantity

Transcription

1 Quantum Lyapunov Conto Based on the Aveage Vaue o an Imaginay Mechanica Quantity Shuang Cong, Fangang Meng, and Sen Kuang Abstact The convegence o cosed quantum systems in the degeneate cases to the desied taget state by using the quantum Lyapunov conto based on the aveage vaue o an imaginay mechanica quantity is studied. On the basis o the existing methods which can ony ensue the singe-conto Hamitonian systems convege towad a set, we design the conto aws to mae the muti-conto Hamitonian systems convege to the desied taget state. The convegence o the conto system is poved, and the convegence to the desied taget state is anayzed. How to mae these conditions o convegence to the taget state to be satisied is poved o anayzed. Finay, numeica simuations o a thee eve system in the degenate case tanseing om an initia eigenstate to a taget supeposition state ae studied to veiy the eectiveness o the poposed conto method. Index Tems quantum Lyapunov conto method, convegence, degeneate I. INTRODUCTION In the ast 3 yeas, the conto theoy o quantum systems has deveoped apidy, and it was widey used in many aeas. One poposed technique is the Lyapunov conto []-[8]. Thee ae mainy thee Lyapunov unctions to be seected: the Lyapunov unction based on the state distance, the state eo and the aveage vaue o an imaginay mechanica quantity. The so-caed imaginay mechanica quantity means that it is a inea Hemitian opeato to be designed and maybe not a physicay meaningu obsevabe such as coodinate and enegy. In ecent yeas, eseach esuts on the convegence o the conto system by using the Lyapunov conto method based on the aveage vaue o an imaginay mechanica quantity ae as oows: The conto system is asymptoticay stabe at the taget state, i i The intena Hamitonian is stongy egua, i.e., the tansition enegies between two dieent eves ae ceay identiied; ii The conto Hamitonians ae u connected, i.e., any two eves ae diecty couped [], [4]-[8]. Howeve, many pactica systems do not satisy these conditions which ae caed in the degeneate cases. Fo these cases, Zhao et a. utiized an impicit Lyapunov conto to sove the pobem o convegence o the singe conto Hamitonian systems govened by the Schödinge equation [8]. Howeve, thei poposed methods ony poved that the singe conto Hamitonian systems wi convege towad a set, but can not ensue be asymptoticay stabe at the desied taget state. The aim o this pape is to mae the muti-conto Hamitonian systems in the degeneate cases convege to an abitay taget state om an abitay initia state. Ou main contibutions ae as oows: i The pobem o convegence to any taget

2 eigenstate o the Schödinge equation o any taget state which commutes with the intena Hamitonian o the quantum Liouvie equation is soved by adding a estiction on the Lyapunov unction and designing the impicit unction petubations. ii The pobem o convegence to the taget supeposition state and the taget state which does not commute with the intena Hamitonian is soved in most cases by intoducing a seies o constant distubances into the conto aws. iii How to mae the conditions o convegence to the taget state to be satisied ae anayzed o poved. II. BILINEAR SCHRÖDINGER EQUATION CASE Conside the N-eve cosed quantum system govened by the oowing biinea Schödinge equation: = ( i ψ( t ( H H v ( t ψ( t = + whee ψ ( t is the quantum state vecto, H is the intena Hamitonian, H,(,, = ae conto Hamitonians, and v (,( t =,, ae conto aws. Two convegence conditions o Hamitonians in [], [4] and [5] ae i The intena Hamitonian is stongy egua, i.e., ω ω,(, (,,,,, {,2,, } ij m i j m i j m N, whee ωm = λ λm, λ is the -th eigenvaue o H coesponding to the eigenstate φ φ ; ii Fo any i j φ, thee exists at east a such that φ H φ. In ode to i j sove the pobem o convegence o the conto system in the degeneate cases to the desied taget state, a seies o petubations ( t, which ae impicit unctions o state ψ ( t and time t, ae intoduced into the conto aws, then ( becomes whee ( t v ( t u (,( t,, = (2 i ψ( t = ( H + H ( ( t + v ( t ψ( t + = = ae the tota conto aws. Ou conto tas is to mae the conto system govened by (2 tanse om an abitay initia pue state ψ to an abitay taget pue state ψ by designing appopiate conto aws u ( t = ( t + v (,( t =,,. In ode to compete this conto tas, isty, the petubations ( and v ( t ae designed. Secondy, the t convegence o the conto system is poved. Thidy, how to mae convegence conditions to be satisied is anayzed.

3 At ist, et us design (,( =,,. Ate intoducing petubations (, = H H ( t t + can be egaded as the new intena Hamitonian o the conto system. In ode to aciitate undestanding the basic idea o this method, we descibe the system in the eigenbasis o H + ( H = t : i ψ ( t = (( H + H ( t + H v ( t ψ( t (3 = = whee ψ = U ψ, H U H U H U H U, = =, U = ( φ,,,,, φ N,,, t, φn,,,, n N ae eigenstates o H + ( H t coesponding to the eigenvaues λn,,,. Accodingy, ψ wi become ψ = U ψ which is aso a unctiona o ( t. The design idea o ( is as oows: ( ae designed to satisy i t = ω { } hods, ωm,,, i, j,,,(, m ( i, j, i, j,, m,2,, N t ω = λ λ ; ii j, o =,,, thee exists at east a m,,,,,,, m,,, ( H, whee ( H is the (j,-th eement o H, thus the conto system can j j convege towad ψ by designing appopiate conto aws u ( t ( t v (,( t,, = + =, thus the conto system can convege towad ψ by designing appopiate conto aws u ( t = ( t + v (,( t =,, ; 2 at the same time, (,( t,, = themseves need convege to zeo, and thei convegent speed must be sowe than that o the conto system to ψ to mae ( t tae eect; 3 ( ψ = must hod to mae the conto system be asymptoticay stabe at the taget state. Reeences [], [4] and [5] poposed the estiction V ( ψ V ( ψ V ( ψ < < to othe mae the system in the non-degeneate cases convege to the taget state ψ om the initia state ψ, whee ψ othe epesents any othe state in the invaiant set in

4 { ψ ψ } E V( = = except the taget state. Howeve, in act it is diicut to design the imaginay mechanica quantity to mae this estiction on the Lyapunov unction be satisied o any initia state and any taget state. Fo the degeneate cases, in ode to mae the the system convege to the taget state, we choose a simpe estiction: V ( ψ V ( ψ < which can be satisied o any initia state and any taget state by designing the imaginay mechanica quantity. In ode to ensue the the system convege to the taget state by adding this etiction, we design a the petubations ( = hods o =,, ony at ψ, t othe i.e., ( ψ =,( =,,, and 2 o ψ ψ, thee exists at east one such that ( ψ. Accoding to the anaysis mentioned above, et us design the speciic =. Since the evoution o the system s state eies on the continuous (,( t,, decease o the Lyapunov unction Vt ( in the Lyapunov conto, we design ( be a monotonicay inceasing unctiona o Vt ( as: ( ψ = C θ ( V( ψ V( ψ (4 whee C, and o =,,, thee exists at east a C >. And θ ( satisies ( θ =, ( s θ > and θ ( s > o evey s >. In this note, the speciic Lyapunov unction based on the aveage vaue o an imaginay mechanica quantity is seected as: V( ψ ψ P ψ,, t = (5 whee P,, = ( ( t,, ( t is a unctiona o ( t and positive deinite. The existence o ( can be estabished by Lemma. Lemma : I t C =, ( ψ =. Ese i C >, θ C ( R + ;[, ], =,, ( is a positive constant satisy θ ( =, θ ( s > and θ ( s > o evey s > θ < (2 CC, C = C 2N +, C = max { P },,, [, ] ψ S, thee is a unique C ( [, ] satisying, and, then o evey

5 ( ψ = C θ ( ψ P ψ ψ P ψ (6,,,, Poo: Assume P,, ae anaytic unctions o the paametes (,,(,, ψ =. P ae bounded on,,,, thus C <. Deine F (,,, ψ = C θ ( ψ P ψ ψ P ψ,,,, whee F(,,, ψ ae egua. Fo a ixed ψ, F( ( ψ,, ( ψ, ψ = hods. Some deductions show that F(,,, ψ hods. Thus accoding to the impicit unction Theoem [8], Lemma is poved. Rema : Fo the sae o simpicity, set (= o some, and othe ( ae equa, denoted by ( t, i.e., set t t (= t (t= θ( ψ P ψ ψ P ψ, =,, ; m ( t =,,, (,, m m (7 whee θ( = θ ( = = θ ( and P ae unctionas o ( t m. Then et us design v ( t to mae Vt ( hods. Setting one can obtain the time deivative o the seected Lyapunov unction as: V iv t ψ H P ψ θ ψ P ψ = ( ( ( P ( P [ P, H + H ( t] = = ( [, ] (+ /( θ ( ψ ψ ψ ψ m n= n, Accoding to Lemma, one can obtain ( ( P ( P ( P (+ θ ψ ψ /( θ ( ψ ψ ψ ψ > hods. In ode to ensue Vt (, v (,( t,, = ae designed as: ( ψ ψ v( t = K i H, P ],( =,, (8 (9 whee K is a constant and K >, and y = ( x,( =,2,, ae monotonic inceasing unctions though the coodinate oigin o the pane x y. Based on LaSae s invaiance pincipe [9], the convegence o the conto system govened by (2 can be obtained as oows:

6 Theoem : Conside the conto system govened by (2 with conto ieds u ( t ( t v (,( t,, = + =, whee ( deined by Lemma and (7, and v ( t deined by (9. I the conto system satisies: i ω ω m,, i, j, { N} i, j,, m,2,,, ω λ,,, λ, m m t,( m, ( i, j, =, whee λ, is the -th eigenvaue o + m n= n coesponding to the eigenstate, H H ( t { N}, thee exits at east one such that ( i, j,2,, (,m-th eement o H U H U iv ( P ( P, m, whee ( mm =, U ( φ,,, φn, φ ; ii Fo any i j H, whee ( m = ; iii m, H is the m n= n ; [ P, H + H ( t] = P is the (,-th eement o U PU, then any tajectoy wi convege towad E i = ψ e θ φ ; θ R, {,, N } Poo: t, ( ψt Without oss o geneaity, assume that o t t t R,(, V =. is satisied. By (8 and (9, one obtains ( V = ψ [ H, P ] ψ = v ( t = As V =, is a constant, denoted by. The state ψ ( t can be witten as N φ =, =. Then ψ ( t can be witten as ( ( H ψ t = c t U φ,. ψ( t c ( t Substituting the soution o (3 with ψ [ H, P ] ψ ψ [ H, P ] ψ = =, gives By conditions i-ii and iv, one can have N = = and ( ( ( ( ( m N iω ( t t,, m e P P c t c t H m, = mm m v t = into ( ( = ( m = { } (2 c ( t c ( t,(, m,, N which impies that thee is at most one c ( t ( {,, N} which is nonzeo. Theoem is poved. Theoem guaantees the conto system conveges to the set E, howeve, it can not guaantee the conto system conveges to the taget state. Fom Theoem, we can see that i the taget state ψ is an eigenstate, ψ is contained in E because

7 o ( ψ =. In ode to mae the system convege to the taget state, on the one hand, as V, we design P to mae V ( ψ V( ψothe < (3 hod, whee ψ othe epesents any othe state in the set E except the taget state. On the othe hand, because / V >, V, hods, we set = α,( < α << when v ( t, ( t = = hods o some time to mae the state tajectoy evove but not stay in E unti i ψ e θ, which is the equivaent state o taget state ψ, is eached. Fom the above anaysis, we can see that i the conto system satisies the conditions i-iv in Theoem and Eq.(3, and at the same time set = α,( < α << when v ( t =, ( t = hods o some time, the conto system can convege to the taget eigenstate om an abitay initia pue state. Next we anayze how to mae these conditions be satisied in detai. Conditions i and ii in Theoem ae associated with H, H,( =,, and ( t. By designing appopiate ( t, these two conditions can be satisied in most cases. Condition iii means that P and + have the same eigenstates. We design the m n= H Hn ( t eigenvaues o P be constant, denoted by P P2,,, PN, and design P as N P j φ j j, φ = j, P = (4 then condition iii can be satisied. I design P P ( j;, j N to mae condition iv hod. Then et us anayze how to mae (3 hod. The eseach esut is given by the oowing Theoem 2. Theoem 2: I one designs Pi > P,( i =,, N, Pi P, then V ( ψ V ( ψothe j < hods, whee P is the eigenvaue o ( P ψ coesponding to ψ. Poo: i Set ψ s ( e θ φ, =. Accoding to Poposition in [8], i one designs = (, then V ( ψ V ( ψs P > P, i =,, N, P P i i < hods. Because o

8 / V >, V, >, V ( ψ ( s V ψothe < hods. Thus V ( ψ V ( ψothe < hods. Theeom 2 is poved. Rema 2: Accoding to the above anaysis and Theoem 2, the design pincipe o the imaginay mechanica quantity is Pi > P,( i =,, N, Pi P and ( P P j. In ode to sove the pobem o convegence to the taget state being a supeposition state, a seies o distubances η whose vaues ae constant ae intoduced into the conto aws. Thus the mechanica equation (2 wi become = (5 i ψ( t = ( H + H ( η + ( t + v ( t ψ( t j Ou basic idea is to design η to mae the taget state ψ be an eigenstate o = H H H η = +. H can be viewed as the new intena Hamitonian o the conto system. I the numbe o the conto Haimtonians is age enough, by designing appopiate η, + = = can be satisied in most ( H H η ψ λ ψ cases, whee λ is the eigenvaue o H coesponding to ψ. Then the design o conto aws and the convegence poo can oow the taget eigenstate cases. One can pove that the designed conto aws ae aso vaid and Theoem and Theoem 2 aso hods with changing H into H. III. QUANTUM LIOUVILLE EQUATION CASE Conside the N-eve cosed quantum system govened by the oowing quantum Liouvie equation: whee ( t v ( t η u (,( t,, = (6 iρ( t = [ H + H ( ( t + v ( t + η, ρ(] t + + = = ae the tota conto aws. The design ideas ae simia to that o Section II. The speciic Lyapunov unction is seected as: V( ρ t( P ρ = (7 whee P,, = ( ( t,, ( t is a unctiona o ( and positive deinite. Fo the sae o simpicity, design ( t as,, t (= t (t= θ( V( ρ V( ρ, =,, ; m ( t =,,, (,, m m (8

9 whee θ ( satisies ( θ =, ( s θ > and θ ( s > o evey s >. Accodingy, becomes P. The existence o ( t can be depicted by Lemma 2. P,, + Lemma 2: I θ C ( R ;[, ], =,, ( is a positive constant satisy ( θ ( s > and ( s θ > o evey s >, and θ (2 C { m } C max P, [, ] θ =, <, C =+ C, =, then o evey ρ, thee is a unique C ( [, ] satisying ( ρ θ( t( P ρ t( P ρ =. The idea o poo is simia to that o Lemma in Section II. Then et us design v ( t such that Vt ( hods. Setting, one can obtain m = n= n [ P, H + H η + H ( t ] = ( ( = (9 V = (+ θ t( P ρ (- θ t( P ( ρ ρ it([ P, H] ρ v( t By θ (2 C < in Lemma 2, t ( P ( t P In ode to ensue Vt ( (+ θ ( ρ (- θ ( ( ρ ρ, v (,( t,, > hods. = ae designed as: ( ρ v ( t K it([ P, H ],(,, = = (2 whee K is a constant and K >, and y = ( x,( =,2,, ae monotonic inceasing unctions which ae though the coodinate oigin o the pane x y. Based on LaSae s invaiance pincipe, the convegence o the conto system can be obtained as oows. Theoem 3: Conside the conto system depicted by (6 with conto aws ( t deined by Lemma 2 and Eq. (8, and v ( t deined by (2. I the conto system satisies: i ω ω { } m,, i, j,,(, m ( i, j, i, j,, m,2,, N, ω,, λ, λ, =, whee λ, is m m the -th eigenvaue o coesponding to the eigenstate + m η + = n= n H H H ( t φ, ; ii j, o =,,, thee exists at east a ( H, whee ( H is the (j,-th eement o = 2 2 with U2 ( φ,,, φn, H U H U j = ; iii j

10 ; iv Fo any j,(, j N m = n= n m [ P, H + H η + H ( t ] =,,, ( P ( P hods, whee ( jj conto system wi convege towad ( Poo: Without oss o geneaity, assume that o and (2, one can get P U PU, P is the (,-th eement o =, then the 2 2 { ρt ρ, (, t ρ t ij } E = U U = = t R t t,( t R, V = V = t([ P, H ] ρ= v ( t = is satisied. By (9 (2 As V =, ae constants, denoted by. The conto system in the eigenbasis o is + m η + = n= n H H H ( t (22 = = i ρ( t = [( H + H ( ( t + η + H v (, t ρ(] t whee ρ = U2 ρu2, H = U2 HU2, H = U2 HU2. Set ρt = ( ρ t. Substituting the soution o Eq. (22 with ( t deined by Eq. (8, =, and v ( t = into t([ P, H ] ρ = t([ P, H ] ρ=, gives m m i( H + Hη+ Hn ( t t i( H+ Hη+ Hn ( t t = n= = n= ρ t t( e e [ P, H ] = (23 P U P U whee =. By condition iii, one can obtain 2 2 Set ( ( ρ ( ( ( = (24 N ω n,,, H P P j= j j jj t j ( H ( P ( P ( ( ρ t ξ =, ( H ( ( ( P ( ( N N P NN ρ ( N ( N t N( N Λ= diag( ω,2,,, ωn, N,, 2 2 2,2,,3, NN,, M = ω ω ω ω ω ω N( N 2 N( N 2 N( N 2,2,,3, NN,, (25 Fo n =, 2, 4,, (24 eads M I ( ξ =. Fo n =, 3, 5,, (24 eads M ΛR ( ξ =. By

11 condition i, and M and Λ ae nonsingua ea matices, one can obtain ξ =. By condition ii and iv, one have ( ρ t j = hods. Theoem 3 is poved. I is age enough, by designing appopiate η, H H, η ρ + = = can be satisied in most cases. Then the taget state ρ is contained in E 2. Fo the specia case that the taget state commutes with the intena Hamitonian, i.e., ρ, H =, set η =. Some anayses show that E 2 has at most N! eements. In ode to mae the system convege to the taget state ρ, on the one hand, we design P to mae V ( ρ V ( ρothe < (26 hod, whee ρ othe epesents any othe state in the set E 2 except the taget state. On the othe hand, we design = α,( < α << when v ( t =, ( t = hods o some time to mae the state tajectoy evove but not stay in E 2 unti ρ is eached. Next we anayze how to mae these conditions be satisied. Fo satisaction o conditions i - iv, one can oow that o Section II. Then et us anayze how to mae (26 hod. Denoting the eigenstates o H + H η = as φi, η,( i {,, N}, = U3ρ U3 can be expessed by a diagona matix, whee U3 φ, η φn, η ρ eseach esut is as oows: Theoem 4: I ( ρ ( ρ,, design Pi Pj hods, whee ( < ii i j N, design P jj i P j > ii i j N, design P jj i Pj ; ese i ( ρ ( ρ,, ρ is the (i,i-th eement o ii ρ. = (,,. The = i j N, > ; i ( ρ ( ρ,, ii <, then V ( ρ < V ( ρ Poo: At ist, popositions and 2 ae poposed, then Theoem 4 ae poved accoding to these two popositions. { } Poposition : I ( ρ,( ρ,,( ρ { P P P },,, N 2 Poo: 22 aanged in a deceasing ode, design aanged in an inceasing ode, then V ( ρ V ( ρ NN < hods. othe jj othe

12 Denote ρs = U3ρsU 3 = diag ( ρ ( ρ ( ρ (,,,, whee ( τ 22( τ NN ( τ { ( τ, 22( τ,, NN( τ } is a pemutation o {, 22,, NN}. At ist, we pove V ( ρ V( ρs <. The Lyapunov unction V( ρ = t( P ρ o = can be witten as V( ρ N = j= = P ρ (27 j jj whee ρ is the (j,j-th eement o ρ = U ρu jj 3 3. Assume ( ρ ( ρ NN >, and < P < < PN. Fo N=2, ( ( V( ρ V( ρ = ( P P ( ρ ρ < (28 2 s whee the subscipt 2 in V ( ρ 2 and V ( ρ s 2 means N = 2. Poposition is tue. Assume Poposition is tue o N-. Then Whee Pj ( ( P = Fo N, N ( ( ( j τ N N s N j= j jj jj( τ = j( j jj( τ V( ρ V( ρ = P ( ρ ρ = ( P P ρ < (29 =. τ jj( τ ( τ ( V( ρ V( ρ ( P P ρ ( P P ρ (3 N N s N = j j( τ j + jj( τ N( = N NN ( τ By (29 and < P < P2 < < PN, one can get V( ρ V( ρ < (3 N s N Because o / V >, V, >, V ( ρ < V ( ρ hods. Thus Poposition is s poved. Poposition 2: I the diagona eements o the diagona taget state {( ρ,( ρ,,( ρ } 22 NN othe ae aanged in a non-deceasing ode with ( ρ = = ( ρ < ( ρ = = ( ρ L L < < ( ρ = = ( ρ, 2 2 2L2 2L2, whee i =, 2,, Q, and j =, 2,, L Q Q QLQ QLQ N, =, = N ij o QL i = ; j, 2,, L2 Q = o i = 2 ; ;, 2,, Q j = L o i Q =. Design { P P P } as,,, N 2 oows:, then V ( ρ V ( ρ P,, P > > P,, P > L Q QL Q < hods. othe

13 Poo: Obviousy, V ( ρ < V ( ρ hods o N=2. Assume that o N-, V ( ρ < V ( ρ s is tue. Then Eq. (29 hods. Fo N, i ( ρ < ( ρ, design ( N ( N NN s P,, P > > P, then (3 hods. I ( ρ = ( ρ = = ( ρ L N QQ Q2Q2 NN, then ( τ Q Q Q( L Q( L NN in (3. Design P Q Q,, P > > P,, L P Q QL Q, then (3 hods. Poposition 2 is poved. Obviousy, accoding to Poposition and Poposition 2, we can obtain Theoem 4. IV. NUMERICAL SIMULATIONS In ode to veiy the eectiveness o the poposed method, conside a thee-eve system with H and H as:.3 H =.5, H, H 2 = =.9 (32 Accoding to H and H, the system is in the degeneate case. Assume that the initia state is an eigenstate as ψ = ( T, and the taget state is a supeposition state as ( ψ = 23 3 T. Accoding to the design ideas in section II, the conto aw is designed as u( t = ( t + v( t + η, ( =,2. Design η = -.377, η2 = to mae the taget state ψ be an eigenstate o 2. ( ψ P ψ ψ P ψ 2 = H H H η = +. H And design = = =, v( t.2 ( i ψ H, P ] ψ =, 2 2 ( ψ ψ =, whee v ( t.2 i H, P ] 3 = P j j φ = j,. Accoding to Theoem 2 in P section II, set P =. and othe two eigenvaues o P ae.4 and.6. In the simuations, the time step size is set as. a.u., and the conto duation is 3 a.u.. The esuts o numeica simuations ae shown in Fig. and Fig.2. Fig. is the popuation evoution cuves o system, i 2,(,2,3 c i = is the popuation o eve i. Fig.2 shows the designed conto ieds. Accoding to numeica esuts, we can see that the poposed method is eective.

14 Fig.. Popuation Fig. 2. Conto ieds V. CONCLUSION In this pape, the Lyapunov conto based on the aveage vaue o an imaginay mechanica quantity has been impoved. By using the poposed method, the quantum Lyapunov conto can compete the state tanse tas om an abitay pue state to an abitay pue state o the Schödinge equation, and om an abitay initia state to an abitay taget state unitaiy equivaent to the initia state o the quantum Liouvie equation in most cases. REFERENCES [] S. Kuang, and S. Cong. Lyapunov conto methods o cosed quantum systems, Automatica, vo. 44, no., pp. 98-8, 28. [2] X. T. Wang, and S. Schime. Anaysis o Lyapunov method o conto o quantum states, IEEE Tansactions on Automatic conto, vo. 55, no., pp , 2. [3] M. Miahimi, P. Rouchon, and G. Tuinici. Lyapunov conto o biinea Schödinge equations, Automatica, vo. 4, pp , 25. [4] S. Givopouos, B. Bamieh. Lyapunov-based conto o quantum systems, IEEE Coneence on Decision and Conto, Maui, Hawaii USA, Decembe 23, pp [5] S. Kuang, S. Cong, and Y. S. Lou. Popuation conto o quantum states based on invaiant subsets unde a diagona Lyapunov unction. IEEE Coneence on Decision and Conto and 28 th Chinese Conto Coneence, Shanghai, P.R. China, Decembe 6-8, 29, pp [6] S. Kuang, and S. Cong. Popuation conto o equiibium states o quantum systems via Lyapunov method. Acta Automatica Sinaca, vo. 36, no. 9, pp , Septembe, 2. [7] S. W. Zhao, H. Lin, J. T. Sun, and Z. G. Xue. An impicit Lyapunov conto o inite-dimensiona cosed quantum systems. Intenationa Jouna o Robust and Noninea conto, vo. 22, pp , 22. [8] S. Kantz, H. Pas, The impicit unction theoem: histoy, theoy, and appications, Boston: Bihause, 22. [9] J. LaSae, and S. Leschetz. Stabiity by Liapunov s Diect Method with Appications. New Yo: Academic Pess, 96.