One Lyapunov control for quantum systems and its application to entanglement generation

Physics Letters A 377 (2013) 851–854

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Physics Letters A www.elsevier.com/locate/pla

One Lyapunov control for quantum systems and its application to entanglement generation Wei Yang, Jitao Sun ∗ Department of Mathematics, Tongji University, Shanghai, 200092, China

a r t i c l e

i n f o

Article history: Received 8 May 2012 Received in revised form 15 January 2013 Accepted 29 January 2013 Available online 31 January 2013 Communicated by A.R. Bishop

a b s t r a c t In this Letter, we investigate the control of finite dimensional ideal quantum systems in which the quantum states are represented by the density operators. A new Lyapunov function based on the Hilbert– Schmidt distance and mechanical quantity of the quantum system is given. We present a theoretical convergence result using LaSalle invariance principle. Applying the proposed Lyapunov method, the generation of the maximally entangled quantum states of two qubits is obtained. © 2013 Elsevier B.V. All rights reserved.

Keywords: Quantum systems Lyapunov function LaSalle’s invariance principle Stability Entanglement generation

1. Introduction

Extending control to the quantum domain has become an important area of research recently [1–12] and references therein. Lyapunov method and LaSalle’s invariance principle are two of the proposed techniques to control quantum systems, and many results about the stability of quantum systems are established [1,13–23] and references therein. When the state of a quantum system is a pure state described by a complex vector |ψ, the system evolution is described by ˙ t ) = H |ψ(t ) [13–16,22,24–26], and in Schrödinger equation i h¯ |ψ( [15], the authors presented a unified form of the Lyapunov functions V (|ψ(t )) = ψ − α ψ f | Q |ψ − α ψ f  to study asymptotic stability. Compared to pure-state quantum systems, quantum systems represented by density operators are more general for practical quantum systems, since they can represent both pure-state systems and mixed-state systems, subject to environmental decoherence or measurements, they also can represent open quantum systems, and the system evolution is mastered by Liouville’s equad tion ih¯ dt ρ (t ) = [ H (t ), ρ (t )] := H (t )ρ (t ) − ρ (t ) H (t ). The Hilbert– Schmidt state distance between final state and system state and V (ρ ) = Tr( Q ρ ) are chosen as Lyapunov functions in [1,19–21,27], and these Lyapunov control designs can be used to entanglement generation by their convergence properties and stability.

In quantum dynamics, entanglement is one of the most astonishing features, since it also can be applied to many fascinating areas [28–31]. The generation of maximally entangled states becomes a crucial task, and continuous feedback control is used in [32–36] to analysis a model consisting of two two-level atoms which are placed in distant cavities and interacting through a radiation field in a dispersive way. Entanglement shared between distant sites is a valuable resource for quantum communications [37]. In this Letter, we introduce a new Lyapunov function to analyze the asymptotic stability of quantum states of ideal quantum systems [38], whose evolution is mastered by Liouville’s equation, and we find that it is more effective than the existing Lyapunov functions. Based on the new Lyapunov function, we generate the maximally entangled quantum states of two qubits. The rest of this Letter is organized as follows, in Section 2, we consider control in the context of density operator representing generic quantum states by introducing a new Lyapunov function. By explicitly characterizing the stationary points and analyzing the property of the designed control function, the asymptotic stability of quantum system is investigated in Section 3. In Section 4, we generate the maximally entangled states of two qubits quantum system by the proposed Lyapunov control. 2. Preliminaries We consider the following n-level quantum system with only one control function, and set the Plank constant h¯ = 1:

*

Corresponding author. Tel.: +86 21 65982341 1307; fax: +86 21 65982341. E-mail addresses: [email protected] (W. Yang), [email protected] (J. Sun).

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i ρ˙ (t ) = H 0 + f (t ) H 1 , ρ (t ) ,

(1)

852

W. Yang, J. Sun / Physics Letters A 377 (2013) 851–854

ρ (t ) is a density operator, represents the mixed state of the quantum system, defined on an n-dimensional Hilbert space H  C n , which is equipped with the scalar product ( A , B ) = Tr( A ∗ B ) and the norm  A 2 = ( A , A ), for any A , B ∈ H. H 0 is the internal Hamiltonian, H 1 represents the interaction energy between the system and the external classical control field f (t ), and is called interaction Hamiltonian. They are both n × n self-adjoint operators in the n-dimensional Hilbert space H and assumed to timeindependent. Denoted by S n the set of quantum states. By choosing a special basis of Hilbert space, we can suppose H 0 diagonal, and set H 0 = diag (a1 , a2 , . . . , an ), with ak  ak+1 and w kl = ak − al  0 when k  l, and H 1 := (h jk )n×n ,

Proof. Necessity: Suppose ρ is a stationary point in the orbit Ou (n) (ρ0 ) = { X ρ X ∗ | X ∈ U (n)}. Let A 1 , A 2 , . . . , An2 be a basis of u (n), a neighborhood of ρ can be given by the set of matrices M = M (t 1 , . . . , tn2 ), defined as

M (t 1 , . . . , tn2 ) = e A 1 t1 · · · e An2 tn2 ρ e − An2 tn2 · · · e − A 1 t1 , 2

(t 1 , . . . , tn2 ) is in an open neighborhood of the origin of R n . All the derivatives of V (ρ (t )) are

∂  A 1 t1 Tr e · · · e A n2 t n2 ρ 2 e − A n2 t n2 · · · e − A 1 t 1 ∂t j − αρd e A 1 t1 · · · e An2 tn2 ρ e − An2 tn2 · · · e − A 1 t1

∗

h jk = hkj , j , k = 1, 2, . . . , n,

− α e A 1 t1 · · · e An2 tn2 ρ e − An2 tn2 · · · e − A 1 t1 ρd   + α2ρ 2 Q 

under this basis. Definition 2.1. (See [20].) The Hamiltonian of the dynamical quantum system is called ideal if (i) H 0 is strongly regular, i.e., w kl = w pq unless (k, l) = ( p , q). (ii) H 1 is fully connected, i.e., bkl = 0 except (possibly) for k = l. In this Letter we will focus on the control of ideal quantum system. We assume that quantum system (1) is operator controllable (complete controllable) [1,2], i.e., L = su (n) or u (n), where L = span{i H 0 , i H 1 } is the linear Lie algebras of matrices, and the Lie bracket [ A , B ] := A B − B A is the standard matrix commutator, { A , B } is another matrix commutator, defined by { A , B } := A B + B A. From the properties of operator controllability, the state ρ (t ) is unitarily equivalent to ρ0 , which means that ρ (t ) is the matrix in the orbit Ou (n) (ρ0 ) = { X ρ0 X ∗ | X ∈ U (n)}. In order to study the stability of quantum systems, we combine the Lyapunov function based on the distance of two quantum states and as that based on the average value of Hermitian operator Q which can be regarded as an observable of a quantum system, and choose the following Lyapunov function:

V (ρ ) = Tr

 

 

ρ (t ) − αρd Q ρ (t ) − αρd



2  = Tr ρ (t ) − αρd Q ,

(2)

Remark 2.1. If ρ is a pure state, i.e., ρ 2 = ρ , and ρ can be represented by |ψψ|, ρd = |ψ f ψ f |, when α = 1, Q = I , then V (|ψ) = 2(1 − |ψ|ψ f |2 ), or α = 0, Q = I − |ψ f ψ f |, V (|ψ) = 1 − |ψ|ψ f |2 , which is used in [14,15] as the Lyapunov function based on the state distance. If α = 0, V (|ψ) = Tr( Q |ψψ|) = ψ| Q |ψ, which is used in [15,26] as the Lyapunov function based on the average value of an imaginary mechanical quantity. When α = 0 or 1, Q = I , V (ρ ) in Eq. (2) describes the commonly used Lyapunov functions in [1,19,20].

= Tr A j ρ − ρ 2 A j − αρd A j ρ + αρd ρ A j   − α A j ρρd + αρ A j ρd Q    = Tr A j ρ , {ρ − αρd , Q } , j = 1, 2, . . . , n2 .

In the following, we set [ Q , H 0 ] = 0, then H 0 = I or Q is also diagonal, and we will consider Q is diagonal below. Since ρd is a real constant diagonal matrix, and [ H 0 , ρd ] = 0, then

V˙ =

d dt

Tr



 

ρ 2 − αρd ρ − αρρd + α 2 ρd2 Q

   = − f Tr −i H 1 {ρ − αρd , Q }, ρ .  



,

where K > 0, y = g (x) is a function whose image passes the origin of the plane x– y monotonically and lies in the quadrant I or III, V˙ will be nonpositive, and V˙ = 0 is equivalent to Tr (−i H 1 [{ρ − αρd , Q }, ρ ]) = 0. When ρ = ρd at one point t 1 ∈ R, f will become 0 since [ρd , Q ] = 0. It means that the control field will disappear automatically when the target state is reached. Theorem 2. If the Hamiltonian of a quantum system is ideal, and [ Q , H 0 ] = 0, the control function f asymptotically drives the state ρ of ρ˙ = −i [ H 0 + f H 1 , ρ ] to a stationary point of V (ρ ). And if the stationary point is in the unitarily equivalent class of ρd , the quantum system will converge asymptotically to the target state ρd . Proof. Let E be the largest invariant set, where dV (ρ ) = 0. E is dt the set of all the Hermitian matrices ρ1 with the same spectrum as ρ0 = ρ (0), satisfies:



Theorem 1. For a given target state ρd , ρ is the stationary point of V (ρ ) if and only if

i.e.,

(3)



f = K g Tr −i H 1 {ρ − αρd , Q }, ρ

Tr −i H 1



(4)

For a stationary point, Eq. (4) should be zero. Since [ρ , {ρ − αρd , Q }]∗ = −[ρ , {ρ − αρd , Q }], i.e., they are skew-Hermitian matrices, then their component along any element of a basis of u (n) is 0. Hence, we have [ρ , {ρ − αρd , Q }] = 0. Sufficiency: If [ρ , {ρ − αρd , Q }] = 0, it means Eq. (4) is zero for all t j , j = 1, . . . , n2 , at the point (0, . . . , 0), then ρ is a stationary point. 2

3. Lyapunov control of ideal quantum systems

ρ , {ρ − αρd , Q } = 0.

2

Choose

where Q is a non-negative definite Hermitian matrix. In order to analyze the “augmented” function V (ρ ) easily, we will constrain α ∈ [0, 1]. We try to find a control function f (t ), such that the solution of system (1) converges to the minimum of Eq. (2).



t 1 =···=tn2 =0

d



E=





e −i H 0 t ρ1 e i H 0 t − αρd , Q , e −i H 0 t ρ1 e i H 0 t











= 0, 

ρ1 ∈ Ou(n) (ρ0 )  Tr −i H 1 e−i H 0 t ρ1 e i H 0 t − αρd , Q , e −i H 0 t ρ1 e i H 0 t



 = 0, ∀t ∈ R .

W. Yang, J. Sun / Physics Letters A 377 (2013) 851–854

ρ1 , since [ Q , H 0 ] = 0,   −i H t   0 0 = Tr −i H 1 e ρ1 e i H 0 t − αρd , Q , e−i H 0 t ρ1 e i H 0 t    = iTr e i H 0 t H 1 e −i H 0 t ρ1 , {ρ1 − αρd , Q } . Let us characterize the set E of

(5)

Taking derivative of Eq. (5) with respect to t at time t = 0, and denoting the ( j , k)-th element of [ρ1 , {ρ1 − αρd , Q }] by ∗ , we have ([ρ1 , {ρ1 − αρd , Q }]) jk = c jk = −ckj



0 = Tr adi H 0 H 1

=i







ρ1 , {ρ1 − αρd , Q }

∗ h jk w jk ckj + h∗jk w jk ckj



field with amplitude A is given to one of two atoms, the output of each cavity enters the other. By eliminating the radiation fields, the internal Hamiltonian becomes H int = 2 J Z ⊗ Z , we apply local laser fields to each atom, then the local Hamiltonian becomes H drv = B ( X ⊗ I + I ⊗ X ), where X , Y , Z are Pauli operators, and the coupling constant B = η J , in order to ensure the derivative of H int remains valid, η should be sufficiently smaller than 1. The spin– spin coupling constant J scales as radiation pressure | A |2 and goes to zero for negligible cavity detuning [34]. For the two two-level atoms or quantum dots, we can choose H 0 = H int , H 1 = H drv , which is called local control [35,36]. The four maximally entangled two-qubit states or Bell states are:

j
= 2i

j
0 = Tr ad2i H 0 H 1

= i2

1 

|ψ +  = √

w jk Re(h jk ckj ),









|φ  = √

2



w 2jk Im(h jk ckj ),

j
...



0 = Tr adniH 0 H 1

= in







ρ1 , {ρ1 − αρd , Q }



∗ , h jk w njk ckj + (−1)n+1 h∗jk w njk ckj

when n even,

(6)

w njk Re (h jk ckj ) = 0,

when n odd.

(7)

j
j
Since the Hamiltonian of the quantum system is ideal, it means w kl = w pq unless (k, l) = ( p , q), and bkl = 0 except (possibly) for k = l. According to the Vandermonde determinant argument on Eqs. (6) and (7), we have c jk = 0 for all j , k = 1, . . . , n, which means [ρ1 , {ρ1 − αρd , Q }] = 0. According to the LaSalle invariance principle [17], the control function f asymptotically drives the state ρ to a stationary point of V (ρ ). 2 Since [ H 0 , Q ] = 0, then Q and H 0 have the same eigenvectors. Let |ψ1 , . . . , |ψn  be the eigenvectors of H 0 , construct λ1 , . . . , λn be the appropriate eigenvalues of Q , such that these eigenvalues fit for the control requirement, and λ1 , . . . , λn correspond appropriately to the eigenvectors of H 0 , then we can construct Q as follows n

λi |ψi ψi |.

2 1 

−

|00 − |11 ,

|ψ  = √

2

 |01 − |10 .

2

w njk Im(h jk ckj ) = 0,

Q =



of H 0 . Take |φ +  as a target state for instance, since the Lyapunov control based on density matrix, we set ρd = |φ + φ + |. Based on the Bell-state basis, the Hamiltonian and the target state can be written as:

i.e.,



 |00 + |11 ,

|φ +  = √

{|0, |1}, to the X -eigenbasis {|+, |−}, where |+ = √1 11 , 2  |− = √1 −11 , and we get the Bell states are the eigenvectors

j


1 

 |01 + |10 ,

A Bell state is defined as a maximally entangled quantum state of two qubits. The qubits are usually thought to be spatially separated. Nevertheless they exhibit perfect correlations which cannot be explained without quantum mechanics. We transform the quantum states space from the Z -eigenbasis 

j


2 1 

−

ρ1 , {ρ1 − αρd , Q }

∗ h jk w 2jk ckj − h∗jk w 2jk ckj

= 2i 3

853

(8)

i =1

Usually, we set the target state ρd = |ψ f ψ f |, which is a pure state, and the eigenvalue of H

0 corresponding to |ψ f  is λ f , n denoted by D = {ρ ∈ S n | ρ = i =1 d i |ψi ψi |} ⊂ E = {ρ ∈ S n | [ρ , {ρ − αρd , Q }] = 0}. Under the conditions V˙  0, and |ψi  are eigenstates of H 0 , we have V (ρ ) < V (ρ0 ) (ρ ∈ D ), and the constructed eigenvalues λi , λ f should be chosen satisfying (1 − 2α )λ f < λi , where λi = λ f . 4. Application of Lyapunov control to entanglement generation In this section, we will consider a model consisting of two twolevel atoms or quantum dots, 1 and 2, placed in distant cavities connected into a closed loop via optical fibers. A coherent input

⎛

⎞

1 ⎜0 ⎜ H0 = 2 J ⎝ 0 0

0 0 0 1 0 0 ⎟ ⎟, 0 −1 0 ⎠ 0 0 −1

⎛

0 ⎜0 ⎜ H 1 = 2η J ⎝ 0 0

0 0 0 1 1 0 0 0

⎞

0 0⎟ ⎟. 0⎠ 0

However, H 0 and H 1 do not satisfy the conditions in Definition 2.1, i.e., the Hamiltonian is not ideal, thus the control function cannot drive every state to the target state. As discussed in [35], we can analyze the generation of entanglement in the subspace E spanned by |++, |−−, then



H0 = 2 J



ρd =

1 0 0 −1

1 0 0 0







,

H 1 = 2η J

0 1 1 0



,

,

and E is a two-dimensional subspace of L = span{i H 0 , i H 1 }, since dim(L) = 3 = 22 − 1, i.e., this subsystem is complete controllable. This Hamiltonian is ideal, from Theorem 2, all the dynamical states in E except |φ +  will be driven to the target state, since |φ +  is in the invariant set. We take the control function f = K Tr(−i H 1 [{ρ − αρd , Q }, ρ ]), where K > 0, let the eigenvalues of Q be λ1 = 0.2, λ2 = 0.8, and α = 1 in this application. The control function steers the system smoothly to the target state as shown in Fig. 1(b). From Eq. (3), any states ρ = (a|++ + b|−−)(a|++ + b|−−)∗ ∈ E = span{|++, |−−}, where a2 + b2 = 1, a, b ∈ C, are not stationary points of V (ρ ) except |φ + φ + | and |φ − φ − |. If the system starts from the state in E, the control designed above produces a control function which steers the system to the target state ρd = |φ + φ + |. Moreover, since

       f (t ) = K Tr −i H 1 {ρ − ρd , Q }, ρ     = K  −i H 1 , {ρ − ρd , Q }, ρ     K  − i H 1  ·  {ρ − ρd , Q }, ρ ,

(9)

854

W. Yang, J. Sun / Physics Letters A 377 (2013) 851–854

Fig. 1. (a) The variation of the control function f (t ); (b) the Lyapunov function V (ρ (t )) = Tr((ρ (t ) − ρd )2 Q ), with the parameters K = 0.2, K = 0.4 and K = 0.8 respectively.

the control function is bounded and influenced by K , and we can choose appropriate K for different models based on experimental conditions and requirements. The control function f (t ) and V (ρ (t )) are plotted in Fig. 1 based on different control gains K = 0.2, 0.4 and 0.8. As illustrated in Fig. 1, we can see that f (t ) and V (ρ (t )) tend to zero within about 15 seconds without strong oscillation. Remark 4.1. The time spent in stabilizing is about 35 seconds in [35], hence, the performance of the proposed method in this Letter is better than that in [35]. 5. Conclusion This Letter has addressed the issue of Lyapunov-based control of ideal quantum systems described by the Liouville equation. By introducing a new Lyapunov function, the stationary points of it have been characterized explicitly, and the asymptotic stability of the controlled system has been investigated with the help of LaSalle invariance principle. Moreover, application of this control method to generate maximally entangled states has also been studied. Based on the proposed method which can control ideal quantum systems described by the Liouville equation in this Letter, we will focus on the control of general quantum systems governed by the Lindblad equation which is more appropriate and other control design methods using different Lyapunov functions. Acknowledgements The authors would like to thank the editor and the reviewers for their constructive comments and suggestions to improve the quality of the Letter. This work is supported by the National Natural Science Foundation of China under Grants 61174039, the Fundamental Research Funds for the Central Universities of China, and China Scholarship Council.

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