Xiaqing SUN,Sen KUANG,Yanan LIU,Juan ZHOU,Shuang CONG

Department of Automation,University of Science and Technology of China,Hefei Anhui 230027,China

Feedback stabilization ofN-dimensional stochastic quantum systems based on bang-bang control

Xiaqing SUN,Sen KUANG?,Yanan LIU,Juan ZHOU,Shuang CONG

Department of Automation,University of Science and Technology of China,Hefei Anhui 230027,China

For anN-dimensional quantum system under the influence of continuous measurement,this paper presents a switching control scheme where the control law is of bang-bang type and achieves asymptotic preparation of an arbitrarily given eigenstate of a non-degenerate and degenerate measurement operator,respectively.In the switching control strategy,we divide the state space into two parts:a set containing a target state,and its complementary set.By analyzing the stability of the stochastic system model under consideration,we design a constant control law and give some conditions that the control Hamiltonian satisfies so that the system trajectories in the complementary set converge to the set which contains the target state.Further,for the case of a non-degenerate measurement operator,we show that the system trajectories in the set containing the target state will automatically converge to the target state via quantum continuous measurement theory;while for the case of a degenerate measurement operator,the corresponding system trajectories will also converge to the target state via the construction of the control Hamiltonians.The convergence of the whole closed-loop systems under the cases of a non-degenerate and a degenerate measurement operator is strictly proved.The effectiveness of the proposed switching control scheme is verified by the simulation experiments on a finite-dimensional angular momentum system and a two-qubit system.

Quantum systems,feedback stabilization,bang-bang control,switching strategy

1 Introduction

Control theory has attained many successful applicationsin the fields ofatom physics,quantum information,quantum chemistry,quantum optics,and so on[1-3].Many classical control methods have been extended to the quantum domain,such as optimal control[4,5],Lyapunov control[6-8],sliding mode control[9,10],H∞control[11],measurement-based feedback con-trol[12-14]and fault-tolerant control[15].It is well known that feedback control may have stronger robustness and better control effect than open-loop control.In the quantum domain,quantum measurement may destroy the state of a system to be measured.It can also be used as an important approach to acquire the information on the controlled system.Actually,the quantum measurement itself can be regarded as a particular control means[16-18].

Since the 1990s,measurement-based feedback control has been widely studied and applied to protect quantum entanglement[19],prepare specific quantum states[20],and so on.For the preparation of quantum states using quantum measurement theory,we usually consider the preparation of a specific eigenstate of a measurement operator.Handel etal.[21]designed a continuous control law based on numerical methods and achieved the feedback stabilization of a desired eigenstate of a non-degenerate measurement operator for spin systems.For high-dimensional systems,direct numerical methods usually lead to expensive computation cost.Mirrahimi and Handel[22]proposed a switching control law by dividing the state space into different sets to reduce the complexity of direct numerical computation,and at the same time achieve the feedback stabilization of an arbitrary eigenstate of the angular momentum operator ofN-dimensional angular momentum systems.Zhou and Kuang[23]proposed a doublechannel control scheme where the control law on one channel is kept as a constant and only the control law on the other channel is designed,and achieved the feedback stabilization preparation of any Bell state of a degenerate measurement operator for two-qubit quantum systems.Then,the scheme has been extended to generalN-qubit systems and used in the feedback stabilization preparation of GHZ entangled states[24].

As an easily implemented control means for many quantum systems,bang-bang control has been utilized in the design of quantum control systems.For instance,via the parameterization of state vectors under complex hyperspherical coordinates,the authors in[25]designed the corresponding bang-bang control sequence which can achieve the desired state transfer.Kuang etal.[26,27]designed an open-loop approximate bang-bang control law based on Lyapunov methods and prepared an eigenstates of the internal Hamiltonians of a closed quantum system.In[28],for a trapped-ion oscillator,Alonso etal.achieved the preparation and manipulation of coherent states with up to 10000 quanta of energy by bang-bang control.For two-qubit and three-qubit systems,Vu etal.[29]designed switching control laws of bang-bang type and achieved the feedback preparation of Bell states for two-qubit systems and GHZ entangled states for three-qubit systems,respectively.In[30],for a non-degenerate measurement operator,Wei etal.designed a switching control law of bang-bang type and realized the feedback generation of symmetric multi-qubit Dicke states.Recently,a feedback control law based on switching between models was proposed to prepare the Bell states of two-qubit systems[31],which is equivalent to a bang-bang control law for a double-channel control model.

This paper further considers the bang-bang control problem of stochastic quantum systems under the influence of measurement feedback.Compared to the existing literature,this paper first extends the bangbang control scheme to generalN-dimensional stochastic quantum systems.Secondly,for the bang-bang control strategy and the stability of the closed-loop system,this paper simultaneously considers the cases of a nondegenerate measurement operator and a degenerate measurement operator.Even for non-degenerate measurementoperators,the design ofthe bang-bang control strategy in this paper is also different from the methods in the existing references since the bang-bang control law in this paper is designed by analyzing the stability of the system with a constant Hamiltonian and using quantum continuous measurement theory,instead of being directly provided or designed via Lyapunov functions.Finally,we give the conditions that the control Hamiltonian satisfies for the global asymptotic stability of the closed-loop systems.

The rest of this paper is organized as follows.In Section 2,we present the stochastic quantum system model under the influence of measurement feedback and describe the control task of this paper.Section 3 analyzes the stability of the system with a constant Hamiltonian;and gives the bang-bang control strategies under a non-degenerate measurement operator and a degenerate measurement operator,including the design of switching control laws and the construction of the control Hamiltonians.In Section 4,the stability of the closed-loop systems with a non-degenerate measurement operator and a degenerate measurement operator is strictly proved.In Section 5,we perform simulation experiments on an angular momentum system with a non-degenerate measurement operator and a two-qubit quantum system with a degenerate measurement op-erator to demonstrate the effectiveness of the control scheme proposed in this paper.Section 6 presents the conclusion.

2 System models and problem description

ConsideranN-dimensionalquantum system.Assume that when one performs proper quantum measurement(e.g.,homodyne measurement)on the observableA,the system dynamics can be described by the following filtering equation[19,32]:

In this paper,we assume that the measurement operatorAis a real-valued diagonal matrix,i.e.,

Under the action of control fields,the system HamiltonianHusually can be written as

whereH0is the free Hamiltonian of the system;Hkcorresponds to the control Hamiltonian of the system;andukis a real-valued control field.

The control task of this paper is to design the control lawukin(4)and give the conditions that the control HamiltonianHksatisfy in order to achieve the convergence of system(1)to a given eigenstate ρdof the measurement operatorA.When the measurement operatorAis non-degenerate,we only need to consider the case wherek=1 for the control task.While when the measurement operatorAis degenerate,two control channels(i.e.,k=2)are used.

3 Switching control strategy

In this section,we first analyze the stability of the system when the HamiltonianHin(1)is a constant matrix,and then give the switching control strategies for a non-degenerate measurement operator and a degenerate measurement operator.

3.1 Stability of the system with constant Hamiltonian

Denote the expectation of the system state as

The dynamics of the average state of system(1)can be written as

We analyze the stability of system(5)via the LaSalle’s invariance principle.We may define the following Lyapunov function:

A direct calculation shows that the time derivative ofQ(ˉρt)is

where?A?Frepresents the Frobenius norm of the matrixA.

Thus,the LaSalle’s invariance principle implies that the state of system(5)will converge to the largest invariant setMcontained in the set

The equilibrium point ρeof system(10)satisfies

Thus,we have the following theorem:

According to Theorem 1,it is difficult to write the general form ofH.However,when the measurement operatorAis non-degenerate,one always can obtain some concrete forms ofHby imposing some special constraints on the HamiltonianH.For instance,each of the following four conditions can guarantee that the condition in Theorem 1 holds:

Example 1Consider a two-qubit system.Assume that the measurement operatorAin Theorem 1 is

Then,[A,ρ]=0 implies that

Thus,the system state in the invariant setMis of the form:

Then,[H,ρ]=0 in Theorem 1 can be expanded as the following set of equations:

According to(17),if we take

Therefore,the constant HamiltonianHof the system can be constructed as

3.2 Design of switching control law

Due to the geometric symmetry of the state space,the eigenstates of the measurement operator which are antipodal with the target state often form the equilibrium points of the closed-loop system.In order to avoid the convergence of the system to its antipodal states,we may design the switching control laws.Let us define the following distance function

and the sets:

Fig.1 The division of the state space.

In what follows,we design the switching control laws for the cases where the measurement operator is nondegenerate and degenerate,respectively.

3.2.1 Switchingcontrollawunderanon-degenerate measurement operator

When the measurement operatorAis nondegenerate,i.e.,

we only use one controlchannel.In this case,the system Hamiltonian can be written as

In fact,due to randomness,after the system state entersS?1-γ,it will exitS?1-γwith a certain probability.Fortunately,those system trajectories leavingS?1-γwill not be too far away from the boundaryS1-γsince the influence of randomness is relatively small.Therefore,we can define two new setsS?1-γ/2andS＞1-γ/2as in[24],and give the following switching control law(also see[30]):

1)If ρt∈S?1-γ,thenu1=0;

2)If ρt∈S＞1-γ/2,thenu1=1;

3)For ρt∈S?1-γ/2∩S＞1-γ,we further consider two specific situations:

?if ρtentersS?1-γ/2∩S＞1-γfromS?1-γ,thenu1=0;

?if ρtentersS?1-γ/2∩S＞1-γfromS＞1-γ/2,thenu1=1.

In this paper,we call the switching control law here under a non-degenerate measurement operator Bang-Bang Switching Control Law I.It should be noted that the switching of the control law depends on which set the current system state is in,and therefore depends on the distance between the system state and the target state.

3.2.2 Switching control law under a degenerate measurement operator

When the measurement operatorAis degenerate,we adopt two control channels.In this case,the system Hamiltonian can be written as

For the system trajectories inS＞1-γ,when we design the constant control lawsu1andu2such that the HamiltonianHin(21)satisfies the condition in Theorem 1,the system almost surely converges intoS?1-γ.For simplicity,we takeu1=1 andu2=0 in this case.

Next,we derive the condition that the system trajectories inS?1-γconverge to the target state and give a switching control law under a degenerate measurement operator.For the system model(1),We consider the following Lyapunov function[31]:

To calculate the infinitesimal generator LV(ρt)ofV(ρt)along the trajectory of system(1),we first calculate dTr(ρtρd)and have

Let the target state ρdbe the eigenstate ofAassociated with the eigenvalue λd.Then,Tr(D[A]ρtρd)=0 holds.Assume that the HamiltonianHsatisfies

then we have

Thus,(23)becomes

According to the Ito formula,we have

Therefore,the infinitesimal generator LV(ρt)ofV(ρt)is

where Tr(H[A]ρtρd)can be calculated as

Substituting(28)into(27)gives

From(29),LV(ρt)=0 means that

Considering that the system trajectory is inS?1-γ,we have

Therefore,Tr(Aρt)= λdholds.

Equation(30)implies that

Denote any state inSdas ρSd,then

Equation(31)implies that

Thus,the system filter(1)can be simplified as

For the system state to converge to the target state,system(32)must have the only equilibrium ρd.That is to say,in the set of all ρ which satisfyAρ = λdρ,if the equation

has the only solution ρ = ρd,then system(1)almost surely converges to the target state.Note that when this condition holds,(24)naturally holds.

Equation(33)gives a condition that the Hamiltonian satisfies when the system trajectories inS?1-γconverge to the target state.For simplicity in the design of the control law,we takeu1=0 andu2=1 in this case.

Thus,similar to the case of a non-degenerate measurement operator,we can give the following switching control strategy to achieve convergence of the system with a degenerate measurement operator to the target state:

We also call the switching control law here under a degenerate measurementoperatorBang-Bang Switching Control Law II.

4 Stability of closed-loop switching systems

In this section,we present the stability results for the whole closed-loop switching systems with a nondegenerate measurement operator and a degenerate measurement operator,respectively.

4.1 Stability under non-degenerate measurement operator

Theorem 2Consider theN-dimensional stochastic quantum system in(1)with the measurement operatorAbeing a non-degenerate diagonal matrix.Assume that the target state ρdis an eigenstate ofAand the HamiltonianH=H0+H1(u1=1)in(20)satisfies one of conditions(12)-(15).Then,with the Bang-Bang Switching Control Law I in Section 3.2.1,the whole closedloop switching system converges to the target state ρdin probability.

The proof includes the following three steps.

Step 1When ρt∈S＞1-γ/2or ρtenters ρt∈S?1-γ/2∩S＞1-γ,the control lawu1=1 makes the system state almost surely enterS?1-γin a finite time.

Step 2When ρt∈S?1-γ,the control lawu1=0 guarantees that the system state stays inS?1-γ/2with probability one.

Step 3For the system states staying inS?1-γ/2,with the control lawu1=0,the system converges to the target eigenstate in probability.

ProofWe first present the proof of Step 1.

Since the system state ρtis a continuous function of timetwhile the distance function

is continuous with respect to the state ρt,we have

then ε＞0.Equation(34)means that there exists a finite timeT＞0 such that

holds fort?T.

According to(35),we have

Equation(36)implies thatthe conclusion in Step 1 holds.

Now we turn to the proof of Step 2.

We firstly calculate the infinitesimal generator of the distance functionV(ρt)in(18).It follows from(18)that

Since ρdis an eigenstate ofA,

hold.Further,considering the fact that the control law isu1=0 when ρtis inS?1-γ,we can write(37)as

In this paper,AandH0are both diagonal matrices.Therefore,ρdis also an eigenstate ofH0,i.e.,[ρd,H0]=0.Substituting[ρd,H0]=0 into(38),we know that the infinitesimal generator ofV(ρt)satisfies

Thus,based on the estimate

from stochastic stability theory[33],we take α =1-γ/2 and have

Equation(40)shows that the probability that the system state entersS?1-γand eventually leaves fromS?1-γ/2is less than 1.That is,the system state will stays inS?1-γ/2with a probability greater than 1-Pafter it entersS?1-γ.Denote this probability asP1.Then,the probability that the system state leaves fromS?1-γ/2isP2=1-P1＜1.

Now,we show that the system state will not keep shuttling betweenS?1-γ/2andS＞1-γ/2forever.Let the number that the system state returns toS＞1-γ/2fromS?1-γ/2bemand denote the probability that this event occurs as P(m),then we have

Now,the proof of Step 3 is presented as follows.

For the system trajectories staying inS?1-γ/2,the control law isu1=0.According to quantum continuous measurement theory,the system state will converge to an eigenstate ofA.Since the measurement operatorAis non-degenerate in this paper,theNdifferent eigenstates ofAare mutually orthogonal.This means that for any eigenstate ρgthat is different from the target state ρd,V(ρg)=1 always holds,i.e.,ρg∈S?1-γ/2.In other words,ρdis the only eigenstate ofAwhich is contained inS?1-γ/2.Quantum continuous measurement theory states that the system state will eventually converge to an eigenstate of the measured physical quantity when the control law is zero.On the other hand,with the designed switching control law,the system state will stays inS?1-γ/2almost surely.Therefore,the system state will converge to the target state ρdin probability. ?

4.2 Stability under degenerate measurement operator

The proof of this theorem still includes three steps as follows:

Step 1When ρt∈S＞1-γ/2or ρtenters ρt∈S?1-γ/2∩S＞1-γ,the control lawu1=1,u2=0 makes the system state almost surely enterS?1-γin a finite time.

Step 2When ρt∈S?1-γ,the control lawu1=0,u2=1 guarantees that the system state stays inS?1-γ/2with probability one.

Step 3For the system states staying inS?1-γ/2,with the control lawu1=0,u2=1,the system converges to the target eigenstate in probability.

The concrete proof process is similar to the case where the measurementoperatoris non-degenerate.For brevity,we omit it here.

5 Numerical examples

In this section,we perform simulation experiments on an angular momentum system with a non-degenerate measurement operator and on a two-qubit system with a degenerate measurement operator,respectively.

5.1 An angular momentum spin system

We consider a 17-dimensional(17=2J+1)angular momentum spin system with the absolute value of the momentumJ=8.When one observes the angular momentum on theZdirection and applies the magnetic field along theYdirection,the corresponding measurement operator and control Hamiltonian can be obtained as[25,34]

We assume the initial state ρ0is and the target state ρdis the last eigenstate of the measurement operator,i.e.,

In simulations,we take Γ =1 and η =1.Using the Bang-Bang Switching Control Law I in Section 3.2.1,we perform three simulation experiments under the same conditions.The simulation results are shown in Figs.2 and 3.

Fig.2 The evolution curves of the distances between the system states and the target state under three sample paths with the same initial state ρ0.

Fig.3 The evolution curves of the control laws associated with the three sample paths,where(a),(b),and(c)correspond to the sample paths 1,2,and 3,respectively.

It can be seen from Fig.2 that the system states under the three sample paths eventually converge to the target state.Fig.3 shows that the switching control law corresponding to each sample path only takes 1 and 0,i.e.,so-called bang-bang control.This is consistent with the theoretical results above.

5.2 A two-qubit system

Now,we consider the two-qubit system in Example 1 where the measurement operator is

Assume that the free Hamiltonian is

The target state is given as

We choose the control Hamiltonians as(also see[29]):

It can be verified that the conditions in Theorem 3 are satisfied.Now,we give an initial state as

We use the Bang-Bang Switching Control Law II in Section 3.2.2 to perform simulation experiments,and the corresponding simulation resultsare shown in Fig.4.It can be seen from Fig.4 that system state eventually converges to the target state and the switching control lawsu1andu2also only take 1 and 0,which is the so-called bang-bang property.

6 Conclusions

For anN-dimensional stochastic quantum system with a non-degenerate or degenerate measurement operator,this paper proposed a switching control law based on the state space division and realized the stabilizing preparation of any eigenstate of the measurement operator.We also gave the conditions on the system Hamiltonian in order to ensure the system stability and proved the stability of the closed-loop system via stochastic stability theory.It should be pointed out that the switching control laws of bang-bang type are not unique for the case of a degenerate measurement operator.For special systems,it is necessary to choose the bang-bang control laws that are more easily realized in physics.

Fig.4 The simulation results under the initial state ρ0,where(a)indicates the distance between the system state and the target state,(b)and(c)are the evolution curves of control laws u1 and u2,respectively.

Acknowledgements

We thank Dr.Daoyi Dong for helpful discussion.

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3 May 2017;revised 17 June 2017;accepted 17 June 2017

DOI 10.1007/s11768-017-7061-6

?Corresponding author.

E-mail:skuang@ustc.edu.cn.

This paper is dedicated to Professor Ian R.Petersen on the occasion of his 60th birthday.This work was supported by the Anhui Provincial Natural Science Foundation(No.1708085MF144)and the National Natural Science Foundation of China(No.61573330).

?2017 South China University of Technology,Academy of Mathematics and Systems Science,CAS,and Springer-Verlag Berlin Heidelberg

Xiaqing SUNreceived a B.Sc.degree in Automation from the Anhui University in 2015.She is currently pursuing a M.Sc.degree with the Department of Automation,University of Science and Technology of China.Her current research interest focuses on quantum feedback control.E-mail:sxqing@mail.ustc.edu.cn.

Sen KUANGreceived a Ph.D.degree in Control Theory and Control Engineering from the University of Science and Technology of China(USTC)in 2007.From 2007 to 2010,he was a post-doctoral fellow with the School ofInformation Science and Technology,USTC.He visited the University of Hong Kong in 2010 and 2015,respectively.From 2014 to 2015,he was a visiting scholar at the University of New South Wales,Canberra,Australia.Currently,he is an associate professor in the Department of Automation,University of Science and Technology of China.His research interests include quantum information and control,quantum machine learning and its applications,and intelligent control.E-mail:skuang@ustc.edu.cn.

Yanan LIUreceived a B.Sc.degree in Measurement&Control Technology and Instrumentation from the Anhui University,Anhui,China,in 2014.She is currently pursuing a M.Sc.degree with the Department of Automation,University of Science and Technology of China.Her current research interests include quantum feedback control and stability analysis.E-mail:liuyn@mail.ustc.edu.cn.

Juan ZHOUreceived a B.Sc.degree in Automation from the Anhui University in 2013,and a M.Sc.degree in Control Theory and Control Engineering from the University of Science and Technology of China in 2016.Her research interests include quantum feedback control and stability analysis.Email:sa130100@mail.ustc.edu.cn.

Shuang CONGreceived a B.Sc.degree from the Beijing University of Aeronautics and Astronautics,Beijing,China,in 1982,and a Ph.D.degree in System Engineering from the University of Rome“La Sapienza,"Rome,Italy,in 1995.She is currently a Professor with the Department of Automation,University of Science and Technology of China,Hefei,China.Her current research interests include advanced control strategies for motion control,fuzzy logic control,neural networks design and applications,robotic coordination control,and quantum systems control.Email:scong@ustc.edu.cn.