Pole placement approach to coherent passive reservoir engineering for storing quantum information

Thien NGUYEN ,Zibo MIAO ,Yu PAN,Nina AMINI,Valery UGRINOVSKII,Matthew R.JAMES

1.ARC Centre for Quantum Computation and Communication Technology,Research School of Engineering,The Australian National University,Canberra,ACT 2601,Australia;

2.QUANTIC lab,INRIA Paris,2 rue Simone Iff,75012 Paris,France;

3.Institute of Cyber-Systems and Control,Zhejiang University,Hangzhou Zhejiang 310027,China;

4.CNRS,Laboratoire des signaux et syst`emes(L2S),CentraleSup′elec,3 rue Joliot Curie,91192 Gif-Sur-Yvette,France;

Pole placement approach to coherent passive reservoir engineering for storing quantum information

Thien NGUYEN1,Zibo MIAO2,Yu PAN3,Nina AMINI4,Valery UGRINOVSKII5?,Matthew R.JAMES1

1.ARC Centre for Quantum Computation and Communication Technology,Research School of Engineering,The Australian National University,Canberra,ACT 2601,Australia;

2.QUANTIC lab,INRIA Paris,2 rue Simone Iff,75012 Paris,France;

3.Institute of Cyber-Systems and Control,Zhejiang University,Hangzhou Zhejiang 310027,China;

4.CNRS,Laboratoire des signaux et syst`emes(L2S),CentraleSup′elec,3 rue Joliot Curie,91192 Gif-Sur-Yvette,France;

Reservoir engineering is the term used in quantum control and information technologies to describe manipulating the environment within which an open quantum system operates.Reservoir engineering is essential in applications where storing quantum information is required.Fromthe controltheory perspective,a quantum systemis capable ofstoring quantuminformation if it possesses a so-called decoherence free subsystem(DFS).This paper explores pole placement techniques to facilitate synthesis of decoherence free subsystems via coherent quantum feedback control.We discuss limitations of the conventional‘open loop’approach and propose a constructive feedback design methodology for decoherence free subsystem engineering.It captures a quite general dynamic coherent feedback structure which allows systems with decoherence free modes to be synthesized from components which do not have such modes.

Open quantum system,decoherence free subsystem,reservoir engineering,coherent feedback control,quantum control

1 Introduction

The environment within which the quantum system operates typically has a continuous degrading effect on the evolution of quantum particles.This effect known asdecoherenceis the reason for the continuous process of degeneration of distinctly quantum states into classical ones[1].On the other hand,when a quantum system possesses a subsystem isolated from the detrimental influence of the environment and probing fields,the quantum information associated with dynamics of such a system is preserved and can be used for quantum computation when needed.In a sense,decoherence free subsystems(DFS)can play roles of memory elements in quantum information processing.This has motivated significant interest in the synthesis of quantum systems with a desired DFS structure.

The problem of DFS synthesis has been found to be nontrivial-it has been shown in[2]that conventional measurement feedback is ineffective in producing quantum systems having a DFS,however certain coherent controllers can overcome this limitation of the measurement-based feedback controllers.The objective of this paper is to put this observation on a solid systematic footing,by developing a quite general constructive coherent synthesis procedure for generating quantum systems with a DFS of desired dimension.

Our particular interest is in a class of quantum linear systems whose dynamics in the Heisenberg picture are described by complex quantum stochastic differential equations expressed in terms of annihilation operators only.Such systems are known to be passive[3].Passivity ensures that the system does not generate energy.In addition,in such systems the notion of system controllability by noise and that of observability from the output field are known to be equivalent[4].Also,one can readily identify uncontrollable and unobservable subspaces of the passive system by analyzing the system in the Heisenberg picture[2].These additional features of annihilation only passive systems facilitate the task of synthesizing decoherence free subsystems by means of coherent feedback.

The focus on a general coherent feedback synthesis is the main distinct feature of this paper which differentiates it from other works of a similar kind,notably from[2,5].The paper[2]presents an analysis of quantum systems equipped with coherent feedback for the purpose of characterizing decoherence free subsystems,quantum nondemolished(QND)variables and measurements capable of evading backaction;in[2]all these characteristics are expressed in geometric termsof(un)controllable and(un)observable subspaces.In contrast,in this paper we propose constructive algebraic conditions for the synthesis of coherent feedback to equip the system with a DFS.These conditions are expressed in terms of linear matrix inequalities(LMIs)and reduce the DFS synthesis problem to an algebraic pole assignment problem.After completing this work we became aware that Nurdin and Gough had also arrived at the pole placement idea[5].However,our results are different in that they are not restricted to interconnected optical cavities considered in[5],and applicable to a coherent feedback interconnection of two general quantum systems of which interconnected optical cavities are a special case;see Section 4.Of course,the generality of our formulation means that the DFS engineering problem in this paper cannot be solved by calculating system poles directly,hence a more general approach is developed in this paper.

Also,the DFS synthesis methodology proposed here extends substantially our preliminary work[6].The controller configuration in that paper was limited to resembling a classical Luenberger observer.It turns out that such a configuration is somewhat restrictive;for example,it is not sufficiently flexible to capture the controller structure analyzed in[2].In this paper,we build our technique using the most general type of dynamic linear passive coherent feedback.We show that the controller structures from[2,6]are in fact special cases of our general setting.In addition,we discuss the conventional open-loop approach to reservoir engineering and show the shortcoming of such approach.A shortened version of this paper has been scheduled for presentation at the 2017 American Control Conference[7].Compared to the conference version,the present version is substantially revised and extended.In particular,it includes background material on quantum passive systems and complete proofs of results.Also,a new example is included to illustrate the possibility of creating a DFS shared by the principal plant and the controller,which appears to be not possible to achieve in simple optical cavity systems.

NotationGiven an underlying Hilbert space ? and an operatorx:? → ?,x＊denotes the operator adjoint tox.In the case of a vector of operators,the vector consisting of the adjoint components ofxis denotedx#,andx?=(x#)T,where T denotes the transpose of a vector.Likewise,for a matrixA,A#is the matrix whose entries are complex conjugate of the corresponding entries ofA,andA?=(A#)T.[x,y]=x y-yxis the commutator of two operators,and in the case wherex,yare vectors of operators,[x,y?]=x y?-(y#xT)T.

2 Background

2.1 Open quantum systems

Open quantum systems are systems that are coupled to an external environment or reservoir[8].The environment exerts an influence on the system,in the form of vectorsW(t),W?(t)consisting of quantum Wiener processes defined on a Hilbert space＆known as the Fock space.The unitary motion of the passive annihilation only system governed by these processes is described by the stochastic differential equation

whereHandLare,respectively,the system Hamiltonian and the coupling operator through which the system couples to the environment.Then,any operatorX:?→?generatesthe evolutionX(t)=jt(X)=U(t)＊(X?I)U(t)in the space of operators on the tensor product Hilbert space ? ? ?,

are the generator and the Lindblad superoperator of the system,respectively[9].The field resulting from the interaction between the system and the environment constitutes the output field of the system

2.2 Linear annihilation only systems

Linear annihilation only systems arise as a particular class of open quantum systems whose operatorsak,k=1,...,n,describe various modes of photon annihilation resulting from interactions between the environment and the system.Such operators satisfy the canonical commutation relations[a j,a＊k]= δjk,where δjkis the Kronecker delta.Taking the system Hamiltonian and the coupling operator of the system to be,respectively,quadratic and linear functions of the vectorX=a=[a1,...,an]T,

whereMis a Hermitiann×nmatrix,andC∈Cm×n,the dynamics and output equations become

where the complex matricesA∈ Cn×n,B∈ Cn×m,andC∈ Cm×nsatisfy

The following fundamental identity then holds[10]

2.3 Passive annihilation only quantum systems

According to[3],passivity of a quantum systemPis defined as a property of the system with respect to an output generated by an exosystemWand applied to input channels of the given quantum system on one hand,and a performance operatorZof the system on the other hand.To particularize the definition of[3]in relation to the specific classofannihilation only systems,we consider a class of exosystems,i.e.,open quantum systems with zero Hamiltonian,an identity scattering matrix and a coupling operatoruwhich couples the exosystem with its input field.The exosystem is assumed to be independent ofPin the sense thatucommutes with any operator from theC＊operator algebra generated byXandX?.The time evolution ofuis however determined by the full interacting systemP?W,and therefore may be influenced byX,X?.

If the output of the exosystemWis fed into the input of the systemPin a cascade or series connection,the resulting systemP?Whas the HamiltonianHP?W=H+Im(u?L),the identity scattering matrix and the field coupling operatorLP?W=L+u[3].The resulting system(P?W)then has the generator GP?W.

Definition1[3]AsystemPwith a performance outputZis passive if there exists a nonnegative observableV(called thestorage observableofP)such that

for some constant λ＞0.The operator

is thesupply ratewhich ensures passivity.

Now supposePis a linear annihilation only system(4).Also,consider a performance output for the systemP?Wto be

withC0∈ Cl×n,D0∈ Cl×m.TakingX=ain(2),the systemP?Wcan be written as

where the complex matricesA∈ Cn×n,B∈ Cn×m,andC∈ Cm×nare the coefficients of the annihilation only systemP.

We further take the storage observableVhaving the formV=a?Pa,and the supply rate having the formr(W)=Z?u+u?Z.Then it can be shown that the systemPis passive with a storage functionVand a supply rater(W)if for some constant λ＞0,

This condition is equivalent to the positive realness condition stated in Theorem 3 of[11](lettingQ=0 in that theorem):

In the special case,whereV=a?a,D0=0[11]andC0=-C,this reduces to the following inequality

as the condition for passivity.Clearly this condition is satisfied in the case of an annihilation only systemPin the light of the identity(7).Hence the annihilation only system(9)is passive with respect to performance outputZ=-Ca,with the storage functionV=a?a.

2.4 Decoherence free subsystems

As mentioned,a decoherence free subsystem represents a subsystem whose variables are not affected by input fields and do not appear in the system output fields;this makesthe DFS isolated from the environment and inaccessible to measurement devices,thus preserving the quantum information carried by the variables of the DFS.In relation to the annihilation only system(5),witha=[a1,...,an]T,a componentajis a decoherencefree mode if the evolution ofajis independent of the inputWand if the system outputYis independent ofa j.The collection of decoherence-free modes forms a subspace,called thedecoherence-free subspace.

An important fact about the existence of a decoherence-free subsystem for linear annihilation only systems follows from the results established in[4]:

Proposition 1The linear annihilation only system(5)has a decoherence-free subsystem if and only if the matrixAhas some of its poles on the imaginary axis,with the remaining poles residing in the open left halfplane of the complex plane.

ProofAccording to[4,Lemma 2],for the system(5),the properties of controllability,observability and Hurwitz stability are equivalent.The statement of the proposition then follows by contraposition,after noting that being passive,the system(5)cannot have eigenvalues in the open right hand-side of the complex plane due to(7). ?

According to Proposition 1,if the system(5)has a DFS,then there must exist a coordinate transformation of the system(5)such that in the new coordinates,the system takes the form,known as the Kalman decomposition:

whereis the corresponding block of the matching partition of the matrix?M.Since?M22is Hermitian and has only realeigenvalues,this implies thatthe matrix?A22can only have imaginary eigenvalues.This observation suggests that engineering a quantum system to have a decoherence free amounts to placing some of the poles of the corresponding system(5)on the imaginary axis.

3 Coherent reservoir engineering

Reservoir engineering refers to the process of determining and implementing coupling operatorsL=[L1;...;Ln]for an open quantum system such that desired behaviors are achieved.Examples of common objectives include quantum computation by dissipation[12],entanglement[13],state preparation[14],and protection of quantum information[15,16].Typically open systems have some unavoidable couplings to the environment,and such channels may lead to loss of energy and quantum coherences.However,in many systems couplings can be engineered at the fabrication stage,providing a resource for tuning the behavior of the system.

In this section,the main results of the paper are presented.With reference to Fig.1,we investigate conditions to enable the synthesis of a quantum coherent controller-system network to generate a DFS in the interconnected system through interactions between the principal quantum system and the controller.

Fig.1 Coherent feedback network for DFS generation.

The quantum linear passive system in Fig.1 is the system of the form(5),and its input fields are further partitioned asW=[wT,uT,fT]T.Here,wrepresents a “natural”environment for the system,andfandurepresent an open-loop and feedback engineered fields,respectively.According to this partitioning,the system evolution is described as

Accordingly,the matrices of the system have dimensions as follows:Ap∈ Cn×n,B1∈ Cn×nw,B2∈ Cn×nu B3∈Cn×nf,andCp∈ Cnw×n(n,nw,nu,n f∈ N).We also use the notationapfor the vectorap(t)=[ap1(t),...apn(t)]Tof the system annihilation operators defined on its underlying Hilbert space ?p.

In terms of the Hamiltonian and coupling operators,the system has the Hamiltonian

whereMis ann×ncomplex Hermitian matrix,and is linearly coupled to the input fields via the coupling operators

where α1∈ Cnw×n,α2∈ Cnu×n,α3∈ Cnf×nare complex matrices.Then the relations(6)specialize as follows:

The starting point of the discussion that follows is the assumption that under the influence of its natural environmentwalone,(i.e.,in the absence of the engineered fieldsfandu),the system does not possess a DFS.Mathematically,this assumption corresponds to the assumption that(Ap,B1)is controllable and(Ap,Cp)is observable,since these properties rule out the existence of a DFS in the plant(11)whenB2=0,B3=0;see Proposition 1 and[2,4].

3.1 Open loop reservoir engineering for DFS generation

In many cases,system couplings can be engineered at a fabrication stage to reduce unavoidable loss of energy due to decoherence[1,12].The process of tuning the system at the fabrication stage does not involve feedback,and we letLp2=0,which corresponds to α2=0 andB2=0 in(11);see Fig.2.

Fig.2 Open loop setup for DFS generation.

Then system(11)reduces to that of the form

Here,wandfsymbolize the natural environment and the fabricated open-loop field,respectively.Accordingly,the coupling operatorLp1corresponds to a fixed coupling with the natural environment,while the couplingLp3corresponds to the engineered coupling.The physical realizability requirement imposes the constraint that

cf.(7).Recall[10]that a quantum stochastic differential equation of the form(14)is said to be(canonically)physically realizable if it preserves the canonical commutation relations,[ap,a?p]=apa?p-(a＊paTp)T=I,and is a representation of an open harmonic oscillator,i.e.,it possesses a Hamiltonian and a coupling operator.The satisfaction of the identity(15)is a necessary and sufficient condition for physical realizability[10,Theorem 5.1].

Theorem 1Suppose(-iM,B1)is controllable.Then a DFS cannot be created by coupling the system to an engineered environment.

ProofTo prove the theorem we will show that the matrixAphas all its eigenvalues in the open left halfplane of the complex plane,and therefore it cannot have a DFS,according to Proposition 1;see[4,Lemma 2].

First consider the system with a fixed coupling with the environment,i.e.,Lp3=0.For this system,the physical realizability properties dictate that

with controllable(Ap1,B1),which has a positive definite solutionP=I.SinceB1B?1?0,according to the inertia theorem[17,Theorem 3],the above observation about the existence of a positive definite solution to the Lyapunov equation implies thatAp1must have all its eigenvalues in the open left half-plane of the complex plane.As a result,if(-iM,B1)is controllable,the corresponding passive quantum system with fixed coupling cannot have a DFS,according to Proposition 1.

Next consider this system when it is coupled to an engineered environment,i.e.,Lp3?0 andB3?0.SinceAp1has been shown to have all eigenvalues in the open left half-plane of the complex plane,there exists a positive definite Hermitian matrixP=P?＞ 0 such that

Next,suppose that(-iM,[B1B3])isnotcontrollable1Here,[B1 B3]is the matrix obtained by concatenating the rows of B1 and B3.,therefore(-iM,B1)is not controllable either.Theorem 1 does not rule out a possibility for a DFS to exist in this case.It is easy to show that

where Cw,Cfare the controllability matrices with respect to the inputswandf,respectively.From this observation,it follows that the dimension of the DFS of system(14)is less or equal to the dimension of each of the decoherence free subsystems arising when the quantum plant is coupled with the fixed and engineered fields only.This leads to the conclusion that coupling the system with additional engineered fields can only reduce the dimension of the DFS.In the remainder of the paper,we will show that using coherent feedback,on the other hand,does allow to create or increase dimension of a DFS.

3.2 Coherent feedback reservoir engineering

In this section we consider a system of the form(11).To simplify the notation we willcombine two static channelswandfinto a single channel,which will again be denoted asw.More precisely,we combine the coupling operatorsLp1andLp3into a single operatorLp1.Then the system(11)reduces to a system of the form

where the new matrixB1is composed of the previous matricesB1andB3,so that using the new notation we have

For a coherent quantum controller for the quantum plant(11),we willconsideranotheropen quantum linear annihilation only system.Such a system willbe assumed to be coupled with three environment noise channels,y′,z′andv.The fieldsy′,z′are to produce output fields which will be used to form the feedback,and the channelvwill be used to ensure that the constructed observer is physically realizable.As is known[10],once physical realizability of the observer is ensured,one can readily construct a scattering matrix,a Hamiltonian and a collection of coupling operators describing the quantum evolution of the controller in the form of a quantum stochastic differential equation(2).Alternatively,a physically realizable coherent controller can be represented in the form of the quantum stochastic differential equation(5)[10],i.e.,in the form

where forphysicalrealizability,the following constraints must be satisfied[10,Theorem 5.1]:

Interconnection between the controller and the plant are through scattering equations relating the output fields of the plant with the input channels of the controllerandvice versa.Specifically,the scattering equation

links the output field of the plantyand the controller environmentzwith the input controller channelsy′,z′.Here,Sis a unitary matrix partitioned as

Likewise,feedback from the controller(19)is via a unitary matrixW,

The matricesAc,G1=-K?,G2=-?K?,G3,and the scattering matricesS,Ware regarded as the controller design parameters.Our objective in this paper is to find a procedure for selecting those parameters so that the resulting coherently interconnected quantum system in Fig.1 possesses a decoherence free subsystem.

To devise the DFS synthesis procedure,we first note that the control system governed byy,z,vand outputucan be represented as

Also,the closed-loop system is described by the quantum stochastic differential equation

with block matricesAcl,Bclpartitioned as shown in(28).

Lemma 1Let

Then forAclto have all eigenvalues on the imaginary axis or in the left half-plane of the complex plane it is necessary and sufficient that the following matrices

have all eigenvalues on the imaginary axis or in the left half-plane of the complex plane.

ProofThe matrixAclhas the same eigenvalues as the matrix

Hence the lemma follows,due to the definition ofAcin(29). □

Theorem 2Suppose matricesS,Ware given.LetG1,G2be such that:

a)The following linear matrix inequality(LMI)inG1,G2is satisfied

where

b)The matrices?AandˇA,defined in equations(30)and(31)respectively,have all their eigenvalues in the closed left half-plane,with at least one of them having eigenvalues on the imaginary axis.

Then a matrixG3can be found such that the closedloop system(27)admits a DFS.

ProofVia the Schur complement,(32)is equivalent to

Therefore one can findG3such that

From this identity and the expression(29),the identity(20)follows.This shows that the feasibility of the LMI(32)ensures thatthe controllersystem(19)can be made physically realizable by appropriately choosingG3.As a result,the closed-loop system,being a feedback interconnection of physically realizable systems,is a physically realizable annihilation only system.Also,condition(b)and Lemma 1 ensure thatAclhas eigenvalues on the imaginary axis.Then it follows from Proposition 1 that the closed-loop system(27)has a DFS. ?

We next demonstrate that our pole assignment problem captured quantum plant-controller DFS architectures considered in[2,6].

3.3 Special case 1:DFS synthesis using a coherent observer[6]

In[6],the DFS synthesis was carried outusing a quantum analog of the Luenberger observer for a class of linear annihilation only systems with a Hamiltonian and a coupling operatordescribed in(12),(13);see Fig.3.This controller structure is a special case of the architecture in Fig.1,when the two channelswandfare combined as per(17),and

With this choice ofSandW,we have from(29)

Fig.3 Coherent plant-observer network considered in[6].

Corollary 1Suppose the pair(Ap,Cp)is observable and the pair(Ap,B2)is controllable.LetG1,G2be such that

a)The following linear matrix inequality(LMI)is satisfied

b)The matrices

have all eigenvalues on the imaginary axis or in the left half-plane of the complex plane,with at least one of them having eigenvalues on the imaginary axis.

Then the closed-loop system admits a DFS.

ProofVia the Schur complement,condition(37)is equivalent to the condition

This ensures that

Therefore,one can findG3such that the controller is physically realizable.The claim then follows from Theorem 2. ?

3.4 Special case 2:coherent feedback DFS generation model from[2]

Consider a system of Fig.1 in whichS=I,W=I,and letG2=0,G3=0.This corresponds to the system shown in Fig.4,which was considered in[2].In this case,the controller matrix becomes

Fig.4 Special case 2:coherent feedback network for DFS generation considered in[2].

Corollary 2Suppose the pair(Ap,Cp)is observable and the pair(Ap,B2)is controllable.LetG1be such that

a)The following equation is satisfied

b)The matrices

have all eigenvalues on the imaginary axis or in the left half-plane of the complex plane,with at least one of them having eigenvalues on the imaginary axis.

Then the closed-loop system admits a DFS.

ProofCondition(42)ensures that

4 Examples

4.1 Example 1

To illustrate the DFS synthesis procedure developed in the previous section,consider a system consisting of two optical cavities interconnected as shown in Fig.3.The system is similar to those considered in[5].

The cavity to be controlled is described by equation(17),with all matrices becoming complex numbers

Here,κ1,κ2are real nonnegative numbers,characterizing the strength of the couplings between the cavity and the input fieldswandu,respectively,andMcharacterizes the Hamiltonian of the cavity.

Clearly,the pair(Ap,Cp)is observable and the pair(Ap,B2)is controllable,therefore the optical cavity cannot have a DFS unless the cavity is lossless.To synthesize a DFS,let us connect this cavity to another optical cavity with the same Hamiltonian,as shown in Fig.3.This corresponds to letting the controller have the coefficients

and letting the scattering matricesSandWbe

Also we must satisfy the LMI condition(37).The matrixRin this example reduces to

Hence,using(50)and(51),the LMI condition(37)reduces to the two following inequalities:

The inequality(52)is the only constraint for the remaining coupling parameter κ4to be determined.Notice that there is an obvious solution to this inequality in the case where κ1= κ2= κ.The solution is κ3= κ4= κ which satisfies both(50)and(52).

The above calculations demonstrate that by placing the pole of the controller on the imaginary axis,one can effectively create a DF mode which did not exist in the original system.This fact has been established previously in[5]by calculating the system poles,whereas we have arrived at this conclusion from a more general Corollary 1,as a special case.

4.2 Example 2

We now present an example in which,the DFS is created which is shared between the controlled system and the controller.The controlled system in thisexample consists of two cavities as shown in Fig.5.

Fig.5 The two-cavity system for Example 2.

Denote the matrices associated of the Hamiltonians corresponding to the each cavity internal dynamics asM1,M2.Also for the convenience of notation,define the complex numbers

associated with the coupling strengths within the cavities.All four constants are assumed to be nonzero.

Then the equations governing the dynamics of the two-cavity system have the form of(17)with

To verify observability ofthe pair(Ap,Cp),we observe that

These observations allow us to apply Corollary 1 to construct a DFS by interconnecting the two-cavity system with a coherent quantum observer,which we now construct.

For simplicity,choose

It remains to show that the LMI condition(37)is satisfied in this example.Noting that with the above choice ofg1,g2,R＜ 0 holds provided|γ4|2＞ 4|γ2|2.Next,the LMI(37)in this example requires that

Using the Schur complement,this requirement is equivalent to

5 Conclusions

In this paper,we have proposed a general coherent quantum controller synthesis procedure for generating decoherence free subspaces in quantum systems.Decoherence free components capable of storing quantum information are regarded to be essential for quantum computation and communication,as quantum memory elements[5].When the feedback loop is in the DFS configuration,the DFS mode is “protected”,which also means that to access dynamics of that mode,the system must be augmented with a mechanism to dynamically change the feedback configuration in order to bring the system in and out of the “DF state”,e.g.,from the above examples,we see that adjusting the values of coupling strengths is one possibility to achieve this.However,this approach is only applicable for experimental systems which have tunable coupling devices available,such as an optical waveguide or a microwave superconducting cavity.Another viable approach for the systems in those examples would be changing the loop configuration by using optical switches to either break the feedback loop or form an additional feedback connection,i.e.,form a double-pass feedback loop;the latter is essentially the approach presented in[5].Ourfuture work willconsider these approaches in greater detail,to obtain general dynamical reading and writing procedures augmenting our general results in a fashion similar to how this has been done in[5]for optical cavity systems.

[1]J.F.Poyatos,J.I.Cirac,P.Zoller.Quantum reservoir engineering with laser cooled trapped ions.Physical Review Letters,1996,77(23):4728-4731.

[2]N.Yamamoto.Coherent versus measurement feedback:Linear systems theory forquantum information.PhysicalReview X,2014,4(4):DOI 10.1103/PhysRevX.4.041029.

[3]M.R.James,J.E.Gough.Quantum dissipative systems and feedback control design by interconnection.IEEE Transactions on Automatic Control,2010,55(8):1806-1821.

[4]J.E.Gough,G.Zhang.On realization theory of quantum linear systems.Automatica,2015,59:139-151.

[5]H.I Nurdin,J.E.Gough.Modular quantum memories using passive linear optics and coherent feedback.Quantum Information&Computation,2015,15(11/12):1017-1040.

[6]Y.Pan,T.Nguyen,Z.Miao,et al.Coherent observer engineering for protecting quantum information.Proceedings of the 35th Chinese Control Conference,Chengdu:IEEE,2016:9139-9144.

[7]T.Nguyen,Z.Miao,Y.Pan,et al.Pole placement approach to coherent passive reservoir engineering for storing quantum information.Proceedings of the American Control Conference,Seattle:IEEE,2017:234-239.

[8]H.-P.Breuer,F.Petruccione.The Theory of Open Quantum Systems.Oxford:Oxford University Press,2002.

[9]H.M.Wiseman,G.J.Milburn.Quantum Measurement and Control.Cambridge:Cambridge university press,2009.

[10]A.I.Maalouf,I.R.Petersen.Bounded real properties for a class of annihilation-operator linear quantum systems.IEEE Transactions on Automatic Control,2011,56(4):786-801.

[11]G.Zhang,M.R.James.Direct and indirect couplings in coherent feedback control of linear quantum systems.IEEE Transactions on Automatic Control,2011,56(7):1535-1550.

[12]F.Verstraete,M.M.Wolf,J.I.Cirac.Quantum computation and quantum-state engineering driven by dissipation.Nature Physics,2009,5(9):633-636.

[13]H.Krauter,C.A.Muschik,K.Jensen,et al.Entanglement generated by dissipation and steady state entanglement of two macroscopic objects.Physical Review Letters,2011,107(8):DOI 10.1103/PhysRevLett.107.080503.

[14]F.Ticozzi,L.Viola.Analysis and synthesis of attractive quantum markovian dynamics.Automatica,2009,45(9):2002-2009.

[15]J.Cohen,M.Mirrahimi.Dissipation-induced continuous quantum error correction for superconducting circuits.Physical Review A,2014,90(6):DOI 10.1103/PhysRevA.90.062344.

[16]Y.Pan,T.Nguyen.Stabilizing quantum states and automatic error correction by dissipation control.IEEE Transactions on Automatic Control,2016:DOI 10.1109/TAC.2016.2622694.

[17]C.-T.Chen.A generalization of the inertia theorem.SIAM Journal on Applied Mathematics,1973,25(2):158-161.

[18]A.Ostrowski,H.Schneider.Some theorems on the inertia of general matrices.Journal of Mathematical analysis and applications,1962,4(1):72-84.

5.School of Engineering and Information Technology,University of New South Wales at the Australian Defence Force Academy,Canberra,ACT 2600,Australia

16 February 2017;revised 7 May 2017;accepted 7 May 2017

DOI 10.1007/s11768-017-7020-2

?Corresponding author.

E-mail:v.ugrinovskii@gmail.com.Tel.:+61 2 6268 8219;fax:+61 2 6268 8443.

This paper is dedicated to Professor Ian R.Petersen on the occasion of his 60th birthday.This work was supported by grants CE110001027 from Australian Research Council Centre of Excellence for Quantum Computation and Communication Technology,FA2386-12-1-4084 from AFOSR,and DP140101779 from Australian Research Council,respectively.N.H.A.acknowledges CNRS funding under the JCJC INS2I 2016 “QIGR3CF”and JCNC INS2I 2017 “QFCCQI”projects.

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

Thien NGUYENis a Ph.D.candidate in the Quantum Cybernetics group at the Australian NationalUniversity.Priorto enrolling at the ANU,he worked for three years as a Quality Engineer for Intel Corporation.He received the B.Sc.degree in Electrical Engineering from Portland State University,Oregon,U.S.A.in 2011,while being an Intel Scholarship recipient.His current research interests include quantum computer architecture,quantum error correction and quantum control.He was the Grand Prize winner of the 2016 Microsoft Quantum Challenge.E-mail:thien.nguyen@anu.edu.au.

Zibo MIAOreceived the B.Eng.degree in Automation from the Honors School,Harbin Institute of Technology,Harbin,China,in 2011.He received the Ph.D.degree in Quantum Cybernetics from the Australian National University,Canberra,Australia,in 2015.He was successful in receiving special funding from the Australian Academy of Science to undertake research in the United States in 2014-2015.In 2015-2016,he was a research fellow with the Department of Electrical and Electronic Engineering,University of Melbourne,Parkville Campus,Melbourne,Australia.Currently,he is a postdoctoral research fellow with the QUANTIC team,INRIA Paris,France.He was selected as Student Best Paper Finalist in the 2015 American Control Conference.His current research interests include quantum engineering,optimal control and stochastic control.E-mail:shenwum@gmail.com.

Yu PANreceived the Ph.D.degree from the Academy of Mathematics and Systems Science,Chinese Academy of Sciences,2012.Before that,he received a B.Sc.degree in Mathematics and Applied Mathematics from Sichuan University.From 2012 to 2015,he was a research fellow in the College of Engineering at the Australian National University,Canberra.He was a research associate in the School of Engineering and Information Technology,at University of New South Wales,Canberra from 2015 to 2016,and a postdoctoral fellow at the Hong Kong Polytechnic University in 2016.He has joined the Institute of Cyber-Systems and Control,Zhejiang University,China in July 2016.His research interests include quantum systems,quantum information and advanced computational intelligence.E-mail:ypan@zju.edu.cn.

Nina AMINIis a CNRS researcher at Laboratory L2S at CentraleSup’elec since October 2014.She did her first postdoc from June 2012 for six months at the College of Engineering and Computer Science,the Australian National University,and her second postdoc at Edward L.Ginzton Laboratory,Stanford University,since December 2012.She received her Ph.D.in Mathematics and Control Engineering from Mines-ParisTech(Ecole des Mines de Paris),in September 2012.Prior to her Ph.D.,she earned a M.Sc.degree in Financial Mathematics and Statistics at ENSAE and the Engineering Diploma of l’Ecole Polytechnique,in 2009.Her research interests include(quantum)control,(quantum)filtering theory,(quantum)probability,and(quantum)information theory.E-mail:nina.amini@lss.supelec.fr.

Valery UGRINOVSKIIreceived the undergraduate degree in Applied Mathematics and the Ph.D.degree in Physics and Mathematics from the State University of Nizhny Novgorod,Russia,in 1982 and 1990,respectively.From 1982 to 1995,he held research positions with the Radiophysical Research Institute,Nizhny Novgorod.From 1995 to 1996,he was a Postdoctoral Fellow at the University of Haifa,Israel.He is currently full Professor in the School of Engineering and Information Technology,at the University of New South Wales Canberra.He has held visiting positions at the Australian National University,Stuttgart University and University of Illinois at Urbana-Champaign.He is an Associate Editor forAutomatica,IEEE Control Systems Letters(IEEE L-CSSandIET Control Theory and Applications.E-mail:v.ugrinovskii@gmail.com.

Matthew R.JAMESwas born in Sydney,Australia,in 1960.He received the B.Sc.degree in Mathematics and the B.E.(Hon.I)in Electrical Engineering from the University of New South Wales,Sydney,Australia,in 1981 and 1983,respectively.He received the Ph.D.degree in Applied Mathematics from the University of Maryland,College Park,U.S.A.,in 1988.In 1988-1989 he was Visiting Assistant Professor with the Division of Applied Mathematics,Brown University,Providence,U.S.A.,and from 1989 to 1991 he was Assistant Professor with the Department of Mathematics,University of Kentucky,Lexington,U.S.A.In 1991 he joined the Australian National University,Australia,where he served as Head of the Department of Engineering during 2001 and 2002.He has held visiting positions with the University of California,San Diego,Imperial College,London,and University of Cambridge.His research interests include quantum,nonlinear,and stochastic control systems.He is a co-recipient(with Drs L.Bouten and R.Van Handel)of the SIAM Journal on Control and Optimization Best Paper Prize for 2007.He is currently serving as Associate Editor for EPJ Quantum Technology and Applied Mathematics and optimization,and has previously served IEEE Transactions on Automatic Control,SIAM Journal on Control and Optimization,Automatica,and Mathematics of Control,Signals,and Systems.He is a Fellow of the IEEE,and held an Australian Research Council Professorial Fellowship during 2004-2008.E-mail:matthew.james@anu.edu.au.