Ian R.PETERSEN

Research School of Engineering,The Australian National University,Canberra ACT 2601,Australia

Time averaged consensus in a direct coupled coherent quantum observer network

Ian R.PETERSEN

Research School of Engineering,The Australian National University,Canberra ACT 2601,Australia

This paper considers the problem of constructing a direct coupling quantum observer for a closed linear quantum system.The proposed distributed observer consists of a network of quantum harmonic oscillators and it is shown that the observer network converges to a consensus in a time averaged sense in which each element of the observer estimates the specified output of the quantum plant.An example and simulations are included to illustrate the properties of the observer network.

Quantum systems,quantum observers,quantum networks

1 Introduction

A number of papers have recently considered the problem of constructing a coherent quantum observer for a quantum system;see[1-4].In the coherent quantum observer problem,a quantum plant is coupled to a quantum observer which is also a quantum system.The quantum observer is constructed to be a physically realizable quantum system so that the system variables of the quantum observer converge in some suitable sense to the system variables of the quantum plant.

In the papers[1,2,4],the quantum plantunderconsideration is a linearquantum system.In recentyears,there has been considerable interestin the modeling and feedback control of linear quantum systems;e.g.,see[5-8].Such linear quantum systems commonly arise in the area of quantum optics;e.g.,see[9,10].For such linear quantum system models an important class of quantum control problems are referred to as coherent quantum feedback control problems;e.g.,see[5,6,11-20].In these coherent quantum feedback control problems,both the plant and the controller are quantum systems and the controller is typically to be designed to optimize some performance index.The coherent quantum observer problem can be regarded as a special case of the coherent quantum feedback control problem in which the objective of the observer is to estimate the system variables of the quantum plant.

In some of the previous papers on quantum observers such as[1-3],the coupling between the plant and the observer is via a field coupling.This leads to an observer structure of the form shown in Fig.1.This enables a one way connection between the quantum plant and the quantum observer.Also,since both the quantum plant and the quantum observer are open quantum systems,they are both subject to quantum noise.

Fig.1 Coherent observer structure with field coupling.

However in the paper[13],a coherent quantum control problem is considered in which both field coupling and direct coupling is considered between the quantum plant and the quantum controller.In this paper,we explore the construction of a coherent quantum observer in which there is only direct coupling between quantum plant and the quantum observer.Furthermore,both the quantum plant and the quantum observer are assumed to be closed quantum systems which means that they are not subject to quantum noise and are purely deterministic systems.This leads to an observer structure of the form shown in Fig.2.It is shown that for the case being considered,a quantum observer can be constructed to estimate some but not all of the system variables of the quantum plant.Also,the observer variables converge to the plant variables in a time averaged sense rather than a quantum expectation sense such as considered in the papers[1,2].

Fig.2 Coherent observer structure with direct coupling.

In this paper,we consider the construction of a direct coupling quantum observer for a linear quantum plant and consider the case in which the quantum observer has the structure of an observer network make up of a collection of observer elements.This observer network is constructed so that the output of each observer element converges to the output of the quantum plant in a time averaged sense.This means that there is a consensus of the observer network element in estimating the output of the quantum plant.In recent years,there has been significant interest in controlling networks of multi-agent systems to achieve a consensus among the agents;e.g.,see[21-25].In particular,some authors have looked at the problem of consensus in distributed estimation problems;e.g.,see[26,27].Furthermore,issues of consensus have been considered in networked quantum systems;see[28-32].This work is motivated by the fact that it is becoming increasingly possible for quantum control experiments to involve the networked interconnection of many quantum elements and these quantum networks will have important applications in problems such as quantum communication and quantum information processing.Also,many macroscopic systems can be regarded as consisting of a large quantum network.These issuesmotivate the direct coupled coherent quantum observer network problem being considered in this paper.

The results presented in this paper build on some of the results presented in the preliminary conference papers[33-35]).However,the results presented here provide a significant generalization compared to the results of[33-35].In particular,in this paperwe allow fora non-zero Hamiltonian forthe quantum plant,whereas in the papers[33-35],the plant Hamiltonian was assumed to be zero.Also,in the paper[33],the quantum observer did not have a network structure and corresponds to a special case of the current paper in which the quantum observer network has only a single element.In addition,the paper[34],restricts attention to quantum observer networks having a simple chain structure and for which the quantum plant and each element of the quantum observer network contains only a single mode.Finally,the paper[35]considers the case in which the quantum plant is a single qubit rather than a quantum linear system as considered in this paper.Also,it is assumed in[35]that each element of the quantum observer network contains only a single mode.

In addition to the papers[33-35],a number of other conference papers have considered problems related to the current problem.The paper[36]considers the case in which the quantum plant is a single qubit and the quantum observer is a single mode quantum linear system.The paper[37]considers the problem of an experimental implementation of the results of[33].The paper[38]considers the problem ofan experimentalimplementation of the results of[33]with the modification that the quantum observer allows for a measurement of its output using Homodyne detection.The paper[39]considers a modification of the results of[33]to allow for a reduced order quantum observer.The paper[40]modifies the approach of[34]to allow for a chain structured observer network which would be more straightforward to implement experimentally than the approach proposed in[34].

2 Quantum systems

In the quantum observer network problem under consideration,both the quantum plant and the quantum observer network are linear quantum systems;see also[5,13,41].We will restrict attention to closed linear quantum systems which do not interact with an externalenvironment.The quantum mechanicalbehaviorofa linear quantum system is described in terms of the systemobservableswhich are self-adjoint operators on an underlying infinite dimensional complex Hilbert space?.The commutator of two scalar operatorsxandyon? is defined as[x,y]=x y-yx.Also,for a vector of operatorsxon ?,the commutator ofxand a scalar operatoryon ? is the vector of operators[x,y]=x y-yx,and the commutator ofxand its adjointx?is the matrix of operators

The dynamics of the closed linear quantum systems under consideration are described by non-commutative differential equations of the form

The initial system variablesx(0)=x0are assumed to satisfy thecommutation relations

where Θ is a real skew-symmetric matrix with components Θjk.In the case of a single quantum harmonic oscillator,we will choosex=(x1,x2)Twherex1=qis the position operator,andx2=pis the momentum operator.The commutation relations are[q,p]=2i.In general,the matrix Θ is assumed to be of the form

whereJdenotes the real skew-symmetric 2×2 matrix

where Θ is defined as in(3);e.g.,see[5].In this case,the system variablesx(t)will satisfy thecommutation relationsat all times:

That is,the system will bephysically realizable;e.g.,see[5].

QuantumplantIn ourproposed directcoupling coherent quantum observer network,the quantum plant is a linear quantum system of the form(1)described by the non-commutative differential equations

Quantum observer networkWe now describe the linear quantum system of the form(1)which will correspond to the quantum observer network;see also[5,13,41].This system is described by non-commutative differential equations of the form

where Θois of the form(3).Furthermore,we will assume that the quantum observer network has a graph structure withNnodes and is coupled to the quantum plant as illustrated in Fig.3.

Fig.3 The graph(G,E)for a typical quantum observer network.

The combined plant observer system is described by a connected graph(G,E)which hasN+1 nodes with node 0 corresponding to the quantum plant and the remaining nodes,labelled 1,2,...,N,corresponding to the observer elements.This corresponds to an observer Hamiltonian of the form

where the vector of observer system variablesxois partitioned according to each element of the quantum observer network as follows:

We assume that the variables for each element of the quantum observer network commute with the variables of all other elements of the quantum observer network;i.e.,

Also,we partition the matrix Θoas

where each matrix Θoiis also of the form(3).

We define a coupling Hamiltonian which defines the coupling between the quantum plant and the quantum observer network:

Furthermore,we write

Note thatRoi∈ Rnoi×noi,Rcij∈ Rnoi×noj,Coi∈ Rmp×noi,and each matrixRoiis symmetric fori=1,2,...,N,j=1,2,...,N.In addition,Rc0j∈ Rnp×noiforj=1,2,...,N.Also,the matricesRcijfori=0,1,...,N,j=1,2,...,Nare such thatRcij?0 if and only if(i,j)∈E,the set of edges for the graph(G,E).

The augmented quantum linear system consisting of the quantum plant and the quantum observer network is described by the total Hamiltonian

Then using(4),it follows that the augmented quantum linear system is described by the equations

whereAa=2ΘaRa,

We now formally define the notion of a direct coupled linear quantum observer network.

Definition 1The matricesRoj,Rcij,Cojfori=0,1,...,N,j=1,2,...,Nand the graph(G,E)define alinear quantum observer networkachieving time-averaged consensus convergence for the quantum plant(6)if the corresponding augmented linear quantum system(12)is such that

3 Constructing a direct coupling coherent quantum observer network

and α0i∈ R2×1fori=1,2,...,mp.This assumption means that the plant variables to be estimated include only one quadrature for each mode of the plant.Also,we assume

fori=1,2,...,mp.Corresponding to the form(15),we can partition the vector of plant variables as

where eachxpiis a 2 by 1 vector of plant variables fori=1,2,...,mp.

In addition,we assume thatRpis of the form

whereM=MT.It thatApin(6)is of the form

Hence,it follows from(6)that

sinceJis skew-symmetric.Therefore

That is,the vector of plant variables to be estimatedzp(t)will remain fixed if the plant is not coupled to the observer network.However,when the plant is coupled to the quantum observer network this may no longer be the case.We will show that if the quantum observer is suitably designed,the plant quantity to be estimatedzp(t)will remain fixed and the condition(14)will be satisfied.

We assume that each element of the observer network is of dimensionnpand that the vector of observer variablesxoican also be partitioned as in(16)as

fori=0,1,...,N.Here,eachxoijis a 2 by 1 vector of observer variables.We also suppose that the matricesRcij,Roifori=0,1,...,N,j=1,2,...,Nare of the form

where αij∈ Rnoi×mp,βij∈ Rnoj×mpand ωi＞ 0 fori=1,2,...,N,j=1,2,...,N.Also,we assume that

forj=1,2,...,Nsuch that(0,j)∈E.In addition,note that αij=0 and βij=0 for(i,j)?E.Furthermore,we assume

fori=1,2,...,N.

We will show that these assumptions imply that the quantityzp(t)=Cpxp(t)will be constant for the augmented quantum system(12).Indeed,the total Hamiltonian(10)will be given by

We will show that these assumptions imply that the quantityzp(t)=Cpxp(t)will be constant for the augmented quantum system(12).Indeed,it follows from(11)-(13)that

However,it follows from(18)that αT0Θpα0=0 and hence,

for allt?0.

Also,it follows from(9)and(11)-(13)that f

orj=1,2,...,N.

To construct a suitable quantum observer network,we will further assume that

fori=1,...,N,j=1,2,...,N,where(i,j)∈E.Here,

Also,we will assume that

forj=1,2,...,Nwhere(0,j)∈E.

In order to construct suitable values for the quantities μijand ωiso that(14)is satisfied,we will require that

forj=1,2,...,N.This condition is equivalent to

for(0,j)∈Eand

for(0,j)?E.

Then,we define

forj=1,2,...,N.It follows from(28)and(24)that

forj=1,2,...,N.

We now write this equation as

whereAois anN×Nblock matrix with blocks

fori=1,2,...,N,j=1,2,...,N.Also,Aois as given in(8)whereRois a symmetricN×Nblock matrix with blocks

fori=1,2,...,N,j=1,2,...,N.

To show that the above candidate quantum observer network leads to the satisfaction of the condition(14),we note that

satisfies(31).Hence,if we can show that

then it will follow from

that(14)is satisfied.

We now show that the symmetric matrixRois positive-definite.

Lemma 1The matrixRois positive definite.

ProofIn order to establish this lemma,let be a non-zero real vector.Then

using(15),(19)and the Cauchy-Schwarz inequality.We now define

fori=1,2,...,N.Again using the Cauchy-Schwarz inequality,it follows that

for 0=1,2,...,N,j=1,2,...,N.Thus,(35)implies

fori=1,2,...,N,j=1,2,...,N.

for(0,j)?E.Hence,we can write

Fig.4 The weighted graph (G,E)in Fig.3.

That is

for allt?0.

Now since Θ andRoare non-singular,

and therefore,it follows from(37)that

asT→∞.Hence,

and hence,it follows from(31)and(33)that

Also,(23)implies

Therefore,condition(14)is satisfied.Thus,we have established the following theorem.

Theorem 1Consider a quantum plant of the form(6)whereRpis of the form(17).Then the matricesRoi,Rcij,Coi,Roifori=1,2,...,N,j=1,2,...,Nand the connected graph(G,E)will define a direct coupled quantum observer network achieving timeaveraged consensus convergence for this quantum plant if the conditions(20)-(22),(25)-(27),(29)and(30)are satisfied.

4 Illustrative example

Fig.5 Quantum observer network.

The augmented plant-observersystem is described by equations(12)and(11).Then,we can write

Thus,the plant variable to be estimatedzp(t)is given by

e1is the first unit vector in the standard basis for RN+1,Φi(t)is theith column of the matrix Φ(t)andxai(0)is theith component of the vectorxa(0).We plot each of the quantitiese1CaΦ1(t),e1CaΦ2(t),...,e1CaΦ2N+2(t)in Fig.6.

From this figure,we can see thate1CaΦ1(t)≡ 1 ande1CaΦ2(t)≡ 0,e1CaΦ2(t)≡ 0,...,e1CaΦ2N+2(t)≡ 0,andzp(t)will remain constant atzp(0)for allt?0.

We now consider the output variables of the quantum observer networkzoi(t)fori=1,2,...,Nwhich are given by

whereei+1is the(i+1)th unit vector in the standard basis for RN+1.We plot each of the quantitiesei+1CaΦ1(t),ei+1CaΦ2(t),...,ei+1CaΦ2N+2(t)in Figs.7-11.

Fig.6 Coefficients defining zp(t).

Fig.7 Coefficients defining zo1(t).

Fig.8 Coefficients defining zo2(t).

Fig.9 Coefficients defining zo3(t).

Fig.10 Coefficients defining zo4(t).

Also,we can consider the spatial average obtained by averaging over each of the distributed observer outputs:

Fig.11 Coefficients defining zo5(t).

Fig.12 Coefficients defining zos(t).

Fig.13 Coefficients defining the time average of zo1(t).

Fig.14 Coefficients defining the time average of zo2(t).

Fig.15 Coefficients defining the time average of zo3(t).

Fig.16 Coefficients defining the time average of zo4(t).

Fig.17 Coefficients defining the time average of zo5(t).

From these figures,we can see that for eachi=1,2,...,N,the time average ofzoi(t)converges tozp(0)ast→ ∞.That is,the distributed quantum observer reaches a time averaged consensus corresponding to the output of the quantum plant which is to be estimated.

5 Conclusions

In this paper we have considered the construction of a direct coupling observer network for a closed quantum linear system in order to achieve a time averaged consensus convergence.We have also presented an illustrative example along with simulations to investigate the consensus behavior of the direct coupling observer network.

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14 February 2017;revised 8 May 2017;accepted 8 May 2017

DOI 10.1007/s11768-017-7019-8

E-mail:i.r.petersen@gmail.com.

This work was supported by the Air Force Office of Scientific Research(AFOSR),under agreement number FA2386-16-1-4065.Some of the research presented in this paper was also supported by the Australian Research Council under grant FL110100020.

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

Ian R.PETERSENwas born in Victoria,Australia.He received a Ph.D in Electrical Engineering in 1984 from the University of Rochester.From 1983 to 1985 he was a Postdoctoral Fellow at the Australian National University.From 1985 until 2016 he was with UNSW Canberra where was most recently a Scientia Professor and an Australian Research Council Laureate Fellow in the School of Engineering and Information Technology.From 2017 he has been a Professor in the Research School of Engineering at the Australian National University.He has served as an Associate Editor for the IEEE Transactions on Automatic Control,Systems and Control Letters,Automatica,and SIAM Journal on Control and Optimization.Currently he is an Editor for Automatica and an Associate Editor for the IEEE Transactions on Control Systems Technology.He is a fellow of IFAC,the IEEE and the Australian Academy of Science.His main research interests are in robust control theory,quantum control theory and stochastic control theory.E-mail:i.r.petersen@gmail.com