The Kalman Decomposition for Linear Quantum Stochastic Systems
Symeon Grivopoulos, Guofeng Zhang, Ian R. Petersen, John Gough
TThe Kalman Decomposition for Linear Quantum Stochastic Systems
Symeon Grivopoulos Guofeng Zhang Ian R. Petersen John Gough Abstract — The Kalman decomposition for Linear QuantumStochastic Systems in the real quadrature operator represen-tation, that was derived indirectly in [1] by the authors, isderived here directly, using the “one-sided symplectic” SVD-likefactorization of [2] on the observability matrix of the system.
I. I
NTRODUCTION
Linear Quantum Stochastic Systems (LQSSs) are a classof models used in linear quantum optics [3], [4], [5], circuitQED systems [6], [7], quantum opto-mechanical systems [8],[9], [10], [11], and elsewhere. The mathematical frameworkfor these models is provided by the theory of quantumWiener processes, and the associated Quantum StochasticDifferential Equations [12], [13], [14]. Potential applicationsof LQSSs include quantum information processing, andquantum measurement and control. In particular, an impor-tant application of LQSSs is as coherent quantum feedbackcontrollers for other quantum systems, i.e. controllers thatdo not perform any measurement on the controlled quantumsystem, and thus, have the potential to outperform classicalcontrollers, see e.g. [15], [16], [17], [18], [19], [20], [21],[10], [22].Controllability (stabilizability) and observability (de-tectability) of a classical linear system are necessary andsufficient conditions for the existence of a stabilizing con-troller for it, and thus, prerequisites for various controldesign methods. These notions, and the related mathemat-ical concepts and techniques, can be transferred essentiallyunchanged to LQSSs, where, again, they are prerequisite forvarious design methods, see e.g. [17], [18], [23]. There is,however, an important difference from the classical case:The allowed state transformations in LQSSs (for the purposeof related state-space decompositions) cannot be arbitrary,but are fundamentally restricted by the laws of quantummechanics. More specifically, in the so called real quadratureoperator representation of an LQSS that is used in thiswork, the only transformations that preserve its structure(see Subsection II-B) are real symplectic ones. Recently,various investigations of controllability and observability forLQSSs have appeared in the literature, see e.g. [24], [25], Symeon Grivopoulos and Ian R. Petersen are with the Schoolof Engineering and Information Technology, UNSW Canberra, Can-berra BC 2610, Australia [email protected],[email protected] Guofeng Zhang is with the Department of AppliedMathematics, The Hong Kong Polytechnic University, Hong Kong
[email protected] John Gough is with the Department of Physics, Aberystwyth University,Wales,cSY23 2BZ, Aberystwyth, UK [email protected]
This work was supported by the Australian Research Council under grantFL110100020 [1]. In [1], the authors of the present work showed that, aKalman decomposition of a LQSS is always possible witha real orthogonal and symplectic transformation. Moreover,they uncovered the following interesting structure in thedecomposition: The controllable/observable ( co ), and uncon-trollable/unobservable subsystems ( ¯ c ¯ o ) are LQSSs in theirown right, as is to be expected from a physics perspec-tive. Furthermore, the states of the controllable/unobservable( c ¯ o ) subsystem are conjugate variables of the states ofthe uncontrollable/observable ( ¯ co ) subsystem. An immediateconsequence of this is that, a c ¯ o subsystem exists if andonly if a ¯ co subsystem does, and they always have the samedimension. This is a consequence of the special structure ofLQSSs.The construction of the Kalman decomposition in [1], isperformed first in the so called creation-annihilation oper-ator representation of a LQSS, where special bases for the co , ¯ c ¯ o , c ¯ o , and ¯ co subspaces are constructed, and the resultis then translated in the real quadrature representation. Weshould point out that the Kalman decomposition of a LQSSin the real quadrature representation offers an advantage overthe corresponding decomposition in the creation-annihilationrepresentation of the LQSS: In the former, the c ¯ o and ¯ co subsystems are separate, as usual, while in the latter, thetwo subsystems are merged, due to the grouping of statesimposed by that representation. In this work, we present aderivation of the Kalman decomposition of a LQSS, directlyin the real quadrature operator representation. This derivationuses the “one-sided symplectic” SVD-like factorization of [2]on the observability matrix of the LQSS, and leads directlyto the desired decomposition. Its value lies in its brevityand directness in uncovering the structure of the Kalmandecomposition of LQSSs.II. B ACKGROUND M ATERIAL
A. Notation and terminology x ∗ denotes the complex conjugate of a complex num-ber x or the adjoint of an operator x , respectively. Fora matrix X = [ x ij ] with number or operator entries, X = [ x ∗ ij ] , X (cid:62) = [ x ji ] is the usual transpose, and X † = ( X ) (cid:62) . The commutator of two operators X and Y is defined as [ X, Y ] = XY − Y X .2) The identity matrix in n dimensions will be denoted by I n , and a r × s matrix of zeros will be denoted by r × s . δ ij denotes the Kronecker delta symbol, i.e. I = [ δ ij ] .We define J k = (cid:16) k × k I k − I k k × k (cid:17) . Also, X X ... X k is thevertical concatenation of the matrices X , X , . . . , X k , a r X i v : . [ qu a n t - ph ] S e p f equal column dimension, ( Y Y . . . Y k ) is the hor-izontal concatenation of the matrices Y , Y , . . . , Y k ofequal row dimension, and diag( Z , Z , . . . , Z k ) is theblock-diagonal matrix formed by the square matrices Z , Z , . . . , Z k .3) For a r × s matrix X , define its (cid:93) - adjoint X (cid:93) , by X (cid:93) = − J s X † J r . The (cid:93) - adjoint satisfies propertiessimilar to the usual adjoint, namely ( x A + x B ) (cid:93) = x ∗ A (cid:93) + x ∗ B (cid:93) , ( AB ) (cid:93) = B (cid:93) A (cid:93) , and ( A (cid:93) ) (cid:93) = A .4) A k × k complex matrix T is called symplectic , if itsatisfies T T (cid:93) = T (cid:93) T = I k . Hence, any symplecticmatrix is invertible, and its inverse is its (cid:93) -adjoint.The set of these matrices forms a non-compact groupknown as the symplectic group. B. Linear Quantum Stochastic Systems
The material in this Subsection is fairly standard, andour presentation aims mostly at establishing notation andterminology. To this end, we follow the papers [23], [26].For the mathematical background necessary for a precisediscussion of LQSSs, some standard references are [12],[13], [14], while for a Physics perspective, see [3], [27].The references [28], [29], [30], [31], [32] contain a lot ofrelevant material, as well.The systems we consider in this work are collections ofquantum harmonic oscillators interacting among themselves,as well as with their environment. The i -th harmonic oscilla-tor ( i = 1 , . . . , n ) is described by its position and momentumvariables, q i and p i , respectively. These are self-adjointoperators satisfying the Canonical Commutation Relations (CCRs) [ q i , q j ] = 0 , [ p i , p j ] = 0 , and [ q i , p j ] = ıδ ij ,for i, j = 1 , . . . , n . As in classical mechanics, the states q i and p i , i = 1 , . . . , n , are called conjugate states. Ifwe define the vectors of operators q = ( q , q , . . . , q n ) (cid:62) , p = ( p , p , . . . , p n ) (cid:62) , and x = (cid:0) qp (cid:1) , the CCRs can beexpressed as [ x, x (cid:62) ] . = xx (cid:62) − ( xx (cid:62) ) (cid:62) = (cid:18) ıI n − ıI n (cid:19) = ı J n . (1)The environment is modelled as a collection of bosonicheat reservoirs. The i -th heat reservoir ( i = 1 , . . . , m ) isdescribed by bosonic field annihilation and creation op-erators A i ( t ) and A ∗ i ( t ) , respectively. The field operatorsare adapted quantum stochastic processes with forwarddifferentials d A i ( t ) = A i ( t + dt ) − A i ( t ) , and d A ∗ i ( t ) = A ∗ i ( t + dt ) − A ∗ i ( t ) . They satisfy the quantum Itˆo products d A i ( t ) d A j ( t ) = 0 , d A ∗ i ( t ) d A ∗ j ( t ) = 0 , d A ∗ i ( t ) d A j ( t ) = 0 ,and d A i ( t ) d A ∗ j ( t ) = δ ij dt . If we define the vector offield operators A ( t ) = ( A ( t ) , A ( t ) , . . . , A m ( t )) (cid:62) , and thevector of self-adjoint field quadratures V ( t ) = 1 √ (cid:18) A ( t ) + A ( t ) ı ( A ( t ) − A ( t ) ) (cid:19) , the quantum Itˆo products above can be expressed as d V ( t ) d V ( t ) (cid:62) = 12 (cid:18) I m ıI m − ıI m I m (cid:19) dt = 12 ( I m + ı J m ) dt. (2) To describe the dynamics of the harmonic oscillators andthe quantum fields, we introduce certain operators. We beginwith the Hamiltonian operator H = x (cid:62) Rx , which specifiesthe dynamics of the harmonic oscillators in the absence ofany environmental influence. R is a n × n real symmetricmatrix referred to as the Hamiltonian matrix. Next, we havethe coupling operator L (vector of operators) that specifiesthe interaction of the harmonic oscillators with the quantumfields. L depends linearly on the position and momentumoperators of the oscillators, and can be expressed as L = L q q + L p p . We construct the real coupling matrix C m × n from L m × nq and L m × np , as C = √ (cid:0) L q + L q L p + L p − ı ( L q − L q ) − ı ( L p − L p ) (cid:1) .Finally, we have the unitary scattering matrix S m × m , thatdescribes the interactions between the quantum fields them-selves.In the Heisenberg picture of quantum mechanics, the jointevolution of the harmonic oscillators and the quantum fieldsis described by the following system of
Quantum StochasticDifferential Equations (QSDEs): dx = ( J R − C (cid:93) C ) xdt − C (cid:93) Σ d V ,d V out = Cxdt + Σ d V , (3)where Σ = 12 (cid:18) S + S ı ( S − S ) − ı ( S − S ) S + S (cid:19) , is a m × m real orthogonal symplectic matrix. The fieldquadrature operators V i out ( t ) describe the outputs of thesystem. (3) is a description of the dynamics of the LQSS inthe real quadrature operator representation, where the states,inputs, and outputs are all self-adjoint operators. We aregoing to use a version of (3) generalized in two ways: First,we replace the real orthogonal symplectic transformation Σ , with a more general real symplectic transformation Σ ,see e.g. [32] for a discussion in the creation-annihilationrepresentation. Second, in the context of coherent quantumsystems in particular, the output of a quantum system maybe fed into other quantum system, so we substitute the moregeneral input and output notations U and Y , for V and V out ,respectively. The resulting QSDEs are the following: dx = ( J R − C (cid:93) C ) xdt − C (cid:93) Σ d U ,d Y = Cxdt + Σ d U , (4)The forward differentials d U and d Y of inputs and outputs,respectively (or, more precisely, of their quadratures), contain“quantum noises”, as well as a “signal part” (linear combi-nations of variables of other systems). One can prove that,the structure of (4) is preserved under linear transformationsof the state ¯ x = T x , if and only if T is real symplectic (with ¯ R = T −(cid:62) RT − , and ¯ C = CT − = CT ). From the pointof view of quantum mechanics, T must be real symplecticso that the transformed position and momentum operatorsare also self-adjoint and satisfy the same CCRs, as one canverify from (1). It is exactly this additional constraint on theallowed state transformations of LQSSs that complicates theconstruction of the Kalman decomposition for these systems.II. T HE K ALMAN D ECOMPOSITION FOR L INEAR Q UANTUM S TOCHASTIC S YSTEMS
System (4) has the standard form of a linear, time-invariant, system with A = J n R − C (cid:93) C , B = − C (cid:93) Σ ,and D = Σ . However, as discussed in Subsection II-B,only linear transformations of the state ¯ x = T x , with T real symplectic, preserve its structure, or, equivalently,preserve the self-adjointness and the CCRs of the states. Inthe following, we prove that there exists a real symplectictransformation of the state that puts (4) in a Kalman-likecanonical form. Before we state and prove this result, weintroduce the conventions used in this work regarding theuncontrollable and observable subspaces. Let C = (cid:0) B AB · · · A n − B (cid:1) , and O = CCA ... CA n − , (5)be the controllability and observability matrices of the system(4). As usual, Im C , and Ker O define the controllable andunobservable subspaces. The uncontrollable and observablesubspaces are defined as the orthogonal complements of Im C , and Ker O in R n , respectively. With this convention,we have the following theorem: Theorem 1:
Given the LQSS (4), there exists a real sym-plectic transformation V such that the following hold:1) The transformed states (cid:0) ˆ q ˆ p (cid:1) = ˆ x = V x = V (cid:0) qp (cid:1) , canbe partitioned as follows: ˆ q = ˆ q k × a ˆ q l × b ˆ q ( n − k − l ) × c , ˆ p = ˆ p k × a ˆ p l × b ˆ p ( n − k − l ) × c , (6)wherea) The states ˆ q a and ˆ p a are both controllable andobservable.b) The states ˆ p b are controllable but unobservable.c) The states ˆ q b are uncontrollable but observable.d) The states ˆ q c and ˆ p c are both uncontrollable andunobservable.2) In the transformed states, (4) takes the form d ˆ x = ˆ A ˆ xdt + ˆ Bd U ,d Y = ˆ C ˆ xdt + Dd U , (7)where ˆ A = A co, A , A co, A ¯ co A , A ¯ c ¯ o, A ¯ c ¯ o, A co, A , A co, A , A A , A , A c ¯ o A , A , A ¯ c ¯ o, A ¯ c ¯ o, , and ˆ B = B co, B co, B c ¯ o , ˆ C = (cid:0) C co, C ¯ co C co, (cid:1) . (cid:3) (8) To prove Theorem 1, we shall need the following lemmas:
Lemma 1:
Let ˜ C = (cid:0) B ( J R ) B · · · ( J R ) n − B (cid:1) , and˜ O = CC ( J R ) ... C ( J R ) n − . (9)Then, Im ˜ C = Im C , and Ker ˜ O = Ker O . (cid:3) This follows from standard results of linear systems theory,since the system (4) can be constructed from a systemwith ( ˜ A, ˜ B, ˜ C, ˜ D ) = ( J R, B, C, D ) , with state feedbackwith gain D − C , or from a system with ( ˜ A, ˜ B, ˜ C, ˜ D ) =( J R, B, C, D ) with output injection with gain − C (cid:93) .Hence, in all of the constructions above, we may use ˜ C and ˜ O in place of C and O . From now on, we shall refer to ˜ C and ˜ O simply as the controllability and observability matrices ofthe system (4). Next, we need another simple fact from linearsystems theory: Lemma 2:
The controllability and observability matricesof a linear time-invariant control system F , C F and O F , re-spectively, transform as follows under a linear transformationof the state x new = V x : C F ,new = V C F , O F ,new = O F V − . (cid:3) (10)The third result we shall make use of, is the following: Lemma 3:
There exists a symplectic matrix T , such that ˜ O = T ˜ C (cid:93) , or, equivalently, ˜ C = ˜ O (cid:93) T . (cid:3) Proof:
Let X , X , . . . , X k be complex matrices of corre-sponding dimensions r × s , . . . , r × s k . Then, (cid:0) X · · · X k (cid:1) (cid:93) = − J s + ... + s k ) (cid:0) X · · · X k (cid:1) † J r = − J s + ... + s k ) X † J r ... X † k J r = − J s + ... + s k ) J s . . . J s k − J s X † J r ... − J s X † k J r = − J s + ... + s k ) diag( J s , . . . , J s k ) X (cid:93) ... X (cid:93)k . Applying the above result to ˜ C , we have that ˜ C (cid:93) = − J nm diag( J m , . . . , J m (cid:124) (cid:123)(cid:122) (cid:125) n ) B (cid:93) (( J R ) B ) (cid:93) ... (( J R ) n − B ) (cid:93) = − J nm diag( J m , . . . , J m ) B (cid:93) B (cid:93) ( J R ) (cid:93) ... B (cid:93) (( J R ) (cid:93) ) n − . owever, B (cid:93) = ( − C (cid:93) D ) (cid:93) = − D (cid:93) C = − D − C , since T (cid:93) = T − for a symplectic T , and ( J R ) (cid:93) = R (cid:93) J (cid:93) =( − J R † J ) ( − J ) = − J R † = − J R , due to the fact that R is real symmetric. Putting everything together, we have that ˜ C (cid:93) = − J nm diag( J m , . . . , J m ) diag( D − , . . . , D − ) × − C − C ( − J R ) ... − C ( − J R ) n − = T − ˜ O , where T − = J nm diag( J m , − J m , . . . , J m , − J m ) × diag( D − , . . . , D − ) . Since each of the matrices J nm , diag( J m , − J m , . . . , J m , − J m ) , and diag( D − , . . . , D − ) is real symplectic, theconclusion of the lemma follows with T = diag( D, . . . , D ) × diag( J m , − J m , . . . , J m , − J m ) J nm . (cid:4) The final result we need is the following “one-sided sym-plectic” SVD from [2]:
Lemma 4: [2, Theorem 3] For any matrix F ∈ R s × r ,there exist an orthogonal matrix Q s × s , and a real symplecticmatrix Z r × r , such that F = Q E Z − , (11)where E s × r = k l r − k − l k l r − k − l Ξ k k I l l k k l (cid:48) , (12)with l (cid:48) = s − k − l , and Ξ k = diag( ξ , . . . , ξ k ) > . (cid:3) Proof of Theorem 1:
We begin by applying Lemma 4to the observability matrix ˜ O nm × n of system (4). Then, ˜ O = Q E Z − as above, with s = 4 nm and r = n , whilethe integers k and l are determined by the lemma. UsingLemma 3, we have that ˜ C = ˜ O (cid:93) T = ( Q E Z − ) (cid:93) T =( Z − ) (cid:93) E (cid:93) Q (cid:93) T = Z E (cid:93) Q (cid:93) T . Now, we perform the statetransformation (cid:0) ˆ q ˆ p (cid:1) = Z − (cid:0) qp (cid:1) . Since Z and Z − are realsymplectic, the transformed system is also of the form (4).According to Lemma 2, the controllability and observabilitymatrices of the transformed system are given by ˆ˜ C = Z − ˜ C = Z − Z E (cid:93) Q (cid:93) T = E (cid:93) Q (cid:93) T , (13) ˆ˜ O = ˜ O ( Z − ) − = Q E Z − Z = Q E. (14)Since Q is of full rank, (14) implies that Ker ˆ˜ O = Ker E .Let e i denote the i -th vector of the standard basis of R n .Then, we conclude that Ker ˆ˜ O = Ker E = span { e k + l +1 , . . . , e n , e n + k +1 , . . . , e n } . From (13), we have that
Im ˆ˜ C = Im E (cid:93) Q (cid:93) T = Im E (cid:93) = Im( − J E (cid:62) J ) = Im J E (cid:62) = Im k
00 0 0 00 0 0 0 − Ξ k − I l = span { e , . . . , e k ,e n +1 , . . . , e n + k , e n + k +1 , . . . , e n + k + l } . The fact that Q and T are of full rank was used in the abovederivation. If we partition the states as in (6), ˆ q = ˆ q k × a ˆ q l × b ˆ q ( n − k − l ) × c , and ˆ p = ˆ p k × a ˆ p l × b ˆ p ( n − k − l ) × c , the calculations of the controllable and unobservable sub-spaces above, along with our convention for the uncontrol-lable and observable subspaces, lead to the following picture:1) The states ˆ q a , ˆ p a , and ˆ p b are controllable, and the states ˆ q b , ˆ q c , and ˆ p c are uncontrollable.2) The states ˆ q c , ˆ p b , and ˆ p c are unobservable, and thestates ˆ q a , ˆ q b , and ˆ p a are observable.Combining the above controllability and observability re-sults, we end up with the classification of states announcedin the statement of the theorem.Hence, the state transformation (cid:0) ˆ q ˆ p (cid:1) = V (cid:0) qp (cid:1) , with V = Z − , essentially puts the system in the Kalman canonicalform. The qualification has to do with the fact that, theusual grouping of states in the Kalman canonical form, ( x co , x c ¯ o , x ¯ co , x ¯ c ¯ o ) , is incompatible with the grouping of thestates of (4) in conjugate pairs of position and momentumcoordinates, (ˆ q, ˆ p ) , that is necessary for the structure of (4)to be preserved. The resolution of this issue is, to modifythe usual Kalman canonical form. To do this, we start fromthe usual Kalman canonical form [33], [34] d x co x c ¯ o x ¯ co x ¯ c ¯ o = A co A A A c ¯ o A A A ¯ co
00 0 A A ¯ c ¯ o x co x c ¯ o x ¯ co x ¯ c ¯ o dt + B co B c ¯ o d U ,d Y = (cid:0) C co C ¯ co (cid:1) x co x c ¯ o x ¯ co x ¯ c ¯ o dt + D d U , (15)and let x co = (cid:0) ˆ q a ˆ p a (cid:1) , x c ¯ o = ˆ p b , x ¯ co = ˆ q b , x ¯ c ¯ o = (cid:0) ˆ q c ˆ p c (cid:1) .Also, partition A co = (cid:0) A co, A co, A co, A co, (cid:1) , A = (cid:0) A , A , (cid:1) , A = (cid:0) A , A , (cid:1) , A = (cid:0) A , A , (cid:1) , A = (cid:0) A , A , (cid:1) , A ¯ c ¯ o = (cid:0) A ¯ c ¯ o, A ¯ c ¯ o, A ¯ c ¯ o, A ¯ c ¯ o, (cid:1) , B co = (cid:0) B co, B co, (cid:1) , and C co = (cid:0) C co, C co, (cid:1) ,ccordingly. Then, by reshuffling the Kalman canonical form,we end up with (7), where ˆ A , ˆ B , and ˆ C are given by (8). (cid:4) Though Theorem 1 constructs one particular Kalmandecomposition of the LQSS (or, equivalently, one particularKalman-like canonical form (7) ), it is easy to generate manymore by use of the following corollary:
Corollary 1:
Let E ∈ R nm × n be the reduced formof the observability matrix ˜ O ∈ R nm × n of system (4),according to Lemma 4, see equation (12). Also, let X ∈ R nm × nm be invertible, and Y ∈ R n × n symplectic, suchthat, X E Y = k l n − k − l k l n − k − l Ξ (cid:48) k k (cid:48) l l (cid:48)(cid:48) k k l (cid:48) , (16)with l (cid:48) = 4 nm − k − l , and every element of the diagonalmatrices Ξ (cid:48) k ∈ R k × k , Ξ (cid:48) l ∈ R l × l , and Ξ (cid:48)(cid:48) k ∈ R k × k , is non-zero. If V is the symplectic transformation to the Kalman-like canonical form in Theorem 1, then the theorem holdsfor V (cid:48) = Y − V , as well. (cid:3) Proof:
We have that ˜ O = Q E Z − = ( Q X − ) ( X E Y ) ( Y − Z − ) . In the proof of Theorem 1, the fact that Q is unitary was usedjust to guarantee that it is of full rank. Also, the exact valuesof the elements of the non-zero diagonal blocks of E wereunimportant. It is straightforward to see that, the proof ofthe theorem follows through using the decomposition above,instead of (12). The conclusion of the corollary follows. (cid:4) IV. A N E XAMPLE
Consider the following 3-mode, 1 input/output LQSS withHamiltonian H = ω q + p ) + λq q + λq q , and coupling operator L = γ √ q + ıp ) . This LQSS models the linearized dynamics of an optome-chanical system where the resonant modes of two opticalcavities, with states ( q , p ) and ( q , p ) , respectively, inter-act with a mechanical mode with states ( q , p ) , of frequency ω . We assume that the cavities are lossless, and that theirinteraction strengths with the mechanical oscillator are equal.The only source of damping in the system is mechanical. The system QSDEs (4), take the following form: dq = 0 ,dq = 0 ,dq = (cid:0) − γ q + ωp (cid:1) dt − γd U ,dp = − λq dt,dp = − λq dt,dp = − (cid:0) λq + λq + ωq + γ p (cid:1) dt − γd U ,d Y = γq dt + d U ,d Y = γp dt + d U . Recall that U and U are the two real quadratures of a singleinput, and similarly for the outputs.Applying the “one-sided symplectic” SVD of [2] to theobservability matrix of the above LQSS, we obtain thesymplectic transformation V that puts the system in theKalman-like canonical form (7): V = − − √ − √ − λa − λa − − / − / λa √ − √ , where a = ω ω + ω + ω + ω + 1 ω + ω + ω + ω + ω + 1 . The new states of the system are given by ˆ q ˆ q ˆ q ˆ p ˆ p ˆ p = p − ( q + q ) √ ( q − q ) − q − λa ( q + q ) λa p − ( p + p ) √ ( p − p ) . ˆ q and ˆ p are the co states, ˆ q and ˆ p are the ¯ co and c ¯ o states, respectively, and ˆ q and ˆ p are the ¯ c ¯ o states. This isconfirmed by the system QSDEs in the transformed states,which take the following form: d ˆ q = (cid:0) − γ q + λb ˆ q + ω ˆ p (cid:1) dt − γd U ,d ˆ q = 0 ,d ˆ q = 0 ,d ˆ p = (cid:0) − ω ˆ q + λa γ q − γ p (cid:1) dt + γd U ,d ˆ p = (cid:0) − λa γ q + λ a ( b + 1) ˆ q − λb ˆ p (cid:1) dt − γλa d U ,d ˆ p = 0 ,d Y = γ ( λa ˆ q − ˆ p ) dt + d U ,d Y = γ ˆ q dt + d U , here b = 1 / ( ω + ω + ω + ω + ω + 1) . We can useCorollary 1, to produce a simpler Kalman decomposition ofthe system. Indeed, with Y = −√ − −√ λa λa − √
00 0 0 0 0 1 , and X = diag( − λγa/ √ b
10 1 01 0 0 , I ) , we obtain the following orthogonal symplectic transforma-tion V (cid:48) = Y − V , that puts the system in the Kalman-likecanonical form (7): V (cid:48) = √ √ √ − √ √ √
00 0 0 √ − √ . The new states of the system are given by ˆ q ˆ q ˆ q ˆ p ˆ p ˆ p = q √ ( q + q ) √ ( q − q ) p ( p + p ) √ ( p − p ) . Again, ˆ q and ˆ p are the co states, ˆ q and ˆ p are the ¯ co and c ¯ o states, respectively, and ˆ q and ˆ p are the ¯ c ¯ o states. Thisis confirmed by the system QSDEs in the transformed states,which take the following form: d ˆ q = (cid:0) − γ q + ω ˆ p (cid:1) dt − γd U ,d ˆ q = 0 ,d ˆ q = 0 ,d ˆ p = (cid:0) − ω ˆ q − √ λ ˆ q − γ p (cid:1) dt − γd U ,d ˆ p = −√ λ ˆ q dt,d ˆ p = 0 ,d Y = γ ˆ q dt + d U ,d Y = γ ˆ p dt + d U . R EFERENCES[1] G. Zhang, S. Grivopoulos, I. R. Petersen, and J. E. Gough, “TheKalman decomposition for linear quantum systems,” 2016. Submittedto the IEEE Transactions on Automatic Control. Preprint availableonline at http://lanl.arxiv.org/abs/1606.05719.[2] H. Xu, “An SVD-like matrix decomposition and its applications,”
Linear Algebra and its Applications , vol. 368, pp. 1 – 24, 2003.[3] C. Gardiner and P. Zoller,
Quantum Noise . Springer-Verlag, Berlin,second ed., 2000. [4] D. Walls and G. Milburn,
Quantum Optics . Springer-Verlag, 2nd ed.,2008.[5] H. Wiseman and G. Milburn,
Quantum Measurement and Control .Cambridge University Press, 2010.[6] A. Matyas, C. Jirauschek, F. Peretti, P. Lugli, , and G. Csaba,“Linear circuit models for on-chip quantum electrodynamics,”
IEEETransactions on Microwave Theory and Techniques , vol. 59, pp. 65–71, 2011.[7] J. Kerckhoff, R. W. Andrews, H. S. Ku, W. F. Kindel, K. Cicak, R. W.Simmonds, , and K. W. Lehnert, “Tunable coupling to a mechanicaloscillator circuit using a coherent feedback network,”
Physical ReviewX , vol. 3, p. 021013, 2013.[8] M. Tsang and C. M. Caves, “Coherent quantum-noise cancellation foroptomechanical sensors,”
Physical Review Letters , vol. 105, p. 123601,2010.[9] F. Massel, T. T. Heikkila, J. M. Pirkkalainen, S. U. Cho, H. Saloniemi,P. J. Hakonen, , and M. A. Sillanpaa, “Microwave amplification withnanomechanical resonators,”
Nature , vol. 480, pp. 351–354, 2011.[10] R. Hamerly and H. Mabuchi, “Advantages of coherent feedbackfor cooling quantum oscillators,”
Physical Review Letters , vol. 109,p. 173602, 2012.[11] C. Dong, V. Fiore, M. C. Kuzyk, , and H. Wang, “Optomechanicaldark mode,”
Science , vol. 338, no. 6114, pp. 1609–1613, 2012.[12] K. Parthasarathy,
An Introduction to Quantum Stochastic Calculus .Birkhauser, 1999.[13] P. Meyer,
Quantum Probability for Probabilists . Springer, second ed.,1995.[14] R. L. Hudson and K. R. Parthasarathy, “Quantum Itˆo’s formulaand stochastic evolutions,”
Communications in Mathematical Physics ,vol. 93, pp. 301–323, 1984.[15] M. Yanagisawa and H. Kimura, “Transfer function approach to quan-tum control-part I: dynamics of quantum feedback systems,”
IEEETransactions on Automatic Control , vol. 48, no. 12, pp. 2107–2120,2003.[16] M. Yanagisawa and H. Kimura, “Transfer function approach toquantum control-part II: control concepts and applications,”
IEEETransactions on Automatic Control , vol. 48, no. 12, pp. 2121–2132,2003.[17] M. James, H. I. Nurdin, and I. Petersen, “ H ∞ control of linear quan-tum stochastic systems,” IEEE Transactions on Automatic Control ,vol. 53, pp. 1787–1803, Sept 2008.[18] H. I. Nurdin, M. R. James, and I. R. Petersen, “Coherent quantumLQG control,”
Automatica , vol. 45, no. 8, pp. 1837 – 1846, 2009.[19] A. I. Maalouf and I. R. Petersen, “Coherent H ∞ control for a classof annihilation operator linear quantum systems,” IEEE Transactionson Automatic Control , vol. 56, no. 2, pp. 309–319, 2011.[20] G. Zhang and M. James, “Quantum feedback networks and control:a brief survey,”
Chinese Science Bulletin , vol. 57, no. 18, pp. 2200–2214, 2012.[21] H. Mabuchi, “Coherent-feedback quantum control with a dynamiccompensator,”
Physical Review A , vol. 78, p. 032323, 2008.[22] O. Crisafulli, N. Tezak, D. B. S. Soh, M. A. Armen, and H. Mabuchi,“Squeezed light in an optical parametric oscillator network withcoherent feedback quantum control,”
Optics Express , vol. 21, no. 15,pp. 3761–3774, 2013.[23] I. R. Petersen, “Quantum linear systems theory,” in
Proceedings of the19th International Symposium on Mathematical Theory of Networksand Systems , (Budapest, Hungary), July 2010.[24] J. E. Gough and G. Zhang, “On realization theory of quantum linearsystems,”
Automatica , vol. 59, pp. 139–151, 2015.[25] M. Guta and N. Yamamoto, “System identification for passive linearquantum systems,”
IEEE Transactions on Automatic Control , vol. 61,no. 4, pp. 921–936, 2016.[26] A. A. J. Shaiju and I. R. Petersen, “A frequency domain condition forthe physical realizability of linear quantum systems,”
IEEE Transac-tions on Automatic Control , vol. 57, pp. 2033–2044, August 2012.[27] C. Gardiner and M. Collett, “Input and output in damped quantumsystems: Quantum stochastic differential equations and the masterequation,”
Physical Review A , vol. 31, no. 6, pp. 3761–3774, 1985.[28] H. I. Nurdin, M. R. James, and A. C. Doherty, “Network synthesisof linear dynamical quantum stochastic systems,”
SIAM Journal onControl and Optimization , vol. 48, no. 4, pp. 2686–2718, 2009.[29] S. C. Edwards and V. P. Belavkin, “Optimal quantum filtering andquantum feedback control,” arXiv:quant-ph/0506018 , August 2005.Preprint.30] J. Gough and M. James, “The series product and its application toquantum feedforward and feedback networks,”
IEEE Transactions onAutomatic Control , vol. 54, pp. 2530–2544, Nov 2009.[31] J. E. Gough, R. Gohm, and M. Yanagisawa, “Linear quantum feedbacknetworks,”
Physical Review A , vol. 78, p. 062104, Dec 2008.[32] J. E. Gough, M. R. James, and H. I. Nurdin, “Squeezing componentsin linear quantum feedback networks,”
Physical Review A , vol. 81,p. 023804, Feb 2010.[33] K. Zhou, J. Doyle, and K. Glover,
Robust and Optimal Control .Prentice Hall, 1996.[34] H. Kimura,
Chain Scattering Approach to H ∞ -Control-Control