aa r X i v : . [ qu a n t - ph ] O c t On Realization Theory of Quantum Linear Systems ∗ John E. Gough † Guofeng Zhang ‡ July 9, 2018
Abstract
The purpose of this paper is to study the realization theory of quantum linear systems. It is shown that for ageneral quantum linear system its controllability and observability are equivalent and they can be checked by means ofa simple matrix rank condition. Based on controllability and observability a specific realization is proposed for generalquantum linear systems in which an uncontrollable and unobservable subspace is identified. When restricted to thepassive case, it is found that a realization is minimal if and only if it is Hurwitz stable. Computational methods areproposed to find the cardinality of minimal realizations of a quantum linear passive system. It is found that the transferfunction of a quantum linear passive system G can be written as a fractional form in terms of a matrix function Σ;moreover, G is lossless bounded real if and only if Σ is lossless positive real. A type of realization for multi-input-multi-output quantum linear passive systems is derived, which is closely related to its controllability and observabilitydecomposition. Two realizations, namely the independent-oscillator realization and the chain-mode realization, areproposed for single-input-single-output quantum linear passive systems, and it is shown that under the assumptionof minimal realization, the independent-oscillator realization is unique, and these two realizations are related to thelossless positive real matrix function Σ. Keyword:
Quantum linear systems, realization theory, controllability, observability.
Linear systems and signals theory has been very useful in the analysis and engineering of dynamical systems. Many funda-mental notions have been proposed to characterize dynamical systems from a control-theoretic point of view. For example,controllability describes the ability of steering internal system states by external input, observability refers to the possi-bility of reconstructing the state-space trajectory of a dynamical system based on its external input-output data. Basedon controllability and observability, Kalman canonical decomposition reveals the internal structure of a linear system.This, in particular minimal realization as a very convenient and yet quite natural assumption, is the basis of widely usedmodel reduction methods such as balanced truncation and optimal Hankel norm approximation. Moreover, fundamentaldissipation theory has been well established and has been proven very effective in control systems design. All of thesehave been well documented, see, e.g., [Kwakernaak & Sivan, 1972]; [Willems, 1972]; [Anderson & Vongpanitlerd, 1973];[Kailath, 1980]; [van der Schaft, 1996]; [Zhou, Doyle & Glover, 1996].In recent years there has been a rapid growth in the study of quantum linear systems. Quantum linear systems and sig-nals theory has been proven very effective in the study of many quantum systems including quantum optical systems, opto-mechanical systems, cavity quantum electro-magnetic dynamical systems, atomic ensembles and quantum memories, see,e.g., [Gardiner & Zoller, 2000]; [Wall & Milburn, 2008]; [Wiseman & Milburn, 2010]; [Stockton, van Handel & Mabuchi, 2004];[Zhang, Chen, Bhattacharya, & Meystre, 2010]; [Massel, et al., 2011]; [Matyas, et al., 2011]; [Tian, 2012]; [Zhang, et al., 2013]; ∗ Corresponding author Guofeng Zhang. Tel. +852-2766-6936. Fax +852-2764-4382. Email: [email protected]. † Institute of Mathematics and Physics, Aberystwyth University, Ceredigion SY23 3BZ, Wales UK. (e-mail: [email protected]). ‡ Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China. (e-mail: [email protected]). H ∞ coherent feedback control of quantum linearpassive systems, [Maalouf & Petersen, 2011b]. For a quantum linear passive system it is shown in [Guta & Yamamoto, 2013,Lemma 3.1] that controllability is equivalent to observability; moreover, a minimal realization is necessarily Hurwitz sta-ble, [Guta & Yamamoto, 2013, Lemma 3.2]. In this paper we explore further controllability and observability of quantumlinear systems. For general quantum linear systems (not necessarily passive), we show that controllability and observabil-ity are equivalent (Proposition 2.3). Moreover, a simple matrix rank condition is established for checking controllabilityand observability. Base on this result, a realization of general quantum linear systems is proposed, in which the un-controllable and unobservable subspace is identified (Theorem 2.6). Theorem 2.6 can be viewed as the complex-domaincounterpart of Theorem 3.1 in [Yamamoto, 2013] in the real domain. However, it is can be easily seen from the proof ofLemma 2.5 that the structure of the unitary transformation involved is better revealed in the complex domain. Restrictedto the passive case, we show that controllability, observability and Hurwitz stability are equivalent to each other (Lemma3.3). Thus, the realization of a quantum linear passive system is minimal if and only if it is Hurwitz stable (Theorem3.5). We also derive formulas for calculating the cardinality of minimal realizations of a given quantum linear passivesystem (Proposition 3.7 for the single-input-single-output case and Proposition 3.8 for the multi-input-multi-output case).Finally we show how a given quantum linear system can be written as a fractional form in term of a matrix function Σ(Proposition 2.8), and for the passive case show that a quantum linear passive system G is lossless bounded real if andonly if the corresponding Σ is lossless positive real (Theorem 3.11).The synthesis problem of quantum linear systems has been investigated in [Nurdin, James & Doherty, 2009], wherethey showed that a quantum linear system can always be realized by a cascade of one-degree-of-freedom harmonicoscillators with possible direct Hamiltonian couplings among them if necessary. Then in [Nurdin, 2010] a necessary andsufficient condition is derived for the realizability of quantum linear systems via pure cascading only. For the passive case,it is shown in [Petersen, 2011] that, under certain conditions on the system matrices, a minimal quantum linear passivesystem can be realized by a cascade of one-degree-of-freedom harmonic oscillators. These restrictions were removed in[Nurdin, 2010] which proves that all quantum linear passive systems can be realised by pure cascading of one-degree-of-freedom harmonic oscillators. Model reduction of quantum linear systems has been studied in, e.g., [Petersen, 2013],and [Nurdin, 2013]. In this paper we propose several realizations of quantum linear passive systems. For the multi-input-multi-output (MIMO) case we show that the proposed realization has a close relationship with controllability andobservably of the quantum linear passive system (Theorem 4.1). In the single-input-single-output (SISO) case, we proposetwo realizations, namely the independent-oscillator realization and the chain-mode realization (Theorem 4.3 and Theorem4.7), and finally we show that if the system is Hurwitz stable, these two realizations are related to the lossless positivereal Σ mentioned in the previous paragraph (Theorem 4.10).The rest of the paper is organized as follows. Section 2 studies general quantum linear systems; specifically, Subsec-tion 2.1 briefly reviews quantum linear systems, Subsection 2.2 investigates their controllability and observability, andSubsection 2.3 presents a fractional form for transfer functions of quantum linear systems. Section 3 studies quantumlinear passive systems, specifically, Subsection 3.1 introduces quantum linear passive systems, Subsection 3.2 investigatestheir Hurwitz stability, controllability and observability, Subsection 3.3 studies minimal realizations of quantum linearpassive systems, and Subsection 3.4 proposes a fractional form for transfer functions of quantum linear passive systems.Section 4 investigates realizations of quantum linear passive systems; specifically, Subsection 4.1 proposes a realizationfor MIMO quantum linear passive systems, Subsections 4.2.1 and 4.2.2 propose an independent-oscillator realization anda chain-mode realization for SISO quantum linear systems respectively, and Subsection 4.2.3 discusses the uniqueness ofthe independent-oscillator realization. Section 5 concludes this paper. Notations. m is the number of input channels, and n is the number of degrees of freedom of a given quantum2inear system, namely, the number of system oscillators. Given a column vector of complex numbers or operators x = [ x · · · x k ] T , define x = [ x ∗ · · · x ∗ k ] T , where the asterisk ∗ indicates complex conjugation or Hilbertspace adjoint. Denote x † = ( x ) T . Furthermore, define a column vector ˘ x to be ˘ x = [ x T ( x ) T ] T . Let I k bean identity matrix and 0 k a zero square matrix, both of dimension k . Define J k = diag( I k , − I k ). Then for a matrix X ∈ C j × k , define X ♭ = J k X † J j . Given two constant matrices U , V ∈ C r × k , define ∆( U, V ) = [
U V ; V U ]. Given twooperators A and B , their commutator is defined to be [ A, B ] = AB − BA . “ ⇐⇒ ” means if and only if. Finally, Spec( X )denotes the set of all distinct eigenvalues of the matrix X , σ ( X ) denotes the diagonal matrix with diagonal entries beingthe non-zero singular values of the matrix X , Ker ( X ) denotes the null space of the matrix X , and Range ( X ) denotesthe space spanned by the columns of the matrix X . We first introduce quantum linear systems in Subsection 2.1, then discuss their controllability and observability inSubsection 2.2, and finally study their transfer functions in Subsection 2.3.
In this subsection quantum linear systems are briefly described in terms of the (
S, L, H ) language, [Gough & James, 2009].More discussions on quantum linear systems can be found in, e.g., [Gardiner & Zoller, 2000]; [Wall & Milburn, 2008];[Wiseman & Milburn, 2010]; [Doherty & Jacobs, 1999]; [Zhang & James, 2012]; [Tezak, et al., 2012].An open quantum linear system G studied in this paper consists of n interacting quantum harmonic oscillators drivenby m input boson fields. Each oscillator j has an annihilation operator a j and a creation operator a ∗ j ; a j and a ∗ j areoperators on the system space h which is an infinite-dimensional Hilbert space. The operators a j , a ∗ k satisfy the canonicalcommutation relations: [ a j , a ∗ k ] = δ jk . Denote a ≡ [ a · · · a n ] T . Then the initial (that is, before the interaction betweenthe system and the input boson fields) Hamiltonian H can be written as H = (1 / ˘a † Ω ˘a , where ˘a = [ a T ( a ) T ] T asintroduced in the Notations part, and Ω = ∆(Ω − , Ω + ) ∈ C n × n is a Hermitian matrix with Ω − , Ω + ∈ C n × n . L in the( S, L, H ) language describes the coupling of the system harmonic oscillators to the input boson fields. The coupling is linear and can be written as L = [ C − C + ] ˘a with C − , C + ∈ C m × n . Finally, in the linear setting S in the ( S, L, H ) languageis taken to be a constant unitary matrix in C m × m .Each input boson field j has an annihilation operator b j ( t ) and a creation operator b ∗ j ( t ), which are operators on aninfinite-dimensional Hilbert space F . Let b ( t ) ≡ [ b ( t ) · · · b m ( t )] T . The operators b j ( t ) and their adjoint operators b ∗ j ( t )satisfy the following commutation relations:[ b j ( t ) , b ∗ k ( r )] = δ jk δ ( t − r ) , ∀ j, k = 1 , . . . , m, ∀ t, r ∈ R . (2.1)For each j = 1 , . . . , m , the j -th input field can also be represented in the integral form B j ( t ) ≡ R t b j ( r ) dr , whose Itoincrement is dB j ( t ) ≡ B j ( t + dt ) − B j ( t ). Denote B ( t ) ≡ [ B ( t ) · · · B m ( t )] T . The gauge process can be defined byΛ jk ( t ) = R t b ∗ j ( r ) b k ( r ) dr , ( j, k = 1 , . . . , m ). The field studied in this paper is assumed to be canonical , that is, the fieldoperators B j ( t ) , B ∗ k ( t ) , Λ rl ( t ) satisfy the following Ito table: × dB k d Λ kl dB ∗ l dtdB i δ ik dB l δ il dt d Λ ij δ jk d Λ il δ jl dB ∗ i dB ∗ j dt G can be described in terms of thefollowing quantum stochastic differential equation (QSDE): dU ( t ) = (cid:8) − (cid:0) L † L/ iH (cid:1) dt + dB † ( t ) L − L † SdB ( t ) + Tr[( S − I ) d Λ T ( t )] (cid:9) U ( t ) , t > , (2.2)3ith U (0) = I being the identity operator. Let X be an operator on the system space h . Then the temporal evolution of X , denoted X ( t ) ≡ U ( t ) ∗ ( X ⊗ I ) U ( t ), is governed by the following QSDE: dX ( t ) = L L,H ( X ( t )) dt + dB † ( t ) S † ( t )[ X ( t ) , L ( t )] + [ L † ( t ) , X ( t )] S ( t ) dB ( t )+Tr[( S † ( t ) X ( t ) S ( t ) − X ( t )) d Λ T ( t )] , (2.3)where the Lindblad operator L L,H ( X ( t )) is L L,H ( X ( t )) ≡ − i [ X ( t ) , H ( t )] + 12 L † ( t )[ X ( t ) , L ( t )] + 12 [ L † ( t ) , X ( t )] L ( t ) . (2.4)Note that X ( t ) is an operator on the joint system-field space h ⊗ F .Let b out,j ( t ) denote the j -th field after interacting with the system, and B out,j ( t ) ≡ R t b out,j ( r ) dr . We have B out,j ( t ) = U ∗ ( t ) ( I ⊗ B j ( t )) U ( t ). Denote B out ( t ) ≡ [ B out , ( t ) , · · · B out ,m ( t )] T . Then in compact form the output field equation is dB out ( t ) = L ( t ) dt + SdB ( t ) . (2.5)Substituting H = (1 / a † Ω˘ a and L = [ C − C + ] ˘a into (2.3) we have a quantum linear system: d ˘a ( t ) = A ˘a ( t ) dt + B d ˘ B ( t ) , (2.6) d ˇ B out ( t ) = C ˘a ( t ) dt + D d ˘ B ( t ) , (2.7)in which A = − C ♭ C − iJ n Ω , B = − C ♭ ∆( S, m × m ) , C = ∆( C − , C + ) ≡ C, D = ∆( S, m × m ) . (2.8)Clearly, the quantum linear system is parameterized by constant matrices S, C,
Ω. In the sequel, we use the notation G ∼ ( S, C,
Ω) for the quantum linear system (2.6)-(2.7) with parameters given in (2.8).For notation’s sake, we introduce the following definition.
Definition 2.1 (2.6)-(2.7) with parameters given in (2.8) is said to be the realization of the quantum linear system G ∼ ( S, C, Ω) . The constant matrices A , B , C , D in (2.8) satisfy the following fundamental relations: A + A ♭ + C ♭ C = 0 , B = −C ♭ D , D ♭ D = I m . (2.9)These equations are often called physically realizability conditions of quantum linear systems. More discussions on physicalrealizability of quantum linear systems can be found in, e.g., [James, Nurdin & Petersen, 2008]; [Zhang & James, 2011];[Zhang & James, 2012]. In this subsection we study controllability and observability of quantum linear systems introduced in Subsection 2.1.Let X be an operator on the system space h . Denote by h X ( t ) i the expected value of X ( t ) with respect to the initialjoint system-field state (which is a unit vector in the Hilbert space h ⊗ F ). Then (2.6)-(2.7) gives rise to the following classical linear system d h ˘a ( t ) i dt = A h ˘a ( t ) i + Bh ˘b ( t ) i , (2.10) d h ˘b out ( t ) i dt = C h ˘a ( t ) i + Dh ˘b ( t i ) . (2.11) Definition 2.2
The quantum linear system G ∼ ( S, C, Ω) is said to be Hurwitz stable (resp. controllable, observable) ifthe corresponding classical linear system (2.10)-(2.11) is Hurwitz stable (resp. controllable, observable). Proposition 2.3
Given a quantum linear system G ∼ ( S, C, Ω) , the following statements are equivalent: (i) G is controllable; (ii) G is observable; (iii) rank( O s ) = 2 n , where O s ≡ CCJ n Ω ... C ( J n Ω) n − . (2.12) Proof. (i) ⇒ (ii). We show this by contradiction. Assume G is not observable. By the classical control theory (see.e.g., [Zhou, Doyle & Glover, 1996, Theorems 3.3]) there exist a scalar λ and a non-zero vector v ∈ C n such that A v = λv and C v = 0. So J n Ω v = iλv and Cv = 0. Let u = J n v and µ = − λ ∗ . Then u † B = − v † C † J m = 0, and u † A = − ( J n v ) † (cid:16) C ♭ C/ iJ n Ω (cid:17) = − ( J n v ) † iJ n Ω = − iv † Ω = − λ ∗ v † J n = µu † . By a standard result in classical control theory, (see. e.g., [Zhou, Doyle & Glover, 1996, Theorems 3.1]), G is not control-lable. We reach a contradiction.(ii) ⇒ (i). This can be established by reversing the proof for (i) ⇒ (ii).(ii) ⇒ (iii). Let v ∈ C n such that O s v = 0. Then C v = Cv = 0 and C ( J n Ω) k v = 0, k = 1 , . . . , n −
1. Moreover, CA v = − C (cid:16) C ♭ C/ iJ n Ω (cid:17) v = − iCJ n Ω v = 0 , CA v = − C (cid:16) C ♭ C/ iJ n Ω (cid:17) v = C ( J n Ω) v = 0 , ... CA n − v = C ( J n Ω) n − v = 0 . But by (ii) G is observable, therefore v = 0. (iii) is established.(iii) ⇒ (ii). This can be established by reversing the proof for (ii) ⇒ (iii).Proposition 2.3 tells us that the controllability and observability of a quantum linear system G ∼ ( S, C,
Ω) areequivalent; moreover they can be determined by checking the rank of the matrix O s .On the basis of Proposition 2.3, we have the following result about the uncontrollable and unobservable subspace ofa quantum linear system. Proposition 2.4
Let C ≡ [ B AB · · · A n − B ] and O ≡ [ C T ( CA ) T · · · ( CA − ) T ] T be the controllabilityand observability matrices of a quantum linear system G ∼ ( S, C, Ω) respectively. Then (in the terminology of modern con-trol theory, [Kwakernaak & Sivan, 1972]; [Anderson & Vongpanitlerd, 1973]; [Kailath, 1980]; [Zhou, Doyle & Glover, 1996])the following statements hold: (i) The unobservable subspace is
Ker ( O ) = Ker ( O s ) , (2.13) where Ker ( X ) denotes the null space of the matrix X , as introduced in the Notations part. (ii) The uncontrollable subspace is
Ker (cid:0) C † (cid:1) = Ker ( O s J n ) . (2.14)5 iii) The uncontrollable and unobservable subspace is
Ker ( O s ) ∩ Ker ( O s J n ) . Proposition 2.4 can be established in the similar way as Proposition 2.3.Propositions 2.3 and 2.4 appear purely algebraic. Nevertheless, they have interesting and important physical conse-quences. We begin with the following lemma.
Lemma 2.5
The dimension of the space
Ker ( O s ) ∩ Ker ( O s J n ) is even. Let the dimension of Ker ( O s ) ∩ Ker ( O s J n ) be l for some nonnegative integer l . There exists a matrix V = [ V V ] with V ∈ C n × l and V ∈ C n × n − l ) such that Range( V ) = Ker ( O s ) ∩ Ker ( O s J n ) , (2.15) V V † = V † V = I n , (2.16) V † J n V = " J l J n − l . (2.17)The proof is given in the Appendix.We are ready to state the main result. Theorem 2.6
Let V be the matrix defined in Lemma 2.5. If Range( V ) is an invariant space under the linear transfor-mation of Ω , then the transformed system " ˘ a DF ˘ a D ≡ V † ˘ a has the following realization: d ˘ a DF ( t ) = − iJ l V † Ω V ˘ a DF ( t ) dt, (2.18) d ˘ a D ( t ) = − (cid:16) ( CV ) ♭ ( CV ) / iJ n − l V † Ω V (cid:17) ˘ a D ( t ) dt − ( CV ) ♭ D d ˇ B ( t ) , (2.19) d ˇ B out ( t ) = ( CV )˘ a D ( t ) dt + D d ˇ B ( t ) . (2.20) Proof.
Because Range( V ) = Ker ( O s ) ∩ Ker ( O s J n ), the coupling operator of the transformed mode [ ˘ a TDF ˘ a TD ] T is C V = [ 0 CV ]. Moreover, because Range( V ) is an invariant space under the linear transformation of Ω, there existsa matrix Y such that Ω V = V Y . We have V † Ω V = Y † V † V = 0 where (2.16) is used. This, together with (2.17), gives V † J n Ω V = V † J n V V † Ω V = " J l V † Ω V J n − l V † Ω V . That is, the transformed system with mode [ ˘ a TDF ˘ a TD ] T has the realization (2.18)-(2.20). Remark 1.
By (2.18), the modes ˘ a DF evolve unitarily as an isolated system. In literature such isolated modes embeddedin an open quantum system is often called decoherence-free modes, see, e.g., [Ticozzi & Viola, 2008], [Ticozzi & Viola, 2009],[Yamamoto, 2013]. Theorem 2.6 can be viewed as the complex-domain counterpart of Theorem 3.1 in [Yamamoto, 2013]in the real domain. However, with the help of the matrix O s , matters are simplified; moreover, it can be seen from theproof of Lemma 2.5 in the Appendix that the structure of the unitary transformation matrix V is better revealed withthe help of O s and in the complex domain.Finally, from the proof of Lemma 2.5 it can be seen that the dimension of the space Ker( C ) is also even. Moreoverwe have the following corollary which shows that under some conditions the unobservable and uncontrollable subspace isexactly Ker( C ). Corollary 2.7
Let the dimension of the space
Ker( C ) be r . Let a matrix T ∈ C n × r be such that Range( T ) = Ker( C ) .If J n T = T J r and Range( T ) is an invariant space under the linear transformation of Ω , then Ker( C ) = Ker( O s ) ∩ Ker( O s J n ) . roof. Clearly, Ker ( O s ) ∩ Ker ( O s J n ) ⊂ Ker( O s ) ⊂ Ker ( C ). Thus it is sufficient to show that Ker ( C ) ⊂ Ker ( O s ) ∩ Ker ( O s J n ). However Range( T ) = Ker ( C ), we show that Range( T ) ⊂ Ker ( O s ) ∩ Ker ( O s J n ). BecauseRange( T ) is invariant with respect to a linear transformation Ω, there exist matrix Y such that Ω T = T Y . This,together with J n T = T J r , gives C ( J n Ω) T = CT J r Y = 0. Similarly, for all k ≥ C ( J n Ω) k T = 0. That is, O s T = 0.Moreover, O s J n T = O s T J r = 0. Consequently Ker( C ) = Range( T ) ⊂ Ker ( O s ) ∩ Ker ( O s J n ). This together withKer ( O s ) ∩ Ker ( O s J n ) ⊂ Ker ( C ) yields Ker ( C ) = Ker ( O s ) ∩ Ker ( O s J n ).Corollary 2.7 can be regarded as the complex-domain counterpart of Proposition 3.1 in [Yamamoto, 2013] in the realdomain. In the frequency domain, the transfer function of the system G ∼ ( S, C,
Ω) is defined to be G ( s ) ≡ D + C ( sI − A ) − B . (2.21)This transfer function has the following fundamental property, see, e.g., [Zhang & James, 2013, Eq. (24)]: G ( iω ) ♭ G ( iω ) = G ( iω ) G ( iω ) ♭ = I m , ∀ ω ∈ R . (2.22)Interestingly, the transfer function G ( s ) of the quantum linear system G ∼ ( S, C,
Ω) can be written into a fractional form.
Proposition 2.8
The transfer function G ( s ) for a Hurwitz stable quantum linear system G ∼ ( S, C, Ω) can be writtenin the following fractional form G ( s ) = ( I − Σ( s ))( I + Σ( s )) − ∆( S, , (2.23) where Σ( s ) ≡ C ( sI + iJ n Ω) − C ♭ , ∀ Re[ s ] > . (2.24) Proof.
Because the system G ( s ) is Hurwitz stable, all the eigenvalues of the matrix A have strictly negative realpart, therefore the matrix sI − A is invertible for all Re[ s ] >
0. Moreover, for all Re[ s ] >
0, by the Woodbury matrixinversion formula,( sI − A ) − = ( sI + iJ n Ω + 12 C ♭ C ) − = ( sI + iJ n Ω) − −
12 ( sI + iJ n Ω) − C ♭ (cid:18) I + 12 C ( sI + iJ n Ω) − C ♭ (cid:19) − C ( sI + iJ n Ω) − . As a result, for all Re[ s ] > I − C ( sI − A ) − C ♭ = I − C (cid:26) ( sI + iJ n Ω) − −
12 ( sI + iJ n Ω) − C ♭ ( I + 12 C ( sI + iJ n Ω) − C ♭ ) − C ( sI + iJ n Ω) − (cid:27) C ♭ = I − s ) + 2Σ( s ) ( I + Σ( s )) − Σ( s )= ( I − Σ( s ))( I + Σ( s )) − , with Σ( s ) as defined in (2.24). Consequently, G ( s ) = ( I − C ( sI − A ) − C ♭ )∆( S,
0) = ( I − Σ( s ))( I + Σ( s )) − ∆( S, . In this section quantum linear passive systems are studied. This type of systems is introduced in Subsection 3.1. Stability,controllability and observability are investigated in Subsection 3.2, while minimal realizations of quantum linear passivesystems are studied in Subsection 3.3. The relation between G and Σ in the passive setting is discussed in Subsection 3.4.7 .1 Quantum linear passive systems If the matrices C + = 0 and Ω + = 0, the resulting system, parameterized by matrices S, C − , Ω − , is often said to be aquantum linear passive system. In this case, it can be described entirely in terms of annihilation operators. Actually aquantum linear passive system has the following form: d a ( t ) = A a ( t ) − C †− SdB ( t ) , (3.1) dB out ( t ) = C − a ( t ) + SdB ( t ) . (3.2)in which A ≡ − C †− C − − i Ω − .In analog to Definition 2.1 for realization of general linear systems we introduce the following realization concept forpassive linear systems. Definition 3.1 (3.1)-(3.2) is said to be the realization of the quantum linear passive system G ∼ ( S, C − , Ω − ) . Clearly, the transfer function of G ∼ ( S, C − , Ω − ) is G ( s ) = S − C − ( sI − A ) − C †− S. (3.3)Define Σ( s ) ≡ C − ( sI + i Ω − ) − C †− . (3.4)Then, in analog to Proposition 2.8, we have G ( s ) = ( I − Σ( s ))( I + Σ( s )) − S. (3.5)In the passive case, Eq. (2.22) reduces to G ( iω ) † G ( iω ) = G ( iω ) G ( iω ) † = I m , ∀ ω ∈ R . (3.6)Because deferent realizations may correspond to the same transfer function (3.3), we introduce the following concept. Definition 3.2
Two realizations are said to be unitarily equivalent if there exists a unitary transformation which trans-forms one to the other.
Clearly, two unitarily equivalent realizations correspond to the same transfer function.
In this subsection we study stability of quantum linear passive systems. In particular, we show that a quantum linearpassive system G ∼ ( S, C − , Ω − ) is Hurwitz stable if and only if it is observable and controllable. Lemma 3.3
The following statements for a quantum linear passive system G ∼ ( S, C − , Ω − ) are equivalent: (i) G is Hurwitz stable; (ii) G is observable; (iii) G is controllable. Proof. (i) → (ii). Clearly, X = I n > A † X + XA + C †− C − = 0 . (3.7)8ccording to [Zhou, Doyle & Glover, 1996, Lemma 3.18], ( C †− C − , A ) is observable, so ( C − , A ) is observable. That is, G is observable.(ii) → (i). Because X = I n > C †− C − ≥ C †− C − , A ) is observable, by [Zhou, Doyle & Glover, 1996,Lemma 3.19], A is Hurwitz stable.The equivalence between (ii) and (iii) has been established in Proposition 2.3. Remark 2.
An alternative proof of the equivalence between (ii) and (iii) is given in [Guta & Yamamoto, 2013, Lemma3.1]. An alternative proof of (ii) → (i) is given in [Guta & Yamamoto, 2013, Lemma 3.2]. In this subsection we study minimal realization of a given quantum linear passive system G ∼ ( S, C − , Ω − ). We firstintroduce the concept of minimal realization. Definition 3.4
If a quantum linear passive system G ∼ ( S, C − , Ω − ) is both controllable and observable, we say itsrealization (3.1)-(3.2) is a minimal realization. The following result is an immediate consequence of Lemma 3.3.
Theorem 3.5 (3.1)-(3.2) is a minimal realization of the quantum linear passive system G ∼ ( S, C − , Ω − ) if and only ifit is Hurwitz stable. In what follows we study the following problem concerning minimal realization.
Problem 3.6
Given a quantum linear passive system G ∼ ( S, C − , Ω − ) which may not be Hurwitz stable, it may havea subsystem ( S, C min , Ω min ) which is Hurwitz stable. In this case, let n min be the number of system oscillators in theminimal realization of ( S, C min , Ω min ) . How to compute n min ? Given a SISO quantum linear passive system G ( s ), let the spectral decomposition of Ω − beΩ − = X ω ∈ spec(Ω − ) ωP ω , where P ω denotes the projection onto the eigenspace of the eigenvalue ω of Ω − . Define σ (Ω − , C − ) ≡ { ω ∈ spec(Ω − ) : C − P ω C †− = 0 } . (3.8)The following result shows that the size of the set σ (Ω − , C − ) is nothing but n min . Proposition 3.7
Given a SISO quantum linear passive system G ∼ ( S, C − , Ω − ) , the number n min of oscillators of aminimal realization ( S, C min , Ω min ) is equal to the size of the set σ (Ω − , C − ) defined in (3.8). The proof is given in the Appendix.
The following result is the MIMO version of Proposition 3.7.
Proposition 3.8
For a MIMO quantum linear passive system G ∼ ( S, C − , Ω − ) , let the distinctive eigenvalues of Ω − be ω , . . . , ω r , each with algebraic multiplicity τ i respectively, i = 1 , . . . , r . Define Λ i = ω i I τ i , i = 1 , . . . , r . Assume Ω − = Λ . . . r . (3.9)9 ccordingly partition C − = [ C C · · · C r ] with C i having τ i columns, i = 1 , . . . , r . Then n min = r X i =1 column rank[ C i ] In particular, if τ i = 1 for all i = 1 , . . . , r , that is, all poles of Ω − are simple poles, then n min = { ω i ∈ spec(Ω − ) : Tr[ C − P ω i C †− ] = 0 } , (3.10) as given in Proposition 3.7. The construction in Proposition 3.8 is essentially the Gilbert’s realization. Its proof follows the discussions in[Kailath, 1980, Sec. 6.1] or [Zhou, Doyle & Glover, 1996, Sec. 3.7]. The details are omitted. G and Σ In this subsection we explore a further relation between a quantum linear passive system G ∼ ( S, C − , Ω − ) and Σ definedin (3.4).We first review the notions of lossless bounded real and lossless positive real. The bounded real lemma for quantumlinear passive systems has been established in [Maalouf & Petersen, 2011a]. Dissipation theory for more general quantumlinear systems has been studied in [James, Nurdin & Petersen, 2008], [Zhang & James, 2011], while the nonlinear casehas been studied in [James & Gough, 2010]. Definition 3.9 (Lossless Bounded Real, [Maalouf & Petersen, 2011a, Definition 6.3].) A quantum linear passive system G = ( S, C − , Ω − ) is said to be lossless bounded real if it is Hurwitz stable and Eq. (3.6) holds. According to Definition 3.9, a Hurwitz stable quantum linear passive system is naturally lossless bounded real, asderived in [Maalouf & Petersen, 2011a].Positive real functions have been studied extensively in classical (namely, non-quantum) control theory, see, e.g.,[Anderson & Vongpanitlerd, 1973]. Here we state a complex-domain version of positive real functions.
Definition 3.10 (Lossless Positive Real.) A function Ξ( s ) is said to be positive real if it is analytic in Re[ s ] > andsatisfies Ξ( s ) + Ξ( s ) † ≥ , ∀ Re[ s ] > . Moreover, Ξ( s ) is called lossless positive real if is positive real and satisfies Ξ( iω ) + Ξ( iω ) † = 0 , (3.11) where iω is not a pole of Ξ( s ) . The following result relates the lossless bounded realness of a quantum linear passive system G ∼ ( S, C − , Ω − ) to thelossless positive realness of Σ( s ) defined in Eq. (3.4). Theorem 3.11
If a quantum linear passive system G ∼ ( S, C − , Ω − ) is minimal, then (i) G ( s ) is lossless bounded real. (ii) Σ( s ) defined in Eq. (3.4) is lossless positive real. roof. (i). Without loss of generality, assume S = I m . Because G ∼ ( I, C − , Ω − ) is minimal, by Theorem 3.5, it isHurwitz stable. Moreover, G ∼ ( I, C − , Ω − ) satisfies Eq. (3.6). Therefore, according to Definition 3.9, G ∼ ( I, C − , Ω − ) islossless bonded real.(ii). Assume iω is not a pole of Σ( s ). Then the matrix iωI + i Ω − is invertible. Note thatΣ( s ) + Σ( s ) † = 12 C − ( sI + i Ω − ) − C †− + 12 C − ( s ∗ I − i Ω − ) − C †− (3.12)= Re [ s ] C − ( sI + i Ω − ) − (cid:0) C − ( sI + i Ω − ) − (cid:1) † , ∀ Re[ s ] > . By (3.12), Σ( iω ) + Σ( iω ) † = 0. Therefore, by Definition 3.10, Σ( s ) is lossless positive real. Remark 3.
In fact it can be shown that in the minimal realization case (i) and (ii) in Theorem 3.11 are equivalent.
Remark 4. Σ( s ) is a linear port-Hamiltonian system, [van der Schaft, 1996, Chapter 4]. However it is worth notingthat a lossless positive real Σ( s ) may not generate a genuine quantum system G ( s ) via G ( s ) = ( I − Σ( s ))( I + Σ( s )) − . Forexample, given Σ( s ) = s +1 s ( s +2) , it is lossless positive real. However, it can be verified that G ( s ) = ( I − Σ( s ))( I + Σ( s )) − =1 − s +1) s + s +2 s +1 is not a genuine quantum linear system. Later in Section 4.2.3 we will give an explicit form of Σ( s ) whichgenerates a genuine quantum linear passive system G , see (4.28) for details.Here we have used the annihilation-operator form to study dissipative properties of quantum linear passive sys-tems. Because the resulting matrices may be complex-valued, they can be viewed as the complex versions of losslessbounded real and lossless positive real in terms of the quadrature form, [James, Nurdin & Petersen, 2008]. In fact, ifthe quantum system is represented in the quadrature form, it is exactly the same lossless bounded real form as that in[Anderson & Vongpanitlerd, 1973, Secs. 2.6 and 2.7] for classical linear systems. In fact, the relation between losslessbounded real and lossless positive real is well-known in electric networks, see. e.g., [Anderson & Vongpanitlerd, 1973]. Several realizations of quantum linear passive systems are proposed in this section. The multi-input-multi-output (MIMO)case is studied in Subsection 4.1. For the single-input-single-output (SISO) case, an independent-oscillator realization isproposed in Subsection 4.2.1, Fig. 2; a chain-mode realization is presented in Subsections 4.2.2, Fig. 3; and the uniquenessof the independent-oscillator realization is discussed in Subsection 4.2.3.
In this subsection a new realization for MIMO quantum linear passive systems is proposed.Before presenting our realizations for quantum linear passive systems, we describe for completeness a realizationproposed in [Nurdin, 2010] and [Petersen, 2011] using the series product to produce a realization of an n -oscillator systemas a cascade of n one-oscillator systems.We begin with the observation that every matrix n × n matrix A admits a Schur decomposition A = U † A ′ U with U unitary and A ′ lower triangular. For a given quantum linear passive system G ∼ ( S, C − , Ω − ), we define a unitarytransform a ′ ≡ U a , such that A ′ = U AU † is lower triangular. Accordingly denote C ′ = C − U † and Ω ′ = U Ω − U † . Thenew system is thus G ′ ∼ ( S, C ′ , Ω ′ ). A standard result from linear systems theory shows that the two systems G and G ′ have the same transfer function. In what follows we show the system G ′ has a cascade realization, Fig. 4.1. Because A ′ = − C ′† C ′ − i Ω ′ is lower triangular, for j < k we have A ′ jk = − C ′† j C ′ k − i Ω ′ jk = 0, so Ω ′ jk = i C ′† j C ′ k . Therefore thelower triangular components are A ′ kj = − C ′† k C ′ j − i Ω ′ kj = − C ′† k C ′ j − i Ω ′∗ jk ≡ − C ′† k C ′ j . Let us now set G ∼ ( S, ,
0) and G k ∼ ( I, C ′ k , Ω ′ kk ) then the new system G ′ has a the cascaded realization G ′ = G n ⊳ · · · ⊳ G ⊳ G , Fig. 4.1. 11igure 1: A quantum linear passive system with n system oscillators is realised as a sequence of n components in series,each one having a one-mode oscillator.Next we present a new realization for MIMO quantum linear passive systems, which may have: 1) a set of inter-connected principal oscillators ˜ a pr that interact with the (possibly part of) environment ˜ b pr ( t ); 2) auxiliary oscillators˜ a aux , and ˜ a aux , ) which only couple to the principal oscillators while otherwise being independent; 3) input-out channels˜ b aux ( t ) that do not couple to the system oscillators. Theorem 4.1
A quantum linear passive system G = ( I, C − , Ω − ) can be unitarily transformed to another one with thecorresponding realization d ˜ a pr ( t ) = − ( σ ( C − ) i ˜Ω )˜ a pr ( t ) dt − i ˜Ω ˜ a aux , ( t ) dt − i ˜Ω ˜ a aux , ( t ) dt − σ ( C − ) d ˜ B in , pr ( t ) , (4.1) d ˜ a aux , ( t ) = − iσ ( ˜Ω )˜ a aux , ( t ) dt − i ˜Ω † ˜ a pr ( t ) dt, (4.2) d ˜ a aux , ( t ) = − i ˜Ω † ˜ a pr ( t ) dt, (4.3) dB out , pr ( t ) = σ ( C − )˜ a pr ( t ) dt + dB in , pr ( t ) , (4.4) dB out , aux ( t ) = dB in , aux ( t ) , (4.5) where ˜Ω = ˜Ω † , ˜Ω = ˜Ω † , and σ ( X ) denotes the diagonal matrix with diagonal entries being the non-zero singular valuesof the matrix X . Clearly, this new realization corresponds the a quantum linear passive system (cid:0) I, ¯ C, ¯Ω (cid:1) with ¯ C ≡ " σ ( C − ) 0 00 0 0 , ¯Ω ≡ ˜Ω ˜Ω ˜Ω ˜Ω † σ ( ˜Ω ) 0˜Ω † . (4.6)The proof is given in the Appendix.The realization (4.1)-(4.5) is in some sense like controllability and observability decomposition of quantum linearpassive systems. In fact by Proposition 2.3 and Theorem 2.6, we have the following result. Corollary 4.2
For the realization (4.1)-(4.5),1. the mode ˜ a pr is both controllable and observable;2. if the system G = ( I, C − , Ω − ) is Hurwitz stable, then ˜Ω = 0 and ˜Ω = 0 .Remark 5. When m = 1, assuming minimal realization, from the proof given in the Appendix it can be seenthat Theorem 4.1 reduces to Theorem 4.3 for the independent-oscillator realization of SISO systems to be discussed inSubsection 4.2.1. In this subsection, two realizations, namely the independent-oscillator realization and the chain-mode realization, of SISOquantum linear passive systems are proposed. 12 . .
Principle Mode
Figure 2: The independent-oscillator realization: the principal mode is coupled to n − Given a SISO quantum linear passive system G ∼ ( I, C − , Ω − ) where C − = [ √ γ . . . √ γ n ] , Ω − = ( ω jk ) n × n , (4.7)we show how to find a unitarily equivalent realization in terms of a single oscillator (the coupling mode c , we also callit the principle mode) which is then coupled to n − c , · · · , c n − . The auxiliary modes are themselvesotherwise independent oscillators, Fig. 2. Theorem 4.3
There exists a unitary matrix T such that the transformed modes c = c c ... c n − ≡ T a (4.8) have the following realizations dc ( t ) = − ( γ/ iω ) c ( t ) dt − n − X j =1 i √ κ j c j ( t ) − √ γdB ( t ) , (4.9) dc j ( t ) = − iω j c j ( t ) dt − i √ κ j c ( t ) dt, j = 1 , . . . , n − , (4.10) dB out ( t ) = √ γc ( t ) + dB ( t ) , (4.11) where γ ≡ n X j =1 γ j , ω ≡ γ n X j,k =1 √ γ j γ k ω jk , (4.12) and the other parameters ω j , κ j ( j = 1 , . . . , n − ) are given in the proof. Proof.
Let R be a unitary matrix whose first row is R j = p γ j /γ , ( j = 1 , . . . n ). Set b ′ j ≡ P nk =1 R jk a k , j = 1 , . . . n .We have [ b ′ j , b ′∗ k ] = δ jk . Clearly L = C − a = √ γb ′ and [ L, b ′∗ j ] = 0 for j = 2 , . . . n . Let us apply a further unitarytransformation V of the form V = " ⊤ n − n − ˜ V with 0 n − the column vector of length n − V unitary in C ( n − × ( n − to be specified later. We set c = c c ... c n − ≡ V b ′ = V R a .
13e have L = √ γc . The Hamiltonian takes the form H = c † V R Ω R † V † c = c † Ω ′ c , whereΩ ′ ≡ " ⊤ n − n − ˜ V R Ω − R † " ⊤ n − n − ˜ V † . As the matrix ˜ V is still arbitrary except being unitary, we may choose it to diagonalize the lower right ( n − × ( n − R Ω R † , and with this choice we obtain Ω ′ of the formΩ IO ≡ ω ε ∗ · · · ε ∗ n − ε ω ε n − ω n − . (4.13)It can be readily verified that ω = γ P njk =1 √ γ j γ k ω jk . Set T = V R and the overall unitary transform is thus c = T a .Finally we may absorb the phases of the ε k into the modes, so without loss of generality we may assume that they arereal and non-negative, say ε k ≡ √ κ k .By Proposition 2.3 and Theorem 2.6, we have the following corollary: Corollary 4.4
For the realization (4.8)-(4.11) constructed in Proposition 4.3, if κ j = 0 , ( j = 1 , . . . , n − ,) then themode c j is neither controllable nor observable. Because the two realizations, G ∼ ( I, C − , Ω − ) with C − , Ω − defined in (4.7) and that in (4.8)-(4.11), are unitarilyequivalent, they have the same transfer function. In what follows we derive their transfer function.The following lemma turns out to be useful. Lemma 4.5
We have the algebraic identity that a b · · · b n b a . . . ... . . . b n a n − row 1 , column 1 = 1 a − P nk =1 b k a k , where ( X ) row 1 , column 1 means the entry on the intersection of the first row and first column of a constant matrix X . The proof is given in the Appendix.We are now ready to present the transfer function.
Corollary 4.6
The SISO quantum linear passive system G ∼ ( I, C − , Ω − ) with C − , Ω − defined in (4.7) has a transferfunction of the form G ( s ) = 1 − γs + γ + iω + P n − k =1 κ k s + iω k . (4.14)The proof follows Theorem 4.3 and Lemma 4.5. Remark 6.
Theorem 4.3 gives an independent-oscillator realization of a quantum linear passive system, Fig. 2.Unfortunately, because the unitary matrices V and R used in the proof of Theorem 4.3 are by no means unique, it isunclear whether this realization is unique or not, that is, whether the parameters ω i and κ j are uniquely determined bythe system parameters γ i and ω jk in (4.7) or not. In Theorem 4.10 to be given in Subsection 4.2.3, we show that theindependent-oscillator realization is unique under the assumption of minimal realization.14 .2.2 Chain-mode realization In the subsection we present the chain-mode realization of SISO quantum linear passive systems.Let G ∼ ( I, C min , Ω min ) be a Hurwitz stable SISO quantum linear system with n min the number of system oscillators.We assume that Ω min is diagonal and the entries of C min are non-negative; specifically, ¯a = ¯ a ...¯ a n min , Ω min = diag (¯ ω , · · · , ¯ ω n min ) , C min = (cid:2) √ ¯ γ , · · · , p ¯ γ n min (cid:3) . (4.15) Remark 7.
Because the matrix Ω min is Hermitian, it can always be diagonalized. Similarly by absorbing phases intosystem oscillators if necessary, the entries of the matrix C min can be taken to be non-negative. Thus, given a Hurwitzstable quantum linear passive system, one can always unitarily transform it to another one corresponding to (4.15).Moreover, by Proposition 3.7, minimality requires that ¯ ω j = ¯ ω k if j = k , and ¯ γ j = 0, j = 1 , . . . , n min .In what follows we unitarily transform the system G ∼ ( I, Ω min , C min ) to a chain-mode realization of an assembly ofinteracting oscillators, Fig. 3. Principal Mode
Figure 3: The Chain-mode realization: the principal mode is coupled to a non-damped mode which in turn is coupled toa finite chain of modes.
Theorem 4.7
For the system G ∼ ( I, C min , Ω min ) defined by (4.15), there exists a unitary transform W such that thetransformed modes ˜ c ˜ c ... ˜ c n min − ≡ W ¯a (4.16) have the following realization: d ˜ c ( t ) = − (¯ γ/ i ˜ ω )˜ c ( t ) dt − i p ˜ κ ˜ c ( t ) dt − √ ¯ γdB ( t ) , (4.17) d ˜ c j ( t ) = − i ˜ ω j ˜ c j ( t ) dt − i p ˜ κ j ˜ c j − ( t ) dt − i p ˜ κ j +1 ˜ c j +1 ( t ) dt, j = 1 , . . . , n min − , (4.18) d ˜ c n min − ( t ) = − i ˜ ω n min − ˜ c n min − ( t ) dt − i p ˜ κ n min − ˜ c n min − ( t ) dt, (4.19) dB out ( t ) = √ ¯ γ ˜ c ( t ) dt + dB ( t ) , (4.20) where the parameters ˜ ω j and ˜ κ j are given respectively in (5.28) and (5.29) in the proof. Remark 8.
In the literature of continued fraction, [Wall, 1948]; [Gautschi, 2004]; [Hughes, Christ & Burghardt, 2009];[Woods, et al., 2014], etc., the matrix
J ≡ ˜ ω √ ˜ κ · · · √ ˜ κ ˜ ω √ ˜ κ √ ˜ κ ˜ ω . . . ...... . . . . . . p ˜ κ n min −
00 0 p ˜ κ n min − ˜ ω n min − p ˜ κ n min − · · · p ˜ κ n min − ˜ ω n min − is often called a Jacob matrix . Clearly J is actually the Hamiltonian matrix for the new system corresponding to therealization (4.17)-(4.20).Because the two realizations, G ∼ ( I, C min , Ω min ) defined by (4.15) and that in (4.17)-(4.20), are unitarily equivalent,they share the same transfer function. Next we study their transfer function.We begin with the following lemma. Lemma 4.8
We have the algebraic identity that a b b a . . .. . . . . . b n b n a n − , column 1 = 1 a − b a − b a − . . . − b n − a n − − b n a n , where ( X ) row 1 , column 1 means the entry on the intersection of the first row and first column of a constant matrix X . The proof is given in the Appendix.Based on Theorem 4.7 and Lemma 4.8, we may derive the transfer function.
Corollary 4.9
The SISO quantum linear passive system G ∼ ( I, C min , Ω min ) has a transfer function in the form of thecontinued fraction expansion G ( s ) = I − ¯ γs + 12 ¯ γ + iω + ˜ κ s + i ˜ ω + . . . + ˜ κ n min − s + i ˜ ω n min − + ˜ κ n min − s + i ˜ ω n min − . (4.21) In Subsection 4.2.1 an independent-oscillator realization for SISO quantum linear passive systems is proposed. From theconstruction it is unclear whether the parameters in this independent-oscillator realization are unique, Remark 6. In thissubsection we show that they are indeed unique if minimality is assumed.
Theorem 4.10
Given a minimal quantum linear passive system G ∼ ( I, C min , Ω min ) in (4.15), its unitarily equivalentindependent-oscillator realization is unique. roof. Firstly, for the minimal realization G ∼ ( I, C min , Ω min ) in (4.15), by (4.12) and (5.30), ω = ˜ ω . Secondly,by (4.14) and (4.21) we see the transfer function takes the form G ( s ) = 1 − γs + γ + iω + ∆( s ) , (4.22)where ∆( s ) ≡ n min − X k =1 κ k s + iω k (4.23)= ˜ κ s + i ˜ ω + ˜ κ s + i ˜ ω + . . . + ˜ κ n min − s + i ˜ ω n min − + ˜ κ n min − s + ˜ ω n min − (4.24)in the independent-oscillator and chain-mode realizations respectively. Replacing s with iω in (4.22), (4.23) and (4.24)we have G ( iω ) = 1 + iγω + ω − γ i − ˆ∆ ( ω ) , (4.25)where ˆ∆ ( ω ) ≡ i ∆( iω ) = n min − X k =1 κ k ω + ω k (4.26)= ˜ κ ω + ˜ ω − ˜ κ ω + ˜ ω − . . . − ˜ κ n min − ω + ˜ ω n min − − ˜ κ n min − ω + ˜ ω n min − (4.27)in the independent-oscillator and chain-mode realizations respectively. By Theorem 4.7, ˜ ω j and ˜ κ j in (4.27) are uniquelydetermined by C min and Ω min , that is, ˆ∆ ( ω ) is unique. On the other hand, because G = ( I, C min , Ω min ) is minimal, in(4.26) ω j = ω k if j = k , and κ i = 0. Clearly, for this single pole fraction form of ˆ∆ ( ω ) in (4.26), κ k and ω k are unique.The proof is completed.We notice that (4.22) implies that Σ ( s ) = 12 γs + iω + ∆ ( s ) (4.28)with ∆( s ) given by (4.24). Remark 9.
Given ∆( s ) in (4.23) and (4.24), by (4.28) an explicit form of Σ( s ) can be constructed, subsequently aquantum linear passive system G ( s ) = ( I − Σ( s ))(( I + Σ( s ))) − can be constructed. According to (4.22), G ( s ) constructedin this way is always a genuine quantum system. In this sense, (4.28) indicates what type of lossless positive real functionscan generate a quantum linear passive system (which is lossless bounded real). In this paper we have studied the realization theory of quantum linear systems. We have shown the equivalence betweencontrollability and observability of general quantum linear systems, and in particular in the passive case they are equivalentto Hurwitz stability. Based on controllability and observability, a special form of realization has been proposed for17eneral quantum linear systems which can be regarded as the complex-domain counterpart of the so-called decoherence-free subspace decomposition studied in [Yamamoto, 2013]. Specific to quantum linear passive systems, formulas forcalculating the cardinality of minimal realizaitons are proposed. A specific realization is proposed for the multi-input-multi-output case which is closely related to controllability and observability decomposition. Finally, two realizations,the independent-oscillator realization and the chain-mode realization, have been derived for the single-input-single-outputcase. It is expected that these results will find applications in quantum systems design.
Acknowledgment
The authors wish to thank Daniel Burgarth for pointing out Reference [Woods, et al., 2014]. The second author wouldlike to thank Runze Cai, Lei Cui and Zhiyang Dong for helpful discussions.
Appendix.
Proof of Lemma 2.5.
We first show that the dimension of the space Ker ( O s ) ∩ Ker ( O s J n ) is even. If a nonzero vector v = " v v ∈ Ker ( O s ) ∩ Ker ( O s J n ) (5.1)with v , v ∈ C n , then C " v v = " C − v + C + v C v + C − v = 0 , C J n " v v = " C − v − C + v C v − C − v = 0 , (5.2)which are equivalent to C " v v = 0 . That is, C " v v = 0 , C J n " v v = 0 ⇐⇒ C " v = 0 , C " v = 0 . (5.3)On the other other hand, by (5.1) we also have C J n Ω " v v = 0 , C J n Ω J n " v v = 0 , (5.4)which are equivalent to C J n Ω " v v = 0 . Therefore we have C J n Ω " v v = 0 , C J n Ω J n " v v = 0 ⇐⇒ C J n Ω " v = 0 , C J n Ω " v = 0 . (5.5)Analogously it can be shown that C ( J n Ω) k " v v = 0 , C J n ( J n Ω) k " v v = 0 ⇐⇒ C ( J n Ω) k " v = 0 , C ( J n Ω) k " v = 0 , k ≥ v = " v v ∈ Ker ( O s ) ∩ Ker ( O s J n ) ⇐⇒ " v , " v ∈ Ker ( O s ) ∩ Ker ( O s J n ) . (5.7)18oreover, it can be readily shown that " v ∈ Ker ( O s ) ∩ Ker ( O s J n ) ⇐⇒ " v ∈ Ker ( O s ) ∩ Ker ( O s J n ) , (5.8) " v ∈ Ker ( O s ) ∩ Ker ( O s J n ) ⇐⇒ " v ∈ Ker ( O s ) ∩ Ker ( O s J n ) , (5.9)As a result, one can choose an orthonormal basis of Ker ( O s ) ∩ Ker ( O s J n ) to be one of the form " v , " v , · · · , " v l , " v l . Therefore, the dimension of the space Ker ( O s ) ∩ Ker ( O s J n ) is even. Here we take it to be 2 l .Secondly, we construct V ∈ C n × l . Noticing Ker (cid:16) O s J n [ I n n ] T (cid:17) = Ker (cid:16) O s [ I n n ] T (cid:17) , we have " v i ∈ Ker ( O s ) ∩ Ker ( O s J n ) ⇐⇒ v i ∈ Ker O s " I n n . Thus it is sufficient to construct the orthonormal basis vectors v , . . . , v l for the space Ker (cid:16) O s [ I n n ] T (cid:17) . This canbe done by the Gram-Schmidt orthogonalisation procedure. Define V ≡ " v · · · v l · · · · · · v · · · v l ∈ C n × l . (5.10)For the above construction, Range( V ) = Ker ( O s ) ∩ Ker ( O s J n ). (2.15) is established.Thirdly, we construct the matrix V . If a normalized vector v l +1 ∈ C n such that for all k = 1 , . . . , l , v † l +1 v k = 0,then ( v l +1 ) † v k = 0. That is, the normalized vectors " v l +1 and " v l +1 are orthogonal to the space Range( V ). Ofcourse " v l +1 and " v l +1 are orthogonal to each other too. By the Gram-Schmidt orthogonalisation procedure anorthonormal basis { v l +1 , . . . , v n } can be found for the orthogonal space of the space spanned by the vectors { v , . . . , v l } . The an orthonormal matrix V can be constructed to be V ≡ " v l +1 · · · v n · · · · · · v l +1 · · · v n ∈ C n × n − l ) . Fourthly, define V ≡ [ V V ]. Clearly, V † V = I n which establishes (2.16).Finally, because V † J n = J l V † , we have V † J n V = " V † J n V V † J n V V † J n V V † J n V = " J l J n − l , which is (2.17). The proof is completed. Proof of Proposition 3.7.
Without loss of generality, assume that Ω − is diagonal. (Otherwise, there exists aunitary matrix T such that ¯Ω = T Ω − T † is diagonal. Correspondingly, denote ¯ P ω = T P ω T † and ¯ C = C − T † . Then¯ C ¯ P ω ¯ C † = C − P ω C † .) Let there be r non-zero entries in the row vector C − . Because Ω − is diagonal, if the i th elementof C − is zero, then the i th column of the matrix in (2.12) is a zero column. As a result, for minimality we need onlyconsider non-zero elements of C − . Without loss of generality, assume C − = [ C C = [ c c · · · c r ] with c i = 0,( i = 1 , . . . , r ). Correspondingly, partition Ω − as Ω − = " Ω
00 Ω , is a r × r square diagonal matrix with ω , . . . , ω r being diagonal entries. Clearly,rank C − C − Ω − ... C − Ω n − − = rank C C Ω ... C Ω r − . (5.11)Notice that C C Ω ... C Ω r − = · · · ω ω · · · ω r ... ... ... ... ω r − ω r − · · · ω r − r c c . . . c r . (5.12)According to Lemma 3.3 and noticing c i = 0 for i = 1 , . . . , r , n min = rank C C Ω ... C Ω r − = rank · · · ω ω · · · ω r ... ... ... ... ω r − ω r − · · · ω r − r . (5.13)Let ℓ be the total number of distinct diagonal entries of the matrix Ω . By a property of the Vandermonde matrices, ℓ = n min . Finally, denote the distinct eigenvalues of Ω by ˆ ω , . . . , ˆ ω ℓ . For each i = 1 , . . . , ℓ , because c i = 0, C − P ˆ ω i C †− = 0.So we have shown that the number n min of system oscillators of a minimal realization ( S, C min , Ω min ) equals the totalnumber of elements of the set σ (Ω − , C − ) defined in (3.8). Proof of Theorem 4.1.
The proof can be done by construction. Let rank( C − ) = r >
0. Firstly, according to[Bernstein, 2009, Theorem 5.6.4] there exist unitary matrices R ∈ C m × m and R ∈ C n × n such that R C − R = " σ ( C − ) r × r
00 0 (5.14)where σ ( C − ) is a diagonal matrix with diagonal entries being singular values of the matrix C − . Partition the matrix R † Ω − R accordingly, and denote ¯Ω = " ˜Ω ˜Ω ˜Ω † ˜Ω ≡ R † Ω − R . (5.15)Define the unitary transformations " ˜ b in , pr ( t )˜ b in , aux ( t ) ≡ R b ( t ) , " ˜ b out , pr ( t )˜ b out , aux ( t ) ≡ R b out ( t ) , " ˜ a pr ( t ) a aux ( t ) ≡ R † a ( t ) , (5.16)where all the first blocks on the left-hand side are a row vector of dimension r . Then G is unitarily equivalent to thefollowing system ˙˜ a pr = − ( σ ( C − ) / i ˜Ω )˜ a pr − i ˜Ω a aux − σ ( C − )˜ b in , pr ( t ) , (5.17)˙ a aux = − i ˜Ω † ˜ a pr − i ˜Ω a aux , (5.18)˜ b out , pr = σ ( C − )˜ a pr + ˜ b in , pr ( t ) , (5.19)˜ b out , aux = ˜ b in , aux ( t ) . (5.20)By Schur decomposition there exists a unitary matrix T ∈ C ( n − r ) × ( n − r ) such that˜Ω = T " σ ( ˜Ω ) 00 0 T † . (5.21)20ccordingly, denote [ ˜Ω ˜Ω ] ≡ ˜Ω T † . As a result, applying the unitary transformation ˜ a pr ˜ a aux , ˜ a aux , ≡ " I r × r T ˜ a pr a aux (5.22)to (5.17)-(5.18) yields the final realization (4.1)-(4.5). Clearly the realization (4.1)-(4.5) corresponds to a quantum linearpassive system whose the parameters are given in (4.6). Proof of Lemma 4.5.
We show this by induction. It is clear true for n = 1, so we the assume it is true for a given n and establish for n + 1. Let us write E ( M ) for the first entry (row 1, column 1) of a matrix M . Let us consider asequence M n = a b · · · b n b a . . . 0... . . . b n a n of matrices, then M n +1 ≡ " M n b n +1 e n b n +1 e ⊤ n a n +1 , where e n = ∈ C n +1 . We recall the Schur-Feshbach inversion formula for a matrix in block form " A A A A − = " Y − − Y − A A − − A − A Y − A − + A − A Y − A A − (5.23)where Y = A − A A − A . From the Schur-Feshbach formula we deduce that E (cid:0) M − n +1 (cid:1) = E (cid:18) ( M n − b n +1 a n +1 e n e ⊤ n ) − (cid:19) . However, the matrix M n − ( b n +1 /a n +1 ) e n e ⊤ n is identical to M n except that we replace the first row first column entry a with a − ( b n +1 /a n +1 ), and by assumption we should then have E (cid:18) ( M n − b n +1 a n +1 e n e ⊤ n ) − (cid:19) = 1 (cid:18) a − b n +1 a n +1 (cid:19) − P nk =1 b k a k . This establishes the formula for n + 1, and so the formula is true by induction. Proof of Proposition 4.7.
The spectral distribution Φ associated with a SISO system G ∼ ( S, C − , Ω − ) is definedthrough the Stieltjes’ integral, i.e., Z ∞−∞ e itω d Φ ( ω ) = 1 C − C †− C − e it Ω − C †− , where the normalization coefficient C − C †− >
0. In particular, in terms of the specific minimal realization G ∼ ( S, C min , Ω min )given in (4.15), we have d Φ ( ω ) = n min X j =1 ¯ γ j ¯ γ δ ( ω − ¯ ω j ) dω ≡ ¯ µ ( ω ) dω, (5.24)where ¯ γ ≡ n min X j =1 ¯ γ j . (5.25)21hat is, the cardinality of the support of d Φ is exactly the number of oscillators n min in the minimal realization of G ∼ ( S, C min , Ω min ). The spectral distribution defined in (5.24) has only finitely many point supports. We define aninner product for polynomials in the field of real numbers in terms of this discrete spectral distribution. More specifically,given two real polynomials P ( ω ) and Q ( ω ), define their inner product with respect to ¯ µ to be h P, Q i ¯ µ ≡ Z ∞−∞ P ( ω ) Q ( ω )¯ µ ( ω ) dω = n min X j =1 ¯ γ j ¯ γ P (¯ ω j ) Q (¯ ω j ) . (5.26)The norm of a polynomial P ( ω ) is of course k P k ≡ p h P, P i ¯ µ . Next we introduce a sequence of n min orthogonalpolynomials { P i } , which are defined via the Gram-Schmidt orthogonalization: P ( ω ) ≡ , P j ( ω ) = ω j − j − X k =0 h ω j , P k i ¯ µ h P k , P k i ¯ µ P k ( ω ) , j = 1 , . . . , n min − , where h ω j , P k i ¯ µ is to be understood as h ω j , P k i ¯ µ = R ∞−∞ ω j P k ( ω )¯ µ ( ω ) dω. It is easy to verify that the above orthogonalpolynomial sequence { P j } n min j =0 satisfies the following three-term recurrence relation, [Gautschi, 2004, Theorem 1.27] P k +1 ( ω ) = ( ω − ˜ ω k ) P k ( ω ) − √ ˜ κ k P k − ( ω ) , k = 0 , . . . , n min − , (5.27)where ˜ κ ≡ k P k and the convention P − ≡ ω k = h ωP k , P k i ¯ µ h P k , P k i ¯ µ , k = 0 , . . . , n min − , (5.28)and ˜ κ k = s h P k , P k i ¯ µ h P k − , P k − i ¯ µ , k = 1 , . . . , n min − . (5.29)(Note that ˜ κ k = 0, k = 0 , . . . , n min − ω = 1¯ γ n min X j =1 ¯ γ j ¯ ω j . (5.30)By normalizing { P j } n min j =0 , that is define ˜ P j ≡ k P j k P j , we can get a set of orthonormal polynomial sequence { ˜ P j } n min j =0 . Wedefine a new set of oscillators to be ˜ c ≡ n min X j =1 r ¯ γ j ¯ γ ˜ P (¯ ω j )¯ a j , (5.31)˜ c k ≡ n min X j =1 r ¯ γ j ¯ γ ˜ P k (¯ ω j )¯ a j , k = 1 , . . . , n min − . (5.32)It can be verified that the transformation (5.31)-(5.32) is unitary. Moreover,˜ c = 1 √ ¯ γ n min X j =1 p ¯ γ j ¯ a j , (5.33)and the canonical commutation relations [˜ c , ˜ c k ] = [˜ c , ˜ c ∗ k ] = 0 , [˜ c j , ˜ c ∗ k ] = δ jk for j, k = 1 , . . . , n min −
1. By (5.33),˜ L = √ ¯ γ ˜ c . (5.34)Define matrices Q = ˜ P (¯ ω ) · · · ˜ P (¯ ω n min )... . . . ...˜ P n min − (¯ ω ) · · · ˜ P n min − (¯ ω n min ) ≡ ˜ P (¯ ω )...˜ P n min − (¯ ω ) (5.35)22nd Γ ≡ diag (cid:16)q ¯ γ ¯ γ , · · · , q ¯ γ n min ¯ γ (cid:17) . It can be shown that n min − X k =0 ¯ γ i ¯ γ ˜ P k (¯ ω i ) ˜ P k (¯ ω j ) = δ ij , i, j = 1 , . . . , n min , (5.36)see, e.g., [Gautschi, 2004, Eq. (1.1.14)]. By (5.36), it can be verified that the inverse matrix of the matrix Q turns out tobe Q − = Γ [ ˜ P (¯ ω ) † . . . , ˜ P n min − (¯ ω ) † ]. Thus we have ˜ c ˜ c ...˜ c n min − = Q Γ ¯ a ¯ a ...¯ a n min . (5.37)With this, the Hamiltonian of the minimal realization can be re-written as n min X j =1 ¯ ω j ¯ a ∗ j ¯ a j = ˜ c ˜ c ...˜ c n min − † (( Q Γ) − ) † Γ ¯ ω . . . ¯ ω n min ( Q Γ) − ˜ c ˜ c ...˜ c n min − . (5.38)Finally, according to (5.35) and (5.36), we have the new Hamiltonian matrix˜ H = (( Q Γ) − ) † Γ ¯ ω
0. . .0 ¯ ω n min ( Q Γ) − (5.39)= ˜ ω √ ˜ κ · · · √ ˜ κ ˜ ω √ ˜ κ √ ˜ κ ˜ ω . . . ...... . . . . . . p ˜ κ n min −
00 0 p ˜ κ n min − ˜ ω n min − p ˜ κ n min − · · · p ˜ κ n min − ˜ ω n min − . With the new coupling operator ˜ J defined (5.34) and new Hamiltonian matrix ˜ H defined in (5.39), the realization(4.17)-(4.20) can be obtained. The proof is completed. Proof of Lemma 4.8.
We again use induction. The formula is clearly true for n = 1. Let us set N n = a b b a . . .. . . . . . b n b n a n and so N n +1 = " N n b n +1 f n b n +1 f ⊤ n a n +1 , where f n = ∈ C n +1 E ( M ) for the first entry (row 1, column 1) of a matrix M . We deduce from the Schur-Feshbach formula(5.23) that E (cid:0) N − n +1 (cid:1) = E (cid:18) ( N n − b n +1 a n +1 f n f ⊤ n ) − (cid:19) . However, the matrix N n − ( b n +1 /a n +1 ) f n f ⊤ n is identical to N n except that we replace the last row, last column entry a n with a n − ( b n +1 /a n +1 ), and if by assumption the relation is true for n we deduce the formula for n + 1. The formula istrue by induction. References [Anderson & Vongpanitlerd, 1973] Anderson, B. D. O. & Vongpanitlerd, S. (1973).
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