Quantum Linear Systems Theory

Transcription

1 RMIT Quantum Linear Systems Theory Ian R. Petersen School of Engineering and Information Technology, University of New South the Australian Defence Force Academy

2 RMIT Acknowledgments Professor Matthew James, Australian National University Dr Hendra Nurdin, Australian National University Ms Aline Maalouf, University of New South Wales A/Prof Elanor Huntington, University of New South Wales The Australian Research Council The Air Force Office of Scientific Research (AFOSR)

3 RMIT Introduction

4 RMIT Introduction Developments in quantum technology and quantum information provide an important motivation for research in the area of quantum feedback control systems.

5 RMIT Introduction Developments in quantum technology and quantum information provide an important motivation for research in the area of quantum feedback control systems. In particular, in recent years, there has been considerable interest in the feedback control and modeling of linear quantum systems.

6 RMIT Introduction Developments in quantum technology and quantum information provide an important motivation for research in the area of quantum feedback control systems. In particular, in recent years, there has been considerable interest in the feedback control and modeling of linear quantum systems. Such linear quantum systems commonly arise in the area of quantum optics.

7 RMIT A linear quantum optics experiment at the University of New South Wales (ADFA). Photo courtesy of Elanor Huntington.

8 RMIT Feedback control of quantum optical systems has potential applications in areas such as quantum communications, quantum teleportation, quantum computing, quantum error correction and gravity wave detection.

9 RMIT Feedback control of quantum optical systems has potential applications in areas such as quantum communications, quantum teleportation, quantum computing, quantum error correction and gravity wave detection. In particular, linear quantum optics is one of the possible platforms being investigated for future secure communication systems and quantum computers.

10 RMIT Feedback control of quantum optical systems has potential applications in areas such as quantum communications, quantum teleportation, quantum computing, quantum error correction and gravity wave detection. In particular, linear quantum optics is one of the possible platforms being investigated for future secure communication systems and quantum computers. Feedback control of quantum systems aims to achieve closed loop properties such as stability, robustness and entanglement.

11 RMIT An important class of linear quantum stochastic models describe the Heisenberg evolution of the (canonical) position and momentum, or annihilation and creation operators of several independent open quantum harmonic oscillators that are coupled to external coherent bosonic fields, such as coherent laser beams.

12 RMIT An important class of linear quantum stochastic models describe the Heisenberg evolution of the (canonical) position and momentum, or annihilation and creation operators of several independent open quantum harmonic oscillators that are coupled to external coherent bosonic fields, such as coherent laser beams. Werner Heisenberg

13 RMIT These linear stochastic models describe quantum optical devices such as optical cavities, linear quantum amplifiers, and finite bandwidth squeezers.

14 RMIT These linear stochastic models describe quantum optical devices such as optical cavities, linear quantum amplifiers, and finite bandwidth squeezers. We consider linear quantum stochastic differential equations driven by quantum Wiener processes.

15 RMIT These linear stochastic models describe quantum optical devices such as optical cavities, linear quantum amplifiers, and finite bandwidth squeezers. We consider linear quantum stochastic differential equations driven by quantum Wiener processes. Norbert Wiener

16 RMIT An important class of quantum feedback control systems involves the use of measurement devices to obtain classical output signals from the quantum system. These classical signals are fed into a classical controller which may be implemented via analog or digital electronics and then the resulting control signal act on the quantum system via an actuator.

17 RMIT An important class of quantum feedback control systems involves the use of measurement devices to obtain classical output signals from the quantum system. These classical signals are fed into a classical controller which may be implemented via analog or digital electronics and then the resulting control signal act on the quantum system via an actuator. Quantum System Actuator Measurement Device Classical Controller

18 RMIT Recent papers on the feedback control of linear quantum systems have considered the case in which the feedback controller itself is also a quantum system. Such feedback control is often referred to as coherent quantum control.

19 RMIT Recent papers on the feedback control of linear quantum systems have considered the case in which the feedback controller itself is also a quantum system. Such feedback control is often referred to as coherent quantum control. Quantum System Coherent quantum feedback control. Coherent Quantum Controller

20 RMIT One motivation for considering such coherent quantum control problems is that coherent controllers have the potential to achieve improved performance since quantum measurements inherently involve the destruction of quantum information

21 RMIT One motivation for considering such coherent quantum control problems is that coherent controllers have the potential to achieve improved performance since quantum measurements inherently involve the destruction of quantum information Also, technology is emerging which will enable the implementation of complex coherent quantum controllers.

22 RMIT One motivation for considering such coherent quantum control problems is that coherent controllers have the potential to achieve improved performance since quantum measurements inherently involve the destruction of quantum information Also, technology is emerging which will enable the implementation of complex coherent quantum controllers. In addition, in many applications, coherent quantum feedback controllers may be preferable to classical feedback controllers due to considerations of speed and ease of implementation.

23 RMIT In a recent paper (James, Nurdin, Petersen, 2008), the coherent quantum H control problem was addressed.

24 RMIT In a recent paper (James, Nurdin, Petersen, 2008), the coherent quantum H control problem was addressed. This paper obtained a solution to this problem in terms of a pair of algebraic Riccati equations.

25 RMIT In a recent paper (James, Nurdin, Petersen, 2008), the coherent quantum H control problem was addressed. This paper obtained a solution to this problem in terms of a pair of algebraic Riccati equations. Also, in a recent paper (Nurdin, James, Petersen, 2009) the coherent quantum LQG problem was addressed.

26 RMIT In a recent paper (James, Nurdin, Petersen, 2008), the coherent quantum H control problem was addressed. This paper obtained a solution to this problem in terms of a pair of algebraic Riccati equations. Also, in a recent paper (Nurdin, James, Petersen, 2009) the coherent quantum LQG problem was addressed. This paper obtained a solution to this problem in terms of a non-convex rank constrained optimization problem.

27 RMIT An example of a coherent quantum H system considered in (Nurdin, James, Petersen, 2008), (Maalouf Petersen 2010) is described by the following diagram:

28 RMIT An example of a coherent quantum H system considered in (Nurdin, James, Petersen, 2008), (Maalouf Petersen 2010) is described by the following diagram: v w y k 1 k a 2 k 3 z u plant 180 Phase Shift controller wc 0 ac kc 2 kc 1

29 RMIT The coherent quantum H control approach of James Nurdin and Petersen (2008) was subsequently implemented experimentally by Hideo Mabuchi of Stanford University:

30 RMIT ag The coherent quantum H control approach of James Nurdin and Petersen (2008) was subsequently implemented experimentally by Hideo Mabuchi of Stanford University:

31 RMIT In general, quantum linear stochastic systems represented by linear Quantum Stochastic Differential Equations (QSDEs) with arbitrary constant coefficients need not correspond to physically meaningful systems.

32 RMIT In general, quantum linear stochastic systems represented by linear Quantum Stochastic Differential Equations (QSDEs) with arbitrary constant coefficients need not correspond to physically meaningful systems. Physical quantum systems must satisfy some additional constraints that restrict the allowable values for the system matrices defining the QSDEs.

33 RMIT In general, quantum linear stochastic systems represented by linear Quantum Stochastic Differential Equations (QSDEs) with arbitrary constant coefficients need not correspond to physically meaningful systems. Physical quantum systems must satisfy some additional constraints that restrict the allowable values for the system matrices defining the QSDEs. In particular, the laws of quantum mechanics dictate that closed quantum systems evolve unitarily, implying that (in the Heisenberg picture) certain canonical observables satisfy the so-called canonical commutation relations (CCR) at all times.

34 RMIT In general, quantum linear stochastic systems represented by linear Quantum Stochastic Differential Equations (QSDEs) with arbitrary constant coefficients need not correspond to physically meaningful systems. Physical quantum systems must satisfy some additional constraints that restrict the allowable values for the system matrices defining the QSDEs. In particular, the laws of quantum mechanics dictate that closed quantum systems evolve unitarily, implying that (in the Heisenberg picture) certain canonical observables satisfy the so-called canonical commutation relations (CCR) at all times. Therefore, to characterize physically meaningful systems, a formal notion of physically realizable quantum linear stochastic systems has been introduced.

35 RMIT Aims

36 RMIT Aims To introduce a class of quantum stochastic differential equations (QSDEs) as models for linear quantum systems.

37 RMIT Aims To introduce a class of quantum stochastic differential equations (QSDEs) as models for linear quantum systems. To introduce a notion of physical realizability for QSDEs and to develop linear systems type results which can be used to characterize the class of physically realizable QSDEs.

38 RMIT Linear Quantum System Models

39 RMIT Linear Quantum System Models We formulate the class of linear quantum system models under consideration.

40 RMIT Linear Quantum System Models We formulate the class of linear quantum system models under consideration. These linear quantum system models take the form of quantum stochastic differential equations which are derived from the quantum harmonic oscillator.

41 RMIT Quantum Harmonic Oscillators

42 RMIT Quantum Harmonic Oscillators We begin by considering a collection of n independent quantum harmonic oscillators which are defined on a Hilbert space H = L 2 (R n, C).

43 RMIT Quantum Harmonic Oscillators We begin by considering a collection of n independent quantum harmonic oscillators which are defined on a Hilbert space H = L 2 (R n, C). Elements of the Hilbert space H, ψ(x) are the standard complex valued wave functions arising in quantum mechanics where x is a spatial variable.

44 RMIT Quantum Harmonic Oscillators We begin by considering a collection of n independent quantum harmonic oscillators which are defined on a Hilbert space H = L 2 (R n, C). Elements of the Hilbert space H, ψ(x) are the standard complex valued wave functions arising in quantum mechanics where x is a spatial variable. Corresponding to this collection of harmonic oscillators is a vector of annihilation operators a = a 1 a 2. a n.

45 RMIT Each annihilation operator a i is an unbounded linear operator defined on a suitable domain in H by (a i ψ)(x) = 1 2 x i ψ(x) ψ(x) x i where ψ H is contained in the domain of the operator a i.

46 RMIT Each annihilation operator a i is an unbounded linear operator defined on a suitable domain in H by (a i ψ)(x) = 1 2 x i ψ(x) ψ(x) x i where ψ H is contained in the domain of the operator a i. The adjoint of the operator a i is denoted a i creation operator. and is referred to as a

47 RMIT Each annihilation operator a i is an unbounded linear operator defined on a suitable domain in H by (a i ψ)(x) = 1 2 x i ψ(x) ψ(x) x i where ψ H is contained in the domain of the operator a i. The adjoint of the operator a i is denoted a i creation operator. and is referred to as a Physically, these operators correspond to the annihilation and creation of a photon respectively.

48 RMIT Canonical Commutation Relations

49 RMIT Canonical Commutation Relations are such that the following canonical commutation relations are satisfied The operators a i and a i [a i,a j] := a i a j a ja i = δ ij where δ ij denotes the Kronecker delta multiplied by the identity operator on the Hilbert space H.

50 RMIT Canonical Commutation Relations are such that the following canonical commutation relations are satisfied The operators a i and a i [a i,a j] := a i a j a ja i = δ ij where δ ij denotes the Kronecker delta multiplied by the identity operator on the Hilbert space H. We also have the commutation relations [a i,a j ] = 0, [a i,a j] = 0.

51 RMIT Canonical Commutation Relations are such that the following canonical commutation relations are satisfied The operators a i and a i [a i,a j] := a i a j a ja i = δ ij where δ ij denotes the Kronecker delta multiplied by the identity operator on the Hilbert space H. We also have the commutation relations [a i,a j ] = 0, [a i,a j] = 0. These commutation relations encapsulate Heisenberg s uncertainty relation.

52 RMIT Notation

53 RMIT Notation For a general vector of operators g = g 1 g 2. g n, on H, we use the notation g # = g 1 g 2... g n, to denote the corresponding vector of adjoint operators.

54 RMIT Also, g T denotes the corresponding row vector of operators g T = [ g 1 g 2... g n ], and g = ( g #) T. Using this notation, the canonical commutation relations can be written as [ [ ] [ ] ] a a a #, a # = = [ a a # ] [ a a # ] [ I 0 0 I ]. ( [ ] # [ ] ) T T a a a # a #

55 RMIT The Quantum State

56 RMIT The Quantum State A state on our system of quantum harmonic oscillators is defined by a density operator ρ which is a self-adjoint positive-semidefinite operator on H with tr(ρ) = 1.

57 RMIT The Quantum State A state on our system of quantum harmonic oscillators is defined by a density operator ρ which is a self-adjoint positive-semidefinite operator on H with tr(ρ) = 1. Corresponding to a state ρ and an operator g on H is the quantum expectation g = tr(ρg).

58 RMIT A state on the system is said to be Gaussian with positive-semidefinite covariance matrix Q C 2n 2n and mean vector α C n if given any vector u C n, [ 1 2 u u ] ] u Q[ T u # ( exp i [ u u ][ ]) T a a # = exp +i [ u u T ][ α α # ] ;.

59 RMIT A state on the system is said to be Gaussian with positive-semidefinite covariance matrix Q C 2n 2n and mean vector α C n if given any vector u C n, [ 1 2 u u ] ] u Q[ T u # ( exp i [ u u ][ ]) T a a # = exp +i [ u u T ][ α α # ] ;. Here, u # denotes the complex conjugate of the complex vector u, u T denotes the transpose of the complex vector u, and u denotes the complex conjugate transpose of the complex vector u.

60 RMIT Note that the covariance matrix Q satisfies Q = [ ][ ] a a a # a #.

61 RMIT Note that the covariance matrix Q satisfies Q = [ ][ ] a a a # a #. In the special case in which the covariance matrix Q is of the form Q = [ I and the mean α = 0, the system is said to be in the vacuum state. ]

62 RMIT Note that the covariance matrix Q satisfies Q = [ ][ ] a a a # a #. In the special case in which the covariance matrix Q is of the form Q = [ I and the mean α = 0, the system is said to be in the vacuum state. In the sequel, it will be assumed that the state on the system of harmonic oscillators is a Gaussian vacuum state. ]

63 RMIT Note that the covariance matrix Q satisfies Q = [ ][ ] a a a # a #. In the special case in which the covariance matrix Q is of the form Q = [ I and the mean α = 0, the system is said to be in the vacuum state. In the sequel, it will be assumed that the state on the system of harmonic oscillators is a Gaussian vacuum state. The state on the system of quantum harmonic oscillators plays a similar role to the probability distribution of the initial conditions of a classical stochastic system. ]

64 RMIT Quantum Weiner Processes

65 RMIT Quantum Weiner Processes The quantum harmonic oscillators described above are assumed to be coupled to m external independent quantum fields modelled by bosonic annihilation field operators A 1 (t), A 2 (t),...,a m (t) which are defined on separate Fock spaces F i defined over L 2 (R) for each field operator.

66 RMIT Quantum Weiner Processes The quantum harmonic oscillators described above are assumed to be coupled to m external independent quantum fields modelled by bosonic annihilation field operators A 1 (t), A 2 (t),...,a m (t) which are defined on separate Fock spaces F i defined over L 2 (R) for each field operator. For each annihilation field operator A j (t), there is a corresponding creation field operator A j (t), which is defined on the same Fock space and is the operator adjoint of A j (t).

67 RMIT The field operators are adapted quantum stochastic processes with forward differentials da j (t) = A j (t + dt) A j (t) and da j(t) = A j(t + dt) A j(t).

68 RMIT The field operators are adapted quantum stochastic processes with forward differentials da j (t) = A j (t + dt) A j (t) and da j(t) = A j(t + dt) A j(t). These differentials have the quantum Itô products: da j (t)da k (t) = δ jk dt; da j(t)da k (t) = 0; da j (t)da k (t) = 0; da j(t)da k(t) = 0.

69 RMIT The field annihilation operators are also collected into a vector of operators defined as follows: A(t) = A 1 (t) A 2 (t). A m (t).

70 RMIT The field annihilation operators are also collected into a vector of operators defined as follows: A(t) = A 1 (t) A 2 (t). A m (t). For each i, the corresponding system state on the Fock space F i is assumed to be a Gaussian vacuum state which means that given any complex valued function u i (t) L 2 (R, C), then ( ) exp i u i (t) da i (t) + i u i (t)da i (t) 0 0 ( = exp 1 ) u(t) 2 dt. 2 0

71 RMIT Hamiltonian, Coupling and Scattering Operators

72 RMIT Hamiltonian, Coupling and Scattering Operators In order to describe the joint evolution of the quantum harmonic oscillators and quantum fields, we first specify the Hamiltonian operator for the quantum system which is a Hermitian operator on H of the form H = 1 2 [ a a T ] M [ a a # ] where M C 2n 2n is a Hermitian matrix of the form [ ] M1 M 2 M = M # 2 M # 1 and M 1 = M 1, M 2 = M T 2.

73 RMIT Hamiltonian, Coupling and Scattering Operators In order to describe the joint evolution of the quantum harmonic oscillators and quantum fields, we first specify the Hamiltonian operator for the quantum system which is a Hermitian operator on H of the form H = 1 2 [ a a T ] M [ a a # ] where M C 2n 2n is a Hermitian matrix of the form [ ] M1 M 2 M = M # 2 M # 1 and M 1 = M 1, M 2 = M T 2. Here, M denotes the complex conjugate transpose of the complex matrix M, M T denotes the transpose of the complex matrix M, and M # denotes the complex conjugate of the complex matrix M.

74 RMIT Also, we specify the coupling operator for the quantum system to be an operator of the form L = [ N 1 N 2 ] [ a a # ] where N 1 C m n and N 2 C m n.

75 RMIT Also, we specify the coupling operator for the quantum system to be an operator of the form L = [ N 1 N 2 ] [ a a # ] where N 1 C m n and N 2 C m n. Also, we write [ L L # ] = N [ a a # ] = [ N1 N 2 N # 2 N # 1 ] [ a a # ].

76 RMIT Also, we specify the coupling operator for the quantum system to be an operator of the form L = [ N 1 N 2 ] [ a a # ] where N 1 C m n and N 2 C m n. Also, we write [ L L # ] = N [ a a # ] = [ N1 N 2 N # 2 N # 1 ] [ a a # ]. In addition, we define a scattering matrix which is a unitary matrix S C m m.

77 RMIT Quantum Stochastic Differential Equations (QSDEs)

78 RMIT Quantum Stochastic Differential Equations (QSDEs) The quantities (S,L,H) define the joint evolution of the quantum harmonic oscillators and the quantum fields according to a unitary adapted process U(t) (which is an operator valued function of time) satisfying the Hudson-Parthasarathy QSDE: du(t) = ((S I) T dλ(t) + da(t) L L da(t) (ih L L)dt)U(t); U(0) = I, where Λ(t) = [Λ jk (t)] j,k=1,...,m.

79 RMIT Here, the processes Λ jk (t) for j,k = 1,...,m are adapted quantum stochastic processes referred to as gauge processes.

80 RMIT Here, the processes Λ jk (t) for j,k = 1,...,m are adapted quantum stochastic processes referred to as gauge processes. The forward differentials dλ jk (t) = Λ jk (t + dt) Λ jk (t) j,k = 1,...,m have the quantum Itô products: dλ jk (t)dλ j k (t) = δ kj dλ jk (t); da j (t)dλ kl (t) = δ jk da l (t); dλ jk da l (t) = δ kl da j(t).

81 RMIT Using the Heisenberg picture of quantum mechanics, the harmonic oscillator operators a i (t) evolve with time unitarily according to for i = 1,2,...,n. a i (t) = U(t) a i U(t)

82 RMIT Using the Heisenberg picture of quantum mechanics, the harmonic oscillator operators a i (t) evolve with time unitarily according to for i = 1,2,...,n. a i (t) = U(t) a i U(t) Also, the linear quantum system output fields are given by for i = 1,2,...,m. A out i (t) = U(t) A i (t)u(t)

83 RMIT We now use the fact that for any adapted processes X(t) and Y (t) satisfying a quantum Itô stochastic differential equation, we have the quantum Itô rule dx(t)y (t) = X(t)dY (t) + dx(t)y (t) + dx(t)dy (t).

84 RMIT We now use the fact that for any adapted processes X(t) and Y (t) satisfying a quantum Itô stochastic differential equation, we have the quantum Itô rule dx(t)y (t) = X(t)dY (t) + dx(t)y (t) + dx(t)dy (t). Using the quantum Itô rule and the quantum Itô products given above, as well as exploiting the canonical commutation relations between the operators in a, the following QSDEs describing the linear quantum system can be obtained.

85 RMIT Quantum Linear System QSDEs

86 RMIT Quantum Linear System QSDEs da(t) = du(t) au(t) a(0) = a; = [ ] [ ] a(t) F 1 F 2 a(t) # dt + [ ] [ ] da(t) G 1 G 2 da(t) # ; da out (t) = du(t) A(t)U(t) = [ H 1 H 2 ] [ a(t) a(t) # ] dt + [ K 1 K 2 ] [ da(t) da(t) # ].

87 RMIT Here F 1 = im F 2 = im ( N 1 N 1 N2 T N # 2 ( N 1 N 2 N2 T N # 1 ) ) ; ; G 1 = N 1 S; G 2 = N T 2 S # ; H 1 = N 1 ; H 2 = N 2 ; K 1 = S; K 2 = 0.

88 RMIT From this, we can write [ ] [ ] [ ] da(t) a(t) da(t) da(t) # = F a(t) # dt + G da(t) # ; [ ] [ ] [ ] da out (t) a(t) da(t) da out (t) # = H a(t) # dt + K da(t) #.

89 RMIT From this, we can write [ ] [ ] [ ] da(t) a(t) da(t) da(t) # = F a(t) # dt + G da(t) # ; [ ] [ ] [ ] da out (t) a(t) da(t) da out (t) # = H a(t) # dt + K da(t) #. Here F = H = [ ] [ ] F1 F 2 G1 G 2 F # 2 F # ; G = 1 G # ; 2 G# 1 [ ] [ ] H1 H 2 K1 K 2 ; K =. H # 2 H# 1 K # 2 K# 1

90 RMIT From this, we can write [ ] [ ] [ ] da(t) a(t) da(t) da(t) # = F a(t) # dt + G da(t) # ; [ ] [ ] [ ] da out (t) a(t) da(t) da out (t) # = H a(t) # dt + K da(t) #. Here F = H = [ ] [ ] F1 F 2 G1 G 2 F # 2 F # ; G = 1 G # ; 2 G# 1 [ ] [ ] H1 H 2 K1 K 2 ; K = H # 2 H# 1 K # 2 K# 1. This is our fundamental quantum linear system model.

91 RMIT Note, the equations for the matrices defining this model can be re-written as F = ijm 1 2 JN JN; [ ] S 0 G = JN 0 S # ; H = N; [ ] S 0 K = 0 S #.

92 RMIT Note, the equations for the matrices defining this model can be re-written as F = ijm 1 2 JN JN; [ ] S 0 G = JN 0 S # ; H = N; [ ] S 0 K = 0 S #. Here J = [ I 0 0 I ].

93 RMIT Example Optical Parametric Oscillator: Squeezer

94 RMIT Example Optical Parametric Oscillator: Squeezer An optical parametric oscillator (OPO) can be used to produce squeezed light in which the quantum noise in one quadrature is squeezed relative to the noise in the other quadrature and yet Heisenberg s uncertainty relation still holds.

95 RMIT Example Optical Parametric Oscillator: Squeezer An optical parametric oscillator (OPO) can be used to produce squeezed light in which the quantum noise in one quadrature is squeezed relative to the noise in the other quadrature and yet Heisenberg s uncertainty relation still holds. An approximate linearized QSDE model of a squeezer is as follows: da = (κa + χa ) dt + 2κdA.

96 RMIT Example Optical Parametric Oscillator: Squeezer An optical parametric oscillator (OPO) can be used to produce squeezed light in which the quantum noise in one quadrature is squeezed relative to the noise in the other quadrature and yet Heisenberg s uncertainty relation still holds. An approximate linearized QSDE model of a squeezer is as follows: da = (κa + χa ) dt + 2κdA. This model is a QSDE quantum linear systems model of the form considered above.

97 RMIT Example Optical Parametric Oscillator: Squeezer An optical parametric oscillator (OPO) can be used to produce squeezed light in which the quantum noise in one quadrature is squeezed relative to the noise in the other quadrature and yet Heisenberg s uncertainty relation still holds. An approximate linearized QSDE model of a squeezer is as follows: da = (κa + χa ) dt + 2κdA. This model is a QSDE quantum linear systems model of the form considered above. The following schematic diagram shows an implementation of a squeezer in the UNSW@ADFA quantum optics laboratory.

98 RMIT homodyne detection G controller laser 1W@532nm 30mW@1064nm seed pump /2 squeezed /2 /4 PBS OPO PPKTP PZT

99 RMIT Generalized Commutation Relations

100 RMIT Generalized Commutation Relations We now consider the case when the initial condition in the QSDE describing our linear quantum system is no longer the vector of annihilation operators but rather a vector of linear combinations of annihilation operators and creation operators defined by ã = T 1 a + T 2 a # where T = [ T1 T 2 T # 2 T # 1 ] C 2n 2n is non-singular.

101 RMIT It follows that where [ [ ] [ ] ] ã ã ã #, ã # = Θ. Θ = TJT [ T1 T = 1 T 2T 2 T 1 T2 T T 2 T1 T T # 2 T 1 T # 1 T 2 T # 2 T 2 T T # 1 T 1 T ].

102 RMIT It follows that where [ [ ] [ ] ] ã ã ã #, ã # = Θ. Θ = TJT [ T1 T = 1 T 2T 2 T 1 T2 T T 2 T1 T T # 2 T 1 T # 1 T 2 T # 2 T 2 T T # 1 T 1 T ]. Θ is referred to as the commutation matrix for the linear quantum system.

103 RMIT In terms of the variables ã(t) = U(t) ãu(t), the QSDEs for the linear quantum system can be written as [ ] dã(t) dã(t) # [ ] da out (t) da out (t) # = F = H [ ] [ ] ã(t) ã(t) # dt + G da(t) da(t) # ; [ ] [ ] ã(t) ã(t) # dt + K da(t) da(t) #. where F = H = [ F1 F2 F # 2 F # 1 [ H1 H2 H # 2 H # 1 ] ] = TFT 1 ; G = [ G1 G2 G # 2 G # 1 = HT 1 ; K = [ K1 K2 K # 2 K # 1 ] ] = TG; = K.

104 RMIT Now, we can re-write the operators H and L defining the above collection of quantum harmonic oscillators in terms of the variables ã as where H = 1 2 [ ã ã T ] M [ ã ã # ], L = Ñ M = ( T ) 1 MT 1, Ñ = NT 1. [ ã ã # ]

105 RMIT Now, we can re-write the operators H and L defining the above collection of quantum harmonic oscillators in terms of the variables ã as where H = 1 2 [ ã ã T ] M [ ã ã # ], L = Ñ M = ( T ) 1 MT 1, Ñ = NT 1. [ ã ã # ] Here M = [ M1 M2 M # 2 M # 1 ], Ñ = [ ] Ñ 1 Ñ 2.

106 RMIT Furthermore, F = iθ M 1 ΘÑ JÑ; [ 2 ] S 0 G = ΘÑ 0 S # ; H = Ñ; [ ] S 0 K = 0 S #.

107 RMIT The above QSDEs define the general class of linear quantum systems being considered.

108 RMIT The above QSDEs define the general class of linear quantum systems being considered. Such quantum systems can be used to model a large range of devices and networks of devices arising in the area of quantum optics including optical cavities, squeezers, optical parametric amplifiers, cavity QED systems, beam splitters, and phase shifters.

109 RMIT Physical Realizability

110 RMIT Physical Realizability Not all QSDEs of the form considered above correspond to physical quantum systems which satisfy all of the laws of quantum mechanics.

111 RMIT Physical Realizability Not all QSDEs of the form considered above correspond to physical quantum systems which satisfy all of the laws of quantum mechanics. For physical systems, the laws of quantum mechanics require that the commutation relations be satisfied for all times.

112 RMIT Physical Realizability Not all QSDEs of the form considered above correspond to physical quantum systems which satisfy all of the laws of quantum mechanics. For physical systems, the laws of quantum mechanics require that the commutation relations be satisfied for all times. This motivates a notion of physical realizability.

113 RMIT Physical Realizability Not all QSDEs of the form considered above correspond to physical quantum systems which satisfy all of the laws of quantum mechanics. For physical systems, the laws of quantum mechanics require that the commutation relations be satisfied for all times. This motivates a notion of physical realizability. This notion is of particular importance in the problem of coherent quantum feedback control in which the controller itself is a quantum system.

114 RMIT Physical Realizability Not all QSDEs of the form considered above correspond to physical quantum systems which satisfy all of the laws of quantum mechanics. For physical systems, the laws of quantum mechanics require that the commutation relations be satisfied for all times. This motivates a notion of physical realizability. This notion is of particular importance in the problem of coherent quantum feedback control in which the controller itself is a quantum system. In this case, if a controller is synthesized using a method such as quantum H control or quantum LQG control, it important that the controller can be implemented as a physical quantum system.

115 RMIT Definition. QSDEs of the form considered above are physically realizable if there exist suitably structured complex matrices Θ = Θ, M = M, Ñ, S such that S S = I, and F = iθ M 1 ΘÑ JÑ; [ 2 ] S 0 G = ΘÑ 0 S # ; H = Ñ; [ ] S 0 K = 0 S # ;

116 RMIT Note that in the above definition, we require that Θ is of the form Θ = TJT = [ T 1 T 1 T T 2 T # 2 T 1 T 2 T T 2 T # 1 T 2 T 1 T T 1 T # 2 T 2 T 2 T T 1 T # 1 ] and M is of the form M = [ M1 M2 M # 2 M # 1 ],

117 RMIT Note that in the above definition, we require that Θ is of the form Θ = TJT = [ T 1 T 1 T T 2 T # 2 T 1 T 2 T T 2 T # 1 T 2 T 1 T T 1 T # 2 T 2 T 2 T T 1 T # 1 ] and M is of the form M = [ M1 M2 M # 2 M # 1 ], The conditions in the above definition require that the QSDEs correspond to a collection of quantum harmonic oscillators with dynamics defined by the operators ( S,L = Ñ [ ã ã # ],H = 1 2 [ ã ã T ] M [ ã ã # ]).

118 RMIT Theorem. The above QSDEs are physically realizable if and only if there exist complex matrices Θ = Θ and S such that S S = I, Θ is of the form above, and FΘ + Θ F + GJ G = 0; [ ] G = Θ H S 0 0 S # ; [ ] S 0 K = 0 S #.

119 RMIT Theorem. The above QSDEs are physically realizable if and only if there exist complex matrices Θ = Θ and S such that S S = I, Θ is of the form above, and FΘ + Θ F + GJ G = 0; [ ] G = Θ H S 0 0 S # ; [ ] S 0 K = 0 S #. Note that the first of these conditions is equivalent to the preservation of the commutation relations for all times.

120 RMIT Remark. In the canonical case when T = I and Θ = J, the physical realizability equations in the above theorem become FJ + J F + GJ G = 0; [ ] G = J H S 0 0 S # ; [ ] S 0 K = 0 S #.

121 RMIT (J, J)-unitary QSDEs

122 RMIT (J, J)-unitary QSDEs We now relate the physical realizability of the above QSDEs to the (J, J)-unitary property of the corresponding transfer function matrix Γ(s) = [ Γ11 (s) Γ 12 (s) Γ 21 (s) Γ 22 (s) ] = H ( si F) 1 G + K.

123 RMIT Definition (Kimura 97). A transfer function matrix Γ(s) of the above form is (J, J)-unitary if Γ(s) JΓ(s) = J for all s C +.

124 RMIT Definition (Kimura 97). A transfer function matrix Γ(s) of the above form is (J, J)-unitary if Γ(s) JΓ(s) = J for all s C +. Here, Γ (s) = Γ( s ) and C + denotes the set {s C : R[s] 0}.

125 RMIT Theorem. (Kimura, 1997) The transfer function matrix of the above form is (J,J)-unitary if and only if K J K = J, and there exists a Hermitian matrix Ψ such that F Ψ + Ψ F + H J H = 0; K J H + G Ψ = 0.

126 RMIT Theorem. (See also Shaiju and Petersen, 2009) Suppose the linear quantum system defined by the above QSDEs is minimal and and that λ i ( F)+λ j ( F) 0 for all eigenvalues λ i ( F), λ j ( F) of the matrix F. Then this linear quantum system is physically realizable if and only if the following conditions hold: (i) The system transfer function matrix Γ(s) is (J, J)-unitary; [ ] (ii) The matrix K is of the form K S 0 = 0 S # where S S = I.

127 RMIT Physical realizability for annihilator operator linear quantum systems

128 RMIT Physical realizability for annihilator operator linear quantum systems An important special case of the linear quantum systems corresponds to the case in which the Hamiltonian operator H and coupling operator L depend only of the vector of annihilation operators a and not on the vector of creation operators a #. This class of linear quantum systems can be used to model interconnections of passive quantum optical devices such as optical cavities, beam splitters, phase shifters and interferometers.

129 RMIT Example

130 RMIT Example Consider an example from quantum optics involving the interconnection of two cavities as shown below.

131 RMIT Example Consider an example from quantum optics involving the interconnection of two cavities as shown below. A out K 1 K 2 A Cavity 1 a γ Cavity 2 a 2

132 RMIT Here K 1 and K 2 are the coupling parameters of the first cavity and γ is the coupling parameter of the second cavity.

133 RMIT Here K 1 and K 2 are the coupling parameters of the first cavity and γ is the coupling parameter of the second cavity. A QSDE describing this quantum system is as follows: d [ a1 a 2 ] = [ K 1 +K 2 2 K 1 K 2 K 1 γ K 2 γ γ 2 [ K1 + ] K 2 da; γ ][ a1 da out = [ K1 + K 2 γ ] [ a 1 a 2 ] dt + da. ] dt a 2

134 RMIT We now consider the question of physical realizability for annihilator operator QSDEs of the form dã(t) = Fã(t)dt + GdA(t) da out (t) = Hã(t)dt + KdA(t)

135 RMIT We now consider the question of physical realizability for annihilator operator QSDEs of the form dã(t) = Fã(t)dt + GdA(t) da out (t) = Hã(t)dt + KdA(t) Definition. The above annihilator operator QSDEs are said to be physically realizable if there exist matrices Θ 1 = Θ 1 > 0, M1 = M 1, Ñ, and S such that S S = I and F = iθ 1 M1 1 2 Θ 1Ñ 1Ñ1; G = Θ 1 Ñ 1 S; H = Ñ1; K = S.

136 RMIT

137 RMIT Theorem. (See also Maalouf and Petersen, 2009) The above annihilator operator QSDEs are physically realizable if and only if there exist complex matrices Θ 1 = Θ 1 > 0 and S such that S S = I and FΘ 1 + Θ 1 F + G G = 0; G = Θ 1 H S; K = S.

138 RMIT In the case of annihilator operator QSDEs, the issue of physical realizability is determined by the lossless bounded real property of the corresponding transfer function matrix Γ(s) = H(sI F) 1 G + K.

139 RMIT In the case of annihilator operator QSDEs, the issue of physical realizability is determined by the lossless bounded real property of the corresponding transfer function matrix Γ(s) = H(sI F) 1 G + K. Definition. The transfer function matrix corresponding to the above annihilator operator QSDEs is said to be lossless bounded real if the following conditions hold: i) F is a Hurwitz matrix; ii) for all ω R. Γ(iω) Γ(iω) = I

140 RMIT The following theorem, which is a complex version of the standard lossless bounded real lemma, gives a state space characterization of the lossless bounded real property.

141 RMIT The following theorem, which is a complex version of the standard lossless bounded real lemma, gives a state space characterization of the lossless bounded real property. Theorem. (Complex Lossless Bounded Real Lemma). Suppose the above annihilator operator QSDEs defines a minimal realization of the transfer function matrix Γ(s) = H(sI F) 1 G + K. Then this transfer function is lossless bounded real if and only if there exists a Hermitian matrix X > 0 such that X F + F X + H H = 0; H K = X G; K K = I.

142 RMIT Combining the above two theorems leads to the following result which provides a complete characterization of the physical realizability property for minimal annihilator operator QSDEs.

143 RMIT Combining the above two theorems leads to the following result which provides a complete characterization of the physical realizability property for minimal annihilator operator QSDEs. Theorem. (See also Maalouf and Petersen, 2009.) Suppose the above annihilator operator QSDEs define a minimal realization of the corresponding transfer function matrix. Then, the QSDEs are physically realizable if and only if the transfer function matrix is lossless bounded real.

144 RMIT Conclusions

145 RMIT Conclusions We have surveyed some recent results in the area of quantum linear systems theory and the issue of physical realizability.

146 RMIT Conclusions We have surveyed some recent results in the area of quantum linear systems theory and the issue of physical realizability. Physical realizability is important in constructing QSDE models which correspond to real physical quantum systems.

147 RMIT Conclusions We have surveyed some recent results in the area of quantum linear systems theory and the issue of physical realizability. Physical realizability is important in constructing QSDE models which correspond to real physical quantum systems. The issue of physical realizability is also important in problems of coherent quantum feedback controller synthesis since the synthesized controller needs to be implemented as a real physical quantum system.

148 RMIT Conclusions We have surveyed some recent results in the area of quantum linear systems theory and the issue of physical realizability. Physical realizability is important in constructing QSDE models which correspond to real physical quantum systems. The issue of physical realizability is also important in problems of coherent quantum feedback controller synthesis since the synthesized controller needs to be implemented as a real physical quantum system. The results surveyed shows that this property of physical realizability is directly connected to the linear systems notions of lossless bounded real systems and (J, J)-unitary systems.