Ian R. Petersen. University of New South Wales, Canberra

Transcription

1 A Direct Coupling Coherent Quantum Observer 1 Ian R. Petersen University of New South Wales, Canberra

2 A Direct Coupling Coherent Quantum Observer 2 This paper considers the problem of constructing a direct coupling quantum observer for a closed linear quantum system. The proposed observer is shown to be able to estimate some but not all of the plant variables in a time averaged sense. A simple example and simulations are included to illustrate the properties of the observer.

3 A Direct Coupling Coherent Quantum Observer 3 In most previous papers on quantum observers the coupling between the plant and the observer is via a field coupling. This leads to an observer structure of the form shown below. quantum plant quantum noise quantum observer quantum noise This enables a one way connection between the quantum plant and the quantum observer. Also, since both the quantum plant and the quantum observer are open quantum systems, they are both subject to quantum noise.

4 A Direct Coupling Coherent Quantum Observer 4 However in a recent paper by Zhang and James, a coherent quantum control problem is considered in which both field coupling and direct coupling is considered between the quantum plant and the quantum controller. In this paper, we explore the construction of a coherent quantum observer in which there is only direct coupling between quantum plant and the quantum observer. Furthermore, both the quantum plant and the quantum observer are assumed to be closed quantum systems which means that they are not subject to quantum noise and are purely deterministic systems. This leads to an observer structure of the form shown below. quantum plant quantum observer It is shown that for the case being considered, a quantum observer can be constructed to estimate some but not all of the system variables of the quantum plant. Also, the observer variables converge to the plant variables in a time averaged sense rather than a quantum expectation sense.

5 A Direct Coupling Coherent Quantum Observer 5 We consider linear non-commutative systems of the form ẋ(t) = Ax(t); x() = x (1) whereais a real matrix inr n n, andx(t) = [x 1 (t)... x n (t)] T is a vector of self-adjoint possibly non-commutative system variables. Herenis assumed to be an even number and n 2 quantum system. is the number of modes in the The initial system variablesx() = x are assumed to satisfy the commutation relations [x j (),x k ()] = 2iΘ jk, j,k = 1,...,n, whereθis a real antisymmetric matrix with componentsθ jk. Here, the matrixθis assumed to be of the formθ = diag(j,j,...,j) wherej denotes the real skew-symmetric2 2matrix J = [ 1 1 ].

6 A Direct Coupling Coherent Quantum Observer 6 This linear quantum system is said to be physically realizable if it ensures the preservation the canonical commutation relations (CCRs): x(t)x(t) T (x(t)x(t) T ) T = 2iΘ for allt. This holds when the system corresponds to a collection of closed quantum harmonic oscillators. Such quantum harmonic oscillators are described by a quadratic Hamiltonian H = 1 2 x()t Rx(), whereris a real symmetric matrix. Theorem 1 The above system is physically realizable if and only if: AΘ+ΘA T =. In this case, the corresponding Hamiltonian matrixris given byr = 1 4 ( ΘA+AT Θ). In addition, for a given Hamiltonian matrixr, the corresponding matrixais given by A = 2ΘR.

7 A Direct Coupling Coherent Quantum Observer 7 Note that the above system cannot be asymptotically stable if it is physically realizable. Since it is not possible for a physically realizable quantum system of the form above to be asymptotically stable, we will need a new notion of convergence for our direct coupled quantum observer.

8 A Direct Coupling Coherent Quantum Observer 8 We first consider general closed linear quantum plants described by non-commutative models of the following form: ẋ p (t) = A p x p (t); x p () = x p ; z p (t) = C p x p (t) wherez p denotes the vector of system variables to be estimated by the observer and A p R n p n p,c p R m p n p. It is assumed that this quantum plant is physically realizable and corresponds to a plant Hamiltonian H p = 1 2 x p() T R p x p () where the symmetric matrixr p is given byr p = 1 4 ( ΘA p +A T pθ). Also, we consider a direct coupled linear quantum observer defined a symmetric matrix R o R n o n o, and matricesr c R n p n o,c o R m p n o.

9 A Direct Coupling Coherent Quantum Observer 9 These matrices define an observer Hamiltonian and a coupling Hamiltonian H o = 1 2 x o() T R o x o (), H c = 1 2 x p() T R c x o ()+ 1 2 x o() T R T c x p (). The matrixc o also defines the vector of estimated variables for the observer as z o (t) = C o x o (t).

10 A Direct Coupling Coherent Quantum Observer 1 The augmented quantum linear system consisting of the quantum plant and the direct coupled quantum observer is then a quantum system of the form described by the total Hamiltonian H a = H p + H c + H o = 1 2 x a() T R a x a () wherex a = [ xp x o ] andr a = [ Rp R c Rc T R o ]. Then, it follows that the augmented quantum linear system is described by the equations [ẋp (t) ẋ o (t) wherea a = 2ΘR a. ] = A a [ xp (t) x o (t) z p (t) = C p x p (t); z o (t) = C o x o (t) ] ; x p () = x p ; x o () = x o ;

11 A Direct Coupling Coherent Quantum Observer 11 Definition 1 The matrices R o R n o n o, R c R n p n o, C o R m p n o define a direct coupled linear quantum observer for the quantum linear plant if the corresponding augmented linear quantum system is such that lim T 1 T T (z p (t) z o (t))dt =.

12 A Direct Coupling Coherent Quantum Observer 12 We now describe the construction of a direct coupled linear quantum observer. We assume that the quantum plant is such thata p =. This corresponds tor p = in the plant Hamiltonian. It follows that the plant system variablesx p (t) will remain fixed if the plant is not coupled to the observer. However, when the plant is coupled to the quantum observer this will no longer be the case. We will show that if the quantum observer is suitably designed, the plant quantity to be estimatedz p (t) will remain fixed and the averaged convergence condition will be satisfied.

13 A Direct Coupling Coherent Quantum Observer 13 We also assume thatm p = n p β = andβ i R 2 1 fori = 1,2,..., n p 2. 2 and the matrixc p is of the formc p = β T where β 1 β 2... np Rn p 2 β n p 2 This assumption means that the plant variables to be estimated include only one quadrature for each mode of the plant. We now suppose that the matricesr o,r c,c o are such thatr c = βα T, α R n o n p 2 and the matrixr o is positive definite. [ ] Θ1 Also, we writeθ = whereθ Θ 1 R n p n p andθ 2 R n o n o. 2 [ ] [ ] βα T 2Θ1 βα Then,R a = αβ T anda R a = 2ΘR a = T o 2Θ 2 αβ T 2Θ 2 R o.

14 A Direct Coupling Coherent Quantum Observer 14 Hence, the augmented system equations describing the combined plant-observer system become ẋ p (t) = 2Θ 1 βα T x o (t); ẋ o (t) = 2Θ 2 αβ T x p (t)+2θ 2 R o x o (t); z p (t) = C p x p (t); z o (t) = C o x o (t). We now use Laplace Transforms to solve these equations. It follows that sx p (s) = 2Θ 1 βα T X o (s)+x p (); sx o (s) = 2Θ 2 αβ T X p (s)+2θ 2 R o X o (s)+x o () and hence, sx o (s) = 4 s Θ 2αβ T Θ 1 βα T X o (s)+ 2 s Θ 2αβ T x p () +2Θ 2 R o X o (s)+x o ().

15 A Direct Coupling Coherent Quantum Observer 15 However, β T Θ 1 β = β T 1 Jβ 1 β T 2 Jβ 2... = β T n p Jβ np sincej is skew-symmetric. Therefore, ( ) X o (s) = (si 2Θ 2 R o ) 1 2 s Θ 2αβ T x p ()+x o ().

16 A Direct Coupling Coherent Quantum Observer 16 Taking the inverse Laplace Transform of this equation, we obtain Also, we obtain x o (t) = e 2Θ 2R o t x o ()+2 = e 2Θ 2R o t x o () t o e 2Θ 2R o (t τ) dτθ 2 αβ T x p () e 2Θ 2R o t ( e 2Θ 2R o t I ) Ro 1 Θ 1 2 Θ 2αβ T x p () = e 2Θ 2R o t ( x o ()+Ro 1 αβ T x p () ) R 1 o αβ T x p (). X p (s) = 4 s 2Θ 1βα T (si 2Θ 2 R o ) 1 Θ 2 αβ T x p () + 2 s Θ 1βα T (si 2Θ 2 R o ) 1 x o () + 1 s x p().

17 A Direct Coupling Coherent Quantum Observer 17 Taking the inverse Laplace Transform of this equation, we obtain t x p (t) = 4Θ 1 βα T e 2Θ 2R o (t τ) τdτθ 2 αβ T x p () o t +2Θ 1 βα T e 2Θ 2R o (t τ) dτx o () +x p () o = 2tΘ 1 βα T R 1 o αβ T x p () +Θ 1 βα T R 2 o Θ 2 αβ T x p () Θ 1 βα T e 2Θ 2R o t R 2 o Θ 2 αβ T x p () +Θ 1 βα T R 1 o Θ 2 x o () Θ 1 βα T e 2Θ 2R o t R 1 o Θ 2 x o () +x p (). We now choose the parameters of the quantum observer so thatc o R 1 o α = I.

18 A Direct Coupling Coherent Quantum Observer 18 It follows that the quantitiesz p (t) = C p x p (t) andz o (t) = C o x o (t) are given by z o (t) = C o e 2Θ 2R o t ( x o ()+R 1 o αβ T x p () ) +z p () and z p (t) = z p () where we have used the fact thatc p Θ 1 β = β T Θ 1 β =. That is, the quantityz p (t) remains constant and is not affected by the coupling to the coherent quantum observer.

19 A Direct Coupling Coherent Quantum Observer 19 Note that the matrixa a will have all purely imaginary eigenvalues. We now verify that the averaged convergence condition is satisfied for this quantum observer. We recall that the quantity 1 2 xt R o x remains constant in time for the linear system: ẋ = 2Θ 2 R o x; x() = x. That is 1 2 x(t)t R o x(t) = 1 2 xt R o x t.

20 A Direct Coupling Coherent Quantum Observer 2 However,x(t) = e 2Θ 2R o t x andr o >. Therefore, it follows that e 2Θ 2R o t x λ max (R o ) λ min (R o ) x for allx andt. Hence, for allt. e 2Θ 2R o t λ max (R o ) λ min (R o )

21 A Direct Coupling Coherent Quantum Observer 21 Now sinceθ 2 andr o are non-singular, T and therefore, it follows that ast. e 2Θ 2R o t dt = 1 2 e2θ 2R o T Ro 1 Θ R 1 o Θ T T e 2Θ 2R o t dt = 1 T 1 2 e2θ 2R o T Ro 1 Θ R 1 o Θ T e2θ 2R o T Ro 1 Θ T 2T R 1 o Θ λ max (R o ) λ min (R o ) R 1 o Θ 1 2 2T R 1 o Θ 1 2

22 A Direct Coupling Coherent Quantum Observer 22 Hence, lim T 1 T T z o (t)dt = z p (). Also, lim T 1 T T z p (t)dt = z p (). Therefore, averaged convergence condition is satisfied. Thus, we have established the following theorem.

23 A Direct Coupling Coherent Quantum Observer 23 Theorem 2 Consider a quantum plant as above wherea p =,C p = β T and β 1 β 2 β =... np Rn p 2. β n p 2 Then the matrices R o >, R c, C o will define direct coupled quantum observer for this quantum plant if R c is of the form R c = Cp T α T where α R n o n p 2 and Co T Ro 1 α = I.

24 A Direct Coupling Coherent Quantum Observer 24 We consider the above result for the single mode case withn p = 2,m p = 1, in which C p = [1]. This means that the variable to be estimated by the quantum observer is the position [ ] qp (t) operator of the quantum plant; i.e.,z p (t) = q p (t) wherex p (t) = p p (t) [ ] [ ] 1 1 By choosingn o = 2,R o = I,C o = [1],β = andα =., the conditions of the above theorem will be satisfied and the observer output variable will be the position operator of the quantum observerq o (t); i.e.,z o (t) = q o (t) where [ ] qo (t) x o (t) =. p o (t) Before the quantum observer is connected to the quantum plant, the quantitiesq p (t) andp p (t) will remain constant since we have assumed thata p =.

25 A Direct Coupling Coherent Quantum Observer 25 Now suppose that the quantum observer is connected to the quantum plant at time t =. Then, the plant position operatorq p (t) will remain constant at its initial value q p (t) = q p () but the plant momentum operatorp p (t) will evolve in an time varying and oscillatory way. In addition, the observer position operatorq o (t) will evolve in an oscillatory way but its time average will converge toq p (). This behavior of the quantum observer is similar to the behavior of quantum measurements. This is not surprising since the behavior of the direct coupled quantum observers considered in this paper and the behavior of quantum measurements are both determined by the quantum commutation relations which are fundamental to the theory of quantum mechanics.

26 A Direct Coupling Coherent Quantum Observer 26 We now present some numerical simulations to illustrate the direct coupled quantum observer. We consider the single mode quantum observer considered above wheren p = 2, m p = 1,n o = 2,A p =,C p = [1],R o = I,C o = [1],β = [ ] 1 α =. As described above, the variable to be estimated by the quantum observer is the position operator of the quantum plant; i.e.,z p (t) = q p (t) wherex p (t) = [ 1 ] and [ qp (t) p p (t) Also, the observer output variable will be the position operator of the quantum observer q o (t); i.e.,z o (t) = q o (t) wherex o (t) = [ qo (t) p o (t) ]. ].

27 A Direct Coupling Coherent Quantum Observer 27 Then the augmented plant-observer system is described by the equations q p (t) ṗ p (t) q o (t) ṗ o (t) = A a q p (t) p p (t) q o (t) p o (t) where A a = [ 2Jβα T 2Jαβ T 2JR o ] =

28 A Direct Coupling Coherent Quantum Observer 28 Then, we can write where Φ(t) = q p (t) p p (t) q o (t) p o (t) = Φ(t) q p () p p () q o () p o () φ 11 (t) φ 12 (t) φ 13 (t) φ 14 (t) φ 21 (t) φ 22 (t) φ 23 (t) φ 24 (t) φ 31 (t) φ 32 (t) φ 33 (t) φ 34 (t) φ 41 (t) φ 42 (t) φ 43 (t) φ 44 (t) Thus, the plant variable to be estimatedq p (t) is given by = ea at. q p (t) = φ 11 (t)q p ()+φ 12 (t)p p ()+φ 13 (t)q o ()+φ 14 (t)p o () and we plot the functionsφ 11,φ 12 (t),φ 13 (t),φ 14 (t) below.

29 A Direct Coupling Coherent Quantum Observer φ 11 (t) φ 12 (t) φ 13 (t).6 φ 14 (t) φ 1i Time From this figure, we can see thatφ 11 (t) 1,φ 12 (t),φ 13 (t),φ 14 (t), andq p (t) will remain constant atq p () for allt.

30 A Direct Coupling Coherent Quantum Observer 3 Also, the other plant variablep p (t) is given by p p (t) = φ 21 (t)q p ()+φ 22 (t)p p ()+φ 23 (t)q o ()+φ 24 (t)p o () and we plot the functionsφ 21,φ 22 (t),φ 23 (t),φ 24 (t) below φ 21 (t) φ 22 (t) φ 2i 2 φ 23 (t) φ 24 (t) Time From this figure, we can see thatp p (t) evolves in a time-varying and oscillatory way when the quantum plant is connected to the quantum observer.

31 A Direct Coupling Coherent Quantum Observer 31 We now consider the output variable of the quantum observerq o (t) which is given by q o (t) = φ 31 (t)q p ()+φ 32 (t)p p ()+φ 33 (t)q o ()+φ 34 (t)p o () and we plot the functionsφ 31,φ 32 (t),φ 33 (t),φ 34 (t) below. 4 3 φ 31 (t) φ 32 (t) φ 33 (t) φ 34 (t) 2 φ 3i Time

32 A Direct Coupling Coherent Quantum Observer 32 To illustrate the time average convergence property of the quantum observer, we now plot the quantities φ ave 31 (T) = 1 T φ ave 32 (T) = 1 T φ ave 33 (T) = 1 T φ ave 34 (T) = 1 T T T T T φ 31 (t)dt φ 32 (t)dt φ 33 (t)dt φ 34 (t)dt below.

33 A Direct Coupling Coherent Quantum Observer av φ 31 (t) av φ (t) 32 av φ (t) 33 av φ 34 (t) 1.5 av φ 3i Time From this figure, we can see that the time average ofq o (t) converges toq p () as t.

34 A Direct Coupling Coherent Quantum Observer 34 To investigate the time average property of the other quantum observer variable, we now plot the quantities φ ave 41 (T) = 1 T φ ave 42 (T) = 1 T φ ave 43 (T) = 1 T φ ave 44 (T) = 1 T T T T T φ 41 (t)dt φ 42 (t)dt φ 43 (t)dt φ 44 (t)dt below.

35 A Direct Coupling Coherent Quantum Observer av φ 41 (t) av φ (t) 42 av φ 43 (t).5 av φ 44 (t) av φ 4i Time

36 A Direct Coupling Coherent Quantum Observer 36 In this paper we have introduced a notion of a direct coupling observer for closed quantum linear systems and given a result which shows how such an observer can be constructed. The main result shows the time average convergence properties of the direct coupling observer. We have also presented an illustrative example along with simulations to investigate the behavior of a direct coupling observer when applied to a simple one mode quantum linear system. Future research in this area might involve extending the class of quantum linear systems for a which a direct coupling observer can be designed and also considering the problem of constructing an observer which is optimal in some sense. Also, future research could investigate the role of direct coupling observers in the design of direct coupling coherent quantum feedback control systems.