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Coherent-ClassicalEstimation for LinearQuantumSystems ⋆

Shibdas Roya,c, Ian R. Petersena, Elanor H. Huntingtonb,c

aSchool of Engineering and Information Technology, University of New South Wales, Canberra

bCollege of Engineering and Computer Science, Australian National University, Canberra

cAustralian Research Council Centre of Excellence for Quantum Computation and Communication Technology, Australia

Abstract

A coherent-classical estimation scheme for a class of linear quantum systems is considered. Here, the estimator is a mixedquantum-classical system that may or may not involve coherent feedback and yields a classical estimate of a variable for thequantum plant. We demonstrate that with no coherent feedback, such coherent-classical estimation provides no improvementover purely-classical estimation, when the quantum plant or the quantum part of the estimator (called the coherent controller)is a passive annihilation operator system. However, when both the quantum plant and the coherent controller are activesystems (that cannot be described by annihilation operators only), coherent-classical estimation without coherent feedbackcan provide improvement over purely-classical estimation in certain cases. Furthermore, we show that with coherent feedback,it is possible to get better estimation accuracies with coherent-classical estimation, as compared to classical-only estimation.

Key words: annihilation-operator, coherent-classical, estimation, Kalman filter, quantum plant.

1 Introduction

Estimation and control problems for quantum systemshave been of significant interest in recent years [3–5, 7,9, 10, 14, 15, 24–28]. An important class of quantumsystems are linear quantum systems [2, 3, 5, 7, 11, 13,14, 17, 19, 21, 22, 24, 25], that describe quantum opticaldevices such as optical cavities [1, 21], linear quantumamplifiers [2], and finite bandwidth squeezers [2]. Muchrecent work has considered coherent feedback control forlinear quantum systems, where the feedback controlleritself is also a quantum system [6–11, 14, 17, 23]. A re-lated coherent-classical estimation problem has been in-troduced by the authors in Refs. [16, 18], where the es-timator consists of a classical part, which produces thedesired final estimate and a quantum part, which mayalso involve coherent feedback. Note that this is differentfrom the problem considered in Ref. [12] which involvesconstructing a quantum observer. A quantum observer isa purely quantum system, that produces a quantum es-timate of a variable for the quantum plant. On the other

⋆ This work was supported by the Australian ResearchCouncil.

Email addresses: roy shibdas@yahoo.co.in (ShibdasRoy), i.r.petersen@gmail.com (Ian R. Petersen),elanor.huntington@anu.edu.au (Elanor H. Huntington).

hand, a coherent-classical estimator is amixed quantum-classical system, that produces a classical estimate of avariable for the quantum plant.

In this paper, we elaborate and build on the results ofthe conference papers [16, 18] to demonstrate two keytheorems, propose three relevant conjectures, and illus-trate our findings with several examples. We show thata coherent-classical estimation scheme not involvingany feedback, where either of the plant and the co-herent controller is a physically realizable annihilationoperator only system, it is not possible to get betterestimates than the corresponding purely-classical es-timation scheme. Although it is possible to get betterestimates than purely-classical estimation with such acoherent-classical estimation scheme in certain cases,when both the plant and the controller are active sys-tems, we observe in examples that for the optimal choiceof the homodyne detection angle, classical-only estima-tion is always superior. Moreover, we demonstrate thata coherent-classical estimation scheme with coherentfeedback provides with higher estimation precision thanclassical-only estimation, if either of the plant and thecontroller can not be defined purely using annihilationoperators. Furthermore, all these results are illustratedwith pertinent examples. In addition, in examples,we observe that if there is any improvement with thecoherent-classical estimation scheme with coherent feed-

Preprint submitted to Automatica 13 February 2015

back over purely-classical estimation for a given plant,we notice that for the optimal choice of the homodyneangle, classical-only estimation is always inferior.

The paper is structured as follows. Section 2 introducesthe class of linear quantum systems considered in thispaper and discusses the notion of physical realizability.Section 3 formulates the problem of the optimal purely-classical estimation of a quantum plant, that is a linearquantum system of the kind of interest here. In Section4, we formulate the optimal coherent-classical estima-tion problem, that does not involve any coherent feed-back. We also present our first key theorem along witha formal proof. Section 5 considers examples involvingdynamic optical squeezers to illustrate the theorem inthe previous section and presents two relevant conjec-tures. Section 6 discusses the coherent-classical estima-tion scheme involving coherent feedback and lays downour second key theorem along with a formal proof. Moreexamples using dynamic squeezers are presented in Sec-tion 7 to support the theorem, before proposing anotherconjecture. Finally, Section 8 concludes the paper withrelevant remarks and possible directions for future work.

2 Linear Quantum Systems and Physical Real-izability

The class of linear quantum systems we consider here aredescribed by the quantum stochastic differential equa-tions (QSDEs) [5, 7, 15, 16, 20]:

[

da(t)

da(t)#

]

= F

[

a(t)

a(t)#

]

dt+G

[

dA(t)

dA(t)#

]

;

[

dAout(t)

dAout(t)#

]

= H

[

a(t)

a(t)#

]

dt+K

[

dA(t)

dA(t)#

]

,

(1)

where

F = ∆(F1, F2), G = ∆(G1, G2),

H = ∆(H1, H2), K = ∆(K1,K2).(2)

Here, a(t) = [a1(t) . . . an(t)]T is a vector of annihila-

tion operators. The adjoint of the operator ai is calleda creation operator, denoted by a∗i . Also, the notation

∆(F1, F2) denotes the matrix

[

F1 F2

F#2 F

#1

]

. Here, F1,

F2 ∈ Cn×n, G1, G2 ∈ Cn×m, H1, H2 ∈ Cm×n, andK1, K2 ∈ Cm×m. Moreover, # denotes the adjoint of avector of operators or the complex conjugate of a com-plex matrix. Furthermore, † denotes the adjoint trans-pose of a vector of operators or the complex conjugatetranspose of a complex matrix. In addition, the vectorA = [A1 . . .Am]T represents a collection of external

independent quantum field operators and the vectorAout represents the corresponding vector of output fieldoperators.

Definition 2.1 (See [7, 15, 16, 20]) A complex linearquantum system of the form (1), (2) is said to be physi-cally realizable if there exists a complex commutation ma-trix Θ = Θ†, a complex Hamiltonian matrix M = M †,and a coupling matrix N such that

Θ = TJT †, (3)

where J =

[

I 0

0 −I

]

, T = ∆(T1, T2) is non-singular, M

and N are of the form

M = ∆(M1,M2), N = ∆(N1, N2) (4)

and

F = −ιΘM − 1

2ΘN †JN,

G = −ΘN †J,

H = N,

K = I.

(5)

Here, ι =√−1 is the imaginary unit, and the commu-

tation matrix Θ satisfies the following commutation re-lation:

[

a

a#

]

,

[

a

a#

]†

=

[

a

a#

][

a

a#

]†

−

[

a

a#

]# [

a

a#

]T

T

= Θ.

(6)

Theorem 2.1 (See [17, 20]) The linear quantum system(1), (2) is physically realizable if and only if there existsa complex matrix Θ = Θ† such that Θ is of the form in(3), and

FΘ+ΘF † +GJG† = 0,

G = −ΘH†J,

K = I.

(7)

If the system (1) is physically realizable, then the matri-ces M and N define a complex open harmonic oscillatorwith a Hamiltonian operator

H =1

2

[

a† aT]

M

[

a

a#

]

,

2

and a coupling operator

L =[

N1 N2

]

[

a

a#

]

.

2.1 Annihilation Operator Only Linear Quantum Sys-tems

Annihilation operator only linear quantum systems are aspecial case of the above class of linear quantum systems,where the QSDEs (1) can be described purely in termsof the vector of annihilation operators [9, 10]. In thiscase, we considerHamiltonian operators of the formH =a†Ma and coupling vectors of the form L = Na, whereM is a Hermitianmatrix andN is a complexmatrix. Thecommutation relation (6), in this case, takes the form:

[

a, a†]

= Θ, (8)

where Θ > 0. Also, the corresponding QSDEs are givenby:

da = Fadt+GdA;

dAout = Hadt+KdA.(9)

Definition 2.2 (See [9, 17]) A linear quantum system ofthe form (9) is physically realizable if there exist complexmatrices Θ > 0, M = M †, N , such that the following issatisfied:

F = Θ

(

−ιM − 1

2N †N

)

,

G = −ΘN †,

H = N,

K = I.

(10)

Theorem 2.2 (See [9, 17]) An annihilation operatoronly linear quantum system of the form (9) is physicallyrealizable if and only if there exists a complex matrixΘ > 0 such that

FΘ+ΘF † +GG† = 0,

G = −ΘH†,

K = I.

(11)

2.2 Linear Quantum System from Quantum Optics

An example of a linear quantum system from quantumoptics is a dynamic squeezer. This is an optical cavitywith a non-linear optical element inside as shown in Fig.1. Upon suitable linearizations and approximations, such

an optical squeezer can be described by the followingquantum stochastic differential equations [16]:

da = −γ

2adt− χa∗dt−√

κ1dA1 −√κ2dA2;

dAout1 =

√κ1adt+ dA1;

dAout2 =

√κ2adt+ dA2,

(12)

where κ1, κ2 > 0, χ ∈ C, and a is a single annihilationoperator of the cavity mode [1, 2]. This leads to a linearquantum system of the form (1) as follows:

[

da(t)

da(t)∗

]

=

[

− γ

2−χ

−χ∗ − γ

2

][

a(t)

a(t)∗

]

dt

−√κ1

[

dA1(t)

dA1(t)#

]

−√κ2

[

dA2(t)

dA2(t)#

]

;

[

dAout1 (t)

dAout1 (t)#

]

=√κ1

[

a(t)

a(t)∗

]

dt+

[

dA1(t)

dA1(t)#

]

;

[

dAout2 (t)

dAout2 (t)#

]

=√κ2

[

a(t)

a(t)∗

]

dt+

[

dA2(t)

dA2(t)#

]

.

(13)

The above quantum system requires γ = κ1+κ2 in orderfor the system to be physically realizable.

Also, the above quantum optical system can be describedpurely in terms of the annihilation operator only if χ = 0,i.e. there is no squeezing, in which case it reduces to apassive optical cavity. This leads to a linear quantumsystem of the form (9) as follows:

da = −γ

2adt−√

κ1dA1 −√κ2dA2;

dAout1 =

√κ1adt+ dA1;

dAout2 =

√κ2adt+ dA2,

(14)

where again the system is physically realizable when wehave γ = κ1 + κ2.

Fig. 1. Schematic diagram of a dynamic optical squeezer[16, 18].

3

3 Purely-Classical Estimation

The schematic diagram of a purely-classical estimationscheme is provided in Fig. 2. We consider a quantumplant, which is a quantum system of the form (1), (2),defined as follows:

[

da(t)

da(t)#

]

= F

[

a(t)

a(t)#

]

dt+G

[

dA(t)

dA(t)#

]

;

[

dY(t)dY(t)#

]

= H

[

a(t)

a(t)#

]

dt+K

[

dA(t)

dA(t)#

]

;

z = C

[

a(t)

a(t)#

]

.

(15)

Here, z denotes a scalar operator on the underlyingHilbert space and represents the quantity to be esti-mated. Also, Y(t) represents the vector of output fieldsof the plant, and A(t) represents a vector of quantumnoises acting on the plant.

In the case of a purely-classical estimator, a quadratureof each component of the vector Y(t) is measured usinghomodyne detection to produce a corresponding classi-cal signal yi:

dy1 = cos(θ1)dY1 + sin(θ1)dY∗1 ;

...

dym = cos(θm)dYm + sin(θm)dY∗m.

(16)

Here, the angles θ1, . . . , θm determine the quadraturemeasured by each homodyne detector. The vector ofclassical signals y = [y1 . . . ym]T is then used as the in-put to a classical estimator defined as follows:

dxe = Fexedt+Gedy;

z = Hexe.(17)

Here z is a scalar classical estimate of the quantity z.The estimation error corresponding to this estimate is

e = z − z. (18)

Fig. 2. Schematic diagram of purely-classical estimation [18].

Then, the optimal classical estimator is defined as thesystem (17) that minimizes the quantity

Jc = limt→∞

〈e∗(t)e(t)〉 , (19)

which is the mean-square error of the estimate. Here,〈·〉 denotes the quantum expectation over the jointquantum-classical system defined by (15), (16), (17).

The optimal classical estimator is given by the standard(complex) Kalman filter defined for the system (15),(16). This optimal classical estimator is obtained fromthe solution to the algebraic Riccati equation:

FaPe + PeF†a +GaG

†a − (GaK

†a + PeH

†a)L

†

× (LKaK†aL

†)−1L(GaK†a + PeH

†a)

† = 0,(20)

where

Fa = F, Ga = G,

Ha = H, Ka = K,

L =[

L1 L2

]

,

L1 =

cos(θ1) 0 . . . 0

0 cos(θ2) . . . 0

. . .

cos(θm)

,

L2 =

sin(θ1) 0 . . . 0

0 sin(θ2) . . . 0

. . .

sin(θm)

.

(21)

Here we have assumed that the quantum noise A ispurely canonical, i.e. dAdA† = Idt and hence K = I.

Equation (20) thus becomes:

FPe + PeF† +GG† − (G+ PeH

†)L†

× L(G+ PeH†)† = 0.

(22)

The value of the cost (19) is given by

Jc = CPeC†. (23)

4 Coherent-Classical Estimation

A schematic diagramof the coherent-classical estimationscheme under consideration is provided in Fig. 3. In this

4

case, the plant output Y(t) does not directly drive a bankof homodyne detectors as in (16). Rather, this outputis fed into another quantum system called a coherentcontroller, which is defined as follows:

[

dac(t)

dac(t)#

]

= Fc

[

ac(t)

ac(t)#

]

dt+Gc

[

dY(t)dY(t)#

]

;

[

dY(t)dY(t)#

]

= Hc

[

ac(t)

ac(t)#

]

dt+Kc

[

dY(t)dY(t)#

]

.

(24)

A quadrature of each component of the vector Y(t) ismeasured using homodyne detection to produce a cor-responding classical signal yi:

dy1 = cos(θ1)dY1 + sin(θ1)dY∗1 ;

...

dym = cos(θm)dYm + sin(θm)dY∗m.

(25)

Here, the angles θ1, . . . , θm determine the quadraturemeasured by each homodyne detector. The vector ofclassical signals y = [y1 . . . ym]T is then used as the in-put to a classical estimator defined as follows:

dxe = Fexedt+ Gedy;

z = Hexe.(26)

Here z is a scalar classical estimate of the quantity z.Corresponding to this estimate is the estimation error(18). Then, the optimal coherent-classical estimator isdefined as the systems (24), (26) which together mini-mize the quantity (19).

We can now combine the quantum plant (15) and thecoherent controller (24) to yield an augmented quantumlinear system defined by the following QSDEs:

Fig. 3. Schematic diagram of coherent-classical estimation[18].

da

da#

dac

da#c

=

F 0

GcH Fc

a

a#

ac

a#c

dt

+

G

GcK

dAdA#

;

dYdY#

=[

KcH Hc

]

a

a#

ac

a#c

dt+KcK

dAdA#

.

(27)

The optimal classical estimator is given by the standard(complex) Kalman filter defined for the system (27),(25). This optimal classical estimator is obtained from

the solution Pe to an algebraic Riccati equation of theform (20), where

Fa =

[

F 0

GcH Fc

]

, Ga =

[

G

GcK

]

,

Ha =[

KcH Hc

]

, Ka = KcK,

L =[

L1 L2

]

,

L1 =

cos(θ1) 0 . . . 0

0 cos(θ2) . . . 0

. . .

cos(θm)

,

L2 =

sin(θ1) 0 . . . 0

0 sin(θ2) . . . 0

. . .

sin(θm)

,

(28)

where since the quantum noiseA is assumed to be purelycanonical, i.e. dAdA† = Idt, we have Ka = KcK = I,which requires Kc = I too, as K = I.

Then, the corresponding optimal classical estimator (26)is defined by the equations:

Fe = Fa − GeLHa;

Ge = (GaK†a + PeH

†a)L

†(LKaK†aL

†)−1;

He =[

C 0]

.

(29)

5

We write:

Pe =

[

P1 P2

P†2 P3

]

, (30)

where P1 is of the same dimension as Pe.

Then, the value of the corresponding cost of the form(19) is then given by

Jc =[

C 0]

Pe

[

C†

0

]

= CP1C†. (31)

Also, we calculate:

GaG†a =

GG† GG†c

GcG† GcG

†c

,

GaK†a + PeH

†a =

G+ P1H† + P2H

†c

Gc + P†2H

† + P3H†c

,

FaPe =

FP1 FP2

GcHP1 + FcP†2 GcHP2 + FcP3

,

PeF†a =

P1F† P1H

†G†c + P2F

†c

P†2F

† P†2H

†G†c + P3F

†c

.

Thus, upon expanding the Riccati equation (20), we getthe following set of equations:

FP1 + P1F† +GG

† − (G+ P1H† + P2H

†c )L

†

× L(G+ P1H† + P2H

†c )

† = 0,

FP2 + P1H†G

†c + P2F

†c +GG

†c − (G+ P1H

†

+ P2H†c )L

†L(Gc + P

†2H

† + P3H†c )

† = 0,

GcHP2 + P†2H

†G

†c + FcP3 + P3F

†c +GcG

†c

− (Gc + P†2H

† + P3H†c )L

†

× L(Gc + P†2H

† + P3H†c )

† = 0.

(32)

Theorem 4.1 Consider a coherent-classical estimationscheme defined by (9) (Aout being Y), (24), (25) and(26), such that the plant is physically realizable, with the

estimation error cost Jc defined in (31). Also, considerthe corresponding purely-classical estimation scheme de-fined by (9), (16) and (17), such that the plant is physi-cally realizable, with the estimation error cost Jc definedin (23). Then,

Jc = Jc. (33)

Proof We first consider the form of the system (15) un-der the assumption that the plant can be characterizedpurely by annihilation operators. This means that theplant is a passive quantum system. A quantum system(1), (2) is characterized by annihilation operators onlywhen F2, G2, H2,K2 = 0.

Then, the equations for the annihilation operators in(15) take the form:

da = F1adt+G1dA;

dY = H1adt+K1dA.(34)

The corresponding equations for the creation operatorsare then:

da∗ = F#1 a∗dt+G

#1 dA#;

dY# = H#1 a∗dt+K

#1 dA#,

(35)

Hence, the plant is described by (15) where:

F =

[

F1 0

0 F#1

]

, G =

[

G1 0

0 G#1

]

,

H =

[

H1 0

0 H#1

]

,K =

[

K1 0

0 K#1

]

.

(36)

Next, we use the assumption that the plant is physicallyrealizable. Then, by applying Theorem 2.2 to (34), thereexists a matrix Θ1 > 0, such that:

F1Θ1 +Θ1F†1 +G1G

†1 = 0,

G1 = −Θ1H†1 ,

K1 = I.

(37)

Hence,

F#1 Θ#

1 +Θ#1 F

T1 +G

#1 G

T1 = 0,

G#1 = −Θ#

1 HT1 ,

K#1 = I.

(38)

Combining (37) and (38), we get:

FΘ+ΘF † +GG† = 0,

G = −ΘH†,

K = I,

(39)

where Θ =

[

Θ1 0

0 Θ#1

]

> 0.

6

Substituting Pe = Θ in the left-hand side of the Riccatiequation (22), we get:

FΘ+ΘF † +GG† − (G+ΘH†)L†L(G+ΘH†)†,

which is clearly zero, owing to (39). Thus, Θ satisfies(22).

Also, it follows from the above that Pe =

[

Θ 0

0 P3

]

sat-

isfies (20) for the case of coherent-classical estimation,since (32) is satisfied, owing to (39), by substitutingP1 = Θ and P2 = 0 to yield:

FΘ+ΘF† +GG

† − (G+ΘH†)L†

× L(G+ΘH†)† = 0,

(G+ΘH†)G†

c − (G+ΘH†)L†

L(Gc + P3H†c )

† = 0,

FcP3 + P3F†c +GcG

†c − (Gc + P3H

†c )L

†

× L(Gc + P3H†c )

† = 0.

(40)

Here, P3 > 0 is the error-covariance of the purely-classical estimation of the coherent controller alone.

Thus, we get Jc = Jc = CΘC†.

Remark 1 We note that the Kalman gain of the purely-classical estimator is zero when Pe = Θ. This impliesthat the Kalman state estimate is independent of the mea-surement. This is consistent with Corollary 1 of Ref. [17],which states that for a physically realizable annihilationoperator quantum system with only quantum noise in-puts, any output field contains no information about theinternal variables of the system.

Remark 2 Theorem 4.1 implies that a coherent-classical estimation of a physically realizable quantumplant, that can be described purely by annihilation opera-tors, performs identical to, and no better than, a purely-classical estimation of the plant. This is so because theoutput field of the quantum plant, as observed above,contains no information about the internal variables ofthe plant and, therefore, serves simply as a quantumwhite noise input for the coherent controller.

5 Dynamic Squeezer Examples

In this section, we consider examples involving dynamicsqueezers. First, we present an example to illustrate The-orem 4.1.

Let us consider the quantum plant to be described bythe QSDEs:

da(t)

da(t)∗

=

− γ

2−χ

−χ∗ − γ

2

a(t)

a(t)∗

dt

−√κ

dA(t)

dA(t)#

;

dY(t)

dY(t)#

=√κ

a(t)

a(t)∗

dt+

dA(t)

dA(t)#

;

z =[

0.2 −0.2]

a

a∗

.

(41)

Here, we choose γ = 4, κ = 4 and χ = 0. Note thatthis system is physically realizable, since γ = κ, and isannihilation operator only, since χ = 0. In fact, this sys-tem corresponds to a passive optical cavity. The matri-ces corresponding to the system (15) are:

F =

[

−2 0

0 −2

]

, G =

[

−2 0

0 −2

]

, H =

[

2 0

0 2

]

,

K = I, C =[

0.2 −0.2]

.

We then calculate the optimal classical-only state esti-mator and the error Jc of (23) for this system using thestandard Kalman filter equations corresponding to ho-modyne detector angles varying from θ = 0◦ to θ = 180◦.

We next consider the case where a dynamic squeezer isused as the coherent controller in a coherent-classicalestimation scheme. In this case, the coherent controller(24) is described by the QSDEs:

da(t)

da(t)∗

=

− γ

2−χ

−χ∗ − γ

2

a(t)

a(t)∗

dt

−√κ

dY(t)

dY(t)#

;

dY(t)

dY(t)#

=√κ

a(t)

a(t)∗

dt+

dY(t)

dY(t)#

.

(42)

Here, we choose γ = 16, κ = 16 and χ = 2, so thatthe system is physically realizable. The matrices corre-sponding to the system (24) are:

Fc =

[

−8 −2

−2 −8

]

, Gc =

[

−4 0

0 −4

]

,

7

Hc =

[

4 0

0 4

]

,Kc = I.

Then, the classical estimator for this case is calculatedaccording to (27), (28), (20), (29) for the homodyne de-tector angle varying from θ = 0◦ to θ = 180◦. The re-sulting value of the cost Jc in (31) alongwith the cost forthe purely-classical estimator case is shown in Fig. 4.

It is clear from this figure that both the classical-onlyand coherent-classical estimators have the same estima-tion error cost for all homodyne angles. This illustratesTheorem 4.1.

Now, we shall consider an example where the controlleris a physically realizable annihilation operator only sys-tem. However, the plant is physically realizable and hasχ 6= 0, i.e. it is not annihilation operator only.

In (41), we choose γ = 4, κ = 4 and χ = 0.5. Note thatthis system is physically realizable, since γ = κ. Next, in(42), we choose γ = 16, κ = 16 and χ = 0. Note that thissystem is also physically realizable, since γ = κ. Also,it is annihilation operator only, since χ = 0. Fig. 5 thenshows a comparison of the estimation error costs.

0 20 40 60 80 100 120 140 160 1800

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

θ (degrees)

Est

imat

ion

err

or

Jc

Jc

Fig. 4. Estimation error vs. homodyne detection angle θ inthe case of an annihilation operator only plant [18].

0 20 40 60 80 100 120 140 160 1800.09

0.092

0.094

0.096

0.098

0.1

0.102

0.104

0.106

0.108

0.11

θ (degrees)

Est

imat

ion

err

or

Jc

Jc

Fig. 5. Estimation error vs. homodyne detection angle θ inthe case of an annihilation operator only controller.

From this figure, clearly the coherent-classical estima-tion error is greater than or equal to the purely-classicalestimation error for all homodyne detector angles. Infact, we observe that the coherent-classical estimator canperform no better than the purely-classical estimator,when the coherent controller can be characterized by itsannihilation operators only. This we present here as aconjecture.

Conjecture 5.1 Consider a coherent-classical estima-tion scheme defined by (15), (24), (25) and (26), wherethe plant is physically realizable and the coherent con-troller is a physically realizable annihilation operator onlysystem, with the estimation error cost Jc defined in (31).Also, consider the corresponding purely-classical estima-tion scheme defined by (15), (16) and (17), such that theplant is physically realizable, with the estimation errorcost Jc defined in (23). Then,

Jc ≥ Jc. (43)

Furthermore, we shall present an example where boththe plant and controller are physically realizable quan-tum systems with χ 6= 0, i.e. none of them are annihila-tion operator only.

In (41), we choose γ = 4, κ = 4 and χ = 1, and in (42),we choose γ = 16, κ = 16 and χ = 4. Fig. 6 then showsa comparison of the estimation error costs.

From this figure, we can see that the coherent-classicalestimator can perform better than the purely-classicalestimator, e.g. for a homodyne detector angle of θ = 40◦.It however appears that for the best choice of homo-dyne detector angle, the classical-only estimator alwaysoutperforms the coherent-classical estimator. We alsopresent this as a conjecture.

Conjecture 5.2 Consider a coherent-classical estima-tion scheme defined by (15), (24), (25) and (26) with an

estimation error cost Jc defined in (31). Also, consider

0 20 40 60 80 100 120 140 160 1800

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

X: 40Y: 0.156

θ (degrees)

Est

imat

ion

err

or

X: 40Y: 0.1443

Jc

Jc

Fig. 6. Estimation error vs. homodyne detection angle θ inthe case of a squeezer plant and a squeezer controller [18].

8

the corresponding purely-classical estimation scheme de-fined by (15), (16) and (17) with an estimation error costJc defined in (23). Then, for the optimal choice of thehomodyne angle θopt, we shall have:

Jc(θopt) ≥ Jc(θopt). (44)

6 Coherent-Classical Estimationwith Feedback

In this section, we consider the case where there is quan-tum feedback from the coherent controller to the quan-tum plant [16]. For this purpose, the quantum plant isassumed to have an additional control input U as de-picted in Fig. 7. The plant (15) then takes the followingform:

da(t)

da(t)#

= F

a(t)

a(t)#

dt

+[

G1 G2

]

dA(t)

dA(t)#

dU(t)dU(t)#

;

dY(t)

dY(t)#

= H

a(t)

a(t)#

dt+[

K 0]

dA(t)

dA(t)#

dU(t)dU(t)#

;

z = C

a(t)

a(t)#

.

(45)

The optimal purely-classical estimator is then obtainedfrom the solution of an algebraic Riccati equation of theform (22), that yields:

FPe + PeF† +G1G

†1 +G2G

†2 − (G1 + PeH

†)

× L†L(G1 + PeH†)† = 0,

(46)

where we have assumedK = I as before. The estimationerror is then given by the cost (23).

Fig. 7. Modified schematic diagram of purely-classical esti-mation [16].

The coherent controller here would have an additionaloutput that is fed back to the control input of the quan-tum plant, as depicted in Fig. 8. The coherent controllerin this case is defined as follows [16]:

dac(t)

dac(t)#

= Fc

ac(t)

ac(t)#

dt

+[

Gc1 Gc2

]

dA(t)

dA(t)#

dY(t)

dY(t)#

;

dY(t)

dY(t)#

dU(t)dU(t)#

=

Hc

Hc

ac(t)

ac(t)#

dt

+

Kc1 Kc2

Kc1 Kc2

dA(t)

dA(t)#

dY(t)

dY(t)#

.

(47)

The quantum plant (45) and the coherent controller (47)can be combined to yield an augmented quantum linearsystem [16]:

da

da#

dac

da#c

=

F +G2Kc2H G2Hc

Gc2H Fc

a

a#

ac

a#c

dt

+

G1 +G2Kc2K G2Kc1

Gc2K Gc1

dAdA#

dAdA#

;

dYdY#

=[

Kc2H Hc

]

a

a#

ac

a#c

dt

+[

Kc2K Kc1

]

dAdA#

dAdA#

.

(48)

9

Fig. 8. Schematic diagram of coherent-classical estimationwith coherent feedback [16].

The optimal coherent-classical estimator is then ob-tained from the solution Pe (given by (30)) to an alge-braic Riccati equation of the form (20), where

Fa =

[

F +G2Kc2H G2Hc

Gc2H Fc

]

,

Ga =

[

G1 +G2Kc2K G2Kc1

Gc2K Gc1

]

,

Ha =[

Kc2H Hc

]

, Ka =[

Kc2K Kc1

]

,

(49)

and L1, L2 and L as in (28). Here, for the coherent con-troller to be physically realizable, we would have:

[

Kc1 Kc2

Kc1 Kc2

]

= I,

which implies Kc1 = Kc2 = I and Kc1 = Kc2 = 0. Theestimation error is then given by the cost (31).

Also, we calculate:

GaG†a =

(G1 +G2)(G1 +G2)† (G1 +G2)G

†c2

Gc2(G1 +G2)† Gc1G

†c1 +Gc2G

†c2

,

GaK†a + PeH

†a =

P2H†c

Gc1 + P3H†c

,

FaPe =

FP1 +G2HP1 FP2 +G2HP2 +G2HcP3

Gc2HP1 + FcP†2 Gc2HP2 + FcP3

,

PeF†a =

P1F† + P1H

†G†2 + P2H

†cG

†2 P1H

†G†c2 + P2F

†c

P†2F

† + P†2H

†G†2 + P3H

†cG

†2 P

†2H

†G†c2 + P3F

†c

.

Thus, upon expanding (20), we get the following set ofequations:

FP1 + P1F† +G1G

†1 +G2G

†2 +G2(HP1 +G

†1)

+ (P1H† +G1)G

†2 +G2HcP

†2 + P2H

†cG

†2

− P2H†cL

†LHcP

†2 = 0,

FP2 + P2F†c + (P1H

† +G1)G†c2 +G2(HcP3 +G

†c2)

+G2HP2 − P2H†cL

†L(Gc1 + P3H

†c )

† = 0,

F3P3 + P3F†c +Gc1G

†c1 +Gc2G

†c2 +Gc2HP2

+ P†2H

†G

†c2 − (Gc1 + P3H

†c )L

†L(Gc1 + P3H

†c )

† = 0.

(50)

Theorem 6.1 Consider a coherent-classical estimationscheme defined by (45), (47), (25) and (26), such thatboth the plant and the coherent controller are physicallyrealizable annihilation operator only systems, with the es-timation error cost Jc defined in (31). Also, consider thecorresponding purely-classical estimation scheme definedby (45), (16) and (17), such that the plant is a physicallyrealizable annihilation operator only system, with the es-timation error cost Jc defined in (23). Then,

Jc = Jc. (51)

Proof The plant (45) may be augmented to account foran unused output Y to recast the QSDE’s in the desiredform, that lends itself appropriately to the physical re-alizability treatment, as follows:

da(t)

da(t)#

= F

a(t)

a(t)#

dt

+[

G1 G2

]

dA(t)

dA(t)#

dU(t)dU(t)#

;

dY(t)

dY(t)#

dY(t)

dY(t)#

=

H

H

a(t)

a(t)#

dt

+

K 0

0 K

dA(t)

dA(t)#

dU(t)dU(t)#

;

z = C

a(t)

a(t)#

.

(52)

Here, K = I for the plant to be physically realizable.Additionally, using the same arguments as in the proof

10

for Theorem 4.1, we must have:

FΘ+ΘF † +G1G†1 +G2G

†2 = 0,

G1 = −ΘH†,

G2 = −ΘH†,

(53)

for the annihilation operator only plant to be physicallyrealizable with the commutation matrix Θ > 0.

Similarly, if the coherent controller (47) is an annihila-tion operator only system, we must have the followingfor it to be physically realizable:

FcΘc +ΘcF†c +Gc1G

†c1 +Gc2G

†c2 = 0,

Gc1 = −ΘcH†c ,

Gc2 = −ΘcH†c ,

(54)

where Θc > 0 is the controller’s commutation matrix.

Clearly, Pe = Θ satisfies the Riccati equation (46), owingto (53), for the purely-classical estimation case. More-

over, it follows from (53) and (54), that Pe =

[

Θ 0

0 Θc

]

satisfies (50), upon substituting P1 = Θ, P2 = 0 and

P3 = Θc. Therefore, Pe satisfies (20) for the coherentclassical estimation case considered here.

Thus, we get Jc = Jc = CΘC†.

Remark 3 Theorem 6.1 implies that a coherent-classical estimation scheme involving coherent feedback,where both the plant and controller are physically re-alizable annihilation operator only quantum systems,performs identical to, and no better than, a purely-classical estimation of the plant. Note that in additionto P2 = 0, we need to have both P1 = Θ and P3 = Θc

for the coherent-classical scheme to be equivalent to theclassical-only scheme.

Remark 4 In the case of coherent-classical estimationnot involving coherent feedback, we observed that sucha scheme cannot outperform purely-classical estimation,when either the plant or the controller is an annihilationoperator only system. By contrast, a coherent-classicalestimation scheme involving coherent feedback cannotperform better than classical-only estimation, when boththe plant and the controller can be purely described byannihilation operators. In other words, it is possible toobtain improvement with the coherent-classical estima-tion scheme involving coherent feedback over the purely-classical estimation scheme, when either of the plant andthe controller has non-zero squeezing, as we shall demon-strate with examples next.

7 More Dynamic Squeezer Examples

In this section, we present examples involving dynamicsqueezers for the case of coherent-classical estimation in-volving coherent feedback. First, we consider an exam-ple to illustrate Theorem 6.1.

In this case, the quantum plant (41) takes the form:

da(t)

da(t)∗

=

− γ

2−χ

−χ∗ − γ

2

a(t)

a(t)∗

dt

−√κ1

dA(t)

dA(t)#

−√κ2

dU(t)dU(t)#

;

dY(t)

dY(t)#

=√κ

a(t)

a(t)∗

dt+

dA(t)

dA(t)#

;

z =[

1√2− 1√

2

]

a

a∗

.

(55)

Here, we choose γ = 4, κ1 = κ2 = 2 and χ = 0. Note thatthis system is physically realizable, since γ = κ1+κ2, andis annihilation operator only, since χ = 0. The matricescorresponding to the system (45) are:

F =

[

−2 0

0 −2

]

, G1 =

[

−1.4142 0

0 −1.4142

]

,

G2 =

[

−1.4142 0

0 −1.4142

]

, H =

[

1.4142 0

0 1.4142

]

,

K = I, C =[

0.7071 −0.7071]

.

We then calculate the optimal classical-only state esti-mator and the error Jc in (23) for this system using thestandard Kalman filter equations corresponding to ho-modyne detector angles varying from θ = 0◦ to θ = 180◦.

The coherent controller (42) in this case takes the form:

11

da(t)

da(t)∗

=

− γ

2−χ

−χ∗ − γ

2

a(t)

a(t)∗

dt

−√κ1

dA(t)

dA(t)#

−√κ2

dY(t)

dY(t)#

;

dY(t)

dY(t)#

=√κ1

a(t)

a(t)∗

dt+

dA(t)

dA(t)#

;

dU(t)dU(t)#

=√κ2

a(t)

a(t)∗

dt+

dY(t)

dY(t)#

.

(56)

Here, we choose γ = 16, κ1 = κ2 = 8 and χ = 0, so thatit is a physically realizable annihilation operator onlysystem. The matrices corresponding to the system (47)are:

Fc =

[

−8 0

0 −8

]

, Gc1 =

[

−2.8284 0

0 −2.8284

]

,

Gc2 =

[

−2.8284 0

0 −2.8284

]

, Hc =

[

2.8284 0

0 2.8284

]

,

Kc1 = I, Kc2 = 0, Hc =

[

2.8284 0

0 2.8284

]

,

Kc1 = 0,Kc2 = I.

Then, the classical estimator for this case is calculatedaccording to (48), (49), (20), (29) for the homodyne de-tector angle varying from θ = 0◦ to θ = 180◦. The re-sulting value of the cost Jc in (31) alongwith the cost forthe purely-classical estimator case is shown in Fig. 9.

0 20 40 60 80 100 120 140 160 1800

0.5

1

1.5

θ (degrees)

Est

imat

ion

err

or

Jc

Jc

Fig. 9. Feedback: Estimation error vs. homodyne detectionangle θ in the case of annihilation operator only plant andcontroller systems.

It is clear from this figure that both the classical-onlyand coherent-classical estimators have the same estima-tion error cost for all homodyne angles. This illustratesTheorem 6.1.

We now want to demonstrate with examples that wheneither of the plant and the controller is not an annihila-tion operator quantum system, the coherent-classical es-timation scheme with coherent feedback can provide im-provement over purely-classical estimation of the plant.We first consider an example where the controller is aphysically realizable annihilation operator only system.However, the plant is physically realizable and hasχ 6= 0,i.e. it is not annihilation operator only.

In (55), we choose γ = 4, κ1 = κ2 = 2 and χ = 0.5,and in (56), we choose γ = 16, κ1 = κ2 = 8 and χ = 0.Fig. 10 then shows a comparison of the estimation errorcosts. Clearly, the coherent-classical estimation error isless than the purely-classical estimation error for all ho-modyne detector angles.

Furthermore, we consider the case where the plant isan annihilation operator only system, but the coherentcontroller is not, i.e. it has χ 6= 0. In (55), we chooseγ = 4, κ1 = κ2 = 2 and χ = 0. Also, in (56), we chooseγ = 16, κ1 = κ2 = 8 and χ = −0.5. Fig. 11 then showsa comparison of the estimation error costs. Clearly, thecoherent-classical estimation error is again less than thepurely-classical estimation error for all homodyne detec-tor angles.

Moreover, we present an example where both the plantand the controller are physically realizable quantum sys-tems with χ 6= 0, i.e. none of them are annihilation op-erator only. In (55), we choose γ = 4, κ1 = κ2 = 2 andχ = 1, and in (56), we choose γ = 16, κ1 = κ2 = 8and χ = −0.5. Fig. 12 then shows a comparison of theestimation error costs. Clearly, in this case the coherent-classical estimation error is less than the purely-classicalestimation error for all homodyne detector angles.

When both the plant and the controller are physically

0 20 40 60 80 100 120 140 160 1800.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

θ (degrees)

Est

imat

ion

err

or

Jc

Jc

Fig. 10. Feedback: Estimation error vs. homodyne detectionangle θ in the case of an annihilation operator only controller.

12

0 20 40 60 80 100 120 140 160 1800.9

0.92

0.94

0.96

0.98

1

1.02

1.04

1.06

1.08

1.1

θ (degrees)

Est

imat

ion

err

or

Jc

Jc

Fig. 11. Feedback: Estimation error vs. homodyne detectionangle θ in the case of an annihilation operator only plant.

0 20 40 60 80 100 120 140 160 1801.4

1.5

1.6

1.7

1.8

1.9

2

2.1

2.2

θ (degrees)

Est

imat

ion

err

or

Jc

Jc

Fig. 12. Feedback: Estimation error vs. homodyne detectionangle θ in the case of a squeezer plant and a squeezer con-troller.

realizable quantum systems with χ 6= 0, i.e. neither ofthem are annihilation operator only, it is possible thatthe coherent-classical estimates are better than the cor-responding purely-classical estimates for only certain ho-modyne angles. An example is shown in Fig. 13, wherewe used γ = 4, κ1 = κ2 = 2 and χ = 0.5 in (55) andγ = 16, κ1 = κ2 = 8 and χ = 0.5 in (56).

However, we observe that if there is any improvementobserved with the coherent-classical estimation (withfeedback) over purely-classical estimation, the coherent-classical scheme is always superior to classical-only es-timation for the best choice of the homodyne detectorangle. This we propose as a conjecture here, that turnsout to be just the opposite of Conjecture 5.2. This is ow-ing to the coherent feedback involved here as opposed tothe earlier open loop case.

Conjecture 7.1 Consider a coherent-classical estima-tion scheme defined by (45), (47), (25) and (26) with an

estimation error cost Jc defined in (31). Also, considerthe corresponding purely-classical estimation scheme de-fined by (45), (16) and (17) with an estimation error costJc defined in (23). Then, if there exists a homodyne angle

θi for which Jc(θi) ≤ Jc(θi), the following always holds

0 20 40 60 80 100 120 140 160 1801.2

1.22

1.24

1.26

1.28

1.3

1.32

1.34

1.36

1.38

1.4

θ (degrees)

Est

imat

ion

err

or

Jc

Jc

Fig. 13. Feedback: Estimation error vs. homodyne detectionangle θ in the case of a squeezer plant and a squeezer con-troller, where it is possible to get better coherent-classicalestimates than purely-classical estimates only for certain ho-modyne detector angles.

for the optimal choice θopt of the homodyne angle:

Jc(θopt) ≤ Jc(θopt). (57)

8 Conclusion

In this paper, we have demonstrated that a coherent-classical estimation scheme without coherent feedbackcannot provide better estimates than a classical-onlyestimation scheme, when either of the plant and thecontroller is assumed to be a physically realizable an-nihilation operator only quantum system. Otherwise,it is possible to get better estimation accuracy for cer-tain homodyne detector angles; however, for the bestchoice of the homodyne angle, we observed that purely-classical estimation is better than coherent-classicalestimation (without feedback). On the other hand, acoherent-classical estimation scheme involving coherentfeedback can be better than the corresponding purely-classical estimation scheme, only if either of the plantand the controller is not annihilation operator only sys-tem. Moreover, in all cases where the coherent-classicalestimation scheme with coherent feedback provideswith improvement over purely-classical estimation, weobserved that classical-only estimation is always worsethan the coherent-classical estimation for the best choiceof the homodyne angle, owing to the coherent feedbackinvolved.

As future work, the conjectures proposed in this pa-per may be proved. Furthermore, a comparison may becarried out between the two flavours of the coherent-classical estimation scheme, i.e. with and without coher-ent feedback, for the same nominal model of the plant.

Acknowledgements

The first author would like to thank Dr. MohamedMabrok for useful discussion related to this work.

13

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