Implementation of a Direct Coupling Coherent Quantum Observer
including Observer Measurements

arXiv:1603.01436v1 [quant-ph] 4 Mar 2016

Ian R. Petersen and Elanor H. Huntington
Abstract— This paper considers the problem of constructing
a direct coupling quantum observer for a quantum harmonic
oscillator system. The proposed observer is shown to be able to
estimate one but not both of the plant variables and produces
a measureable output using homodyne detection.

I. I NTRODUCTION
A number of papers have recently considered the problem
of constructing a coherent quantum observer for a quantum
system; e.g., see [1]–[4]. In the coherent quantum observer
problem, a quantum plant is coupled to a quantum observer
which is also a quantum system. The quantum observer is
constructed to be a physically realizable quantum system
so that the system variables of the quantum observer converge in some suitable sense to the system variables of the
quantum plant. The papers [4]–[7] considered the problem
of constructing a direct coupling quantum observer for a
given closed quantum system. In [4], the proposed observer
is shown to be able to estimate some but not all of the
plant variables in a time averaged sense. Also, the paper
[8] shows that a possible experimental implementation of
the augmented quantum plant and quantum observer system
considered in [4] may be constructed using a non-degenerate
parametric amplifier (NDPA) which is coupled to a beamsplitter by suitable choice of the NDPA and beamsplitter
parameters.
One important limitation of the direct coupled quantum
observer results given in [4]–[8] is that both the quantum
plant and the quantum observer are closed quantum systems.
This means that it not possible to make an experimental measurement to verify the properties of the quantum observer.
In this paper, we address this difficulty by extending the
results of [4] to allow for the case in which the quantum
observer is an open quantum linear system whose output
can be monitored using homodyne detection. In this case, it
is shown that similar results can be obtained as in [4] except
that now the observer output is subject to a white noise
perturbation. However, by suitably designing the observer,
it is shown that the level of this noise perturbation can
be made arbitrarily small (at the expense of slow observer
This work was supported by the Australian Research Council (ARC)
and the Chinese Academy of Sciences Presidents International Fellowship
Initiative (No. 2015DT006).
Ian R. Petersen is with the School of Engineering and Information Technology, University of New South Wales at the Australian Defence Force Academy, Canberra ACT 2600, Australia.

i.r.petersen@gmail.com
Elanor H. Huntington is with the College of Engineering and Computer
Science, The Australian National University, Canberra, ACT 0200, Australia. Email: Elanor.Huntington@anu.edu.au.

convergence). Also, the results of [8] are extended to show
that a possible experimental implementation of the augmented quantum plant and quantum observer system may
be constructed using a non-degenerate parametric amplifier
(NDPA) which is coupled to a beamsplitter by suitable choice
of the NDPA and beamsplitter parameters. In this case, the
NDPA contains an extra field channel as compared to the
result in [8] and this extra channel is used for homodyne
detection in the observer.
II. D IRECT C OUPLING C OHERENT Q UANTUM O BSERVER
WITH O BSERVER M EASUREMENT
In this section, we extend the theory of [4] to the case of a
direct coupled quantum observer which is also coupled to a
field to enable measurements to be made on the observer. In
our proposed direct coupled coherent quantum observer, the
quantum plant is a single quantum harmonic oscillator which
is a linear quantum system (e.g., see [9]–[13]) described by
the non-commutative differential equation
ẋp (t)
zp (t)

=
=

0; xp (0) = x0p ;
Cp xp (t)

(1)

where zp (t) denotes the system variable to be estimated by
the observer and Cp ∈ R1×2 . This quantum plant corre
qp
sponds to a plant Hamiltonian Hp = 0. Here xp =
pp
where qp is the plant position operator and pp is the plant
momentum operator. It follows from (1) that the plant system
variables xp (t) will remain fixed if the plant is not coupled
to the observer.
We now describe the linear quantum system which will
correspond to the quantum observer; see also [9]–[13]. This
system is described by a quantum stochastic differential
equation (QSDE) of the form
dxo

= Ao xo dt + Bo dw;

xo (0) = x0o ;

= Co xo dt + dw;
= Kyo
(2)


dQ
where dw =
is a 2 × 1 vector of quantum noises
dP
expressed in quadrature form corresponding to the input field
for the observer and dyo is the corresponding output field;
e.g., see [9], [11]. The observer output zo (t) will be a real
scalar quantity obtained by applying homodyne detection
2×2
2×2
to the observer output field.
 Ao ∈ R , Bo ∈ R ,
qo
is a vector of self-adjoint
Co ∈ R2×2 . Also, xo (t) =
po
system variables corresponding to the observer position and
dyo
zo (t)

momentum operators; e.g., see [9]. We assume that the
plant variables commute with the observer variables. The
system dynamics (2) are determined by the observer system
Hamiltonian and coupling operators which are operators
on the underlying Hilbert space for the observer. For the
quantum observer under consideration, this Hamiltonian is
a self-adjoint operator given by the quadratic form: Ho =
1
T
2 xo (0) Ro xo (0), where Ro is a real symmetric matrix.
Also, the coupling operator L is defined by a matrix Wo ∈
R2×2 so that


L + L∗
= Wo xo .
(3)
L−L∗
i

Then, the corresponding matrices Ao , Bo and Co in (2) are
given by
1
Ao = 2JRo + WoT JWo , Bo = JWoT J, Co = Wo (4)
2
where


0
1
J=
;
−1 0
e.g., see [9], [11]. Furthermore, we will assume that the
quantum observer is coupled to the quantum plant as shown
in Figure 1. In addition, we define a coupling Hamiltonian
Quantum
Observer

Quantum
Plant

Homdyne
Detector
yo

zo

zp
Fig. 1.

Plant Observer System.

Then, we can write the QSDEs describing the closed loop
system as follows:
dxp
dxo

dyo

= 2Jαβ T xo dt;
1
= 2ωo Jxo dt + 2JβαT xp dt + JWoT JWo xo dt
2
+JW T Jdw

 κo
√
2ωo
−2
xo dt + 2JβαT xp dt − κdw;
=
κ
−2ωo − 2
= Wo xo dt + dw
√
=
κxo dt + dw;
(9)

e.g., see [9], [11]. Now it follow from (1) and (8) that
zp = αT xp .
Hence, it follows from the first equation in (9) that
dzp = 2αT Jαβ T xo dt = 0.
That is, the quantity zp remains constant even after the quantum plant is coupled to the quantum observer. In addition,
we can re-write the remaining equations in (9) as
 κ

√
−2
2ωo
dxo =
xo dt + 2Jβzp dt − κdw;
−2ωo − κ2
√
(10)
dyo =
κxo dt + dw;
To analyse the system (10), we first calculate the steady
state value of the quantum expectation of the observer
variables as follows:
 κ
−1
−2
2ωo
< x̄o > = −2
Jβzp
−2ωo − κ2


4
κ
4ωo
Jβzp .
=
κ2 + 16ωo2 −4ωo κ
Then, we define the quantity

which defines the coupling between the quantum plant and
the quantum observer:
T

Hc = xp (0) Rc xo (0).
The augmented quantum linear system consisting of the
quantum plant and the quantum observer is then a linear
quantum system described by the total Hamiltonian
Ha
where
xa =



= Hp + Hc + Ho
1
=
xa (0)T Ra xa (0)
2
xp
xo



, Ra =



0
RcT

Rc
Ro

(5)


,

(6)

and the coupling operator L defined in (3). Extending the
approach used in [4], we assume that we can write
Rc = αβ T ,

(7)

√
Ro = ωo I, Wo = κI where α ∈ R2 , β ∈ R2 , ωo > 0 and
κ > 0. In addition, we assume
α = CpT .

(8)

4
x̃o = xo − < x̄o >= xo − 2
κ + 16ωo2



κ
−4ωo

4ωo
κ



Jβzp .

We can now re-write the equations (10) in terms of x̃0 as
follows

 κ
√
2ωo
−2
xo dt + 2Jβzp dt − κdw
dx̃o =
−2ωo − κ2
 κ

2ωo
−2
=
x̃o dt
−2ωo − κ2
 κ
−1
 κ
−2
2ωo
2ωo
−2
Jβzp dt
−2
−2ωo − κ2
−2ωo − κ2
√
+2Jβz dt − κdw
 κ p

√
−2
2ωo
=
x̃o dt − κdw;
κ
−2ωo − 2
−1
 κ
√
√
2ωo
−2
Jβzp dt + dw
dyo =
κx̃o dt − 2 κ
−2ωo − κ2
 κ
−1
√
−2
2ωo
= −2 κ
Jβzp dt + dwout (11)
−2ωo − κ2
where
dwout =

√

κx̃o dt + dw.

We now look at the transfer function of the system

 κ
√
2ωo
−2
x̃o dt − κdw;
dx̃o =
κ
−2ωo − 2
√
dwout =
κx̃o dt + dw
(12)
which is given by
G(s) = −κ



s + κ2
2ωo

−2ωo
s + κ2

−1

.

It is straightforward to verify that this transfer function is
such that
G(jω)G(jω)† = I
for all ω. That is G(s) is all pass. Also, the matrix

− κ2
2ωo
is Hurwitz and hence, the system (12)
−2ωo − κ2
will converge to a steady state in which dwout represents
a standard quantum white noise with zero mean and unit
intensity. Hence, at steady state, the equation
−1
 κ
√
2ωo
−2
Jβzp dt + dwout (13)
dyo = −2 κ
−2ωo − κ2
shows that the output field converges to a constant value plus
zero mean white quantum noise with unit intensity.
We now consider the construction of the vector K defining
the observer output zo . This vector determines the quadrature
of the output field which is measured by the homodyne
detector. We first re-write equation (13) as
dyo = ezp dt + dwout
where
√
e = −2 κ



− κ2
−2ωo

2ωo
− κ2

−1

Jβ

(14)

However, if we choose
K=

eT
kek2

(16)

1
. Hence, this value of
then (15) is satisfied and kKk = kek
K must be the optimal K.
We now consider the special case of ωo = 0. In this case,
we obtain

 2
√
4
0
e = 2 κ κ 2 Jβ = √ Jβ.
0 κ
κ

Hence, as κ → 0, kek → ∞ and therefore kKk → 0. This
means that we can make the noise level on our measurement
arbitrarily small by choosing κ > 0 sufficiently small.
However, as κ gets smaller, the system (12) gets closer to
instability and hence, takes longer to converge to steady state.
III. A P OSSIBLE I MPLEMENTATION OF THE P LANT
O BSERVER S YSTEM
In this section, we describe one possible experimental
implementation of the plant-observer system given in the previous section. The plant-observer system is a linear quantum
system with Hamiltonian
1
Hc + Ho = xTp Rc xo + xTo Ro xo
(17)
2
and coupling operator defined so that


L + L∗
∗
= Wo xo .
L−L
i

Furthermore,
we assume that Rc = αβ T , Ro = ωo I, Wo =
√
κI where α ∈ R2 , β ∈ R2 , ωo > 0 and κ > 0.
In order to construct a linear quantum system with a
Hamiltonian of this form, we consider an NDPA coupled
to a beamsplitter as shown schematically in Figure 2; e.g.,
see [14].

is a vector in R2 . Then

out
B2

B2

dzo = Kezp dt + Kdwout .
Hence, we choose K such that

A

Ke = 1

NDPA
B1

(15)

and therefore
dzo = zp dt + dn

a,b

out
A

out
B1

where
dn = Kdwout
will be a white noise process at steady state with intensity
kKk2 . Thus, to maximize the signal to noise ratio for our
measurement, we wish to choose K to minimize kKk2
subject to the constraint (15). Note that it follows from (15)
and the Cauchy-Schwartz inequality that
1 ≤ kKkkek
and hence
kKk ≥

1
.
kek

B1

A
Beamsplitter

Fig. 2.

NDPA coupled to a beamsplitter.

A linearized approximation for the NDPA is defined by a
quadratic Hamiltonian of the form
1
ı
H1 = (ǫa∗ b∗ − ǫ∗ ab) + ωo b∗ b
2
2

where a is the annihilation operator corresponding to the
first mode of the NDPA and b is the annihilation operator
corresponding to the second mode of the NDPA. These
modes will be assumed to be of the same frequency but with
a different polarization with a corresponding to the quantum
plant and b corresponding to the quantum observer. Also, ǫ
is a complex parameter defining the level of squeezing in the
NDPA and ωo corresponds to the detuning frequency of the b
mode in the NDPA. The a mode in the NDPA is assumed to
be tuned. In addition, theNDPA
corresponds
to a vector of

√
κ1 a
√
coupling operators L =  κ2 b . Here κ1 > 0, κ2 > 0,
√
κ3 b
κ3 > 0 are scalar parameters determined by the reflectance
of the mirrors in the NDPA.
From the above Hamiltonian and coupling operators, we
can calculate the following quantum stochastic differential
equations (QSDEs) describing the NDPA:




da
db



=

0
ǫ

2

ǫ
2

0
γ1
2



a∗
b∗



dt



a
dt
−
γ2
1
b
0
2 + 2 ıωo


 dA
 √
κ1 0
0
 dB1  ;
√
√
−
0
κ2
κ3
dB2



 √


dA
κ1 0
√
a
 0
κ 
dt +  dB1  ;
√ 2
b
κ3
dB2
0
(18)


dAout
 dB1out  =
dB2out


0

where γ1 = κ1 and γ2 = κ2 + κ3 .
We now consider the equations defining the beamsplitter


A
B1



=



cos θ
−eıφ sin θ

e−ıφ sin θ
cos θ



Aout
B1out



where θ and φ are angle parameters defining the beamsplitter;
e.g., see [15]. This implies


Aout
B1out



=



cos θ
eıφ sin θ

−e−ıφ sin θ
cos θ



A
B1



.

Substituting this into the second equation in (18), we obtain


−ıφ





cos θ
−e
sin θ
dA
dB1
eıφ sin θ cos θ




 √
a
dA
κ1 0
√
dt +
=
b
dB1
0
κ2

and hence



−e−ıφ sin θ
cos θ − 1
 √

κ1 0
√
=
κ2
0

cos θ − 1
eıφ sin θ


dA
dB1

a
dt.
b

We now assume that cos θ 6= 1. It follows that we can write


dA
=
dB1


1
cos θ − 1 e−ıφ sin θ
2(1 − cos θ) −eıφ sin θ cos θ − 1


 √
a
κ1 0
√
dt.
×
b
κ2
0
Substituting this into the first equation in (18), we obtain


da
=
db
 ∗ 

a
0 2ǫ
dt
ǫ
0
b∗
2


 γ1
a
0
2
dt
−
γ2
1
b
0
2 + 2 ıωo
 √


1
cos θ − 1 e−ıφ sin θ
κ1 0
√
−
κ2
−eıφ sin θ cos θ − 1
2(1 − cos θ) 0

 √

a
κ1 0
√
×
dt
b
0
κ2


0
dB2 ;
− √
κ3
√
dB2out = κ3 b + dB2 .
These QSDEs can be written



da
 db 




 da∗  = F 
db∗




dB2out
= H
out∗

dB2

in the form

a


b 
dB2

dt + G
;
a∗ 
dB2∗
∗
b

a


dB2
b 

dt +
dB2∗
a∗ 
∗
b
F2
F1#



√
κ1 κ2 e−ıφ sin θ
2(1−cos θ)
− κ23 − 21 ıωo



where the matrix F is given by F =

F1

=



F2

=





0

√

0
ǫ
2

κ1 κ2 eıφ sin θ
2(1−cos θ)

ǫ
2
.



F1
F2#

−

0

Also, the matrix G is given by

0
 √κ3
G = −
 0
0

and the matrix H is given by

√
0
κ3
H=
0 0


0

0
,

0
√
κ3
0
0

0
√
κ3



.

and

,

It now follows from the proof of Theorem 1 in [16] that we
can construct a Hamiltonian for this system of the form


a
  b 
1 ∗ ∗

a b a b M
H=
 a∗ 
2
b∗

where the matrix R is given by
R =
=
Rc =

where the matrix M is given by

M1
M2

=

ı
2

"

=

ı
2



0
0
ǫ

√
κ κ2 e−ıφ sin θ
− 1 1−cos
θ

√
κ1 κ2 eıφ sin θ
1−cos θ

ǫ
0



−ıωo

#

and δ =

.

Also, we can construct the coupling operator for this
system in the form


a
 b 


L = N1 N2 
 a∗ 
b∗


N1 N2
is given by
where the matrix N =
N2# N1#
N = H.
Hence,
N1 =



0

√

κ3



Then we calculate

H

=

=

1
qp
2
1 T
xp
2

pp

xTo

0
1
0
1


0
ı 
.
0 
−ı



qp
  pp 

qo po R 
 qo 
po



xp
R
xo

ℜ(ǫ) + ℜ(δ)
ℑ(ǫ) − ℑ(δ)



Hence,

1
ωo xTo xo + xTp Rc xo .
2

Comparing this with equation (7), we require that


−ℑ(ǫ) − ℑ(δ) ℜ(ǫ) + ℜ(δ)
= αβ T
ℜ(ǫ) − ℜ(δ)
ℑ(ǫ) − ℑ(δ)

(19)

and the condition (15) to be satisfied in order for the system
shown in Figure 2 to provide an implementation of the
augmented plant-observer system.
We first observe that the matrix on the right hand side of
equation (19) is a rank one matrix and hence, we require that


−ℑ(ǫ) − ℑ(δ) ℜ(ǫ) + ℜ(δ)
det
= |δ|2 − |ǫ|2 = 0.
ℜ(ǫ) − ℜ(δ)
ℑ(ǫ) − ℑ(δ)
That is, we require that
√

κ1 κ2

sin θ
= |ǫ|.
1 − cos θ

sin θ
Note that the function 1−cos
θ takes on all values in (−∞, ∞)
for θ ∈ (0, 2π) and hence, this condition can always be
satisfied for a suitable choice of θ. This can be seen in Figure
sin θ
3 which shows a plot of the function f (θ) = 1−cos
θ.

, N2 = 0.

We now wish to calculate the Hamiltonian H in terms of the
quadrature variables defined such that




a
qp
 b 


 ∗  = Φ  pp 
 a 
 qo 
b∗
po

where the matrix Φ is given by

1 ı
 0 0
Φ=
 1 −ı
0 0

√
κ1 κ2 eıφ sin θ
.
1−cos θ

=

,

−ℑ(ǫ) − ℑ(δ)
ℜ(ǫ) − ℜ(δ)

H=

40

30

20

10

f(θ)

and J =

M1 M2
M2# M1#


ı
JF − F † J
M=
2


I 0
. Then, we calculate M
 0 −I
where



†
Φ
 MΦ

0
Rc
,
RcT ωo I

0

−10

−20

−30

−40

0

1

2

Fig. 3.

3

θ (rad)

4

5

6

7

Plot of the function f (θ).

Furthermore, we will assume without loss of generality
that θ ∈ (0, π) and hence we obtain our first design equation
|ǫ|
sin θ
.
= √
1 − cos θ
κ1 κ2

(20)

In practice, this ratio would be chosen in the range of
|ǫ|
√
κ1 κ2 ∈ (0, 0.6) in order to ensure that the linearized model
which is being used is valid.
We now construct the vectors α and β so that condition
(19) is satisfied. Indeed, we let
#
"


1
−ℑ(ǫ) − ℑ(δ)
, β=
.
α=
ℜ(ǫ)−ℜ(δ)
ℜ(ǫ) + ℜ(δ)
−ℑ(ǫ)−ℑ(δ)
For these values of α and β, it is straightforward to verify
that (19) is satisfied provided that |ǫ| = |δ|. With this value
of β, we now calculate the quantity e defined in (14) as
follows:


4
κ3
4ωo
e =
Jβ
κ23 + 16ωo2 −4ωo κ3



4
ℜ(ǫ) + ℜ(δ)
κ3
4ωo
.
= − 2
ℑ(ǫ) + ℑ(δ)
κ3 + 16ωo2 −4ωo κ3
Then, the vector K defining the quadrature measured by the
homodyne detector is constructed according to the equation
(16).
In the special case that ωo = 0, this reduces to


4
ℜ(ǫ) + ℜ(δ)
e=−
.
κ3 ℑ(ǫ) + ℑ(δ)
In terms of complex numbers e = e(1) + ıe(2), we can write
this as
4
e = − (ǫ + δ).
κ3
Then, in terms of complex numbers K = K(1) + ıK(2), the
formula (16) becomes
K=

κ3 (ǫ + δ)
κ3
e
=−
=−
|e|2
4(ǫ + δ)(ǭ + δ̄)
4(ǭ + δ̄)

¯ denotes complex conjugate. Also, as noted in
where (·)
Section II, the steady state measurement noise intensity is
given by
κ3
1
=−
kek
4|ǫ + δ|

which approaches zero as κ3 → 0. However, this is at the
expense of increasingly slower convergence to steady state.
IV. C ONCLUSIONS

In this paper, we have shown that a direct coupling
observer for a linear quantum system can be implemented
in the case that the observer can be measured using a
Homodyne detection measurement. This would allow the
plant observer system to be constructed experimentally and
the performance of the observer could be verified using the
measured data.
R EFERENCES
[1] Z. Miao and M. R. James, “Quantum observer for linear quantum
stochastic systems,” in Proceedings of the 51st IEEE Conference on
Decision and Control, Maui, December 2012.
[2] I. Vladimirov and I. R. Petersen, “Coherent quantum filtering for
physically realizable linear quantum plants,” in Proceedings of the
2013 European Control Conference, Zurich, Switzerland, July 2013,
arXiv:1301.3154.

[3] Z. Miao, L. A. D. Espinosa, I. R. Petersen, V. Ugrinovskii, and M. R.
James, “Coherent quantum observers for finite level quantum systems,”
in Australian Control Conference, Perth, Australia, November 2013.
[4] I. R. Petersen, “A direct coupling coherent quantum observer,” in Proceedings of the 2014 IEEE Multi-conference on Systems and Control,
Antibes, France, October 2014, also available arXiv 1408.0399.
[5] ——, “A direct coupling coherent quantum observer for a single
qubit finite level quantum system,” in Proceedings of 2014 Australian
Control Conference, Canberra, Australia, November 2014, also arXiv
1409.2594.
[6] ——, “Time averaged consensus in a direct coupled distributed
coherent quantum observer,” in Proceedings of the 2015 American
Control Conference, Chicago, IL, July 2015.
[7] ——, “Time averaged consensus in a direct coupled coherent quantum
observer network for a single qubit finite level quantum system,” in
Proceedings of the 10th ASIAN CONTROL CONFERENCE 2015, Kota
Kinabalu, Malaysia, May 2015.
[8] I. R. Petersen and E. H. Huntington, “A possible implementation
of a direct coupling coherent quantum observer,” in Proceedings of
2015 Australian Control Conference, Gold Coast, Australia, November
2015.
[9] M. R. James, H. I. Nurdin, and I. R. Petersen, “H ∞ control of
linear quantum stochastic systems,” IEEE Transactions on Automatic
Control, vol. 53, no. 8, pp. 1787–1803, 2008, arXiv:quant-ph/0703150.
[10] I. R. Petersen, “Quantum linear systems theory,” in Proceedings of the
19th International Symposium on Mathematical Theory of Networks
and Systems, Budapest, Hungary, July 2010.
[11] H. I. Nurdin, M. R. James, and A. C. Doherty, “Network synthesis
of linear dynamical quantum stochastic systems,” SIAM Journal on
Control and Optimization, vol. 48, no. 4, pp. 2686–2718, 2009.
[12] J. Gough and M. R. James, “The series product and its application to
quantum feedforward and feedback networks,” IEEE Transactions on
Automatic Control, vol. 54, no. 11, pp. 2530–2544, 2009.
[13] G. Zhang and M. James, “Direct and indirect couplings in coherent
feedback control of linear quantum systems,” IEEE Transactions on
Automatic Control, vol. 56, no. 7, pp. 1535–1550, 2011.
[14] H. Bachor and T. Ralph, A Guide to Experiments in Quantum Optics,
2nd ed. Weinheim, Germany: Wiley-VCH, 2004.
[15] L. Mandel and E. Wolf, Optical Coherence and Quantum Optics.
Cambridge, UK: Cambridge University Press, 1995.
[16] A. J. Shaiju and I. R. Petersen, “A frequency domain condition for the
physical realizability of linear quantum systems,” IEEE Transactions
on Automatic Control, vol. 57, no. 8, pp. 2033 – 2044, 2012.

