Performance Analysis and Coherent Guaranteed Cost Control for
Uncertain Quantum Systems Using Small Gain and Popov Methods

arXiv:1508.06377v2 [] 21 Mar 2016

Chengdi Xiang, Ian R. Petersen and Daoyi Dong
Abstract— This paper extends applications of the quantum
small gain and Popov methods from existing results on robust
stability to performance analysis results for a class of uncertain
quantum systems. This class of systems involves a nominal linear quantum system and is subject to quadratic perturbations in
the system Hamiltonian. Based on these two methods, coherent
guaranteed cost controllers are designed for a given quantum
system to achieve improved control performance. An illustrative
example also shows that the quantum Popov approach can
obtain less conservative results than the quantum small gain
approach for the same uncertain quantum system.

I. INTRODUCTION
Due to recent advances in quantum and nano-technology,
there has been a considerable attention focusing on research
in the area of quantum feedback control systems; e.g., [1][24]. In particular, robust control has been recognized as
a critical issue in quantum control systems, since many
quantum systems are unavoidably subject to disturbances and
uncertainties in practical applications; e.g., [1], [2] and [3].
A majority of papers in the area of quantum feedback control
only consider the case in which the controller is a classical
system. In this case, analog and digital electronic devices
may be involved and quantum measurements are required
in the feedback loop. Due to the limitations imposed by
quantum mechanics on the use of quantum measurement, recent research has considered the design of coherent quantum
controllers to achieve improved performance. In this case, the
controller itself is a quantum system; e.g., [1], [4] and [5].
In the linear case, the quantum system is often described
by linear quantum stochastic differential equations (QSDEs)
that require physical realizability conditions in terms of
constraints on the system matrices to represent physically
meaningful systems.
As opposed to using QSDEs, the papers [6], [7] have
introduced a set of parameterizations (S, L, H) to represent
a class of open quantum systems, where S is a scattering
matrix, L is a vector of coupling operators and H is a
Hamiltonian operator. The matrix S, together with the vector
L, describes the interface between the system and the field,
and the operator H represents the energy of the system. The
advantage of using a triple (S, L, H) is that this framework
automatically represents a physically realizable quantum
system. Therefore, in this paper, a coherent guaranteed cost
Chengdi
School of
New South
berra ACT

Xiang, Ian R. Petersen and Daoyi Dong are with the
Engineering and Information Technology, University of
Wales at the Australian Defence Force Academy, Can2600, Australia. {elyssaxiang, i.r.petersen,
daoyidong}@gmail.com This work was supported by the Australian
Research Council.

controller is designed based on (S, L, H) and the physical
realizability condition does not need to be considered.
The small gain theorem and the Popov approach are two
of the most important methods for the analysis of robust
stability in classical control. The paper [8] has applied the
small gain method to obtain a robust stability result for
uncertain quantum systems. This result gives a sufficient
condition for robust stability in terms of a strict bounded real
condition. The small gain method has also been extended
to the robust stability analysis of quantum systems with
different perturbations and applications (e.g., see [9], [10],
[11] and [12]). The paper [13] has introduced a quantum
version of the Popov stability criterion in terms of a frequency domain condition which is of the same form as the
classical Popov stability condition (e.g., see [25]). The Popov
approach has also been used to analyze the robust stability
of an optical cavity containing a saturated Kerr medium
[14]. Also, the paper [13] has shown that the frequency
domain condition obtained in [13] is less conservative than
the stability result using the small gain theorem [8].
In this paper, we extend the quantum small gain method
in [8] and the Popov type approach in [13] from robust
stability analysis to robust performance analysis for uncertain
quantum systems. We assume that the system Hamiltonian
H can be decomposed in terms of a known nominal Hamiltonian H1 and a perturbation Hamiltonian H2 , i.e., H =
H1 + H2 . The perturbation Hamiltonian H2 is contained in
a set of Hamiltonians W. We consider uncertain quantum
systems where the nominal system is a linear system and
the perturbation Hamiltonian is quadratic. Moreover, a coherent controller is designed using the small gain approach
and the Popov approach for the uncertain quantum system,
where a guaranteed bound on a cost function is derived in
terms of linear matrix inequality (LMI) conditions. Although
preliminary versions of the results in this paper have been
presented in the conference papers [22] and [23], this paper
presents complete proofs of the main results and modifies the
example in [22] for a consistent performance comparison of
the proposed two methods.
The remainder of the paper proceeds as follows. In Section II, we define the general class of quantum systems
under consideration and specify the nominal system as a
linear quantum system. We then present a set of quadratic
Hamiltonian perturbations in Section III. In Section IV, a
performance cost function for the given system is defined.
Moreover, a small gain approach and a Popov type method
are used to analyze the performance of the given system.
In Section V, a quantum controller is added to the original

system to stabilize the system and also to achieve improved
performance. Also, corresponding approaches are used to
construct a coherent guaranteed cost controller in terms of
LMI conditions. In Section VI, an illustrative example is
provided to demonstrate the method that is developed in
this paper. A performance comparison between these two
methods is also shown in the illustrative example. We present
some conclusions in Section VII.
II. QUANTUM SYSTEMS
In this section, we describe the general class of quantum
systems under consideration, which is defined by parameters
(S, L, H). Here H = H1 + H2 , H1 is a known self-adjoint
operator on the underlying Hilbert space referred to as the
nominal Hamiltonian and H2 is a self-adjoint operator on
the underlying Hilbert space referred to as the perturbation
Hamiltonian contained in a specified set of Hamiltonians
W; e.g., [6], [7]. The set W can correspond to a set of
exosystems (see, [6]). The corresponding generator for this
class of quantum systems is given by
G(X) = −i[X, H] + L(X)

(1)

where L(X) = 12 L† [X, L] + 12 [L† , X]L. Here, [X, H] =
XH −HX describes the commutator between two operators
and the notation † refers to the adjoint transpose of a vector
of operators. Based on a quantum stochastic differential
equation, the triple (S, L, H), together with the corresponding generators, defines the Heisenberg evolution X(t) of an
operator X [6]. The results presented in this paper will build
on the following results from [13].
Lemma 1: [13] Consider an open quantum system defined by (S, L, H) and suppose there exist non-negative selfadjoint operators V and W on the underlying Hilbert space
such that
G(V ) + W ≤ λ
(2)
where λ is a real number. Then for any plant state, we have
Z
1 T
lim sup
hW (t)idt ≤ λ.
(3)
T →∞ T 0
Here W (t) denotes the Heisenberg evolution of the operator
W and h·i denotes quantum expectation; e.g., see [13] and
[6].
In this paper, the nominal system is considered to be
a linear quantum system. We assume that H1 is in the
following form



1 †
a
a aT M
(4)
H1 =
a#
2
where M ∈ C2n×2n is a Hermitian matrix and has the
following form with M1 = M1† and M2 = M2T


M1 M2
M=
.
(5)
M2# M1#
Here a is a vector of annihilation operators on the underlying
Hilbert space and a# is the corresponding vector of creation
operators. In the case of matrices, the notation † refers to

the complex conjugate transpose of a matrix. In the case
of vectors of operators, the notation # refers to the vector
of adjoint operators and in the case of complex matrices,
this notation refers to the complex conjugate matrix. The
canonical commutation relations between annihilation and
creation operators are described in the following way
 

a
a#

†

†  
 
a
a
a
=
,
a#
a#
a#
 
T T
# 
a
a
−
a#
a#
=J
(6)



I
0
where J =
[2].
0 −I
The coupling vector L is assumed to be of the form





 a
a
L = N1 N2
=
Ñ
(7)
a#
a#
where N1 ∈ Cm×n and N2 ∈ Cm×n . We also write



 


N1 N2
L
a
a
=N
=
.
L#
a#
a#
N2# N1#

(8)

When the nominal Hamiltonian H is a quadratic function
of the creation and annihilation operators as shown in (4)
and the coupling operator vector is a linear function of
the creation and annihilation operators, the nominal system
corresponds to a linear quantum system (see, [2]).
We consider self-adjoint “Lyapunov” operators V of the
form


 †

a
T
a
a
V =
P
(9)
a#
where P ∈ C2n×2n is a positive definite Hermitian matrix
of the form


P 1 P2
P =
.
(10)
P2# P1#
We then consider a set of non-negative self-adjoint operators P defined as


V of the form (9) such that P > 0 is a
P=
. (11)
Hermitian matrix of the form (10)
III. PERTURBATIONS OF THE HAMILTONIAN
In Section II, we introduced the nominal linear quantum system. This section defines the quadratic Hamiltonian
perturbations (e.g., see [8], [18]) for the quantum system
under consideration. We first define two general sets of
Hamiltonians in terms of a commutator decomposition, and
then present two specific sets of quadratic Hamiltonian
perturbations.

When the matrix ∆ is subject to the norm bound

A. Commutator Decomposition
For the set of non-negative self-adjoint operators P and
given real parameters γ > 0, δ ≥ 0, a particular set of
perturbation Hamiltonians W1 is defined in terms of the
commutator decomposition
[V, H2 ] = [V, z † ]w − w† [z, V ]

(12)

for V ∈ P, where w and z are given vectors of operators.
W1 is then defined in terms of sector bound condition:
w† w ≤

1 †
z z + δ.
γ2

(13)

We define


H2 : ∃ w, z such that (13) is satisfied
W1 =
. (14)
and (12) is satisfied ∀ V ∈ P
B. Alternative Commutator Decomposition
Given a set of non-negative operators P, a self-adjoint
operator H1 , a coupling operator L, real parameters β ≥ 0
γ > 0, and a set of Popov scaling parameters Θ ⊂ [0, ∞),
we define a set of perturbation Hamiltonians W2 in terms of
the commutator decompositions [13]
[V − θH1 , H2 ] = [V − θH1 , z † ]w − w† [z, V − θH1 ],
L(H2 ) ≤ L(z † )w + w† L(z) + β[z, L]† [z, L]

We define


H2 ≥ 0 : ∃ w, z such that (15) and (16)
W2 =
.
are satisfied ∀ V ∈ P, θ ∈ Θ
(17)
C. Quadratic Hamiltonian Perturbation
We consider a set of quadratic perturbation Hamiltonians
that is in the form



1 †
ζ
T
ζ ζ
∆
(18)
H2 =
ζ#
2
where ζ = E1 a + E2 a# and ∆ ∈ C2m×2m is a Hermitian
matrix of the form


∆1 ∆2
∆=
(19)
∆#
∆#
2
1
with ∆1 = ∆†1 and ∆2 = ∆T2 .
Since the nominal system is linear, we use the relationship:

 




E1 E2
ζ
a
a
z=
=
=E
.
ζ#
a#
a#
E2# E1#
(20)
Then



1 †
a
a aT E † ∆E
H2 =
.
(21)
a#
2

(22)

where k.k refers to the matrix induced norm, we define


H2 of the form (18) and (19) such that
W3 =
.
condition (22) is satisfied
(23)
In [8], it has been proven that for any set of self-adjoint
operators P,
W3 ⊂ W1 .
(24)
When the matrix ∆ is subject to the bounds
0≤∆≤

4
I,
γ

(25)

we define


H2 of the form (18) and (19) such that
W4 =
.
condition (25) is satisfied
(26)
In [13], it has been proven that if [z, L] is a constant vector,
then for any set of self-adjoint operators P,
W4 ⊂ W2 .

(27)

IV. PERFORMANCE ANALYSIS

(15)

for V ∈ P and θ ∈ Θ, where w and z are given vectors of
operators. Note that (12) and (15) correspond to a general
quadratic perturbation of the Hamiltonian. This set W2 is
then defined in terms of the sector bound condition
1
1
1
(16)
(w − z)† (w − z) ≤ 2 z † z.
γ
γ
γ

2
,
γ

k∆k ≤

In this section, we provide several results on performance
analysis for quantum systems subject to a quadratic perturbation Hamiltonian. Also, the associated cost function is
defined in the following way:


Z

1 T  †
a
T
idt
(28)
h a a
R
J = lim sup
a#
T →∞ T 0
where R > 0. We denote

W = a†

aT




R

a
a#


.

(29)

In order to prove the following theorems on performance
analysis, we require some algebraic identities.
Lemma 2: (See Lemma 4 of [8]) Suppose V ∈ P, H1 is
of the form (4) and L is of the form (7). Then



†
a
a
(P JM − M JP )
,
(30)
[V, H1 ] =
a#
a#
1
L(V ) = −
2



a
a#

†

†

†



(N JN JP + P JN JN )

I 0
†
+ Tr(P JN
N J),
0 0

 
† 



a
a
a
a
[
,
P
] = 2JP
.
a#
a#
a#
a#
Lemma 3: For V ∈ P and z defined in (20),


a
[z, V ] = 2EJP
,
a#

a
a#





†

[V, z ][z, V ] = 4



a
a#

†

†

P JE EJP



a
a#

(31)
(32)

(33)


,

(34)

†


a
a
†
E
.
z z=
E
a#
a#
Proof: The proof follows from Lemma 2.

Proof: Since V ∈ P and H2 ∈ W1 ,



†

(35)
2

Lemma 4: (See Lemma 5 of [13]) For z defined in (20)
and L being of the form (7),




a
a
] = EJΣÑ T
(36)
,
Ñ
[z, L] = [E
a#
a#
is a constant vector, where



0 I
.
(37)
I 0
Lemma 5: (See Lemma 6 of [13]) For z defined in (20),
H1 defined in (4) and L being of the form (7), we have


1
a
− i[z, H1 ] + L(z) = E(−iJM − JN † JN )
(38)
a#
2
Σ=

and




a
i[z, V ] = 2iEJP
.
(39)
a#
Now we present two theorems (Theorem 1 and Theorem 2)
which can be used to analyze the performance of the given
quantum systems using a quantum small gain method and a
Popov type approach, respectively.
A. Performance Analysis Using the Small Gain Approach
Theorem 1: Consider an uncertain quantum system
(S, L, H), where H = H1 + H2 , H1 is in the form of (4), L
is of the form (7) and H2 ∈ W3 . If F = −iJM − 12 JN † JN
is Hurwitz,
"
#
†
F † P + P F + γE2 τE2 + R 2P JE †
<0
(40)
2EJP
−I/τ 2
has a solution P > 0 in the form of (10) and τ > 0, then
Z
1 T
J = lim sup
hW (t)idt
T →∞ T 0


Z

1 T  †
δ
a
= lim sup
h a aT R
idt ≤ λ̃ + 2
#
a
τ
T →∞ T 0
(41)
where


I 0
λ̃ = Tr(P JN
N J).
(42)
0 0
In order to prove this theorem, we need the following lemma.
†



Lemma 6: Consider an open quantum system (S, L, H)
where H = H1 + H2 and H2 ∈ W1 , and the set of nonnegative self-adjoint operators P. If there exists a V ∈ P
and real constants λ̃ ≥ 0, τ > 0 such that
− i[V, H1 ] + L(V ) + τ 2 [V, z † ][z, V ] +

1 †
z z + W ≤ λ̃,
γ2τ 2
(43)

then
lim sup
T →∞

1
T

Z

T

hW (t)idt ≤ λ̃ +
0

δ
, ∀t ≥ 0.
τ2

(44)

G(V ) = −i[V, H1 ] + L(V ) − i[V, z † ]w + iw† [z, V ]. (45)
Also,
0 ≤ (τ [V, z † ] −

i †
i
w )(τ [V, z † ] − w† )†
τ
τ

(46)
w† w
= τ [V, z ][z, V ] + i[V, z ]w − iw [z, V ] + 2 .
τ
Substituting (45) into (46) and using the sector bound
condition (13), the following inequality is obtained:
2

†

†

†

G(V ) ≤ −i[V, H1 ]+L(V )+τ 2 [V, z † ][z, V ]+

1 †
δ
z z+ 2 .
γ2τ 2
τ
(47)

Hence,

δ
.
(48)
τ2
Consequently, the conclusion in the lemma follows from
Lemma 1.
2
Proof of Theorem 1: Using the Schur complement [26],
the inequality (40) is equivalent to
G(V ) + W ≤ λ̃ +

F † P + P F + 4τ 2 P JE † EJP +

E†E
+ R < 0.
γ2τ 2

(49)

If the Riccati inequality (49) has a solution P > 0 of the
form (10) and τ > 0, according to Lemma 2 and Lemma 3,
we have
1
− i[V, H1 ] + L(V ) + τ 2 [V, z † ][z, V ] + 2 2 z † z + W =
γ τ
!


†
†
2
†
F
P
+
P
F
+
4τ
P
JE
EJP
a
a
†
a#
a#
+ γE2 τE2 + R


I 0
+ Tr(P JN †
N J).
0 0
(50)
Therefore, it follows from (40) that condition (43) is satisfied
with


I 0
λ̃ = Tr(P JN †
N J) ≥ 0.
(51)
0 0
Then, according to the relationship (24) and Lemma 6, we
have
Z
1 T
lim sup
hW (t)idt
T →∞ T 0


Z

δ
1 T  †
a
h a aT R
idt ≤ λ̃ + 2 .
= lim sup
#
a
T
τ
T →∞
0
(52)
2
B. Performance Analysis Using the Popov Approach
Theorem 2: Consider an uncertain quantum system
(S, L, H), where H = H1 + H2 , H1 is in the form of (4), L
is of the form (7) and H2 ∈ W4 . If F = −iJM − 21 JN † JN
is Hurwitz, and


P F + F †P + R
−2iP JE † + E † + θF † E †
<0
2iEJP + E + θEF
−γI
(53)

has a solution P > 0 in the form of (10) for some θ ≥ 0,
then
Z
1 T
hW (t)idt
J = lim sup
T →∞ T 0


Z

1 T  †
a
= lim sup
h a aT R
idt ≤ λ (54)
a#
T →∞ T 0
where




4θ #
Ñ ΣJE † EJΣÑ T .
γ
(55)
In order to prove this theorem, we need the following lemma.
λ = Tr(P JN †

I
0

0
0

N J) +

Lemma 7: (See Theorem 1 of [13]) Consider a set of nonnegative self-adjoint operators P, an open quantum system
(S, L, H) and an observable W , where H = H1 + H2 and
H2 ∈ W2 defined in (17). Suppose there exists a V ∈ P and
real constants θ ∈ Θ, λ ≥ 0 such that
1
(i[z, V − θH1 ] + θL(z) + z)†
γ
× (i[z, V − θH1 ] + θL(z) + z) + θβ[z, L]† [z, L] + W ≤ λ.
(56)
− i[V, H1 ] + L(V ) +

Then
lim sup
T →∞

1
T

Z

T

hW (t)idt ≤ λ.

(57)

0

Here W (t) denotes the Heisenberg evolution of the operator
W and h·i denotes quantum expectation.
Proof of Theorem 2: Using the Schur complement, (53) is
equivalent to
1
(−2iP JE † + E † + θF † E † )
γ
× (2iEJP + E + θEF ) + R < 0.

P F + F †P +

(58)

According to Lemma 2 and Lemma 5, we have
1
(i[z, V − θH1 ] + θL(z) + z)†
γ
4θ
× (i[z, V − θH1 ] + θL(z) + z) + [z, L]† [z, L] + W
γ


†

†


PF + F P
a
a
1
†
†
† † 

+ γ (−2iP JE + E + θF E )
=
a#
a#
×(2iEJP + E + θEF ) + R


4θ
I 0
+ Tr(P JN †
N J) + Ñ # ΣJE † EJΣÑ T .
0 0
γ
(59)
− i[V, H1 ] + L(V ) +

From this and using the relationship (27), Lemma 4 and
Lemma 7, we obtain
Z
1 T
lim sup
hW (t)idt
T →∞ T 0

† 

Z
(60)
1 ∞ a
a
h
R
idt
= lim sup
#
#
a
a
T →∞ T 0
≤λ

where
λ = Tr(P JN

†



I
0

0
0


N J) +

4θ #
Ñ ΣJE † EJΣÑ T .
γ
(61)
2

V. COHERENT GUARANTEED COST
CONTROLLER DESIGN
In this section, we design a coherent guaranteed cost
controller for the uncertain quantum system subject to a
quadratic perturbation Hamiltonian to make the control system not only stable but also to achieve an adequate level of
performance. The coherent controller is realized by adding a
controller Hamiltonian H3 . H3 is assumed to be in the form



1 †
a
a aT K
(62)
H3 =
a#
2
where K ∈ C2n×2n is a Hermitian matrix of the form


K1 K2
K=
(63)
K2# K1#
and K1 = K1† , K2 = K2T . Associated with this system is
the cost function J



†
Z
1 ∞ a
a
2
(R
+
ρK
)
idt (64)
J = lim sup
h
a#
a#
T →∞ T 0
where ρ ∈ (0, ∞) is a weighting factor. We let

†


a
a
2
.
(R
+
ρK
)
W =
a#
a#

(65)

The following theorems (Theorem 3 and Theorem 4)
present our main results on coherent guaranteed cost controller design for the given quantum system using a quantum
small gain method and a Popov type approach, respectively.
A. Coherent Controller Design Using the Small Gain Approach
Theorem 3: Consider an uncertain quantum system
(S, L, H), where H = H1 + H2 + H3 , H1 is in the form
of (4), L is of the form (7), H2 ∈ W3 and the controller
Hamiltonian H3 is in the form of (62). With Q = P −1 ,
Y = KQ and F = −iJM − 21 JN † JN , if there exist a
matrix Q = q ∗ I (q is a constant scalar and I is the identity
matrix), a Hermitian matrix Y and a constant τ > 0, such
that


1
A + 4τ 2 JE † EJ
Y
qR 2
qE †


Y
−I/ρ
0
0

 < 0 (66)
1


qR 2
0
−I
0
2 2
qE
0
0
−γ τ I
where A = qF † + F q + iY J − iJY , then the associated cost
function satisfies the bound
Z
1 T
lim sup
hW (t)idt
T →∞ T 0

†


Z
1 ∞ a
a
2
= lim sup
h
(R
+
ρK
)
idt (67)
a#
a#
T →∞ T 0
δ
≤ λ̃ + 2
τ

where




I 0
†
λ̃ = Tr(P JN
N J).
(68)
0 0
Proof: Suppose the conditions of the theorem are satisfied.
Using the Schur complement, (66) is equivalent to


1
A + 4τ 2 JE † EJ + ρY Y qR 2
qE †
1

 < 0.
qR 2
−I
0
2 2
qE
0
−γ τ I
(69)
Applying the Schur complement again, it follows that (69)
is equivalent to


A + 4τ 2 JE † EJ + ρY Y + q 2 R
qE †
< 0 (70)
qE
−γ 2 τ 2 I

Hence, we minimize ξ subject to (79) and (66) in Theorem
3. This is a standard LMI problem.

and (70) is equivalent to
qF † + F q + iY J − iJY + 4τ 2 JE † EJ
+ ρY Y + q 2 (

E†E
+ R) < 0.
γ2τ 2

(71)

Substituting Y = Kq = qK † into (71), we obtain
q(F − iJK)† + (F − iJK)q + 4τ 2 JE † EJ
+ q2 (

E†E
+ R + ρK 2 ) < 0.
γ2τ 2

(72)

Since P = Q−1 , premultiplying and postmultiplying this
inequality by the matrix P , we have
(F − iJK)† P + P (F − iJK) + 4τ 2 P JE † EJP

We know that P = Q−1 = q −1 I and apply the Schur
complement to inequality (77), so that we have

1 
−ξ + τδ2 B 2
≤0
(78)
1
B2
−q


I 0
†
where B = Tr(JN
N J). Applying the Schur
0 0
complement again, it is clear that (78) is equivalent to

1 
1
B2
−ξ δ 2
1
 δ 2 −τ 2
(79)
0  ≤ 0.
1
B2
0
−q

B. Coherent Controller Design Using the Popov Approach
Theorem 4: Consider an uncertain quantum system
(S, L, H), where H = H1 + H2 + H3 , H1 is in the form
of (4), L is of the form (7), H2 ∈ W4 , the controller
Hamiltonian H3 is in the form of (62). With Q = P −1 ,
Y = KQ and F = −iJM − 21 JN † JN , if there exist a
matrix Q = q ∗ I (q is a constant scalar and I is the identity
matrix), a Hermitian matrix Y and a constant θ > 0, such
that

1 
A
B†
Y
qR 2
 B
−γI
0
0 

<0
(80)
 Y
0
−I/ρ
0 
1
0
0
−I
qR 2

(73)
E†E
+ R + ρK 2 < 0.
2
2
γ τ
It follows straightforwardly from (73) that F − iJK is
Hurwitz. We also know that
where A = F q + qF † − iJY + iY J and B = 2iEJ + Eq +
1 †
2
†
− i[V, H1 + H3 ] + L(V ) + τ [V, z ][z, V ] + 2 2 z z + W θEF q − iθEJY , then the associated cost function satisfies
γ τ
the bound




†

(F − iJK)† P + P (F − iJK)
Z
a
 +4τ 2 P JE † EJP + E † E/(γ 2 τ 2 )  a#
1 T
=
lim
sup
hW (t)idt
a#
a
+R + ρK 2
T →∞ T 0





†
Z
I 0
1 ∞ a
a
2
+ Tr(P JN †
N J).
(74)
(R
+
ρK
)
idt ≤ λ
=
lim
sup
h
0 0
a#
a#
T →∞ T 0
(81)
According to the relationship (24) and Lemma 6, we have
Z T
1
where
hW (t)idt
lim sup


T
T →∞
0
4θ
I 0
†


†
Z ∞ 
λ
=
Tr(P
JN
N J) + Ñ # ΣJE † EJΣÑ T .
1
a
a
2
0
0
(75)
γ
= lim sup
h
(R + ρK )
idt
a#
a#
(82)
T →∞ T 0
δ
Proof: Suppose the conditions of the theorem are satisfied.
≤ λ̃ + 2
Using the Schur complement, (80) is equivalent to
τ
where



1 
A + γ1 B † B
Y
qR 2
I 0
†
λ̃ = Tr(P JN
N J).
(76)

0 0
(83)
Y
−I/ρ
0  < 0.
1
2
2
qR
0
−I
Remark 1: In order to design a coherent controller which
minimizes the cost bound (67) in Theorem 3, we need to We then apply the Schur complement to inequality (83) and
obtain
formulate an inequality
"
#
1


A + γ1 B † B + ρY Y qR 2
δ
I
0
< 0.
(84)
1
Tr(P JN †
N J) + 2 ≤ ξ.
(77)
qR 2
−I
0 0
τ
+

Also, (84) is equivalent to
†

F q + qF − iJY + iY J+
1
(−2iJE † + qE † + θqF † E † + iθY JE † )
γ
× (2iEJ + Eq + θEF q − iθEJY )

(85)

2
Remark 2: For each fixed value of θ, the problem is an
LMI problem. Then, we can iterate on θ ∈ [0, ∞) and choose
the value which minimizes the cost bound (82) in Theorem
4.
VI. ILLUSTRATIVE EXAMPLE

+ q 2 R + ρY Y < 0.
Substituting Y = Kq = qK † into (85), we obtain
(F − iJK)q + q(F − iJK)†
1
+ (−2iJE † + qE † + θq(F − iJK)† E † )
γ
× (2iEJ + Eq + θE(F − iJK)q)

(86)

+ q 2 (R + ρK 2 ) < 0.
Since P = Q−1 , premultiplying and postmultiplying this
inequality by the matrix P , we have
P (F − iJK) + (F − iJK)† P
1
+ (−2iP JE † + E † + θ(F − iJK)† E † )
(87)
γ
× (2iEJP + E + θE(F − iJK)) + R + ρK 2 < 0.
It follows straightforwardly from (87) that F − iJK is
Hurwitz. We also know that
− i[V, H1 + H3 ] + L(V )
1
+ (i[z, V − θ(H1 + H3 )] + θL(z) + z)†
γ
× (i[z, V − θ(H1 + H3 )] + θL(z) + z)
4θ
+ [z, L]† [z, L] + W
γ


† 
a
a
M̃
=
a#
a#


4θ
I 0
+ Tr(P JN †
N J) + Ñ # ΣJE † EJΣÑ T
0 0
γ
(88)
where
M̃ =P (F − iJK) + (F − iJK)† P
1
+ (−2iP JE † + E † + θ(F − iJK)† E † )
γ
× (2iEJP + E + θE(F − iJK))

(89)

+ R + ρK 2 .
According to the relationship (27) and Lemma 7, we have
Z
1 T
lim sup
hW (t)idt
T →∞ T 0

†


Z
(90)
1 ∞ a
a
2
= lim sup
(R
+
ρK
)
idt
h
#
#
a
a
T
T →∞
0
≤λ
where
λ = Tr(P JN †



I
0

0
0


N J) +

4θ #
Ñ ΣJE † EJΣÑ T .
γ
(91)

In order to illustrate our methods and compare their
performance, we use the same quantum system considered in
[23] as an example. The system corresponds to a degenerate
parametric amplifier and its (S, L, H) description has the
following form
√
1
(92)
H = i((a† )2 − a2 ), S = I, L = κa.
2
We let the perturbation Hamiltonian be




1 †
a
1
0.5i
a aT
H2 =
(93)
−0.5i
1
a#
2
and the nominal Hamiltonian be




1 †
a
−1
0.5i
a aT
H1 =
−0.5i −1
a#
2

(94)

so that H1 + H2 = H. The corresponding parameters considered in Theorem 1, Theorem 2, Theorem 3 and Theorem
4 are as follows:


 √

−1
0.5i
κ √0
M=
,N =
,
−0.5i −1
0
κ
 κ

(95)
−2 + i
0.5
F =
,
E
=
I
0.5
− κ2 − i
and


∆=

1
0.5i
−0.5i
1


.

(96)

To illustrate Theorem 1 and Theorem 3, we consider
H2 ∈ W3 . Hence, γ = 1 is chosen to satisfy (22). The
performance using the small gain approach for the uncertain
quantum system is shown in Figure 1. In Figure 1, the dashed
line represents the cost bound for the linear quantum system
considered in Theorem 1 as a function of the parameter
κ. The solid line shows the system performance with the
coherent controller designed in Theorem 3. Compared to
the performance without a controller, the coherent controller
can guarantee that the system is stable for a larger range
of the damping parameter κ and gives the system improved
performance.
Now we illustrate one approach to realizing the desired
controller. For instance, when κ = 4.5, by using the controller design method in Theorem 3, we have the desired
controller Hamiltonian as




1 †
a
0
−0.5i
T
a a
.
(97)
H3 =
0.5i
0
a#
2
This controller Hamiltonian can be realized by connecting
the degenerate parametric amplifier with a static squeezer as
shown in Figure 2. This static squeezer is a static Bogoliubov
component which corresponds to the Bogoliubov transformation [27], [28]. Also, we have the following definition:

Definition 1: (see [27], [28]) A static Bogoliubov component is a component
that
the
 Bogoliubov
 implements


du(t)
dy(t)
, where B =
=B
transformation:
#
du# (t)
 dy (t)

B1 B2
, B † JB = J.
B2# B1#
 5

− 34
4
To realize H3 in (97), we let the matrix B =
5
− 34
4
†
which satisfies the Bogoliubov condition B JB = J, and
κ̃ = 31 . Therefore, the overall Hamiltonian of the closed
loop system is H = H1 + H2 + H3 , which achieves the
controller design goal. Detailed procedure regarding how to
get the matrix B can be found in appendix.

Fig. 3.

Performance versus θ

In Figure 4, the dashed line shows the performance for
the given system considered in Theorem 2 and the solid line
describes the cost bound for the linear quantum system with
the coherent controller considered in Theorem 4. As can be
seen in Figure 4, the system with a controller has better
performance than the case without a controller.

Fig. 1. Guaranteed cost bounds for an uncertain quantum system with a
controller (solid line) and without a controller (dashed line) using the small
gain approach

Fig. 2.

Degenerate parametric amplifier coupled to a static squeezer.

To illustrate Theorem 2 and Theorem 4, we consider H2 ∈
W4 . Hence, γ = 2 is chosen to satisfy (25). The results
using the Popov approach are shown in Figure 3 and Figure
4. Figure 3 demonstrates how to choose the value of θ. We
consider the same example as above with κ = 3.8 and iterate
on θ ∈ [0, 1]. Figure 3 shows the cost bound for this quantum
system obtained in Theorem 4 as a function of the parameter
θ. It is clear that the minimal cost bound is achieved when
θ = 0.1. Therefore, we choose θ = 0.1 for κ = 3.8 and use
a similar method to choose θ for other values of κ.

Fig. 4. Guaranteed cost bounds for the uncertain quantum system with a
controller (solid line) and without a controller (dashed line) using the Popov
approach

Also, we can observe that the method in Theorem 3 can
only make the quantum system stable for κ > 4 in the
example. Therefore, compared with the results in Figure 1,
the Popov method obtains a lower cost bound and a larger
range of robust stability as shown in Figure 4. This is as
expected, since the Popov approach allows for a more general
class of Lyapunov functions than the small gain approach.

VII. CONCLUSION
In this paper, the small gain method and the Popov
approach, respectively, are used to analyze the performance
of an uncertain linear quantum system subject to a quadratic
perturbation in the system Hamiltonian. Then, we add a
coherent controller to make the given system not only stable
but also to achieve improved performance. By an illustrative
example, we have also shown that the Popov method adds
to a considerable improvement over the small gain method
in terms of system performance. Future work will include
the extension of these approaches to nonlinear uncertain
quantum systems [24].
A PPENDIX
The detailed procedure regarding how to realize the desired controller is shown below. We consider a degenerate
parametric amplifier (DPA) as an example. Based on the
(S, L, H) description in (92), we can calculate the following
quantum stochastic differential equations [1] describing the
DPA:



  κ
a(t)
da(t)
1
−2
dt
=
1 − κ2
a# (t)
da# (t)
 √


dA1 (t)
κ √0
−
;
κ
0
dA#
1 (t)
√
dAout
1 (t) = κa(t)dt + dA1 (t).
We have known that when κ = 4.5 and using the controller
design method in Theorem 3, we have the desired controller
Hamiltonian as in (97).
Next, we show how to realize this controller Hamiltonian
by connecting this degenerate parametric amplifier with
a static squeezer as shown in Fig. 2. The corresponding
quantum stochastic differential equations for this DPA is as
follows:

  κ+κ̃


da(t)
a(t)
− 2
1
=
dt
da# (t)
a# (t)
1
− κ+κ̃
2
 √


dA1 (t)
κ √0
−
0
κ
dA#
1 (t)
 √


(98)
dA2 (t)
κ̃ √0
;
−
dA#
0
κ̃
2 (t)
√
out
dA1 (t) = κa(t)dt + dA1 (t);
√
dAout
κ̃a(t)dt + dA2 (t).
2 (t) =
We have known that this static squeezer is a static Bogoliubov component which satisfies Definition 1. According to
the Definition 1, we have the following relation:




dAout
dA2 (t)
2 (t)
=B
.
(99)
dA#
dAout#
(t)
2
2 (t)
According to (98), we have

  √


dAout
a(t)
κ̃ √0
2 (t)
=
dt
a# (t)
dAout#
(t)
0
κ̃
2


dA2 (t)
+
.
dA#
2 (t)

Substituting (99) into (100), we obtain that
  √



dA2 (t)
a(t)
κ̃ √0
−1
=
B
dt
a# (t)
dA#
0
κ̃
2 (t)


dA2 (t)
+
.
dA#
2 (t)
Hence,
(B −1 − I)



dA2 (t)
dA#
2 (t)

 √


=

κ̃ √0
0
κ̃



a(t)
a# (t)


dt.

(101)
We now assume inverse of B −1 − I exists. It follows (101)
that we can write


 √


dA2 (t)
a(t)
κ̃ √0
−1
−1
= (B − I)
dt.
a# (t)
dA#
0
κ̃
2 (t)
(102)
Substituting (102) into first equation in (98), we get

  κ+κ̃


a(t)
da(t)
− 2
1
dt
=
a# (t)
da# (t)
1
− κ+κ̃
2
 √


dA1 (t)
κ √0
−
0
κ
dA#
1 (t)
 √

κ̃ √0
−
(B −1 − I)−1
0
κ̃


 √
a(t)
κ̃ √0
dt
×
a# (t)
κ̃
0
 κ+κ̃


a(t)
− 2
1
=
dt
a# (t)
1
− κ+κ̃
2
 √


dA1 (t)
κ √0
−
0
κ
dA#
1 (t)



a(t)
κ̃ 0
−1
−1
− (B − I)
dt.
0 κ̃
a# (t)
(103)
According to Definition 1, the B matrix satisfies the
following relation:

 †



B1 B2
I 0
I 0
B1 B2T
JB † JB =
0 −I
0 −I
B2# B1#
B2† B1T




B1
B2
B1†
B2T
=
†
−B2# −B1#
−B2 −B1T


B1† B2 − B2T B1#
B1† B1 − B2T B2#
= I.
=
−B2† B1 + B1T B2# −B2† B2 + B1T B1#
In our case that B is a 2 × 2 matrix, we have
B1† B2 − B2T B1# = 0; −B2† B1 + B1T B2# = 0.
Moreover, we need to have the following relation:
2
2
2
2
B1† B1 −B2T B2# = −B2† B2 +B1T B1# = (B1x
+B1y
)−(B2x
+B2y
) = I.

where B1 = B1x + iB1y , B2 = B2x + iB2y . Thus, we can
assume that
2
2
2
2
B1x
+ B1y
= cosh(r)2 ; B2x
+ B2y
= sinh(r)2 .

(100)

Hence, we may write the matrix B in the following form:

B=

cosh(r)cos(α) + icosh(r)sin(α)
sinh(r)cos(β) − isinh(r)sin(β)

sinh(r)cos(β) + isinh(r)sin(β)
cosh(r)cos(α) − icosh(r)sin(α)


.

Since B, I are 2 × 2 matrices, we have

1
B1 − 1
(B −1 − I)−1 =
B2#
2 − 2B1x

B2
B1# − 1


.

Therefore, the last term on the right side of equation (103)
can be expressed as:
κ̃
−1
−1
κ̃(B
− I)
=
2 − 2cosh(r)cos(α)


cosh(r)cos(α) − 1
0
×(
0
cosh(r)cos(α) − 1


icosh(r)sin(α)
sinh(r)cos(β) + isinh(r)sin(β)
+
)
sinh(r)cos(β) − isinh(r)sin(β)
−icosh(r)sin(α)


κ̃
κ̃
−2
0
+
=
0
− κ̃
2 − 2cosh(r)cos(α)
2


icosh(r)sin(α)
sinh(r)cos(β) + isinh(r)sin(β)
×
.
sinh(r)cos(β) − isinh(r)sin(β)
−icosh(r)sin(α)
(104)

Substituting (104) into (103), we have
 


−κ
1
da(t)
a(t)
2
=
dt
κ
#
#
1
−2
da (t)
a (t)
 √


dA1 (t)
κ √0
−
0
κ
dA#
1 (t)
κ̃
×
−
2 − 2cosh(r)cos(α)


icosh(r)sin(α)
sinh(r)cos(β) + isinh(r)sin(β)
sinh(r)cos(β) − isinh(r)sin(β)
−icosh(r)sin(α)


a(t)
×
dt.
a# (t)
(105)



Therefore, the closed loop system with DPA and a static
squeezer has the dynamical equation (105). The difference
between this closed loop system (105) and the original
uncertain quantum system (98) is the addition of the last term
on right side of (105) which corresponds to the controller
Hamiltonian H3 . To realize our desired controller Hamiltonian H3 as in (97), the following relation is required:
κ̃
−
×
2 − 2cosh(r)cos(α)

icosh(r)sin(α)
sinh(r)cos(β) − isinh(r)sin(β)


0
−0.5
= −iJK =
.
−0.5
0

sinh(r)cos(β) + isinh(r)sin(β)
−icosh(r)sin(α)



(106)

When α = 0, β = 0, sinh(r) = − 34 , cosh(r) = 54 , κ̃ = 13 ,
the relationship (106) holds. In this case, we may have B
5
− 43
4
matrix as B =
which satisfies the Bogoliubov
3
5
−4
4
condition B † JB = J.
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