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Workshop On Uncertain Dynamical Systems Udine 2011 1

Robust Linear Quantum Systems Theory Robust Linear Quantum Systems Theory

Ian R. Petersen†

† School of Engineering and Information Technology, University of New South Wales @ the Australian

Defence Force Academy

Workshop On Uncertain Dynamical Systems Udine 2011 2

Acknowledgments Acknowledgments

Professor Matthew James, Australian National University Dr Hendra Nurdin, Australian National University Dr Aline Maalouf, University of New South Wales Prof Elanor Huntington, University of New South Wales

• Dr. A.J. Shaiju, IIT Madras
• Dr. Igor Vladimirov, University of New South Wales
• Mr. Shanon Vuglar, University of New South Wales

The Australian Research Council The Air Force Office of Scientific Research (AFOSR)

Workshop On Uncertain Dynamical Systems Udine 2011 3

Introduction Introduction

This presentation surveys some recent results on the theory of

robust control for quantum linear systems.

Quantum linear systems are a class of systems whose dynamics,

which are described by the laws of quantum mechanics, take the specific form of a set of linear quantum stochastic differential equations (QSDEs).

Such systems commonly arise in the area of quantum optics and

related disciplines. Systems whose dynamics can be described or approximated by linear QSDEs include interconnections of optical cavities, beam-splitters, phase-shifters, optical parametric amplifiers,

• ptical squeezers, and cavity quantum electrodynamic systems.

Workshop On Uncertain Dynamical Systems Udine 2011 4

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

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With advances in quantum technology, the feedback control of such

quantum systems is generating new challenges in the field of control theory.

Potential applications of such quantum feedback control systems

include quantum computing, quantum error correction, quantum communications, gravity wave detection, metrology, atom lasers, and superconducting quantum circuits.

A recently emerging approach to the feedback control of quantum

linear systems involves the use of a controller which itself is a quantum linear system.

This approach to quantum feedback control, referred to as coherent

quantum feedback control, has the advantage that it does not destroy quantum information, is fast, and has the potential for efficient implementation.

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Quantum System Coherent Quantum Controller

Coherent quantum feedback control.

The presentation discusses recent results concerning the synthesis

• f H∞ optimal controllers for linear quantum systems in the

coherent control case.

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An example of a coherent quantum H∞ system considered in

(Nurdin, James, Petersen, 2008), (Maalouf Petersen 2010) is described by the following diagram:

controller plant Phase Shift

u y w a v z k1 k2 k3 kc1 kc2 180◦ ac wc0

Workshop On Uncertain Dynamical Systems Udine 2011 8

The coherent quantum H∞ control approach of James Nurdin and

Petersen (2008) was subsequently implemented experimentally by Hideo Mabuchi of Stanford University:

Workshop On Uncertain Dynamical Systems Udine 2011 9

In general, quantum linear stochastic systems represented by linear

QSDEs with arbitrary constant coefficients need not correspond to physically meaningful systems.

Physical quantum systems must satisfy some additional constraints

that restrict the allowable values for the system matrices defining the QSDEs.

In particular, the laws of quantum mechanics dictate that closed

quantum systems evolve unitarily, implying that (in the Heisenberg picture) certain canonical observables satisfy the so-called canonical commutation relations (CCR) at all times.

Therefore, to characterize physically meaningful systems, a formal

notion of physically realizable quantum linear stochastic systems has been introduced.

Workshop On Uncertain Dynamical Systems Udine 2011 10

Quantum Harmonic Oscillators Quantum Harmonic Oscillators

We formulate a class of linear quantum system models.

These linear quantum system models take the form of QSDEs which

are derived from the quantum harmonic oscillator.

We begin by considering a collection of n independent quantum

harmonic oscillators which are defined on a Hilbert space

H = L2(Rn, C).

Elements of the Hilbert space H, ψ(x) are the standard complex

valued wave functions arising in quantum mechanics where x is a spatial variable.

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Corresponding to this collection of harmonic oscillators is a vector of

annihilation operators

a =     a1 a2

. . .

an     .

Each annihilation operator ai is an unbounded linear operator

defined on a suitable domain in H by

(aiψ)(x) = 1 √ 2xiψ(x) + 1 √ 2 ∂ψ(x) ∂xi

where ψ ∈ H is contained in the domain of the operator ai.

The adjoint of the operator ai is denoted a∗

i and is referred to as a

creation operator.

These correspond to the annihilation and creation of a photon.

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Canonical Commutation Relations Canonical Commutation Relations

The operators ai and a∗

i are such that the following canonical

commutation relations are satisfied

[ai, a∗

j] := aia∗ j − a∗ jai = δij

where δij denotes the Kronecker delta multiplied by the identity

• perator on the Hilbert space H.

We also have the commutation relations

[ai, aj] = 0, [a∗

i , a∗ j] = 0.

These commutation relations encapsulate Heisenberg’s uncertainty

relation.

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Notation Notation

For a general vector of operators

g =     g1 g2

. . .

gn     ,

• n H, we use the notation

g# =     g∗

1

g∗

2

. . .

g∗

n

    ,

to denote the corresponding vector of adjoint operators.

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Also, gT denotes the corresponding row vector of operators

gT = g1 g2 . . . gn

• , and g† =
• g#T .

Using this notation, the canonical commutation relations can be

written as

a a#

• ,

a a# † = a a# a a# † − a a# # a a# T T =

• I

0 −I

• = J.

Workshop On Uncertain Dynamical Systems Udine 2011 15

Quantum Weiner Processes Quantum Weiner Processes

The quantum harmonic oscillators described above are assumed to

be coupled to m external independent quantum fields modelled by bosonic annihilation field operators A1(t), A2(t), . . . , Am(t) which are defined on separate Fock spaces Fi defined over L2(R) for each field operator.

For each annihilation field operator Aj(t), there is a corresponding

creation field operator A∗

j(t), which is defined on the same Fock

space and is the operator adjoint of Aj(t).

The field annihilation operators are also collected into a vector of

• perators defined as follows:

A(t) =     A1(t) A2(t)

. . .

Am(t)     .

Workshop On Uncertain Dynamical Systems Udine 2011 16

Hamiltonian, Coupling and Scattering Operators Hamiltonian, Coupling and Scattering Operators

In order to describe the joint evolution of the quantum harmonic

• scillators and quantum fields, we specify the Hamiltonian operator

for the quantum system which is a Hermitian operator on H. This

• perator describes the internal dynamics of the quantum system.

Also, we specify the coupling operator vector for the quantum

system L, which is a vector of operators on H. These operators define the interaction between the quantum system and the light fields which interact with it.

In addition, we define a scattering matrix which is a unitary matrix

S ∈ Cm×m. This matrix describes the interactions between the

light fields.

Workshop On Uncertain Dynamical Systems Udine 2011 17

Robust Stability of Uncertain Quantum Systems Robust Stability of Uncertain Quantum Systems

We consider an open quantum system defined by the parameters

(S, L, H) where H = H1 + H2.

H1 corresponds to the nominal (known) part of the Hamiltonian and

H2 corresponds to the uncertain (unknown) part of the Hamiltonian.

The corresponding generator for this quantum system is given by

G(X) = −i[X, H] + L(X)

where L(X) = 1

2L†[X, L] + 1 2[L†, X]L.

This defines the dynamics of the open quantum system.

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We first assume that there exist operator column vectors z and w

such that

[V, H2] = [V, z†]w − w†[z, V ]

for all non-negative self-adjoint operators V .

Also, we assume the sector bound condition:

w†w ≤ 1 γ2 z†z.

Let V be any non-negative self-adjoint operator and consider G(V ). Then

G(V ) = = −i[V, H] + L(V ) = −i[V, H1] − i[V, H2] + L(V ) = −i[V, H1] + L(V ) − i[V, z†]w + iw†[z, V ]. (1)

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Now [V, z†] = V z† − z†V and hence

[V, z†]† = zV − V z = [z, V ] since V is self adjoint.

Therefore,

0 ≤

• [V, z†] − iw†

[V, z†] − iw†† = [V, z†][z, V ] + i[V, z†]w − iw†[z, V ] + w†w

and hence

−i[V, z†]w + iw†[z, V ] ≤ [V, z†][z, V ] + w†w.

Substituting this into (1), it follows that

G(V ) ≤ −i[V, H1] + L(V ) + [V, z†][z, V ] + w†w ≤ −i[V, H1] + L(V ) + [V, z†][z, V ] + 1 γ2 z†z

using the sector bound condition.

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Using this inequality, we obtain the following result.

• Theorem. Suppose that the open quantum system (S, L, H) satisfies

the above conditions. Also suppose there exists a non-negative self- adjoint operator V and real constants c > 0, λ ≥ 0 such that

−i[V, H1] + L(V ) + [V, z†][z, V ] + 1 γ2 z†z + cV ≤ λ.

Then V (t) ≤ e−ct V + λ

c , ∀t ≥ 0.

• Proof. If the conditions of the theorem are satisfied, then it follows

from the previous inequality that

G(V ) + cV ≤ λ.

Then the result of the theorem follows from a result of (James, Gough, 2010). Note < · > denotes quantum expectation.

Workshop On Uncertain Dynamical Systems Udine 2011 21

Linear Uncertain Quantum Systems Linear Uncertain Quantum Systems

We now specialize the previous result to the case of linear uncertain

quantum systems.

In this case, we assume that the nominal Hamiltonian is of the form

H1 = 1 2

• a† aT

M a a#

• where M ∈ C2n×2n is a Hermitian matrix of the form

M = M1 M2 M #

2

M #

1

• and M1 = M †

1 , M2 = M T 2 .

Workshop On Uncertain Dynamical Systems Udine 2011 22

In addition, we assume the coupling operator vector L is of the form

L = N1 N2 a a#

• where N1 ∈ Cm×n and N2 ∈ Cm×n.

Also, we write

L L#

• = N

a a#

• =

N1 N2 N #

2

N #

1

a a#

• .

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In addition we assume that V is of the form

V =

• a† aT

P a a#

• where P ∈ C2n×2n is a positive-definite Hermitian matrix of the

form

P = P1 P2 P #

2

P #

1

• and P1 = P †

1 , P2 = P T 2 .

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Also, we assume H2 is of the form

H2 = 1 2

• ζ† ζT

∆(t) ζ ζ#

• where ∆(t) ∈ C2m×2m is a Hermitian matrix of the form

∆(t) = ∆1(t) ∆2(t) ∆2(t)# ∆1(t)#

• and ∆1(t) = ∆1(t)†, ∆2(t) = ∆2(t)T .

Also, ζ = E1a + E2a#.

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We let

w = 1 2 ∆1(t) ∆2(t) ∆2(t)# ∆1(t)# ζ ζ#

• = 1

2

• ∆1(t)ζ + ∆2(t)ζ#

∆2(t)#ζ + ∆1(t)#ζ#

• and

z = ζ ζ#

• =

E1 E2 E#

2 E# 1

a a#

• = E

a a#

• .

Hence, H2 = w†z = 1

2

• a† aT

E†∆(t)E a a#

• .

From this it follows that for any self-adjoint operator V

[V, H2] = [V, z†]w − w†[z, V ]

The sector bound condition is equivalent to ∆(t) ≤ 2

γ .

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Now we calculate −i[V, H1]:

−i[V, H1] = −i1 2

• a† aT

P a a#

• ,
• a† aT

M a a#

• =

a a# † [PJM − MJP] a a#

• .

Also, we calculate

L(V ) = 1 2L†[V, L] + 1 2[L†, V ]L = tr

• PJN †
• I 0

0 0

• NJ
• −1

2 a a# † N †JNJP + PJN †JN a a#

• .

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In addition, we calculate

[z, V ] = 2EJP a a#

• .

and therefore,

[V, z†][z, V ] = 4 a a# † PJE†EJP a a#

• .

Also,

z†z = a a# † E†E a a#

• .

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Hence, we obtain

−i[V, H1] + L(V ) + [V, z†][z, V ] + z†z γ2 + cV = a a# † F †P + PF + 4PJE†EJP + E†E γ2 + cP a a#

• +λ

where F = −iJM − 1

2JN †JN and

λ = tr

• PJN †
• I 0

0 0

• NJ
• .

Then, the condition in the above theorem is satisfied if and only if the

following LMI is satisfied:

F †P + PF + 4PJE†EJP + E†E γ2 < 0.

Workshop On Uncertain Dynamical Systems Udine 2011 29

It follows from the bounded real lemma that this LMI will have a

solution P > 0 if and only if the following H∞ norm bound condition is satisfied:

• E (sI − F)−1 D
• ∞ < γ

2

where D = iJE†.

In this case, it follows from the theorem and P > 0 that

a(t) a#(t) † a(t) a#(t)

• ≤ e−ct

a(0) a#(0) † a(0) a#(0)

• λmax[P]

λmin[P] + λ cλmin[P] ∀t ≥ 0.

Workshop On Uncertain Dynamical Systems Udine 2011 30

Quantum Stochastic Differential Equations Quantum Stochastic Differential Equations

Given (S, L, H1) as above, we can construct the corresponding

QSDE model as follows:

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)#

• ,

where

F = −iJM − 1 2JN †JN; G = −JN † S 0 −S#

• ;

H = N; K = S 0 S#

• ;

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Also, we can introduce a change of variables to convert these into

real QSDEs in terms of position and momentum operators:

• q

p

• = Φ

a a#

• ;
• Q

P

• = Φ

A A#

• ;
• Qout

Pout

• = Φ

Aout Aout#

• where the unitary matrices Φ have the form Φ =

1 √ 2

• I

I −iI iI

• .

Then

• dq

dp

• = A
• p

q

• dt + B2
• dQ

dP

• ;
• dQout

dPout

• = C2
• q

p

• dt + D22
• dP

dQ

• ,

where A = ΦFΦ−1, B2 = ΦGΦ−1, C2 = ΦHΦ−1,

D22 = ΦKΦ−1 are real matrices.

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If we define B1 = ΦDΦ−1 and C1 = ΦEΦ−1, then our previous

robust stability condition becomes

• C1 (sI − A)−1 B1
• ∞ < γ

2 .

We can then set up a corresponding quantum H∞ control problem

to achieve this defined by the controlled quantum uncertain system

• dq

dp

• = A
• p

q

• dt + B1dw + B2du;

dz = C1

• q

p

• dt; dy = C2
• q

p

• dt + D22du;

A recent result (James, Nurdin, Petersen, 2008) shows that this

problem can be solved via the two Riccati equation method (with the addition of some small perturbations to make the problem non-singular and some loop shifting to deal with the D22 term).

Workshop On Uncertain Dynamical Systems Udine 2011 33

Physical Realizability Physical Realizability

Not all QSDEs satisfy the laws of quantum mechanics. This motivates a notion of physical realizability.

• Definition. QSDEs of the form considered above are physically real-

izable if there exist suitably structured complex matrices Θ = Θ†,

M = M †, N, S such that S†S = I, and F = −iΘM − 1 2ΘN †JN; G = −ΘN † S 0 −S#

• ;

H = N; K = S 0 S#

• ;

Physical realizability means that the QSDEs correspond to a

quantum harmonic oscillator.

Workshop On Uncertain Dynamical Systems Udine 2011 34

• Theorem. The above QSDEs are physically realizable if and only if there

exist complex matrices Θ = Θ† and S such that S†S = I, Θ is of the form above, and

FΘ + ΘF † + GJG† = 0; G = −ΘH† S 0 −S#

• ;

K = S 0 S#

• .

Note that the first of these conditions is equivalent to the

preservation of the commutation relations for all times.

Workshop On Uncertain Dynamical Systems Udine 2011 35

(J, J)-unitary Transfer Function Matrices (J, J)-unitary Transfer Function Matrices

We now relate the physical realizability of the above QSDEs to the

(J, J)-unitary property of the corresponding transfer function matrix Γ(s) =

• Γ11(s) Γ12(s)

Γ21(s) Γ22(s)

• = H (sI − F)−1 G + K.
• Definition. A transfer function matrix Γ(s) of the above form is (J, J)-

unitary if

Γ(s)∼JΓ(s) = J

for all s ∈ C.

Here, Γ∼(s) = Γ(−s∗)†.

Workshop On Uncertain Dynamical Systems Udine 2011 36

• Theorem. (Shaiju and Petersen) Suppose the linear quantum system

defined by the above QSDEs is minimal and that λi(F)+λ∗

j(F) = 0 for

all eigenvalues λi(F), λj(F) of the matrix F . Then this linear quantum system is physically realizable if and only if the following conditions hold: (i) The system transfer function matrix Γ(s) is (J, J)-unitary; (ii) The matrix K is of the form K =

S 0 S#

• where S†S = I.

In solving the quantum H∞ control problem, if the controller is to be

a coherent controller, implemented as a quantum system itself, then it must be physically realizable.

A result of (James, Nurdin, Petersen, 2008) showed that any LTI

controller (such as obtained using the two Riccati solution to the

H∞ control problem) can be made physically realizable by suitably

adding quantum noises.

Workshop On Uncertain Dynamical Systems Udine 2011 37

Robust Stabilization of an Optical Parametric Oscillator Robust Stabilization of an Optical Parametric Oscillator

An optical parametric oscillator (OPO) can be used to produce

squeezed light in which the quantum noise in one quadrature is squeezed relative to the noise in the other quadrature and yet Heisenberg’s uncertainty relation still holds.

The following is a schematic diagram of an OPO.

MgO:LiNbO 3 nonlinear optical material second harmonic generator

• utput beam

partially reflective mirror fully reflective mirror Laser and

• ptical isolator

Optical Cavity

Workshop On Uncertain Dynamical Systems Udine 2011 38

An approximate linearized QSDE model of an OPO is as follows:

da = −(κ 2 + iδ(t))adt + χa∗dt − √κdA; dAout = √κadt + dA

where δ(t) represents the mismatch between the OPO resonant frequency and the frequency of the driving laser, which will be treated as an uncertain parameter.

The uncertainty Hamiltonian is H2 = 1

2δ(t)a∗a.

Also, H1 = 1

2iχ

• (a∗)2 − a2

, L = √κa, S = I.

This corresponds to M1 = 0, M2 = iχ, N1 =

• (κ), N2 = 0;

E1 = 1, E2 = 0, ∆1 = δ, ∆2 = 0.

Furthermore, we choose the parameters values κ = 2000,

χ = 2000, γ = 20 and set up a corresponding H∞ control

problem.

Workshop On Uncertain Dynamical Systems Udine 2011 39

Converting to real coordinates, solving the H∞ control problem,

converting back to the original coordinates, we obtain the controller

dxc = Fcxcdt + Gcdy; du = Hcxcdt, Fc = 106 ×

• −1.0487 −0.0875

−0.0875 −1.0487

• ; Gc =
• 978.8771 43.7221

43.7221 978.8771

• ;

Hc =

• 978.8990 43.7231

43.7231 978.8990

• ;

We now consider whether this controller can be implemented as a

coherent controller (with minimum number of additional noises).

Do there exist matrices Gco, Hco, such that the system

dxc = Fcxcdt + Gcdy + Gcod ˜ w; du = Hcxcdt + d ˜ w; d˜ u = Hcoxcdt + dy

is physically realizable; i.e., (J, J)-unitary?

Workshop On Uncertain Dynamical Systems Udine 2011 40

In a recent result (Vuglar, Petersen, 2011), it is shown that this holds

if and only if the Riccati equation

F †

c X + XFc + H† cJHc + XGcJG† c = 0

has a solution X = TJT † where T is non-singular and then,

Gco = −X−1HcJ.

In this example, we find X =

• 0.6466

−0.6466

• and then

Gco = 103 ×

• −1.5140 0.0676

0.0676 −1.5140

• .