A feedback control strategy using the Lyapunov method for quantum systems by Haseena Ahmed A Creative Component submitted to my graduate committee in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Major: Applied Mathematics Program of Study Committee: Prof. Domenico D’Alessandro, Major Professor Prof. Michael Smiley Prof. Murti Salapaka Iowa State University Ames, Iowa 2005 Copyright c Haseena Ahmed, 2005. All rights reserved.
Transcript
A feedback control strategy using the Lyapunov method for quantum systems
by
Haseena Ahmed
A Creative Component submitted to my graduate committee
in partial fulfillment of the requirements for the degree of
MASTER OF SCIENCE
Major: Applied Mathematics
Program of Study Committee:Prof. Domenico D’Alessandro, Major Professor
First and foremost, I am very grateful to my major professor Dr. Domenico D’Alessandro
for accepting me as his student and for all his help during my study. His guidance, patience
and support were a tremendous help to accomplish my goals.
I would also like to thank my committee members Dr. Michael Smiley and Dr. Murti
Salapaka for their efforts and contributions to this work. The courses I took with Dr. Smiley
were of great help in my research.
My special thanks to our Graduate Coordinator, Dr. Paul Sacks for his advice, Melanie
Erickson for her encouragement and my friend Jaemin Shin for taking out the time to go over
my write up and make suggestions. On a personal note, I would like to thank my husband
Ankit Saran for being there for me.
1
Introduction
Control theory is an interdisciplinary field which studies how to modify the behavior of
dynamical systems. A control law (also called control input) is a function which appears in
the equations describing the dynamical behavior of a system. Control theory studies how
to design such control laws to make the system behave in a desired way. Quantum control
in particular aims at finding ways to manipulate the time evolution of systems which evolve
according to the laws of quantum mechanics. The ability to control quantum systems opens
doors to many new applications in quantum physics, quantum optics, quantum chemistry and
quantum computation.
The most effective strategies in classical control applications involve feedback control. Feed-
back is a mechanism present in virtually any mechanical, electrical and biological system.
Feedback controllers determine the value of the control input to a system by continuously
monitoring the state of the system. The implementation of classical feedback control for quan-
tum systems, however, poses severe challenges since quantum measurements tend to destroy
the state of the system according to the laws of quantum mechanics (see e.g. chapter III of
[4]). Several attempts have been made to obtain feedback control of quantum systems which
are based on the generalized measurement theory (see e.g. [11]). In these cases the modifica-
tion of the state by measurement is assumed to be mild and slow enough so that it does not
compromise the effect of the control.
In this work, we present a strategy for steering the state of a quantum system from an initial
value to a desired final value. This strategy consists of a feedback controller which calculates
the control law from the knowledge of the state. We do not however make any attempt to
apply generalized measurement feedback in the implementation of this law. The control law
2
presented will be a nominal feedback in that, in the actual implementation, the system will
be simulated and, this way, a control law will be obtained to be applied in open loop on the
actual system.
This creative component is organized as follows. In Chapter 1, we introduce some basic
notations and concepts from quantum mechanics that will be used in the rest of this work.
Dirac’s notation is used in the study and the model of the dynamics of the system under control
is given by the Schrodinger equation. In order to understand the precise meaning of distance
between two states, different notions of distance are discussed.
In Chapter 2, we introduce the Lyapunov method of control. The properties of Lyapunov
functions and the notion of stability in the sense of Lyapunov are discussed for autonomous
systems. Standard stability and asymptotic stability theorems of Lyapunov are stated and
illustrated with examples.
In Chapter 3, we present a feedback control strategy for quantum systems based on the
Lyapunov method of control. We study the stability properties and design a control function.
We show, using the LaSalle’s Invariance Principle, that with a few conditions on the structure
of the system, the control law achieves the goal of steering the state of the quantum system
from an initial state to a desired final state. The results are illustrated by some simulations.
3
CHAPTER 1. Quantum Mechanics Preliminaries
1.1 Finite Dimensional Quantum Systems
We recall some definitions and concepts from quantum mechanics in order to introduce the
class of systems of interest.
Definition 1.1.1 A Hilbert SpaceH is a complex vector space with an inner product namely
an operation (·, ·) : H×H −→ C which satisfies the following properties:
1. (y, x) = (x, y)∗
2. (x + y, z) = (x, z) + (y, z)
3. (αx, y) = α(x, y) for any scalar α. Also, (x, αy) = α∗(x, y)
4. (x, x) ≥ 0 and (x, x) = 0 ⇔ x = 0
The norm of a vector x is defined as ‖x‖ =√
(x, x). The Hilbert space is complete in this
norm.
Definition 1.1.2 Kets: The state of a quantum system is represented by a vector in a Hilbert
space. These state vectors are called kets and they are denoted using Dirac’s notation as |ψ(t)〉,where t denotes the time. This ket contains complete information about the physical state.
|ψ〉 and α|ψ〉, for any α ∈ C, α 6= 0 represent the same physical state and hence we usually
assume that its norm is equal to 1. The vector space spanned by kets is called the ket space.
Definition 1.1.3 Observable: Every measurable quantity of a quantum system is called an
observable and is represented by a linear operator X in the ket space. It acts on a ket to yield
another ket, X|ψ〉.
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Definition 1.1.4 Inner Product: With each pair of kets |φ〉 and |ψ〉 taken in this order, we
associate a complex number which is their inner product (|φ〉, |ψ〉). The notation (|φ〉, |ψ〉) is
equivalent to 〈φ|ψ〉. In general, the result of the product is a complex number. Two kets |φ〉and |ψ〉 are said to be orthogonal if 〈φ|ψ〉 = 0.
Definition 1.1.5 Bras: To any ket |ψ〉 is associated a linear operator H −→ C which, when
applied to the vector |φ〉, gives the number 〈ψ|φ〉. Such a linear operator is called the bra
associated to the ket |ψ〉 and is denoted by 〈ψ|. It is a vector in the dual space of H and the
vector space spanned by bras is called the bra space.
Definition 1.1.6 Outer Product: The outer product of a ket |ψ〉 and a bra 〈φ| is denoted
by |ψ〉〈φ|. It is a linear operator from H → H which, when acting on a general ket |ς〉 gives
|ψ〉〈φ|ς〉. Since 〈φ|ς〉 is a number, |ψ〉〈φ| acting on a general ket |ς〉 yields another ket.
Example 1.1.7 Consider a two-dimensional Hilbert space H2 with basis vectors |0〉 and |1〉.Let |ψ1〉 = |0〉 and |ψ2〉 = 1√
2|0〉 + i√
2|1〉 be any two vectors in H2. The outer product of the
ket |ψ1〉 and the bra 〈ψ2| is given by
|ψ1〉〈ψ2| = ( |0〉)( 1√2〈0| − i√
2〈1| )
=1√2|0〉〈0| − i√
2|0〉〈1|
Definition 1.1.8 Density Matrix: Kets represent systems whose state is perfectly known.
However in practice it is not so and systems are described by a statistical mixture of states. A
density matrix is an operator of the form ρ.=
∑
j
pj |ψj〉〈ψj |. It represents the state of a system
which is in state |ψ1〉 with probability p1, or in state |ψ2〉 with probability p2 and so on. Here,∑
j
pj = 1.
Mixed states are described by density operators of this form with more than one pj
different from zero. A special case is the one of pure states which are such that pj = 1 for
some j. They are described by density operators of the form ρ = |ψj〉〈ψj |.Density matrices are Hermitian and positive semi-definite. They have trace equal to one
since the kets are assumed to have unit norm. For pure states, ρ2 = ρ (see [5] for proof). Also
for pure states, rank(ρ) = trace(ρ) = 1.
5
Example 1.1.9 Refer to example (1.1.7), and suppose that the system is in state |ψ1〉 with
probability 12 and in state |ψ2〉 with probability 1
2 , the density operator is given by
ρ =12|ψ1〉〈ψ1|+ 1
2|ψ2〉〈ψ2|
=12|0〉〈0|+ 1
2(
1√2|0〉+
i√2|1〉)( 1√
2〈0| − i√
2〈1|)
=34|0〉〈0| − i
4|0〉〈1|+ i
4|1〉〈0|+ 1
4|1〉〈1|
In the basis { |0〉, |1〉}, the density operator can be written in the matrix form as
ρ =
34 − i
4
i4
14
The following are pure states of the system
ρ1 = |ψ1〉〈ψ1| = |0〉〈0|
ρ2 = |ψ2〉〈ψ2| = 12|0〉〈0| − i
2|0〉〈1|+ i
2|1〉〈0|+ 1
2|1〉〈1|
1.2 Model of quantum systems with a control variable
We will assume the Hilbert space to be finite dimensional. The state |ψ(t)〉 of a system
evolves according to the Schrodinger equation:
i~d
dt|ψ(t)〉 = H(t)|ψ(t)〉
or
i~|ψ〉 = H|ψ〉
H : H → H is a linear operator in general called the Hamiltonian operator associated with
the system. The system is linear, but time varying. In finite dimensional systems, H(t) is
represented by a Hermitian matrix and hence has real eigenvalues. H(t) is assumed to be
the sum of a term H0 which is called unperturbed Hamiltonian and Hc(t) which is called the
interaction Hamiltonian. Hc(t) has the form Hc(t) =r∑
l=1
Hl ul(t), where ul(t) are time varying
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scalar controls and the way they act is specified by the corresponding Hermitian matrix Hl. ~
is the Planck’s constant and i is the imaginary unit.
The evolution of the Schrodinger equation is referred to as a quantum trajectory. In ap-
propriate units we can set ~ = 1 and we will study the following model of a quantum system
given by the re-scaled Schrodinger equation:
i| ˙ψ(t)〉 = H(t)|ψ(t)〉, |ψ(t0)〉 = ψ0 (1.1)
We can convert this differential equation with kets into one with density matrices as follows.
From the definition (1.1.8) of density matrices, we have that ρ(t) =n∑
j=1
pj |ψj(t)〉〈ψj(t)| where
n is the dimension of the system. Then, ρ0 = ρ(t0) =n∑
j=1
pj |ψj(t0)〉〈ψj(t0)| and
˙ρ(t) =n∑
j=1
pj |ψj(t)〉〈ψj(t)|+n∑
j=1
pj |ψj(t)〉〈ψj(t)|
=n∑
j=1
− ipj H(t)|ψj(t)〉〈ψj(t)|+n∑
j=1
pj |ψj(t)〉i〈ψj(t)|H(t) from (1.1)
= −iH(t)ρ(t) + iρ(t)H(t)
= i[ρ,H(t)]
where the notation [A,B] = AB −BA. This is known as the Liouville’s equation. Also,
˙ρ(t) = −i(H0 + Hc)ρ(t) + iρ(t)(H0 + Hc)
= −iH0ρ(t)− iHcρ(t) + iρ(t)H0 + iρ(t)Hc
= i[ρ(t), H0] + i[ρ(t),Hc]
= i[ρ(t), H0] + ir∑
l=1
[ρ(t),Hl]ul (1.2)
ρ(t) varies in a set of density matrices, i.e., a set of Hermitian, positive semi-definite matrices
of trace equal to 1. We will use the above equation to derive controls to steer the system from
a given initial density matrix to a target final density matrix ρf . It can be shown easily that
the solution to (1.2) exists and is unique.
7
1.3 Notion of the distance between the states of a quantum system
The aim of the control law is to drive the system, evolving according to Liouville’s equation,
to a final density matrix ρf . To do this, we shall construct in the following chapters a suitable
state feedback, that ensures the asymptotic stability of the closed loop control system by letting
the distance between the actual and final density matrices decrease asymptotically. Hence it
is very important that we know how the distance between two density matrices is defined (see
[3]).
Consider two quantum states described by the density matrices ρ1 and ρ2. The definition
of the distance between these two states can be different depending on the metrics of the space.
Definition 1.3.1 For any matrix A, the trace norm is defined as ‖A‖tr =∑
i
σi(A) where
σi(A) are the singular values of A. This norm leads to the trace distance
Dtr(ρ1, ρ2) = tr√
(ρ1 − ρ2)2
Definition 1.3.2 The Frobenius norm, also called the Euclidean norm, is defined as ‖A‖F =√√√√n∑
i,j=1
|aij |2 and it leads to the Hilbert-Schmidt distance
DHS(ρ1, ρ2) =√
tr{(ρ1 − ρ2)2}
Definition 1.3.3 A commonly used distance for pure states is the Bures distance given by
DBures(ρ1, ρ2) =
√2(1−
√tr{√ρ1ρ2
√ρ1})
In all the above definitions, tr(·) is the trace of the matrix.
8
CHAPTER 2. The Lyapunov Method of Control
Nonlinear systems are very complex systems and in most cases it is not possible to compute
the exact solution of the nonlinear differential equations governing these systems. Hence simple
tools are needed to analyze their behavior which do not require the computation the exact
solution. The Lyapunov method is one such tool. It was introduced at the end of the 19th
century to study the stability of equilibrium points or more in general that of trajectories.
Lyapunov theory requires one to search for a function called the Lyapunov function V (x)
which satisfies some specified properties. In addition to giving us criteria for stability, asymp-
totic stability and instability of solutions, the method also gives a way of estimating the region
of asymptotic stability (see e.g. section 5.5 of [10]).
Lyapunov theory can be used not only for analysis of dynamical systems but also for syn-
thesis of control laws that, for instance, guarantee stability of the system under consideration.
The choice V (x), often suggests a feedback control u = u(x) such that the closed-loop sys-
tem x = f(x, u) has globally asymptotically stable trajectory. We will study more about the
Lyapunov theory in this chapter. Let us first introduce some basic definitions.
Definition 2.0.4 Positive Definite Functions: A real scalar function h(x) is said to be
positive definite in some closed bounded region D of the state space containing a point x if for
all x in D,
1. h(x) is continuously differentiable with respect to x.
2. h(x) = 0
3. h(x) > 0, for all x 6= x
9
h(x) is positive semidefinite on D if in addition to conditions 1 and 2, h(x) ≥ 0, for all x
hold.
Definition 2.0.5 Negative Definite Functions: A real scalar function h(x) is said to be
negative definite in some, closed, bounded region D if −h(x) is positive definite in D. h(x) is
negative semidefinite on D if −h(x) is positive semidefinite in D.
Example 2.0.6 h(x) = x21 +x2
2 is positive definite on R2 since h(x) > 0 for all x except x = 0
for which h(x) = 0.
Example 2.0.7 h(x) = (x1 + 2x2)2 is positive semidefinite on R2 since h(x) ≥ 0 for all x. It
is not positive definite since there are some nonzero x, for example x1 = 2, x2 = −1 for which
h(x) = 0.
2.1 Stability in the sense of Lyapunov
Consider a nonlinear autonomous (time invariant) system of the form:
x(t) = f(x), x(t0) = x0 (2.1)
Let x be its only equilibrium point. That is, a point such that f(x) = 0. We will assume that
f : D → Rn is locally Lipschitz in x and D ⊂ Rn is a domain containing x. This guarantees
the existence and uniqueness of its solutions.
Let x(t) be the solution of (2.1) existing for t ≥ t0. An equilibrium solution is stable if
all solutions starting at nearby points stay nearby. An equilibrium solution is asymptotically
stable if all solutions starting at nearby points not only stay nearby but also tend to the
equilibrium solution as time approaches infinity. Formally, we can state [9]:
1. An equilibrium point x = x is said to be stable if for every ε > 0, there exists a
δ = δ(ε) > 0 such that if ‖x0 − x‖ < δ, then ‖x(t)− x‖ < ε for t ≥ t0.
2. An equilibrium point x = x is said to be unstable if it is not stable.
10
3. An equilibrium point x = x is said to be asymptotically stable if it is stable and if δ
can be chosen such that ‖x0 − x‖ < δ implies
limt→∞ ‖x(t)− x‖ = 0
4. An equilibrium point x = x is said to be globally asymptotically stable if it is
asymptotically stable and every solution of the initial value problem (2.1) converges to
x as t → +∞.
In all the above definitions ‖ · ‖ is any norm on Rn.
2.2 Lyapunov stability criterion
Consider an autonomous system given by equation (2.1). We will assume that the region D
under consideration contains an equilibrium point x in its interior. We will present a criterion
for the stability and asymptotic stability of the equilibrium point x. For a system given by
(2.1), the Lyapunov theorem can be stated as follows:
Theorem 1 Let V : D → R be a continuously differentiable function such that
V (x) = 0 and V (x) > 0 in D − {x}
V (x) ≤ 0 in D
Then, x = x is stable. Moreover, if
V (x) < 0 in D − {x}
then x = x is asymptotically stable.
The Lyapunov stability theorem gives sufficient conditions for stability and asymptotic
stability. To demonstrate the application of the Lyapunov method, consider the following
simple example.
11
Example 2.2.1 Consider a scalar system given by the following differential equation:
x1 = −x2 , x2 = −x1
The only equilibrium point for this system is given by the point x = 0. Let us use the Lyapunov
function candidate V (x) = x212 + x2
22 . Note that V (0) = 0 and V (x) > 0 when x 6= 0. We have
that
V (x) = x1 x1 + x2 x2 = −x21 − x2
2 < 0, ∀x 6= 0
The assumptions of Theorem 1 are satisfied and hence the origin is asymptotically stable.
Another tool for proving asymptotic stability is the LaSalle’s Invariance Principle. It is usually
used when the Lyapunov function fails to satisfy the negative definiteness condition of Theorem
1. The LaSalle’s Invariance Principle is stated as follows
Theorem 2 Let x = x be an equilibrium point for (2.1). Let V : D → R be a continuously
differentiable positive definite function on a domain D containing the point x such that V (x) ≤0 in D. Let S = {x ∈ D | V (x) = 0} and suppose that no solution can stay identically in S,
other than the solution x(t) = x. Then x is asymptotically stable.
The following example from [9] illustrates an application of the above theorem.
Example 2.2.2 Consider the pendulum equation with friction given by
x1 = x2 , x2 = −a sinx1 − b x2
with a > 0, b > 0. This has x = 0 as the only equilibrium point on −π < x1 < π. Using
V (x) = a(1− cosx1) + x222 as the Lyapunov function candidate, we get
V (x) = a x1sinx1 + x2x2 = a x2sinx1 + x2(−a sinx1 − b x2) = −b x22 ≤ 0
This is not negative definite since V (x) = 0 for x2 = 0 irrespective of the value of x1.
Now set S = {x ∈ R2 | V (x) = 0} = {x1 − axis }. We have that if x2 = 0 ⇒ x2 = 0 and
from the given system of equations we have that sinx1(t) = 0. Hence on −π < x1 < π the
system can maintain the V (x) = 0 condition only at x = 0. Hence by LaSalle’s Principle we
have that x = 0 is asymptotically stable.
12
Many more applications of the Lyapunov Theorems to classical systems can be found in
any standard ODE/nonlinear systems texts such as [8],[9],[10].
Lypapunov theorems can be applied to verify stability without solving the differential
equation. The consideration of physical variables such as the energy, or geometric variables such
as the distance between two states, often suggests the form of a suitable Lyapunov function.
In the next Chapter, we are going to use Lyapunov theory to determine a feedback strategy
to drive the state of a quantum system to a desired value.
13
CHAPTER 3. The Feedback Control Strategy
In this chapter, we propose a feedback control law and then show that the system with
the proposed control is asymptotically stable (see section 2.1) by making use of the Lyapunov
theorems stated in the previous chapter.
3.1 The control problem for a 2-level system
We have seen in section 1.2 that the dynamics of the system we want to control is modeled
by the Liouville’s equation
ρ(t) = i[ρ(t), H0] + ir∑
l=1
[ρ(t), Hl]ul (3.1)
Our aim is to drive the state of (3.1) from a given density matrix to a target final density
matrix ρf which is its equilibrium point by using some external controls.
We shall restrict ourselves to the application of the Lyapunov method of control for a two-
level system. H0 and Hl will then be 2 × 2 matrices. Two level quantum systems are the
simplest interesting quantum systems with important applications. An example is given by a
spin 12 particle controlled by a time varying electromagnetic field. The state of the spin of the
particle is given by a density matrix. The components of the electromagnetic field are varied
(which is the control function u(x) ) in order to change the direction of the spin. Define
E1 =
1 0
0 0
and E2 =
0 0
0 1
These matrices represent the spin up and spin down states. Consider a problem of driving the
system to a desired density matrix say ρf = E2 with one control u1. Then we have that
ρ(t) = i[ρ(t),H0] + i[ρ(t),H1]u1 (3.2)
14
Here we assume H0 to be diagonal, say H0 =
λ1 0
0 λ2
. This is because the component
of the magnetic field in the z-direction is kept constant. We assume a time varying magnetic
field in the x-direction so that H1 =
0 1
1 0
.
3.2 The feedback control approach
In order to apply the Lyapunov theorems we need to choose an appropriate Lyapunov
function. We use V (ρ) = 12 [tr(E1ρ)]2 as a Lyapunov function candidate where tr(·) is the
trace of the matrix. Note that V (ρ) = 0 when tr(E1ρ) = 0. That is, when tr((I − E2)ρ) = 0.
This happens when ρ = E2 which is the desired final density matrix. Note that the Lyapunov
function we use here is obtained from the Hilbert Schmidt distance defined in (1.3.2) as follows:
Let ρ1 = E2 and ρ2 = ρ. Then,
D2HS = tr(E2 − ρ)2 = tr(E2
2 − E2ρ− ρE2 + ρ2)
= tr(E22)− tr(E2ρ)− tr(ρE2) + tr(ρ2)
= 1− 2 tr(E2ρ) + 1 since tr(ρE2) = tr(E2ρ)
= 2− 2 tr((I −E1)ρ)
= 2− 2 tr(ρ) + 2 tr(E1ρ)
= 2 tr(E1ρ)
= 2√
(2V )
which is why the Lyapunov function is chosen as V (ρ) = 12 [tr(E1ρ)]2 = 1
8D4HS . In order to use
the Lyapunov theorems differentiate V (ρ) to get
V (ρ) =12· 2 tr(E1ρ) · tr(E1ρ)
= tr(E1ρ) · tr( iE1[ρ,H0] + iE1[ρ,H1]u1)
= i tr(E1ρ) · tr(E1[ρ,H0]) + iu1 tr(E1ρ) · tr(E1[ρ,H1]) since trace is linear
15
If ρ =
ρ11 ρ12
ρ∗12 1− ρ11
, where ρ∗12 is the conjugate of ρ12, we can evaluate E1[ρ,H0] as
[ρ,H0] = ρH0 −H0ρ
=
ρ11 ρ12
ρ∗12 1− ρ11
λ1 0
0 λ2
−
λ1 0
0 λ2
ρ11 ρ12
ρ∗12 1− ρ11
=
λ1ρ11 λ2ρ12
λ1ρ∗12 λ2(1− ρ11)
−
λ1ρ11 λ1ρ12
λ2ρ∗12 λ2(1− ρ11)
=
0 −(λ1 − λ2)ρ12
(λ1 − λ2)ρ∗12 0
Hence we have that,
E1[ρ,H0] =
1 0
0 0
0 −(λ1 − λ2)ρ12
(λ1 − λ2)ρ∗12 0
=
0 −(λ1 − λ2)ρ12
0 0
so that tr(E1[ρ,H0]) = 0. The derivative V (ρ) of the Lyapunov function reduces to
V (ρ) = i u1 tr(E1ρ) · tr(E1[ρ,H1])
If we now choose u1 = i tr(E1ρ) · ( tr(E1[ρ,H1]) )∗, we have that
V (ρ) = −|tr(E1ρ) · tr(E1[ρ, H1])|2
≤ 0
V (ρ) is negative semidefinite and hence ρ(t) = ρf is stable by Theorem 1. Note that tr(E1ρ)
is a real number and equals ρ11 but tr(E1[ρ,H1]) is a complex number and it is shown to be
equal to (ρ12 − ρ∗12) in the next section. Hence we need to take the conjugate of tr(E1[ρ,H1])
while defining the control function u1.
In order to show asymptotic stability, we apply the LaSalle’s Invariance Principle. Note
that V = 0 if and only if tr(E1ρ) = 0 or tr(E1[ρ,H1]) = 0. Let us now handle these two cases
separately.
16
Case 1: tr(E1ρ) = 0
This condition gives
tr
1 0
0 0
ρ11 ρ12
ρ∗12 1− ρ11
= 0
⇒ tr
ρ11 ρ12
0 0
= 0
⇒ ρ11 = 0
Let S1 = {ρ : tr(E1ρ) = 0}. Hence for ρ to belong to S1, ρ needs to have the form
0 ρ12
ρ∗12 1
since ρ11 = 0. But ρ is positive semidefinite which requires ρ12 = 0. So we
get that ρ has to have the form
0 0
0 1
which is the desired state ρf .
Case 2: tr(E1[ρ,H1]) = 0
Let S2 = {ρ : tr(E1[ρ,H1]) = 0}. We have that
E1[ρ,H1] =
1 0
0 0
ρ11 ρ12
ρ∗12 1− ρ11
0 1
1 0
−
0 1
1 0
ρ11 ρ12
ρ∗12 1− ρ11
=
1 0
0 0
ρ12 − ρ∗12 2ρ11 − 1
1− 2ρ11 ρ∗12 − ρ12
=
ρ12 − ρ∗12 2ρ11 − 1
0 0
Hence, tr([E1, ρ]H1) = 0 implies that (ρ12 − ρ∗12) = 0 ⇒ Im(ρ12) = 0 So, for ρ to belong to
S2 it should have the form ρ =
ρ11 α
α 1− ρ11
where α is real. Assume now that ρ varies
in S2. Since the control u1 = 0, the density matrix satisfies ρ = i[ρ,H0]. A solution of this
differential equation with initial condition ρ0 has the form ρ(t) = e−iH0tρ0eiH0t. This can be
verified as
ρ = −iH0e−iH0tρ0e
iH0t + e−iH0tρ0iH0eiH0t = −iH0ρ + iρH0 = i[ρ,H0]
17
We now try to show that a solution starting from ρ ∈ S2 does not stay in S2, so that we
can apply the LaSalle’s Invariance Principle (Theorem 2) and obtain that the only density
matrices which give the condition that V ≡ 0 is when ρ ∈ S1. So, beginning with ρ0 = ρ =
ρ11 α
α 1− ρ11
, we obtain that ρ(t) has the form
ρ(t) = e−iH0tρ eiH0t
=
e−iλ1t 0
0 e−iλ2t
ρ11 α
α 1− ρ11
eiλ1t 0
0 eiλ2t
=
ρ11e−iλ1t αe−iλ1t
αe−iλ2t (1− ρ11)e−iλ2t
eiλ1t 0
0 eiλ2t
=
ρ11 αe−i(λ1−λ2)t
αei(λ1−λ2)t 1− ρ11
ρ(t) has off-diagonal elements which have imaginary parts which implies that starting from
ρ ∈ S2, ρ(t) does not belong to S2. Hence S2 is not an invariant set.
Hence we can conclude that no solution can stay identically in S = {ρ : V (ρ) = 0} other
than the target density matrix ρf = E2. Hence obtain that ρf is asymptotically stable.
Vettori [1] gives the asymptotic convergence of state vectors to the desired value evolving
according the general Schrodinger equation (1.1) by using the Lyapunov function method.
3.3 Matlab Simulations
In the following pages a few Matlab simulations are presented for different initial conditions
of the density matrix ρ and different time intervals t. We need that the density matrix satisfying
(3.2) converges to ρf = E2 =
0 0
0 1
, which requires that ρ11, Re(ρ12) and Im(ρ12) all
converge to zero.
Figures 3.1-3.6 give plots of ρ11, Re(ρ12) and Im(ρ12) for the initial condition ρ11(0) = 12 ,
Re(ρ12) = 0 and Im(ρ12) = 12 for time intervals t = [0, 50] and t = [0, 1000]. Figures 3.7-3.12
give plots ρ11, Re(ρ12) and Im(ρ12) for the initial condition ρ11(0) = 12 , Re(ρ12) = 0 and
18
Im(ρ12) = 14 for time intervals t = [0, 50] and t = [0, 1000]. Note that all the initial density
matrices have trace equal to 1. Figures 3.7-3.12 illustrate that we do not have convergence if
