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Automatica 49 (2013) 955–959
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Brief paper
Reachable set of open quantum dynamics for a single spin in
Markovian environment✩
Haidong Yuan 1
Department of Applied Mathematics, Hong Kong Polytechnic University, Hung Hom, Hong Kong
article
info
Article history:
Received 22 November 2011
Received in revised form
15 August 2012
Accepted 23 October 2012
Available online 13 February 2013
abstract
In this article, we study the control of a single spin in the Markovian environment. Given the initial
state, we compute all the possible states to which the spin can be driven to at any given time, under
the assumption that fast coherent control is available.
© 2013 Elsevier Ltd. All rights reserved.
Keywords:
Quantum control
Bilinear control system
Reachable set
1. Introduction
Control of quantum systems has important applications in
physics and chemistry. In particular, the ability to steer the state
of a quantum system (or an ensemble of quantum systems) from
a given initial state to a desired target state forms the basis
of spectroscopic techniques such as nuclear magnetic resonance
(NMR) (Ernst, Bodenhausen, & Wokaun, 1987), electron spin
resonance (ESR) spectroscopy (Schweiger, 1990), laser coherent
control (Warren, Rabitz, & Dahleh, 1993) and quantum computing
(Cory, Fahmy, & Havel, 1997; Gershenfeld & Chuang, 1997;
Wegrzyn, Klamka, Znamirowski, Winiarczyk, & Nowak, 2004). In
reality, the systems are always coupled to the environment, which
leads to the open system dynamics. In this article, we are interested
in finding all possible states that the system can be steered to at
any given time. In control language, this is the problem of finding
the reachable set of the controlled system at any given time, which
has a close connection with time optimal control of the system.
Quite a few works have been done on time optimal control of
closed quantum systems (Carlini, Hosoya, Koike, & Okudaira, 2006,
2007; Khaneja, Brockett, & Glaser, 2001; Yuan, Glaser, & Khaneja,
2007; Yuan & Khaneja, 2005; Yuan, Zeier, & Khaneja, 2008; Zeier,
Yuan, & Khaneja, 2008) and on state transferring of some open
✩ The material in this paper was partially presented at the 48th IEEE Conference
on Decision and Control (CDC) and 28th Chinese Control Conference (CCC), December 16–18, 2009, Shanghai, China. This paper was recommended for publication
in revised form by Associate Editor James Lam under the direction of Editor Ian R.
Petersen.
E-mail address: [email protected].
1 Tel.: +852 34003751; fax: +852 23629045.
0005-1098/$ – see front matter © 2013 Elsevier Ltd. All rights reserved.
doi:10.1016/j.automatica.2013.01.005
quantum systems (Altafini, 2004; Bonnard, Chyba, & Sugny, 2009;
Bonnard & Sugny, 2009). Some qualitative descriptions of the
reachable set have been obtained (Altafini, 2004), while here we
will give a platform that enables a quantitative description, provide
a frame for general time optimal control problem. This result is also
expected to be helpful for quantum error correcting (Shor, 1995)
as it gives information on when the errors may reach a certain
threshold.
The article is organized as follows: Section 2 introduces the
dynamical evolution for open quantum systems and the dynamical
equation for the spectra of the system; Section 3 gives analytical
descriptions to the reachable set of two widely used physical model
on the spin system, namely the phase damping and the amplitude
damping; Section 4 concludes.
2. Open quantum system dynamics
The density matrix, which is the state of an N-dimensional
quantum system, denoted as ρ(ρ ∈ CN ×N ), is semi-positive definite and Tr(ρ) = 1. Its dynamics is described by the Lindblad equation, which takes the form
ρ̇ = −i[H (t ), ρ] + L(ρ).
(1)
Here H (t ) is the Hamiltonian of the system, which can be represented as a Hermitian matrix. −i[H (t ), ρ] is called the unitary part
of the evolution, as the solution to ρ̇ = −i[H (t ), ρ] is captured by
unitary conjugation, i.e., ρ(t ) = U ρ U Ď , where U ∈ SU (N ). L(ρ) is
the dissipative part of the evolution. The term L(ρ) is linear in ρ
and is given by the Lindblad form (Alicki & Lendi, 1986; Lindblad,
1976):
L(ρ) =

αβ

aαβ
Ď
Fα ρ Fβ −
1
2
Ď

{Fβ Fα , ρ} ,
956
H. Yuan / Automatica 49 (2013) 955–959
where {, } is the anti-commute operation ({A, B} = AB + BA), {Fα }
are the Lindblad operators which forms a linear basis for traceless
operators on the density matrix, and the coefficients {aαβ }, which
indicate the coupling strengths of the system with the environment, can form a Hermitian matrix A = (aαβ ) which is known as
the GKS (Gorini, Kossakowski and Sudarshan) matrix (Gorini, Kossakowski, & Sudarshan, 1976).
The problem we address in this paper is to compute all the
density matrices the system can be steered to under the Lindblad
equation of motion given by Eq. (1).
Eq. (4) is then
Λ̇(t ) = diag(U Ď L(U ΛU Ď )U )



Ď
= diag
aαβ U Ď Fα U ΛU Ď Fβ U
αβ

(2)
(3)
Consider now the change in the spectrum under the evolution
of Eq. (3), if the spectrum is not degenerate, the change of the
spectrum on the right hand side is, to first order in δ t, just the
diagonal, i.e.
Λ(t + δ t ) = diag(Λ − i[U Ď H (t )U , Λ(t )]δ t
+ U Ď L(U ΛU Ď )U δ t ) + O(δ t 2 ).
As Λ(t ) is a diagonal matrix, for any matrix P , P Λ(t ) and Λ(t )P
have the same diagonal entries, so the diagonal part of −i[U Ď (t )
H (t )U (t ), Λ(t )] is zero. Thus we get
Λ̇(t ) = diag(U Ď L(U ΛU Ď )U ).
(4)
The only density matrix of two-level system which has a degenerate spectrum is ρ = 21 I, at this point, the Lindblad equation (1)
becomes
ρ̇ = L(ρ),
which does not depend on the coherent control and with proper
parametrization this point lies at the boundary of the state space
and does not affect the dynamics.
For a two-level (N = 2) system, we can take the Lindblad
operators {Fα } as normalized Pauli spin operators √1 {σx , σy , σz },
2
where σx :=
0
1
1
0
, σy :=
0
i
−i
0

, and σz :=

1
0
0

−1 . The GKS
matrix is now a 3 × 3 matrix

A=
axx
ayx
azx
axy
ayy
azy
axz
ayz
azz
Ď
− {U Fβ UU Fα U , Λ}
,
2
(6)
for the last step we just used the fact that Fβ
is a Pauli matrix which
is Hermitian. Now substitute U Ď Fα U =
γ cαγ Fγ , where C =
c
cxy
cyy
czy
xx
cxz
cyz
czz

∈ SO(3) is the adjoint representation of U, into



1
′
Eq. (6), we obtain Λ̇(t ) = diag
αβ aαβ Fα ΛFβ − 2 {Fβ Fα , Λ} ,

where a′αβ =
γ ,µ cγ α aγ µ cµβ forms the transformed GKS matrix,
′
′
i.e., A = (aαβ ) = C T AC .
As the trace of Λ(t ) is 1, we can write Λ(t ) = 21 I +λ(t )σz , where


λ(t ) ∈ 0, 12 ; from the dynamics of Λ(t ), we obtain the dynamics
for λ(t ) in the non-degenerate case: λ̇(t ) = −(a′xx + a′yy )λ(t ) +
i
(a′xy − a′yx ).
2
To compute the reachable set of λ governed by this dynamics,
cyx
czx
we will just need to know the range of the right side of the above
equation, which can be written as Tr(M (t )A′ ) = Tr(M (t )C T AC ),
where
−λ(t )
M (t ) = 

i
−
i

0
2
−λ(t )
2
0
0

,
0
(7)
0
i.e.,
λ̇(t ) = Tr(M (t )C T AC ).
(8)
As we assume the coherent control can generate any unitary operator in negligible time, C then can take all elements in
SO(3). Since SO(3) is connected and compact, the range of Tr(M (t )
C T AC ) is thus just an interval [µ(λ(t )), ν(λ(t ))], where µ(λ(t )) =
minC ∈SO(3) Tr(M (t )C T AC ) and ν(λ(t )) = maxC ∈SO(3) Tr(M (t )C T AC ).
We now just need to describe µ(λ(t )) and ν(λ(t )), as the solution
of
λ̇(t ) = µ(λ(t )),
λ̇(t ) = ν(λ(t ))
(9)
will thus give the minimum and maximum values of λ(t ) at any
given time t under this dynamics.
At the degenerate point ρ = 12 I , λ = 0, ρ̇ = L(ρ), since
L(ρ) is Hermitian
and traceless, we can write it as L(ρ) =


x − iy
−z
z
x + iy
L(ρ)δ t are
λ̇ =

1
2
. It is easy to check that the eigenvalues of
±

x2
+
y2
+
z2
1
I
2
+
δ t; thus at the degenerate point,
x2 + y2 + z 2 . Computing x, y, z from the GKS matrix, we
can get

.
Ď

Ď

aαβ U Ď Fα U ΛU Ď Fβ U

1

+ U L(U ΛU )U δ t )U .



αβ
ρ(t + δ t ) ≈ ρ(t ) + (−i[H (t ), ρ(t )] + L(ρ(t )))δ t .
Substitute ρ(t ) = U (t )Λ(t )U Ď (t ), we get
ρ(t + δ t ) ≈ ρ(t ) + (−i[H (t ), ρ] + L(ρ(t )))δ t
= U (Λ(t ) − i[U Ď H (t )U , Λ(t )]δ t

Ď
2
= diag
We assume that the coherent control can be executed on a
time scale much shorter than that of dissipation, and the control
Hamiltonian H (t ) can produce any unitary transformation U ∈
SU (N ) on the system, i.e., any unitary transformation can be
produced on the system in negligible time compared to that of
the dissipation. This assumption is widely met in various physical
systems, for example, in nuclear magnetic resonance, the time
scale for control is ∼10−3 s while the time scale for dissipation
is ∼1 s. Under this assumption, the reachable set of the density
matrix is completely captured by its diagonal form, or ‘spectrum’,
Λ, where ρ = U ΛU Ď . If a spectrum form Λ can be reached at time
T , then all the elements on the iso-spectral orbit {U ΛU Ď } can also
be reached; thus, we will reformulate the control problem in terms
of the spectrum. In Sklarz, Tannor, and Khaneja (2004), this kind of
reformulation has been used to find the optimal cooling strategy.
In the following, we will give the equation for the spectrum of ρ .
For the purpose of this paper, we will confine our attention to twolevel (single spin) systems.
The evolution of the density matrix ρ is governed by the
Lindblad equation (1), thus
Ď
Ď Ď
− {U Fβ UU Fα U , Λ}
2.1. The dynamics of the spectrum
Ď

1
(5)
λ̇ =

|axy − ayx |2 + |axz − azx |2 + |ayz − azy |2 .
(10)
H. Yuan / Automatica 49 (2013) 955–959
957
3. Reachable set on single spin systems
3.2. Amplitude damping
The range of forms like Tr(MC T AC ), which sometimes is denoted as the C-numerical range, has been studied in mathematics
(Gustafson, 1997) and has been recently introduced to solve problems of closed quantum dynamical systems (Dirr, Helmke, Kleinsteuber, Glaser, & Schulte-herbrüggen, 2006; Schulte-herbrüggen,
Dirr, Helmke, & Glaser, 2008), where numerical approaches have
been discussed. In this section, we will give analytical solutions
to the most widely used physics models on single-spin system,
namely the phase damping model and the amplitude damping
model.
The following theorem on C-numerical range will be used in our
analysis:
If a two level system is initially at the state |1⟩, it will gradually
decay to the ground state |0⟩; this process is usually called
amplitude damping, or in atomic physics, spontaneous emission
(Nielsen & Chuang, 2000). The Lindblad equation for this process is
given by
Theorem 1 (Brockett, 1988). If A, B ∈ RN ×N are both symmetric matrices, then the range of Tr(BC T AC ), where C ∈ SO(N ), is an interval
I = [µ, ν] with
µ=
N

aj b N − j ,
ν=
j=1
N

L(ρ) =
γ
2


1
σ− ρσ+ − {σ+ σ− , ρ} ,
(14)
2
where
σ+ = σx + iσy =
σ− = σx − iσy =



0
0
2
,
0
0
2
0
.
0
(15)

(16)
In this case, the GKS matrix is
aj b j ,

A=γ
j =1
1
−i
0
where a1 ≥ a2 ≥ · · · ≥ aN and b1 ≥ b2 ≥ · · · ≥ bN are the
eigenvalues of A and B, respectively.
i
1
0
0
0
0

= γ (B1 + iB2 ).
(17)
When λ = 0,
λ̇(t ) = Tr(M (t )C T AC ).
3.1. Phase damping
(18)
Note that Tr(B1 C B2 C ) = 0, so we have
T
Phase damping is a decoherence process that results in the loss
of coherence between different basis states. It can be caused by
random phase shifts of the system due to its interaction with the
environment (Nielsen & Chuang, 2000). The Lindblad equation for
the phase damping model is ρ̇ = −i[H (t ), ρ] + L(ρ), where the
γ
dissipative part takes the form L(ρ) = − 4 [σz , [σz , ρ]], in this case
0
the GKS matrix is A =
0
0
0
0
0
0
0

γ
0
0
0
0
1
0
0
0
0
0

, then we can write
2
2
2
c21
+ c22
+ c23
= 1,
λ̇(t ) = Tr(−λ(t )B1 C T AC ).
(12)
When λ = 0, from Eq. (10), we get λ̇ = 0, which is actually
the same as Eq. (12), so in this case, Eq. (12) gives a complete
description of the dynamics of the spectrum.
Applying Theorem 1 and noticing that the eigenvalues of A and
−λ(t )B1 are {γ , 0, 0} and {0, −λ(t ), −λ(t )} respectively, it is then
easy to see the range of Tr(−λ(t )B1 C T AC ) is [−γ λ(t ), 0].
At a given time T , the minimum value of λ(T ) is then given by
the solution of the equation λ̇(t ) = −γ λ(t ), which is λ(T ) =
exp(−γ T )λ(0). And the maximum value of λ(T ) is given by the
solution of λ̇(t ) = 0, which is λ(T ) = λ(0). So in the case of phase
damping, the reachable set of the density matrix at any given time
T is
ρ(T ) =
U
2
I + λ(T )σz
+
2
2
2
c11
+ c12
+ c13
= 1,
= −Tr(C T ACB2 ) = −Tr(B2 C T AC ),
thus Tr(B2 C T AC ) = 0, and Tr(M (t )C T AC ) = Tr(−λ(t )B1 C T AC ), so

= γ [−λ(t )(
2
c11
(11)
Tr(B2 C T AC ) = Tr(C T AT CBT2 )
1

T

2
c12
+
2
c21
2
+ c22
) + (c12 c21 − c11 c22 )].
(19)
We now need to find the range of the above expression. First
note that as C ∈ SO(3),
M = −λ(t )B1 − 2i B2 . As B2 is a skew-symmetric matrix and C T AC is
a symmetric matrix in this case, using the fact that Tr(P ) = Tr(P T ),
we have
 
i
2
, which is a real symmetric
matrix. When λ ̸= 0, from Eq. (8), we get that
λ̇(t ) = Tr(M (t )C T AC ).
1 0 0 
0
Let B1 = 0 1 0 , B2 = −1

−λ(t )B1 − B2 C γ (B1 + iB2 )C
2


1
T
T
= γ −λ(t )Tr(B1 C B1 C ) + Tr(B2 C B2 C )
Tr(MC AC ) = Tr
T
Ď
U |λ(T )
2
2
2
2
1 ≤ c11
+ c12
+ c21
+ c22
≤ 2,
(20)
also
1 2
2
2
2
(c11 + c12
+ c21
+ c22
) ≥ c12 c21 − c11 c22
2
1 2
2
2
2
+ c12
+ c21
+ c22
),
(21)
≥ − (c11
2
combine
 1  the inequalities of (20), (21) and the fact that λ(t ) ∈
0, 2 , we get

2
1
2
 

1
2
2
2
2
− λ(t ) ≥
− λ(t ) (c11
+ c12
+ c21
+ c22
)
2
2
2
2
2
≥ −λ(t )(c11
+ c12
+ c21
+ c22
)
+ (c12 c21 − c11 c22 )


1
2
2
2
2
≥ − − λ(t ) (c11
+ c12
+ c21
+ c22
)
2

≥ −2
1
2

+ λ(t ) ,
(22)
2
2
2
2
i.e., the range of γ [−λ(t )(c11
+ c12
+ c21
+ c22
) + (c12 c21 − c11 c22 )]
is

∈ [exp(−γ T )λ(0), λ(0)], U ∈ SU (2) .
2
2
and 0 ≤ c13
+ c23
≤ 1, so we have
(13)





1
1
+ λ(t ) , 2γ
− λ(t ) ,
−2 γ
2
2
(23)
958
H. Yuan / Automatica 49 (2013) 955–959
where the lower bound is achieved by setting c11 = c22 = 1, c12 =
c21 = 0 and the upper bound is achieved by setting c11 = c22 =
0, c12 = c21 = 1.
When λ = 0, it is at the degenerate point, from Eq. (10) we get
λ̇ = 2γ .
(24)
 Now
 we will calculate the range of λ(T ), which is a subset of
0, 12 . If λ(0) ̸= 0, we start with the dynamics of Eq. (18), in this
case we first solve the following equations separately:
Thus the reachable set for this system is
ρ(T ) =
 
U
1
2
2
2
which gives the minimum and maximum values of λ(T ) respectively under this dynamics. It is easy to get



1
λ(T ) ∈ exp (−2γ T )
+ λ(0)
2



1
1
1
− , exp(−2γ T ) λ(0) −
+
.
2
2
2
If the minimum value
1
2

1
+ λ(0) − > 0,
2



1
λ(T ) ∈ exp(−2γ T )
+ λ(0)
2



1
1
1
− , exp(−2γ T ) λ(0) −
+
.
2
2
2
If the minimum value

1
2

1
+ λ(0) − ≤ 0,
2
as the derivative of λ(t ) lies in the interval of

−2γ

1
2



1
+ λ(t ) , 2γ
− λ(t )
2
which contains 0 as λ(t ) ∈ 0, 21 , we can always use coherent
control to change the derivative proportionally such that λ(T ) = 0,
so the range of λ(T ) would be




0, exp(−2γ T ) λ(0) −
1
2


+
1

2
in this case. Thus the range of λ(T ) is





1
max exp(−2γ T )
+ λ(0) − , 0 ,
2
2



1
1
exp(−2γ T ) λ(0) −
+
.
1
2
2
If λ(0) = 0, it will deviate from 0 a little bit under the dynamics
of Eq. (24), then it evolves according to Eq. (18), and the maximum

value λ(T ) takes the same form exp(−2γ T ) λ(0) − 21 + 12 , as
the integral does not change by the derivative on a single point.
The minimum value is 0, as after it deviates from 0, we can
control
to
always use coherent


 drive it back. So the range is
0, exp(−2γ T ) λ(0) − 12 + 21 , and it can also be written as




1
+ λ(0) − , 0 ,
2
2



1
1
exp(−2γ T ) λ(0) −
+
.
max exp(−2γ T )

1
2
2
(25)
The intuitive interpretation of this is as follows. For times short
compared with γ1 , the reachable spectra are close to the original
spectrum. For longer times, however, one can play the tendency of
the system to relax off against the ability to perform unitary control
to manipulate the spectrum in any desired faction so that for T ≫
1
, essentially all possible states can be reached. This is different
γ
from the case of phase damping: there the time evolution is unital
and tends inevitably toward the fully mixed state; coherent control
can only delay the process and achieve a variety of less than fully
mixed states at various times along the way. The presence of
relaxation in addition to decoherence allows a richer set of states
to be attained by playing decoherence (which drives the system to
a fully mixed state) and relaxation (which drives the system to a
pure state) off against each other.
4. Conclusion
then the range of λ(T ) is just
exp(−2γ T )
2


1
1
exp(−2γ T ) − + λ(0) +
.

2

1
λ̇(t ) = −2γ
+ λ(t ) ,
2


1
λ̇(t ) = 2γ
− λ(t ) ,

I + λ(T )σz U Ď |U ∈ SU (2), λ(T )





1
1
∈ max 0, exp(−2γ T )
+ λ(0) −
,

exp(−2γ T )

2
Control of open quantum systems is an important problem for
a wide variety of physics, chemistry and engineering applications.
This paper analyzed the problem of controlling open quantum
systems in cases where fast coherent control of the system is
available. This coherent control can be used to ‘present’ various
aspects of the system’s state to the environmental interaction.
Because of the presence of fast coherent control, the quantity
of interest under control is the spectrum of the density matrix.
We analyzed the reachability of various spectral forms for twolevel systems and derived general formulas for reachability in the
presence of pure decoherence and of decoherence and relaxation.
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Haidong Yuan received his B.E. in Electrical Engineering
from Tsinghua University in 2001, M.S. in Engineering
Science in 2002 and Ph.D. in Applied Mathematics in 2006,
both from Harvard University. After completing his Ph.D.,
Dr. Yuan worked as postdoctoral fellow at Massachusetts
Institute of Technology and Harvard University. Starting
from August 2012, he is with the Department of Applied
Mathematics at The Hong Kong Polytechnic University
as an Assistant Professor. His research interests are
in the areas of dynamical systems and control theory,
information, optimization, signals and systems, current
interests are on control and system problems in the context of quantum science
and technology, including control and optimization problems in NMR (Nuclear
Magnetic Resonance) spectroscopy, magnetic resonance imaging, atomic system
and superconducting quantum interference devices.