Decoherence Control and Purification of Two-dimensional Quantum Density Matrices under Lindblad Dissipation

0.35 MB·English

by  Patrick Rooney

Decoherence Control and Purification of
Two-dimensional Quantum Density Matrices under
Lindblad Dissipation
Patrick Rooney, Anthony Bloch and Chitra Rangan ∗†
January 4, 2012
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0
2 Abstract

n Controlofquantumdissipativesystemscanbechallengingbecausecontrolvariablesaretypicallypart
a ofthesystemHamiltonian,whichcanonlygeneratemotionalongunitaryorbitsofthesystem. Totransit
J
between orbits, one must harness the dissipation super-operator. To separate the inter-orbit dynamics

1 from the Hamiltonian dynamics for a generic two-dimensional system, we project the Lindblad master

equation onto the set of spectra of the density matrix, and we interpret the location along the orbit to
]
h be a new control variable. The resulting differential equation allows us to analyze the controllability
p of a general two-dimensional Lindblad system, particularly systems where the dissipative term has an
- anti-symmetric part. We extend this to categorize the possible purifiable systems in two dimensions.
t
n
a
1 Introduction
u
q
[ Recentdecadeshaveseentheapplicationofmathematicalcontroltheorytoquantumsystemsinbothphysics

1 and chemistry, as technological advances have allowed for greater precision in manipulation of these systems
v [1][2][3][4][5]. One particular area of interest is the possible construction of quantum computers, which have
9 the power to perform algorithms not accessible to conventional computers. A major experimental obstacle
9
to any implementation of such a computer, however, is the decoherence of the system under influence of the
3
environment. While much progress has been made on the control of closed quantum systems [6][7][8], work
0
. on open quantum systems has proved more challenging [9][10][11][12][13].
1
OneimportantissueisthatcontrolsarenearlyalwaysintheformofHamiltonianoperators. Thistypeof
0
control is unable to directly affect the purity of a state [14] or transfer the state between unitary orbits. To
2
1 controlpurity, onemustusethedissipativedynamicstomovebetweenorbits. Tothisend, wewishtoderive
: a differential equation that captures only the inter-orbit dynamics, and collects the remaining dynamics
v
(along the orbits) into a new control variable. This can be done if we assume arbitrary control over the
i
X Hamiltoniandynamics. Theresultingdifferentialequationcantellushowthelocationalongtheorbitaffects
r themotionbetweenorbits. Inthispaper,weshowthatthiscanbedonefortwo-dimensionalsystemssubject
a
to Lindblad dissipation, and the formalism can be extended to consider the problem of purification (for
relatedworkintwodimensions, see[9][10]). ThepreliminariesofLindbladdissipationareoutlinedinsection
2,andtheprojectionoftheLindbladdifferentialequationontothesetofunitaryorbitsisdiscussedinsection
3. In section 4, we analyze the controllability of this equation for various choices of system parameters − in
particular, the case where the anti-symmetric part of the dissipation is non-zero. In section 5, we present a
theorem that specifies necessary and sufficient conditions for purifiability.
∗P.R.andA.B.areattheDepartmentofMathematics,UniversityofMichigan,AnnArbor,MI48109,[email protected]
(P.R.)[email protected](A.B.). ResearchispartlysupportedbyNSF.
†C.R. is at the Department of Physics, University of Windsor, ON, N9B 3P4. Canada, [email protected]. Research is
supportedbyNSERC,Canada.
1
2 Preliminaries
The state of a closed quantum system is described by a norm-one vector in a complex Hilbert space that
evolves according to the Schro¨dinger equation:
d
|ψ(t)(cid:105)=−iH|ψ(t)(cid:105)
dt
In order for the norm to be preserved, the Hamiltonian operator H must be Hermitian. An open quantum
system, on the other hand, is described by a trace-one, positive-semidefinite operator ρ on the Hilbert space,
known as the density operator (or density matrix when working in finite dimensions, as we shall). The
interpretation of this matrix is the following: an eigenvalue of ρ is the probability that the system is in
the corresponding eigenstate. Since the matrix is trace-one and positive-semidefinite, these eigenvalues are
non-negative real numbers that sum to one. A state |ψ(cid:105) in the closed system becomes a rank-one projection
operator|ψ(cid:105)(cid:104)ψ|1. TheSchr¨odingerequation,whenextendedtothedensitymatrix,becomesthevonNeumann
equation:
d
ρ(t)=[−iH,ρ(t)]
dt
Certain relevant quantities are invariant under the von Neumann equation. The density matrix at any
time can be written ρ(t) = U(t)ρ(0)U−1(t), where U(t) is unitary. Since matrices at different times are
(cid:112)
similar, the eigenvalues are constant. The purity of the system, which is defined to be tr(ρ2), is also
invariantsinceitisthe2-normofthevectorofeigenvalues. Thishasimplicationsforquantumcontrol. Since
control variables typically appear in the Hamiltonian only, the control dynamics cannot directly alter the
probabilities, or purify the state (i.e. achieve a purity of one).
However, a system that interacts with the environment will have non-Hamiltonian dynamics. In general,
this will be an integro-differential equation, but if one assumes the dynamics depends only the present state
and not its history (i.e. the Markovian condition) and there is not explicit time dependence, the resulting
differential equation is the Lindblad equation[15][16]:
M (cid:18) (cid:19)
d (cid:88) 1
|ρ(t)(cid:105)=[−iH,ρ(t)]+ L ρL†− {L†L ,ρ} (1)
dt j j 2 j j
j=1
The Lindblad operators {L } can be taken to be traceless, as adding a multiple of the identity aI to L is
j j
equivalent to adding an operator i(a¯L −aL†) to the Hamiltonian. An alternate equation, known as the
2 j j
Lindblad-Kossakowski equation, chooses a basis {l } of the set of traceless n-dimensional matrices that is
j
orthonormal relative to the inner product (A,B)=tr(A†B):
n2−1 (cid:18) (cid:19)
d (cid:88) 1
|ρ(t)(cid:105)=[−iH,ρ(t)]+ a l ρl† − {l†l ,ρ} (2)
dt jk j k 2 k j
j,k=1
wherethecoefficientsa ’sformapositive-semidefinitematrix,knownastheGorini-Kossakowski-Sudarshan
jk
matrix.
ALindbladoperatorcanbethoughtasastochasticjumpwithrecoil. UndertheinfluenceofoneLindblad
operator, apurestate|ψ(cid:105)(cid:104)ψ|intimeδtbecomesamixtureoftwostates, M |ψ(cid:105)(cid:104)ψ|M†+M |ψ(cid:105)(cid:104)ψ|M†. Here,
√ 1 1 2 2
M =L δt and M =I− 1L†Lδt. In other words, |ψ(cid:105) jumps to the state √ 1 L|ψ(cid:105) with probability
1 2 2 (cid:104)ψ|L†L|ψ(cid:105)
(cid:112)
(cid:104)ψ|L†L|ψ(cid:105)δt. This is a jump because the δt appears in the probability only, meaning the destination
state does not approach the original state as δt → 0+. Conversely, the second state in the mixture is
√ 1 (I−1L†Lδt)|ψ(cid:105),whichisinfinitesimallyclosetotheoriginalstate. Inotherwords,depending
1−(cid:104)ψ|L†L|ψ(cid:105)δt 2
on L, there may be an infinitesimal recoil needed to compensate for the jump process. When L†L is a
multipleoftheidentity(forexample,whentheLindbladoperatorisamultipleofaPaulimatrix),thesecond
state reduces to the original state |ψ(cid:105), so that the jump is recoil-less.
1The bra-ket notation prescribes that a vector be written as |a(cid:105) and its dual as (cid:104)a|. Inner products are written (cid:104)a|b(cid:105) and
outerproducts(orrank-onematrices)|a(cid:105)(cid:104)b|.
2
3 Projection of dynamics in two dimensions
Asmentionedintheintroduction,controlofopenquantumsystemstypicallyinvolvescontrolvariablesinthe
Hamiltonian. Hamiltonian operators, however, can only move states along unitary orbits, and not between
orbits. The goal of this paper is to isolate the between-orbit dynamics for a generic two-dimensional system
under Lindblad dissipation. Our starting point is the following control system:
(cid:18) (cid:19)
d (cid:88) 1 (cid:88) 1
ρ= [−iu σ ,ρ]+ a σ ρσ − {σ σ ,ρ} (3)
dt j j 2 jk j k 2 k j
j=x,y,z j,k=x,y,z
where {σ :j =x,y,z} are the Pauli matrices. The controls {u } are unbounded and may take any value in
j j
R. NotethatwehavechosenoursetofcontrolHamiltonianstospansu(2). Inotherwords,wecanmakeany
unitaryoperatoruptoanon-physicalphasedifference,andthereforewecanmovebetweenanytwostateson
(cid:80)
a given unitary orbit in arbitrary time. We are neglecting any drift Hamiltonian H =c I + c σ ,
0 0 j=x,y,z j j
since the component along the identity matrix does not contribute to the dynamics, and the components
along the Pauli matrices can be treated by re-calibrating the control variables: u →u −c .
j j j
The density operator can be written in terms of the Pauli matrices: ρ= 1(I+(cid:80) n σ ), where the
2 j=x,y,z j j
n ’s are components of the Bloch vector, such that i.e. n2 +n2+n2 ≤1. Substituting this expressions into
j x y z
the equation (3), we get:
1(cid:88)dnjσ =(cid:88)[−iu σ ,1n σ ]+ 1(cid:88)a [σ ,σ ]+ 1(cid:88)a n (σ σ σ − 1{σ σ ,σ })
2 dt j j j 2 k k 4 jk j k 4 jk l j l k 2 k j l
j j,k jk jkl
(cid:88) −i 1 1(cid:88) 1
= ( u n + a )[σ ,σ ]+ a n (σ σ σ − {σ σ ,σ })
2 j k 4 jk j k 4 jk l j l k 2 k j l
j,k jkl
The Pauli matrices obey the relations
(cid:88)
[σ ,σ ]=2i (cid:15) σ
j k l l
l
{σ ,σ }=2δ I
j k jk
1
σ σ σ − {σ σ ,σ }=δ σ +δ σ −2δ σ
j l k 2 k j l kl j jl k jk l
Using these relations, the Lindblad-Kossakowski equation above becomes:
1(cid:88)dnlσ = (cid:88)(cid:15) u n σ +(cid:88) 1ia (cid:15) σ + 1(cid:88)(a (n σ +n σ )−2a n σ )
2 dt l jkl j k l 2 jk jkl l 4 jl j l l j jj l l
l j,k,l j,k,l jl
If we define b =(cid:80)ia (cid:15) , and aS = ajk+akj, we have
l jk jkl jk 2
(cid:88)dnlσ =2(cid:88)(cid:15) u n σ +(cid:88)b σ +(cid:88)(aSn σ −aS n σ )
dt l jkl j k l l l jl j l jj l l
l j,k,l l jl
In vector notation, we can write:
d(cid:126)n
=(cid:126)b+(cid:126)u×(cid:126)n+(AS −tr(AS)I)(cid:126)n (4)
dt
where AS is the matrix with elements aS.
ij
Now we want to decompose this equation into dynamics along and between unitary orbits. ρ has eigen-
values 1±r, where r :=|(cid:126)n| and eigenvectors
2
(cid:114)
1+n n +in
|ψ (cid:105):= z|1(cid:105)+ x y |2(cid:105)
± 2 (cid:112)2(1+n )
z
Note the spectra correspond one-to-one with the values of r, the Bloch radius. It follows that the unitary
orbits are concentric spheres, except for the completely mixed state, which corresponds to the point r = 0.
So we can parametrize the orbits by r, which lives on the closed interval [0,1], and characterize the motion
3
along orbits with the unit vector nˆ =(cid:126)n/r. We must be careful with respect to the innermost orbit however.
nˆ is not defined there, which means that the differential equations which we will derive for r and nˆ will have
solutions that exist for finite times (those solutions correspond to trajectories of ρ that pass through the
completely mixed state).
Since r2 =(cid:126)n·(cid:126)n , 2rdr =2(cid:126)n· d(cid:126)n and therefore dr =nˆ· d(cid:126)n. So:
dt dt dt dt
dr
=nˆ·(cid:126)b+nˆ·((cid:126)u×(cid:126)n)+nˆ·(AS −tr(AS)I)(cid:126)n
dt
The middle term vanishes, the first term is constant in r and the third is linear in r. We can write:
dr
=nˆ·(cid:126)b+r(nˆ·(ASnˆ)−tr(AS)) (5)
dt
To find the ODE for nˆ, we use (cid:126)n=rnˆ, which gives dnˆ = 1(d(cid:126)n − drnˆ). So we get:
dt r dt dt
dnˆ 1
=2(cid:126)u×nˆ+ ((cid:126)b−((cid:126)b·nˆ)nˆ)+(AS −nˆ·(ASnˆ))nˆ (6)
dt r
Our goal here is to view equation (5) as a control ODE where nˆ is the control. This view requires that we
have full control over nˆ, and we claim that we do, in terms specified by the following lemma.
Lemma 3.1. Let S be the sphere centered at the origin with radius one, let B be the associated closed ball,
and let B∗ be the closed ball with the origin removed. Let nˆ(t) be a piecewise differentiable function from a
timeinterval[0,T]ontoS suchthatthecorrespondingsolutionr(t)ofequation(5)iscontainedintheinterval
(0,1]. Then there are piecewise continuous control functions u (t), u (t) and u (t) such that equation (4)
x y z
has the piecewise differentiable solution (cid:126)n(t)=r(t)nˆ(t) on B∗.
Proof. First re-write equation (6):
(cid:18) (cid:19)
1 dnˆ 1
(cid:126)u×nˆ = − ((cid:126)b−((cid:126)b·nˆ)nˆ)−(AS −nˆ·(ASnˆ))nˆ
2 dt r
Any equation of the form (cid:126)x×(cid:126)a =(cid:126)b, where(cid:126)a·(cid:126)b = 0, has solution (cid:126)x =(cid:126)a×(cid:126)b. It follows that we can choose
the controls to be:
(cid:18) (cid:19)
1 dnˆ 1
(cid:126)u(t)=nˆ× − ((cid:126)b−((cid:126)b·nˆ)nˆ)−(AS −nˆ·(ASnˆ))nˆ
2 dt r
(cid:18) (cid:19)
1 1
= nˆ(t)×nˆ˙ − nˆ(t)×(cid:126)b−nˆ(t)×(ASnˆ(t))
2 r(t)
Since nˆ(t), nˆ˙(t) and r(t) are piecewise continuous, so is (cid:126)u(t).
Note that the prescription for (cid:126)u(t) is unbounded as r → 0 because of the middle term. This is because
the system cannot approach the completely mixed state from any direction: when (cid:126)n=(cid:126)0, d(cid:126)n is fixed to be(cid:126)b
dt
regardless of the controls (cid:126)u(t).
Wefinishthissectionbywritingdownanalternateversionof(5)intermsoftheeigenvaluesofAS,which
allows us to specify a given system in terms of six real parameters. Let a ≥ a ≥ a be the eigenvalues of
1 2 3
AS. Let {b : j = 1,2,3} and {n : j = 1,2,3} be the components of(cid:126)b and (cid:126)n relative to the intrinsic axes
j j
of AS (whereas the subscripts x, y and z denote the components relative to the eigenvectors of the Pauli
matrices). This gives:
3 3
dr (cid:88) (cid:88)
= b n −r a (1−n2) (7)
dt j j j j
j=1 j=1
The six parameters obey the following inequality, which arises from the positive semi-definiteness of A:
a b2+a b2+a b2 ≤4a a a (8)
1 1 2 2 3 3 1 2 3
Thepositivesemi-definitenessofAalsoensuresthepositivesemi-definitenessofAS,sowealsohavea ,a .a ≥
1 2 3
0.
4
4 Controllability analysis
For a fixed r, the right-hand side of equation (7) can be seen as a map from S2, the set of available controls,
to the set of possible values of r˙. Since this is a smooth map from a compact set to R, the image should be
a closed finite interval. To analyze the controllability of (7), we define functions f (r) and f (r) to be the
M m
right and left endpoints, respectively, of this interval. That is, f (r) is the maximum possible rate at which
M
r can increase, and f m) the minimum, for a given value of r. It is clear that (7) is controllable on a closed
(
subinterval of (0,1) if f > 0 and f < 0 everywhere on the subinterval. To steer between two points r
M m i
and r , we choose our controls so that r˙(t)=f (r(t)) if r <r , or r˙(t)=f (r(t)) if r >r .
f M i f m i f
Some properties of f and f can be gleaned from inspection of the differential equation, which we
M m
collect into a proposition:
Proposition 4.1. If f (r):=sup{r˙(r)} and f (r):=inf{r˙(r)},
M m
1. f and f are non-increasing.
M m
2. lim f (r)=|(cid:126)b| and lim f (r)=−|(cid:126)b|.
r→0+ M r→0+ m
3. f (1)≤0.
M
4. f (r) ≤ 0 for all r and system parameters. f (r) = 0 for r > 0 only for the trivial where a = 0
m m 1
(which requires that all a ’s and b ’s are zero.
j j
5. If(cid:126)b has non-zero magnitude, f (r) has an isolated intercept r ∈(0,1].
M T
Proof. 1. If a control vector nˆ∗ achieves the maximum r˙ at r = r∗, then choosing that control for all
r < r∗ can only achieve a larger or equal r˙, since the coefficient of r in the differential equation,
(cid:80)3 a (1−n2),mustbenon-negative. Similarly,ifacontrol(cid:126)n∗ achievestheminimumatr =r∗,then
j=1 j j
choosingthatcontrolforallr >r∗ canonlyachieveasmallerorequalr˙. Furthermore, ifa anda are
1 2
positive, the coefficient of r cannot be made zero, so in this case, we can strengthen “non-increasing”
to “decreasing”.
2. As r →0+, the linear term in (7) can be neglected, and we must extremize(cid:126)b·(cid:126)n. The range of this is
clearly [−|(cid:126)b|,|(cid:126)b|]
3. Since r cannot exceed one, r˙| ≤0.
r=1
4. Non-positivityfollowsfrom1)and2). Ifa >0,r˙canbealwaysmadenegativebychoosing(cid:126)n=(cid:104)0,0,1(cid:105).
1
5. Non-zero(cid:126)b implies that at a and a are positive, which means that f is strictly decreasing on (0,1).
1 2 M
This, together with 2) and 3) imply the existence of r .
T
Corollary 4.2. If(cid:126)b is nonzero, there is an interval (0,r ), which we call a trap, inside of which the system
T
is controllable. Outside of the trap, on [r ,1], the system is one-way controllable; that is, r can be steered
T i
to r in finite time if and only if r ≤r .
f f i
Proof. The statements in the proposition imply that f (r) < 0 < f (r) on (0,r ), in which case we can
m M T
steer r to r ≥ r by choosing the control that satisfies r˙ = f (r ) provided r < r . Conversely, to steer
i f i M f f T
r to r ≤ r , we can choose the control that satisfies r˙ = f (r ). On the interval [r ,1], f (r) ≤ 0, so r
i f i m f T M i
cannot be steered to r >r , but can be steered to r <r by choosing the control that satisfies r˙ =f (r ),
f i f i m i
which must be negative.
In the case that |(cid:126)b|=0, there is no trap: r˙ ≤0 for all r, and in fact we can say that
−r(a +a )≤r˙ ≤−r(a +a ) (9)
1 2 2 3
where we can achieve the upper and lower bounds by choosing(cid:126)n to be (cid:104)±1,0,0(cid:105) and (cid:104)0,0,±1(cid:105), respectively.
In the case that a = a = 0, the decay of r may be halted, but otherwise r will decay exponentially to
2 3
zero at a rate above or equal to a +a . It is evident, then, that the presence of an asymmetric part in the
2 3
dissipative term (represented by(cid:126)b) significantly enhances the possibility of control.
5
In order to calculate f and f for given r, we can use the method of Lagrange multipliers. In some
M m
cases,wecansolvetheresultingequationsanalytically,butingeneralonemustfindtherootsofasixth-order
polynomial, so we must resort to numerics. Before considering the general case, we will look at a particular
case that can be treated analytically. We consider the possibility that a two-level system can undergo one of
twoprocessesrepresentedbytheraisingandloweringoperatorsσ andσ atratesα andα , respectively.
+ − + −
IfoneconstructstheLindbladequationusingthisscenario,andexpressesitinthebasisofthePaulimatrices,
one finds that a = a = |α+−α−|, a = 0, b = b = 0 and b = α −α . The fact that(cid:126)b has only one
1 2 2 3 1 2 3 + −
non-zero component simplifies the equations so that we can treat the system analytically.
If we apply the method of Lagrange multipliers to the right-hand side of (7) and set b = b = 0 and
1 2
a =a , we get:
1 2
2ra n =2λn
1 1 1
2ra n =2λn
1 2 2
b =2λn
3 3
n2+n2+n2 =1
1 2 3
where λ is the Lagrange multiplier. This has the following solutions:
nˆ =(cid:104)0,0,±1(cid:105) (10)
(cid:28) (cid:29)
b
nˆ = n ,n , 3 (11)
1 2 2a r
1
where n and n in (11) can be any pair that satsifies the normalization condition. Solutions (11) do not
1 2
exist for all r, since the magnitude of n must not exceed one. They exist only on [|b3|,1]. To determine
3 2a1
which solutions correspond to f and f , we substitute back into (7). Solutions (10) give
M m
r˙ =±|b |−2a r (12)
3 1
and solutions (11) give
|b |2
r˙ = 3 −ra (13)
4a r 1
1
We can easily conclude that f (r)=−|b |−2a r. Furthermore, the right-hand side of (13) is greater than
m 3 1
or equal to those of (12), but since it has a limited interval of definition, we have:
(cid:40)
|b |−2a r, r ∈(0,|b3|)
fM(r)= |b33|2 −ra1 , r ∈(|b32|,a11) (14)
4a1r 1 2a1
It happens that r in this case coincides with the point at which f switches between (10) and (11), i.e.
T M
r = |b3|. This is not a general phenomenon, however: if a > 0, the switching point and the trap radius
T 2a1 3
would not coincide. Fig. 1 depicts these solutions for a =a =10 and b =12.
1 2 3
Moregenerally,onecanperformthisanalyticaltreatmentinthefollowingcases: (1)if(cid:126)bhasonenon-zero
component, (2) if(cid:126)b has two non-zero components, and the corresponding a ’s are equal, and (3)(cid:126)b has three
j
non-zero components, and a = a = a . If the system does not fall into any of those three categories,
1 2 3
Lagrange multipliers lead to either a fourth-degree polynomial in λ (technically solvable, but inordinately
messy) or a sixth-degree polynomial (generally not solvable). The fourth-degree polynomial arises in the
cases (1)(cid:126)b has two non-zero components but corresponding a ’s are not equal and (2)(cid:126)b has three non-zero
j
components and a = a > a or a > a = a . The sixth-degree polynomial arises if(cid:126)b has three non-zero
1 2 3 1 2 3
components and a >a >a .
1 2 3
In those cases, we can find the real roots of the polynomial numerically. Then we can compute the
corresponding values of r˙, choose the maximum and minimum values, and assign the values to f and f .
M m
In fig. 2, f and f are shown for a particular system that required solving a sixth-order polynomial. We
M m
have computed the curves for 10,000 points apiece. r can be found by numerically interpolating f . For
T M
the case depicted in fig. 2, r was computed to be 0.544387876644064 (to machine precision).
T
6
20
15
Maximum achievable dr/dt
10
5
0
−5
dt
dr/ −10
−15
−20
−25
−30 Minimum achievable dr/dt
−35
0 0.2 0.4 0.6 0.8 1
r
Figure 1: Maximum and minimum achievable dr/dt vs. r for a case that can be solved analytically. System
parameters: a = a = 10, a = 0, b = b = 0, b = 12. Solid lines represent f and f . Blue and purple
1 2 3 1 2 3 M m
indicate solutions (10) and (11), respectively. Dotted lines indicate where these solutions do not coincide
with f .
M
10
Maximum achievable dr/dt
5
0
−5 Trap radius is ~0.544
dt
dr/
−10
−15
−20 Minimum achievable dr/dt
−25
0 0.2 0.4 0.6 0.8 1
r
Figure 2: Maximum and minimum achievable dr/dt vs. r for a case that must be solved numerically. The
trap radius is where the maximum achievable dr/dt passes from positive to negative. System parameters:
√ √
a =10, a =5, a =0.3, b =0.15 0.6, b =0.9, b =3 6
1 2 3 1 2 3
7
5 Purifiable systems
An important goal in quantum control is purification: the process of steering a mixed state to a pure state,
(cid:112)
whichcanbecharacterizedbyapurity tr(ρ2)=1. Alternatively, asystemispureiftheleadingeigenvalue
is one, with remaining eigenvalues being zero. In terms of the above analysis, we say a system is purifiable
if and only if the trap radius r =1. In other words, the function f (r) has an isolated intercept at r =1.
T M
Thissectionisdevotedtoprovingatheoremthatcharacterizesthepossiblepurifiablesystems. First,wewill
use the following lemma:
Lemma 5.1. r˙ = 0 at r = 1 if and only the system is in a state that is an eigenvector of all contributing
Lindblad operators.
Proof. Because the Bloch radius can be written r = λ −λ , where λ ≥ λ are the eigenvalues of ρ, we
+ − + 2
can write r =(cid:104)ψ |ρ|ψ (cid:105)−(cid:104)ψ |ρ|ψ (cid:105). Differentiating this, we get an alternative expression for r˙:
+ + − −
dr
=(cid:104)ψ˙ |ρ|ψ (cid:105)−(cid:104)ψ˙ |ρ|ψ (cid:105)+(cid:104)ψ |ρ˙|ψ (cid:105)−(cid:104)ψ |ρ˙|ψ (cid:105)+(cid:104)ψ |ρ|ψ˙ (cid:105)−(cid:104)ψ |ρ|ψ˙ (cid:105)
dt + + − − + + − − + + − −
=λ ((cid:104)ψ˙ |ψ (cid:105)+(cid:104)ψ |ψ˙ (cid:105))−λ ((cid:104)ψ˙ |ψ (cid:105)+(cid:104)ψ |ψ˙ (cid:105))+(cid:104)ψ |ρ˙|ψ (cid:105)−(cid:104)ψ |ρ˙|ψ (cid:105)
+ + + + + − − − − − + + − −
=(cid:104)ψ |ρ˙|ψ (cid:105)−(cid:104)ψ |ρ˙|ψ (cid:105)
+ + − −
where in the last step, the normalization of the vectors makes the quantities in parentheses vanish. Now,
if the dissipation is characterized by a collection of Lindblad operators {L }’s, which are not necessarily
j
orthogonal we can use (1) to specify ρ˙:
dr
=(cid:104)ψ |[−iH,ρ]|ψ (cid:105)−(cid:104)ψ |[−iH,ρ]|ψ (cid:105)
dt + + − −
(cid:18)
(cid:88) 1 1
+ (cid:104)ψ |L ρL†|ψ (cid:105)− (cid:104)ψ |L†L ρ|ψ (cid:105)− (cid:104)ψ |ρL†L |ψ (cid:105)
+ j j + 2 + j j + 2 + j j +
j
(cid:19)
1 1
−(cid:104)ψ |L ρL†|ψ (cid:105)+ (cid:104)ψ |L†L ρ|ψ (cid:105)+ (cid:104)ψ |ρL†L |ψ (cid:105)
− j j − 2 − j j − 2 − j j −
TheHamiltoniantermsvanishsincetheyarediagonalelementsofaskew-symmetricmatrix. Weareinterested
in r˙ when r =1, so insert ρ=|ψ (cid:105)(cid:104)ψ |. We get:
+ +
dr (cid:88)(cid:16) (cid:17)
= (cid:104)ψ |L |ψ (cid:105)(cid:104)ψ |L†|ψ (cid:105)−(cid:104)ψ |L†L |ψ (cid:105)−(cid:104)ψ |L |ψ (cid:105)(cid:104)ψ |L†|ψ (cid:105)
dt + j + + j + + j j + − j + + j −
j
If we insert the identity operator between L† and L in the middle term, we get the expression:
j j
dr (cid:88)
=−2 |(cid:104)ψ |L |ψ (cid:105)|2
dt − j +
j
For r˙ to vanish, we need |(cid:104)ψ |L |ψ (cid:105)|2 to vanish for each L . This is only possible however if |ψ (cid:105) is an
− j + j +
eigenvectorofeachL ,sinceotherwiseL |ψ (cid:105)wouldhavesomecomponentinthe|ψ (cid:105)direction. Thisproves
j j + −
the lemma.
This leads to the following theorem:
Theorem 5.2. A two-level Lindblad system is purifiable if and only if one of the following characterizations
hold:
• There is one Lindblad operator, and it is singular.
• There is one Linblad operator and it is non-singular with non-orthogonal eigenvectors.
• There is no more than one singular Lindblad operator and any number of non-singular operators. All
share a common eigenvector.
• There are any number of non-singular Lindblad operators that share a common eigenvector.
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Proof. We are required to show two things to prove a system is purifiable: f (1) = 0, and a > 0. a ≥ 0.
M 2 2
The latter ensures that f is strictly decreasing rather than constant in r. When combined with the former
M
condition, this implies that f is positive for all r <1, and therefore controllable.
M
It follows from the lemma that a system is purifiable only if all contributing Lindblad operators share a
common eigenvector, or else f (1) will be strictly negative. This is only a necessary condition however and
M
notasufficientone,sincetheconditionimpliesonlythatf (1)=0. Wealsorequirethata >0. Soconsider
M 2
the case a =0. This implies that a and(cid:126)b are also zero (due to (8)), so that A has only one non-zero entry
2 3
in its natural basis. We claim that A in this form corresponds to a non-singular operator with orthogonal
(cid:80)
eigenvectors. Itisarank-onerealpositivematrix,andthereforecanbewrittenA= m m forsome
ij=x,y,z i j
real 3-vector m(cid:126). When one diagonalizes the Lindblad equation however, this results in a single Lindblad
(cid:80)
operator L = m σ . This operator is Hermitian and traceless, however, so neglecting the trivial
j=x,y,z j j
zero operator, it is non-singular with orthogonal eigenvectors.
In other words, as long as the system obeys the terms of the lemma, and does not consist of a single
Hermitianoperator, thesystemispurifiable. Thefirsttwocasesinthetheoremcovertheremainingpossible
single-operator cases. The remaining two cases can be seen by noting that two singular operators cannot
share eigenvectors, since they have only one (we consider two operators that are multiples of each other to
be essentially one process). The third case covers the possibility of one singular operator: it has only one
eigenvector, and that eigenvector must be shared with the other non-singular operator. The fourth case in
the theorem covers the possibility of no singular operators but more than one non-singular operator. Note
thatthenon-singularoperatorsinthethirdandfourthcasesneednothavenon-orthogonaleigenvectors.
6 Conclusions
Wehaveshownthattheinter-orbitdynamicsofacontrolledquantumsystemcanbeisolatedfromtheintra-
orbit dynamics by projecting onto the set of spectra of the density matrix. If one makes certain assumptions
about the controllability of the system along the orbits, the position of the system along the orbit can be
viewedasanewcontrolvariable,sincetheintra-orbitdynamicscanbemadearbitrarilyfasterthantheinter-
orbit dynamics. In two dimensions, we have derived a equation describing this inter-orbit dynamics, where
the new control is the normalized Bloch vector, and the most general Lindblad system can be described
by six real parameters: three describing the symmetric part of the dissipation, and three describing the
anti-symmetric part.
We have analyzed the controllability of a general two-dimensional system under Lindblad dissipation,
particularly for dissipation with non-zero anti-symmetric part. For systems of this type, there exists a trap,
or a subinterval of the state space where each state is reachable from any other, but from which states may
not escape. The size of this trap can be calculated analytically for certain simple cases, but in general must
be calculated numerically. We have shown how this can be done using the method of Lagrange multipliers,
and shown results for a particular generic system.
Furthermore, we have applied this formalism to categorize the set of purifiable systems. A necessary
condition for purifiability is that all Lindblad operators share a common eigenvector. To strengthen this to
a sufficient condition, one must eliminate the case of a single Hermitian Lindblad operator.
Theimmediatedirectionoffutureworkistoapplythisformalismtothreeandhigherdimensionalsystems.
It is well-known that the structure of density matrices is richer and less well-understood than the case for
two dimensions [12]. For one, the set of pure states no longer constitutes the boundary of the set, but a
(measure-zero) subset of the boundary. Currently we are studying how our formalism translates to higher
dimensions and what obstructions are posed by the richer geometry. Furthermore, we would like to know
whetherthesetofpurifiablehigher-dimensionalsystemscanbecategorizedasithasbeendoneinthispaper.
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