On completely positive non-Markovian evolution of a d-level system

0.13 MB·English

by  Andrzej Kossakowski

On completely positive non-Markovian evolution
of a d-level system
8
0
0 Andrzej Kossakowski
2
Institute of Physics
n
Nicholaus Copernicus University
a
J 87-100 Torun, Poland

7 kossak@fizyka.umk.pl

and
]
h Rolando Rebolledo ∗
p
- Laboratorio de An´alisis Estoc´atico
t
n Facultad de Matem´aticas
a
Pontificia Universidad Cat´olica de Chile
u
q Casilla 306, Santiago 22, Chile
[ [email protected]

1
v
7
5 Abstract
0
A sufficient condition for non-Markovian master equation which en-
1
. sures thecomplete positivity of theresulting timeevolution is presented.
1
0
8 1 Introduction
0
:
v Anopensystemisonecoupledtoanexternalenvironment[1,2]. Theinteraction
i
X between the system and its environment leads to phenomena of decoherence
and dissipation, and for this reason recently receive intense consideration in
r
a quantum information, where decoherence is viewed as a fundamental obstacle
totheconstructionofquantuminformationprocessors[3]. Inprinciple,the von
Neumann equation for the total density matrix of the system and the reservoir
provides complete predictions for all the observables. However, this equation
is in practice impossible to solve since all degrees of freedom of the reservoir
have to be taken into account. Main efforts have focused in deducing the time
evolutionofthereducedstatedensitymatrix. Thisistheaimofthewell-known
exact theory of subsystem dynamics due to Nakajima-Zwanzig ([4, 5]) which
relies in a generalized (non-Markovian)master equation approach.
∗PartiallysupportedbyPBCT-ACT13andDireccio´ndeRelacionesInternacionales-PUC
1
TheNakajima-Zwanzigprojectionoperatormethodmakespossibletoderive
an exact equation for the reduced density from the von Neumann equation of
the composed system. The resulting generalized master equation -an integrod-
ifferential equation- is mostly of formal interest since such an exact equation
can almost never be even written down explicitely in the closed form. In con-
trast,whenonemakestheMarkovianapproximation,i.e.,whenoneneglectsthe
reservoir memory effects, the resulting Markovian master equation [6, 7] takes
asimpleformandtherequired[8]completepositivity oftheresultingtime evo-
lution is maintained. The main goal of the theory of open quantum systems is
a non-Markovian description of the dynamics which at the same time include
reservoir memory effects and retain complete positivity.
A variety of non-Markovian master equations have been proposed (cf. [2,
9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29,
30]). However,the complete positivity of the resulting time evolutionis still an
important problem to be investigated.
In the present paper a sufficient condition for non-Markovian master equa-
tions is given, which ensures that the resulting time evolution is completely
positive. It is shown that this condition is rather difficult to verify in practice.
The main reason for that is related to the normalization condition of the time
evolution. This difficulty can be overcomed if one looks first for completely
positive unnormalized solutions to non-Markovian master equations, while the
normalization is imposed separately.
2 Notations
Let Cd be a d-dimensional Hilbert space with the scalar product , and ele-
h· ·i
ments e,x,y,z,....
The C∗–algebra of linear operators on Cd will be denoted by M . Elements
d
of M will be denoted by a,b,c,... and the unit of M is 1 . The M is the
d d d d
Hilbert space under the scalar product a,b =tr(a∗b).
The C∗–algebraof linearmaps fromhM iinto M willbe denoted by L(M ),
d d d
its elements are A,B,C,... and the identity map in L(M ) will be denoted by
d
id. The conjugation (duality) # in L(M ) is defined by the relation:
d
·
A#a,b = a,Ab , (2.1)
h i h i
for all a,b M .
d
∈
This operation endows the following property: the relations
A1 =1 , L1 =0, (2.2)
d d d
and
tr A#a =tr(a), tr L#a =0, (2.3)
are equivalent. (cid:0) (cid:1) (cid:0) (cid:1)
TheconeofallcompletelypositivemapsonM willbedenotedby +(M ).
d d
Finally, if A L(M ), t 0, then the Laplace transform of ABwill be
t d t
denoted by Aˆ . ∈ ≥
p
2
3 Non-Markovian master equations
The reduced dynamics can be studied equivalently in the Schr¨odinger or the
Heinsenberg pictures. Suppose that A : M M describes the reduced
t d d
→
dynamics in the Heisenberg picture, then it should satisfy the following condi-
tions: A +(M ), A 1 =1 , for all t 0, and A =lim A =id. In the
t d t d d 0 t→0 t
∈B ≥
Schr¨odinger picture these relations are given in terms of A#, t 0.
t ≥
In the present section, the reduced dynamics is investigated under the as-
sumption that A is the solution of a non-Markovian master equation of the
t
form:
dA t
t =LA + dsL A , (3.1)
dt t Z t−s s
0
with the initial condition A =id, where
0
1
La=i[h,a]+Fa F(1 ),a , (3.2)
d
− 2{ }
and h = h∗ M , F +(M ), that is, L is the generator of a completely
d d
∈ ∈ B
positive semigroup.
The normalization condition A 1 =1 . implies the equality
t d d
L 1 =0. (3.3)
t d
Anon-Markovianmasterequationoftheform(3.1)canbeeasilyderivedfrom
the Heisenberg equation for the composed system by the Nakajima-Zwanzig
methodundertheassumptionoffactorizationoftheinitialstateofthecomposed
system and the invariance of the initial reservoir state under the reservoir free
evolution, c.f. [2]. In this case, La=i[h,a] only, with h=h∗ M .
d
∈
Taking the Laplace transform of (3.1) one finds:
(idp L Lˆ )Aˆ =id. (3.4)
p p
− −
The equality before implies that both relations below:
Aˆ =(pid L Lˆ )−1 (3.5)
p p
− −
and
Aˆ (id L Lˆ )−1 =id, (3.6)
p p
− −
hold.
It follows from (3.6) that equation (3.1) can also be written in the form
dA t
t =A L+ dsA L , (3.7)
dt t Z s t−s
0
and consequently, the dual dynamics becomes:
dA# t
t = LA#+ dsL #A#. (3.8)
dt − t Z t−s s
0
3
This means that in the case of non-Markovian master equations there is an
analogy to the Markoviancase.
To find conditions on L and L that ensure that the time evolution A
t t
resulting from (3.1) is completely positive for all t 0 is the fundamental
≥
problem of non-Markovian master equations. The main result of the current
paper can be summarized in the following theorem.
Theorem 1 Let us suppose that A is the solution of the equation (3.1), where
t
L has the form
t
L =B +Z , (3.9)
t t t
where B +(M ) for all t 0,
t d
∈B ≥
1
Z a= B (1 ),a +i[h ,a], (3.10)
t t d t
−2{ }
and h = h∗, then A is completely positive for all t 0 if the solution of the
t t t ≥
normalization equation
dN t
t =LN + dsZ N , (3.11)
dt t Z t−s s
0
with the initial condition N =id, is completely positive for all t 0.
0
≥
Proof. It follows from (3.1) that the Laplace transform Aˆ of A is given by
p t
the formula
Aˆ =(idp L Zˆ Bˆ )−1, (3.12)
p p p
− − −
and satisfies the equation
Aˆ =(idp L Zˆ )−1+(idp L Zˆ )−1Bˆ Aˆ . (3.13)
p p p p p
− − − −
It follows from (3.13) and (3.11) that (3.1) can be written in the form:
t t−u
A =N + du dsN B A . (3.14)
t t t−u−s u s
Z Z
0 0
If N is completely positive for all t 0, then iterating (3.14) it is easy to see
t
≥
thatA iscompletelypositiveaswellforallt 0,sinceB +(M ),provided
t t d
≥ ∈B
the iteration procedure converges.
Inordertoanalyzetheproblemsrelatedtothesolutionofthenormalization
equation let us consider the non-Markovianmaster equation of the form
dA t
t = dsk(t s)(B id)A , (3.15)
dt Z − t−s− s
0
where B +(M ) and B (1 )=1 for all t 0, and k(t) 0.
t d t d d
∈B ≥ ≥
The normalization equation takes the form
dN t
t = dsk(t s)N , (3.16)
dt −Z − s
0
4
with the initial condition N =id.
0
The solution of (3.16) has the form
N =f(t)id, (3.17)
t
where f(t) satisfies the equation
df(t) t
= dsk(t s)f(s), (3.18)
dt −Z −
0
and f(0)=1.
As a particular case, let us choose k(t) in the Lidar-Shabani form, cf. [28],
ie.,
k(t)=κ2e−2κγt. (3.19)
In this case one easily finds
e−κγt cos(κt 1 γ2)+ γ sin(κt 1 γ2) , if 0 γ <1 ,
 (cid:20) − √1−γ2 − (cid:21) ≤
p p
f(t)= e−κt(1+κt), if γ =1,
e−κγt coshκt γ2 1)+ γ sinh(κt γ2 1) ,if γ >1.
 (cid:20) p − √γ2−1 p − (cid:21) (3.20)
It follows from (3.17) that N is completely positive if and only if f(t) 0 for
t
≥
all t 0 for all t 0, and (3.20) shows that f(t) 0 for all t 0 if and only if
≥ ≥ ≥ ≥
γ 1.
≥
The above example clearly indicates that the structure of non-Markovian
master equations is much more complicated than the Markovianones.
4 Modified non-Markovian master equations
The time evolution (in the Heisenberg picture) is given by the family of maps
A : M M , t 0, such that A +(M ), for all t 0, (complete pos-
t d d t d
itivity co→ndition), ≥A (1 ) = 1 for a∈ll Bt 0, (normaliza≥tion condition) and
t d d
≥
A :=lim A =id. Insection3ithasbeenshownthatifA satisfiesequation
0 t↓0 t t
(3.1),thenthenormalizationconditioncanbeimposedwithnotrouble. Indeed,
if (3.2), (3.9) and (3.10) are satisfied, then the normalization condition is triv-
ially fulfilled. On the other hand, complete positivity of A leads to complete
t
positivityofsolutionsto the normalizationequation(3.10)whichisaverydiffi-
cult problem. However one can circumvent the above difficulty in the following
manner. LetV ,t 0be the family ofcomplete positivemaps onM suchthat
t d
lim V =id. If≥V (1 )>0 for all t 0, then the maps A , t 0, defined as
t→0 t t d t
≥ ≥
A (a)=V (1 )−1/2V (a)V (1 )−1/2, (4.1)
t t d t t d
are completely positive and normalized.
5
LetV ,t 0bethesolutionofthefollowingmodifiednon-Markovianmaster
t
≥
equation:
dV t
t =PV + dsB V , (4.2)
dt t Z t−s s
0
with the initial condition lim V =id, where P is a completely positive map
t→0 t
and B +(M ) for all t 0.
t d
∈B ≥
The resolvent of (4.2),
V =(idp P B )−1, (4.3)
p p
− −
b b
satisfies the equation
V =(idp P)−1+(idp P)−1B V , (4.4)
p p p
− −
b b b
which is the integral form of (4.2).
Iteration of (4.4) yields that V is completely positive since exp(tP) and B
t t
are completely positive.
If the solution of (4.2) satisfies the condition V (1 ) > 0 for all t 0, then
t d
≥
A , t 0, defined through (4.1) gives the correct time evolution, i.e., it is
t
≥
completely positive and normalized.
The above approach contains as a special case the semigroup form of the
dynamics. Let us consider the equation
dV t
t =LV +λ2 dse(t−s)LV , (4.5)
dt t Z s
0
where L is the generator of a completely positive semigroup. One easily finds
the solution of (4.5) which is of the form:
V =cosh(λt)etL, (4.6)
t
and V (1 ) = 1 cosh(λt). The corresponding normalized evolution A has the
t d d t
form
A =etL, (4.7)
t
that is, it is a semigroup.
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