Initial Correlations in Open Quantum Systems Constructing Linear Dynamical Maps and Master Equations Alessandra Colla1Niklas Neubrand1and Heinz-Peter Breuer1 2

2025-05-05 0 0 688.02KB 12 页 10玖币

侵权投诉

Initial Correlations in Open Quantum Systems: Constructing Linear Dynamical Maps

and Master Equations

Alessandra Colla,1Niklas Neubrand,1and Heinz-Peter Breuer1, 2

1Institute of Physics, University of Freiburg, Hermann-Herder-Straße 3, D-79104 Freiburg, Germany

2EUCOR Centre for Quantum Science and Quantum Computing,

University of Freiburg, Hermann-Herder-Straße 3, D-79104 Freiburg, Germany

(Dated: October 25, 2022)

We investigate the dynamics of open quantum systems which are initially correlated with their

environment. The strategy of our approach is to analyze how given, ﬁxed initial correlations modify

the evolution of the open system with respect to the corresponding uncorrelated dynamical behavior

with the same ﬁxed initial environmental state, described by a completely positive dynamical map.

We show that, for any predetermined initial correlations, one can introduce a linear dynamical map

on the space of operators of the open system which acts like the proper dynamical map on the set

of physical states and represents its unique linear extension. Furthermore, we demonstrate that this

construction leads to a linear, time-local quantum master equation with generalized Lindblad struc-

ture involving time-dependent, possibly negative transition rates. Thus, the general non-Markovian

dynamics of an open quantum system can be described by means of a time-local master equation

even in the case of arbitrary, ﬁxed initial system-environment correlations. We present some illus-

trative examples and explain the relation of our approach to several other approaches proposed in

the literature.

I. INTRODUCTION

The theory of open quantum systems has since its con-

ception gained more and more traction as a powerful tool

for studying quantum systems [1]. Its formulation for the

case of systems weakly coupled to Markovian baths has

been extended to arbitrary coupling and non-Markovian

behavior with numerous techniques, which brought to

the formulation of popular exact master equations such

as the Nakajima-Zwanzig equation [2, 3] and the time-

convolutionless (TCL) master equation [4, 5]. An often

made assumption in many approaches is that of factor-

izing initial condition between the system and the en-

vironment. Such a requirement has been highly criti-

cized, either on the grounds of it being unphysical [6–8],

or just too restrictive, and the question of initial corre-

lation has become of importance [9–11]. For example,

the treatment of the dynamics of open systems including

correlations in the initial state leads to a general method

for the local detection of correlations between the sys-

tem and an inaccessible environment [12, 13] which has

been realized experimentally both in trapped ion and in

photonic systems [14–16]. Several eﬀorts have thus been

made to extend the theory to allow for correlated initial

states, both in terms of examining conditions for which

the resulting dynamical map is completely positive [17–

23] and of ﬁnding alternative methods for the dynami-

cal description of the reduced system [24–28]. The ap-

proaches are numerous and varied, and the wide array of

papers on the subject paints the picture of a complicated

subject matter. Considering initial correlations can mean

very diﬀerent things, depending on what one assumes of

the initial system-environment state: should the envi-

ronment be in a ﬁxed, speciﬁc initial state, or should it

depend on the correlations? Are the system and the en-

vironment initially in a separable, classically correlated,

entangled state? Diﬀerent choices lead to diﬀerent re-

duced dynamics, and may induce limitations such as loss

of complete positivity or a restriction of the set of initial

system states that can be studied. Trying to avoid these

choices to allow for any initial total states might instead

result in other losses, like that of a unique dynamical map

[26]. Ultimately, the decision should be made based on

which questions one is trying to answer or which advan-

tage one wants to gain, carefully taking into account the

consequent drawbacks.

A possible limitation of initial correlations between the

system and the environment is that they may impose

non-linearity of the equations describing the evolution of

the reduced system, the dynamical map and the master

equation, depending on the initial conditions chosen – for

example, such non-linearity may appear in the shape of

additional inhomogeneous terms [29, 30]. However, dy-

namical maps in the case of entangled initial state have

been extended to linear maps on matrices by perform-

ing a basis transformation [31, 32], showing that such

a linearized map is, as a consequence, no longer com-

pletely positive. Some groups, primarily Dominy, Sha-

bani and Lidar (DSL) [33, 34], have then more rigorously

approached the subject of initial correlations by develop-

ing general frameworks for which the linearity of the dy-

namical map can be preserved or obtained. The linearity

property, in fact, comes with several advantages, such as

the operator-sum representation due to Choi [35] of linear

and Hermiticity preserving superoperators and the subse-

quent criterion for such a superoperator to be completely

positive, the renowned Kraus representation [36]. More-

over, a dynamical map which is linear – even if not com-

pletely positive (CP) – automatically leads, through its

inverse, to an exact generator of the evolution which can

be put into a time-local generalized Lindblad form with

time-dependent, possibly negative rates [37, 38]. Master

arXiv:2210.13241v1 [quant-ph] 24 Oct 2022

equations of this form have a wide range of applications,

e.g. in non-Markovian stochastic unravelings [39–42] and

recently proposed formulations of quantum thermody-

namics [43], and make a strong case for why ﬁnding a

linear dynamical map should be preferred. But there is

a trade-oﬀ: in order to maintain linearity, these frame-

works have to sacriﬁce the universality of states studied,

both in the sense of what class of total initial states are

considered, and of how many initial reduced states can be

described via the same evolution. While the DSL frame-

work has been proven to be the most general for the study

of initial correlations while maintaining linearity [44], in

its generality one might ﬁnd it diﬃcult to get a feeling

for the operational procedure involved in practice, and

for how big are the sacriﬁces made for the safeguard of

linearity.

In this work, we focus on a view of initial correla-

tions which is not intended to keep track of the kind

of correlations initially present, but rather focuses on

the diﬀerence between uncorrelated and correlated ini-

tial states, by looking at how the evolution depends on

the initial correlation operator. While this has been seen

in other approaches to give rise to an aﬃne dynamical

map [29, 30], this context can be easily embedded into

a formalism which is in spirit analogous to the DSL ap-

proach. In fact, as we shall see through an explicit pre-

scription, it is always possible to construct a linear dy-

namical map for the reduced system, as a consequence

of the fact that the dynamical map itself acts linearly

on the set of density matrices (it does so even on the

set of trace 1 matrices) and can thus be extended to a

linear map on the set of all operators. The resulting

evolution is then comprised of the usual “uncorrelated”,

completely positive and trace preserving (CPT) dynam-

ical map plus an extra part which depends on the initial

correlations. From this we can construct the associated

master equation, which is once again linear, and exists

any time the inverse of the original uncorrelated dynam-

ical map exists. Thanks to the linearity of the master

equation, this can be put in generalized Lindblad form,

with initial correlations contributions appearing only in

the dissipator. In our view, our results show how initial

correlations do not represent such an added conceptual

challenge with respect to the uncorrelated initial state,

contrary to what is often held, the only drawbacks being

the possible loss of complete positivity and the restriction

on the initial reduced states that can be studied with the

same dynamical map.

The structure of the paper is the following: in Sec. II

we review how the assumption of a ﬁxed initial corre-

lation operator leads to an aﬃne dynamical map, and

formally deﬁne the domain of validity of such map. In

Sec. III we explicitly construct the unique linear exten-

sion for the dynamical map and establish a formal cri-

terion for complete positivity. In Sec. IV we deduce the

linear master equation accounting for initial correlations,

and study the structural change with respect to the un-

correlated case. Sec. V contains an application of the

proposed approach to the Jaynes-Cummings model. We

make concluding remarks in Sec. VI.

II. INITIAL CORRELATIONS AND AFFINE

DYNAMICAL MAPS

For an uncorrelated initial state of the total system

ρSE (0) = ρS(0) ⊗ρEthe dynamical map Φtdescribing

the evolution of the reduced system is determined by two

factors only:

1. the unitary evolution for the total system Ut

2. the initial environmental state ρE

as one can see directly from the deﬁnition of the density

matrix of the reduced system

ρS(t)χ=0

= Φt[ρS(0)] = TrE{UtρS(0) ⊗ρEU†

t},(1)

where the superscript χ= 0 denotes the absence of ini-

tial correlations. Throughout the paper, we assume the

Hilbert space of the reduced system to be ﬁnite dimen-

sional. The map Φtis linear, trace preserving and always

completely positive (CPT), thus admitting a Kraus rep-

resentation

Φt[ρS(0)] = X

Ωi(t)ρS(0)Ω†

i(t),(2)

through the set of time dependent operators Ωisatisfying

PiΩ†

iΩi=Iat all times. For a correlated initial state

ρSE (0) 6=ρS(0) ⊗ρEmost of these results fail. However,

one can analogously study the general case and compare

it to its uncorrelated counterpart by dividing any ini-

tial total system state into its corresponding uncorrelated

state and a correlation operator χ:

ρSE (0) = ρS(0) ⊗ρE+χ , (3)

where ρS(0) = TrE{ρSE (0)},ρE= TrS{ρSE (0)}are the

respective reduced states and χhas the property of being

Hermitian and of yielding the null operator for both par-

tial traces, TrE{χ}= 0,TrS{χ}= 0. Now, the dynam-

ical map depends additionally on the initial correlation

operator through an extra term:

Φχ

t[ρS(0)] = Φt[ρS(0)] + Iχ

t,(4)

with Iχ

t= TrE{UtχU †

t}. In the special case where χ

commutes with the Hamiltonian generating the unitary

evolution Ut, namely when the correlation operator is left

invariant by the evolution, the dynamical map above is

identical to the uncorrelated Φt.

Our strategy in order to deal with correlated initial

states is to determine the reduced dynamics through a

dynamical map determined by

1. the unitary evolution for the total system Ut

2. the initial environmental state ρE

3. the correlation operator χ,

eﬀectively regarding the initial correlation operator as a

parameter to be taken independently of the reduced sys-

tem state, analogously to what is done to ρEin the un-

correlated case. Note that these assumptions are also at

the core of [29], where the parameters describing the en-

vironment and the correlations at time zero should be de-

termined (ﬁxed). Indeed one can always do so; however,

the set of initial reduced states whose evolution can be

adequately described is then limited to the set of “phys-

ical” states Pχ

E(HS), dependent on ρEand χ, for which

the total operator (3) is still a proper state of the total

Hilbert space. A general and explicit characterization of

this set is not straightforward, as it heavily depends on

the interplay between the chosen environmental state and

the correlation operator. Nonetheless, a formal deﬁnition

for Pχ

E(HS)can be given by introducing an assignment

map [9]. This map is designed to map a reduced state

into a unique system-environment state. For our case,

the assignment map depends on the choice of initial en-

vironmental state and correlations:

αχ

E:B(HS)−→ B(HSE )(5)

X7−→ X⊗ρE+χ . (6)

Using the notation S(HSE )to denote the convex set

of states of the full Hilbert space, the physical domain

Pχ

E(HS)can be identiﬁed as the preimage of S(HSE )

under the assignment map αχ

E, i.e.

Pχ

E(HS)=(αχ

E)−1[S(HSE )] .(7)

In more explicit terms, the physical domain corresponds

to the elements of S(HS)for which the corresponding

total operator is positive, i.e. any state ρS∈ S(HS)

satisfying the condition

ρS⊗ρE+χ≥0.(8)

Naturally, for χ= 0 one recovers Pχ=0

E(HS) = S(HS).

Once χis set, then the dynamical map (4) becomes

an aﬃne map on the space of bounded operators B(HS),

with a CPT linear component and a traceless oﬀset term.

It is thus still trace-preserving, but not necessarily com-

pletely positive. While it is true that formally Φχ

tis well

deﬁned on all B(HS), it only assumes physical meaning

when acting on elements of Pχ

E(HS). As we will see in

the next section, this allows us to replace (4) with an

equivalent linear map deﬁned on all bounded operators

and that acts as the proper dynamical map on Pχ

E(HS).

III. LINEAR DYNAMICAL MAP FOR INITIAL

CORRELATIONS

Let us deﬁne the set of all bounded operators of the

reduced Hilbert space that have trace one, A1(HS) :=

{X∈ B(HS)|Tr{X}= 1}. This is an aﬃne subspace of

B(HS), since all aﬃne combinations (combinations with

coeﬃcients λi∈Csuch that Piλi= 1) of trace one

operators Xi∈ A1(HS)are still of trace one:

TrnX

λiXio=X

λi= 1 .(9)

We recall that an aﬃne map can also be deﬁned when

acting on an aﬃne space as a map that is linear under

aﬃne combinations; i.e., a map facting on an aﬃne

space Ais aﬃne if and only if, for any set of elements

Ai∈ A and any set of coeﬃcients λisuch that Piλi= 1,

it follows that

fX

λiAi=X

λif(Ai).(10)

It is known that such an aﬃne map can be uniquely ex-

tended to a linear map on the smallest linear space con-

taining A, denoted by Span(A). It follows from this rea-

soning that the dynamical map (4), which acts linearly

over all trace one operators A1(HS), can be uniquely

extended to a linear map on Span(A1(HS)) = B(HS),

the full space of bounded operators for the reduced sys-

tem. These are straightforward mathematical results,

whose proofs we nonetheless report for completeness in

Appendix A, both in abstract terms and for the speciﬁc

case studied. The important point is that the dynamical

map for initial correlations (4) can be substituted with

a unique equivalent linear map on all bounded operators

that still describes the proper evolution of all relevant

states. One can easily check that the following map

Ψχ

t[X]=Φt[X] + Iχ

tTr{X}(11)

has the wanted properties of being linear and extending

(4) to all X∈ B(HS), and must therefore be its unique

linear extension.

Let us examine its features. Like in the aﬃne case, the

tracelessness of Iχ

tguarantees that the map is trace pre-

serving. In general, with respect to the previous uncor-

related version, it instead loses the property of complete

positivity and possibly even positivity. Still, the map

evolves any element of the physical domain to a proper

state of the reduced system, Ψχ

t[Pχ

E(HS)] ⊂ S(HS), and

it is written as a sum of a completely positive, correlation

independent part and a term depending on initial corre-

lations. Since this second term is linear and Hermiticity

preserving, it admits a pseudo-Kraus representation of

the following form [35]:

Iχ

tTr{ρS(0)}=X

fi(t)Fi(t)ρS(0)F†

i(t),(12)

with the extra condition Pifi(t)F†

i(t)Fi(t) = 0 and

where the coeﬃcients fi(t)can be negative. From the

spectral decomposition of Iχ

Iχ

t=X

aj(t)|ϕj(t)ihϕj(t)|,(13)

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

InitialCorrelationsinOpenQuantumSystems:ConstructingLinearDynamicalMapsandMasterEquationsAlessandraColla,1NiklasNeubrand,1andHeinz-PeterBreuer1,21InstituteofPhysics,UniversityofFreiburg,Hermann-Herder-Straÿe3,D-79104Freiburg,Germany2EUCORCentreforQuantumScienceandQuantumComputing,UniversityofFreibur...

展开>> 收起<<

Initial Correlations in Open Quantum Systems Constructing Linear Dynamical Maps and Master Equations Alessandra Colla1Niklas Neubrand1and Heinz-Peter Breuer1 2.pdf

共12页,预览3页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Initial Correlations in Open Quantum Systems Constructing Linear Dynamical Maps and Master Equations Alessandra Colla1Niklas Neubrand1and Heinz-Peter Breuer1 2

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: