Non-Markovian dynamics in open quantum systems

Heinz-Peter Breuer

Physikalisches Institut,Universitat Freiburg,Hermann-Herder-Straße 3,D-79104 Freiburg,Germany

Elsi-Mari Laine

QCD Labs, COMP Centre of Excellence,Department of Applied Physics,Aalto University,P.O. Box 13500,FI-00076 AALTO,Finland

Jyrki Piilo

Turku Centre for Quantum Physics,Department of Physics and Astronomy,University of Turku,FI-20014 Turun yliopisto,Finland

Bassano Vacchini

Dipartimento di Fisica,Universita degli Studi di Milano,Via Celoria 16, I-20133 Milan,ItalyINFN, Sezione di Milano,Via Celoria 16, I-20133 Milan,Italy

(Dated: May 7, 2015)

The dynamical behavior of open quantum systems plays a key role in many applica-tions of quantum mechanics, examples ranging from fundamental problems, such as theenvironment-induced decay of quantum coherence and relaxation in many-body sys-tems, to applications in condensed matter theory, quantum transport, quantum chem-istry and quantum information. In close analogy to a classical Markovian stochasticprocess, the interaction of an open quantum system with a noisy environment is oftenmodeled phenomenologically by means of a dynamical semigroup with a correspondingtime-independent generator in Lindblad form, which describes a memoryless dynamicsof the open system typically leading to an irreversible loss of characteristic quantum fea-tures. However, in many applications open systems exhibit pronounced memory effectsand a revival of genuine quantum properties such as quantum coherence, correlationsand entanglement. Here, recent theoretical results on the rich non-Markovian quantumdynamics of open systems are discussed, paying particular attention to the rigorousmathematical definition, to the physical interpretation and classification, as well as tothe quantification of quantum memory effects. The general theory is illustrated bya series of physical examples. The analysis reveals that memory effects of the opensystem dynamics reflect characteristic features of the environment which opens a newperspective for applications, namely to exploit a small open system as a quantum probesignifying nontrivial features of the environment it is interacting with. This article fur-ther explores the various physical sources of non-Markovian quantum dynamics, suchas structured environmental spectral densities, nonlocal correlations between environ-mental degrees of freedom and correlations in the initial system-environment state, inaddition to developing schemes for their local detection. Recent experiments addressingthe detection, quantification and control of non-Markovian quantum dynamics are alsobriefly discussed.

PACS numbers: 03.65.Yz, 03.65.Ta, 03.67.-a, 42.50.-p, 42.50.Lc

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CONTENTS

I. Introduction 2

II. Definitions and measures for quantum non-Markovianity 3A. Basic concepts 3

1. Open quantum systems and dynamical maps 32. Divisibility and time-local master equations 4

B. Classical versus quantum non-Markovianity 51. Classical stochastic processes and the Markov

condition 52. Non-Markovianity in the quantum regime 6

C. Quantum non-Markovianity and information flow 61. Trace distance and distinguishability of quantum

states 72. Definition and quantification of memory effects 73. Generalizing the trace distance measure 94. Connection between quantum and classical

non-Markovianity 10D. Alternative approaches 10

1. Quantum non-Markovianity and CP-divisibility 112. Monotonicity of correlations and entropic

quantities 12

III. Models and applications of non-Markovianity 12A. Prototypical model systems 12

1. Pure decoherence model 132. Two-level system in a dissipative environment 143. Single qubit with multiple decoherence channels 154. Spin-boson model 16

B. Applications of quantum non-Markovianity 161. Analysis and control of non-Markovian dynamics 172. Open systems as non-Markovian quantum probes 18

IV. Impact of correlations and experimental realizations 18A. Initial system-environment correlations 19

1. Initial correlations and dynamical maps 192. Local detection of initial correlations 19

B. Nonlocal memory effects 20C. Experiments on non-Markovianity and correlations 21

1. Control and quantification of memory effects inphotonic systems 21

2. Experiments on the local detection of correlations 223. Non-Markovian quantum probes detecting nonlocal

correlations in composite environments 22

V. Summary and outlook 23

Acknowledgments 24

References 24

I. INTRODUCTION

The observation and experimental control of charac-teristic quantum properties of physical systems is of-ten strongly hindered by the coupling of the systemto a noisy environment. The unavoidable interactionof the quantum system with its surroundings generatessystem-environment correlations leading to an irretriev-able loss of quantum coherence. Realistic quantum me-chanical systems are thus open systems governed by anon-unitary time development which describes all fea-tures of irreversible dynamics such as the dissipationof energy, the relaxation to a thermal equilibrium or a

stationary nonequilibrium state and the decay of quan-tum coherences and correlations (Alicki and Lendi, 1987;Breuer and Petruccione, 2002; Davies, 1976).

There is a well-established treatment of the dynam-ics of open quantum systems in which the open system’stime evolution is represented by a dynamical semigroupand a corresponding master equation in Lindblad form(Gorini et al., 1976; Lindblad, 1976). If one adopts a mi-croscopic system-environment approach to the dynamicsof open systems, such a master equation may be derived,for example, with the help of the Born-Markov approx-imation by assuming a weak coupling between the sys-tem and its environment. However, it turns out that formany processes in open quantum systems the approxi-mations underlying this approach are not satisfied andthat a description of the dynamics by means of a dy-namical semigroup fails. Typically, this is due to the factthat the relevant environmental correlation times are notsmall compared to the system’s relaxation or decoherencetime, rendering impossible the standard Markov approx-imation. The violation of this separation of time scalescan occur, for example, in the cases of strong system-environment couplings, structured or finite reservoirs,low temperatures, or large initial system-environmentcorrelations.

If the dynamics of an open quantum system substan-tially deviates from that of a dynamical semigroup oneoften speaks of a non-Markovian process. This termrefers to a well-known concept of the theory of classicalstochastic processes and is used to loosely indicate thepresence of memory effects in the time evolution of theopen system. However, the classical notions of Marko-vianity and non-Markovianity cannot be transferred tothe quantum regime in a natural way since they are basedon a Kolmogorov hierarchy of joint probability distribu-tions which does not exist in quantum theory (Accardiet al., 1982; Lindblad, 1979; Vacchini et al., 2011). There-fore, the concept of a quantum non-Markovian processrequires a precise definition which cannot be based onclassical notions only. Many important questions needto be discussed in this context: How can one rigorouslydefine non-Markovian dynamics in the quantum case,how do quantum memory effects manifest themselves inthe behavior of open systems, and can such effects beuniquely identified experimentally through monitoring ofthe open system? The definition of non-Markovianityshould provide a general mathematical characterizationwhich does not rely on any specific representation or ap-proximation of the dynamics, e.g. in terms of a pertur-bative master equation. Moreover, definitions of non-Markovianity should lead to quantitative measures forthe degree of non-Markovianity, enabling the compari-son of the amount of memory effects in different physicalsystems.

Recently, a series of different proposals to answer thesequestions has been published, rigorously defining the bor-

3

der between the regions of Markovian and non-Markovianquantum dynamics and developing quantitative measuresfor the degree of memory effects (see, e.g., (Breuer et al.,2009; Chruscinski et al., 2011; Chruscinski and Manis-calco, 2014; Luo et al., 2012; Rivas et al., 2010; Wolf et al.,2008), the tutorial paper (Breuer, 2012) and the recentreview by Rivas et al. (2014)). Here, we describe anddiscuss several of these ideas, paying particular attentionto those concepts which are based on the exchange of in-formation between the open system and its environment,and on the divisibility of the dynamical map describingthe open system’s time evolution. We will also explainthe relations between the classical and the quantum no-tions of non-Markovianity and develop a general classi-fication of quantum dynamical maps which is based onthese concepts.

The general theory will be illustrated by a series ofexamples. We start with simple prototypical modelsdescribing pure decoherence dynamics, dissipative pro-cesses, relaxation through multiple decay channels andthe spin-boson problem. We then continue with thestudy of the dynamics of open systems which are coupledto interacting many-body environments. The examplesinclude an Ising and a Heisenberg chain in transversemagnetic fields, as well as an impurity atom in a Bose-Einstein condensate (Apollaro et al., 2011; Haikka et al.,2012, 2011). The discussion will demonstrate, in partic-ular, that memory effects of the open system dynamicsreflect characteristic properties of the environment. Thisfact opens a new perspective, namely to exploit a smallopen system as a quantum probe signifying nontrivial fea-tures of a complex environment, for example the criticalpoint of a phase transition (Gessner et al., 2014a; Smirneet al., 2013). Another example to be discussed here isthe use of non-Markovian dynamics to determine nonlo-cal correlations within a composite environment, carryingout only measurements on the open system functioningas quantum probe (Laine et al., 2012; Liu et al., 2013a;Wißmann and Breuer, 2014).

A large variety of further applications of quantummemory effects is described in the literature. Interestedreaders may find examples dealing with, e.g., phenomeno-logical master equations (Mazzola et al., 2010), chaoticsystems (Znidaric et al., 2011), energy transfer processesin photosynthetic complexes (Rebentrost and Aspuru-Guzik, 2011), continous variable quantum key distribu-tion (Vasile et al., 2011b), metrology (Chin et al., 2012),steady state entanglement (Huelga et al., 2012), Coulombcrystals (Borrelli et al., 2013), symmetry breaking (Chan-cellor et al., 2013), and time-invariant quantum discord(Haikka et al., 2013).

The standard description of the open system dynamicsin terms of a dynamical map is based on the assumptionof an initially factorizing system-environment state. Theapproach developed here also allows to investigate theimpact of correlations in the initial system-environment

state and leads to schemes for the local detection ofsuch correlations through monitoring of the open system(Gessner and Breuer, 2011; Laine et al., 2010b).

In recent years, several of the above features of non-Markovian quantum dynamics have been observed exper-imentally in photonic and trapped ion systems (Cialdiet al., 2014; Gessner et al., 2014b; Li et al., 2011; Liuet al., 2011; Smirne et al., 2011; Tang et al., 2014). Webriefly discuss the results of these experiments whichdemonstrate the transition from Markovian to non-Markovian quantum dynamics, the non-Markovian be-havior induced by nonlocal environmental correlations,and the local scheme for the detection of nonclassical ini-tial system-environment correlations.

II. DEFINITIONS AND MEASURES FOR QUANTUMNON-MARKOVIANITY

A. Basic concepts

1. Open quantum systems and dynamical maps

An open quantum system S (Alicki and Lendi, 1987;Breuer and Petruccione, 2002; Davies, 1976) can be re-garded as subsystem of some larger system composed ofS and another subsystem E, its environment, see Fig. 1.The Hilbert space of the total system S + E is given bythe tensor product space

HSE = HS ⊗HE , (1)

where HS and HE denote the Hilbert spaces of S andE, respectively. Physical states of the total system arerepresented by positive trace class operators ρSE on HSEwith unit trace, satisfying ρSE > 0 and trρSE = 1. Givena state of the total system, the corresponding states ofsubsystems S and E are obtained by partial traces overHE and HS , respectively, i.e., we have ρS = trEρSE andρE = trSρSE . We denote the convex set of physical statesbelonging to some Hilbert space H by S(H).

environment! open systemHilbert space HE

Hamiltonian HE

environment state E

Hilbert space HS

Hamiltonian HS

system state S

interaction HI

FIG. 1 (Color online) Sketch of an open quantum systemdescribed by the Hilbert space HS and the Hamiltonian HS ,which is coupled to an environment with Hilbert space HE

and Hamiltonian HE through an interaction Hamiltonian HI .

We suppose that the total system S +E is closed andgoverned by a Hamiltonian of the general form

H = HS ⊗ IE + IS ⊗HE +HI , (2)

4

where HS and HE are the free Hamiltonians of systemand environment, respectively, and HI is an interactionHamiltonian. The corresponding unitary time evolutionoperator is thus given by

U(t) = exp(−iHt) (~ = 1). (3)

The dynamics of the total system states is obtained fromthe von Neumann equation,

d

dtρSE(t) = −i[H, ρSE(t)], (4)

which yields the formal solution

ρSE(t) = UtρSE(0) ≡ U(t)ρSE(0)U†(t). (5)

An important concept in the theory of open quantumsystems is that of a dynamical map. To explain this con-cept we assume that the initial state of the total systemis an uncorrelated tensor product state

ρSE(0) = ρS(0)⊗ ρE(0), (6)

which leads to the following expression for the reducedopen system state at any time t > 0,

ρS(t) = trEU(t)ρS(0)⊗ ρE(0)U†(t)

. (7)

Considering a fixed initial environmental state ρE(0) andany fixed time t > 0, Eq. (7) defines a linear map

Φt : S(HS) −→ S(HS) (8)

on the open system’s state space S(HS) which maps anyinitial open system state ρS(0) to the corresponding opensystem state ρS(t) at time t:

ρS(0) 7→ ρS(t) = ΦtρS(0). (9)

Φt is called quantum dynamical map. It is easy to verifythat it preserves the Hermiticity and the trace of oper-ators, and that it is a positive map, i.e., that it mapspositive operators to positive operators. Thus, Φt mapsphysical states to physical states.

A further important property of the dynamical mapΦt is that it is not only positive but also completely pos-itive. Maps with this property are also known as tracepreserving quantum operations or quantum channels inquantum information theory. Let us recall that a linearmap Φ is completely positive if and only if it admits aKraus representation (Kraus, 1983), which means thatthere are operators Ωi on the underlying Hilbert spaceHS such that ΦA =

∑i ΩiAΩ†i , and that the condition

of trace preservation takes the form∑i Ω†iΩi = IS . An

equivalent definition of complete positivity is the follow-ing. We consider for any number n = 1, 2, . . . the tensorproduct spaceHS⊗Cn which represents the Hilbert spaceof S combined with an n-level system. We can define a

map Φ⊗ In operating on the combined system by linearextension of the relation (Φ ⊗ In)(A ⊗ B) = (ΦA) ⊗ B.This map Φ⊗ In thus describes a quantum operation ofthe composite system which acts nontrivially only on thefirst factor representing subsystem S. One then definesthe map Φ to be n-positive if Φ ⊗ In is a positive map,and completely positive if Φ⊗ In is a positive map for alln. We note that the existence of maps which are positivebut not completely positive is closely connected to theexistence of entangled states. We note further that posi-tivity is equivalent to 1-positivity, and that for a Hilbertspace with finite dimension NS = dimHS complete pos-itivity is equivalent to NS-positivity.

If the time parameter t now varies over some time in-terval from 0 to T , where T may be finite or infinite, weobtain a one-parameter family of dynamical maps,

Φ = Φt | 0 6 t 6 T,Φ0 = I , (10)

where I denotes the unit map, and the initial environ-mental state ρE(0) used to construct Φt is still kept fixed.This family contains the complete information on the dy-namics of all initial open system states over the time in-terval [0, T ] we are interested in.

2. Divisibility and time-local master equations

Let us suppose that the inverse of Φt exists for all timest > 0. We can then define a two-parameter family ofmaps by means of

Φt,s = ΦtΦ−1s , t > s > 0, (11)

such that we have Φt,0 = Φt and

Φt,0 = Φt,sΦs,0. (12)

The existence of the inverse for all positive times thus al-lows us to introduce the very notion of divisibility. WhileΦt,0 and Φs,0 are completely positive by construction, themap Φt,s need not be completely positive and not evenpositive since the inverse Φ−1

s of a completely positivemap Φs need not be positive. The family of dynamicalmaps is said to be P-divisible if Φt,s is positive, and CP-divisible if Φt,s is completely positive for all t > s > 0.A simple example for a CP-divisible process is providedby a semigroup Φt = eLt with a generator L in Lindbladform. In this case we obviously have Φt,s = eL(t−s) whichis trivially completely positive. As we will discuss lateron there are many physically relevant models which giverise to dynamical maps which are neither CP-divisiblenor P-divisible.

An interesting property of the class of processes forwhich Φ−1

t exists is given by the fact that they alwayslead to a time-local quantum master equation for the

5

open system states with the general structure

d

dtρS = KtρS (13)

= −i [HS(t), ρS ]

+∑i

γi(t)[Ai(t)ρSA

†i (t)−

1

2

A†i (t)Ai(t), ρS

].

The form of this master equation is very similar to thatof a Lindblad master equation, where however the Hamil-tonian contribution HS(t), the Lindblad operators Ai(t)(which can be supposed to be linearly independent), andthe decay rates γi(t) may depend on time since the pro-cess does not represent a semigroup, in general. Notethat Eq. (13) involves a time-dependent generator Kt,but no convolution of the open system states with a mem-ory kernel as in the Nakajima-Zwanzig equation (Naka-jima, 1958; Zwanzig, 1960). Master equations of the form(13) can be derived employing the time-convolutionlessprojection operator technique (Chaturvedi and Shibata,1979; Shibata et al., 1977).

It is an important open problem to formulate generalnecessary and sufficient conditions under which the mas-ter equation (13) leads to a completely positive dynamics.If the process represents a semigroup, the HamiltonianHS , the operators Ai and the rates γi must be time-independent and a necessary and sufficient condition forcomplete positivity of the dynamics is simply that all de-cay rates are positive, γi > 0. This is the famous Gorini-Kossakowski-Sudarshan-Lindblad theorem (Gorini et al.,1976; Lindblad, 1976). However, we will see later by sev-eral examples that in the time-dependent case the ratesγi(t) can indeed become temporarily negative withoutviolating the complete positivity of the dynamics.

On the other hand, for divisible quantum processes itis indeed possible to formulate necessary and sufficientconditions. In fact, the master equation (13) leads to aCP-divisible dynamics if and only if all rates are positivefor all times, γi(t) > 0, which follows from a straight-forward extension of the Gorini-Kossakowski-Sudarshan-Lindblad theorem. Moreover, the master equation (13)leads to a P-divisible dynamics if and only if the weakerconditions ∑

i

γi(t)|〈n|Ai(t)|m〉|2 > 0 (14)

hold for all orthonormal bases |n〉 of the open systemand all n 6= m. This statement can be obtained by apply-ing a characterization of the generators of positive semi-groups due to Kossakowski (1972a,b).

B. Classical versus quantum non-Markovianity

1. Classical stochastic processes and the Markov condition

In classical probability theory (Gardiner, 1985; vanKampen, 1992) a stochastic process X(t), t > 0, tak-

ing values in a discrete set xii∈N is characterizedby a hierarchy of joint probability distributions Pn =Pn(xn, tn;xn−1, tn−1; . . . ;x1, t1) for all n ∈ N and timestn > tn−1 > . . . > t1 > 0. The distribution Pn yields theprobability that the process takes on the value x1 at timet1, the value x2 at time t2, . . . , and the value xn at timetn. In order for such a hierarchy to represent a stochasticprocess the Kolmogorov consistency conditions must besatisfied which, apart from conditions of positivity andnormalization, require in particular the relation∑

xm

Pn(xn, tn; . . . ;xm, tm; . . . ;x1, t1)

= Pn−1(xn, tn; . . . ;x1, t1), (15)

connecting the n-point probability distribution Pn to the(n− 1)-point probability distribution Pn−1.

A stochastic process X(t) is said to be Markovian ifthe conditional probabilities defined by

P1|n(xn+1, tn+1|xn, tn; . . . ;x1, t1)

=Pn+1(xn+1, tn+1; . . . ;x1, t1)

Pn(xn, tn; . . . ;x1, t1)(16)

satisfy the relation

P1|n(xn+1, tn+1|xn, tn; . . . ;x1, t1)

= P1|1(xn+1, tn+1|xn, tn). (17)

This is the classical Markov condition which means thatthe probability for the stochastic process to take the valuexn+1 at time tn+1, under the condition that it assumedvalues xi at previous times ti, only depends on the lastprevious value xn at time tn. In this sense the pro-cess is said to have no memory, since the past historyprior to tn is irrelevant to determine the future once weknow the value xn the process assumed at time tn. Notethat Eq. (17) imposes an infinite number of conditionsfor all n-point probability distributions which cannot bechecked if only a few low-order distributions are known.

The Markov condition (17) substantially simplifies themathematical description of stochastic processes. In fact,one can show that under this condition the whole hi-erarchy of joint probability distributions can be recon-structed from the initial 1-point distribution P1(x0, 0)and the conditional transition probability

T (x, t|y, s) ≡ P1|1(x, t|y, s) (18)

by means of the relations

Pn(xn, tn;xn−1, tn−1; . . . ;x1, t1)

=

n−1∏i=1

T (xi+1, ti+1|xi, ti)P1(x1, t1) (19)

and

P1(x1, t1) =∑x0

T (x1, t1|x0, 0)P1(x0, 0). (20)

6

For a Markov process the transition probability has toobey the Chapman-Kolmogorov equation

T (x, t|y, s) =∑z

T (x, t|z, τ)T (z, τ |y, s) (21)

for t > τ > s. Thus, a classical Markov process isuniquely characterized by a probability distribution forthe initial states of the process and a conditional tran-sition probability satisfying the Chapman-Kolmogorovequation (21). Indeed, the latter provides the necessarycondition in order to ensure that the joint probabilitiesdefined by (19) satisfy the condition (15), so that theyactually define a classical Markov process. Provided theconditional transition probability is differentiable withrespect to time (which will always be assumed here) oneobtains an equivalent differential equation, namely theChapman-Kolmogorov equation:

d

dtT (x, t|y, s) (22)

=∑z

[Wxz(t)T (z, t|y, s)−Wzx(t)T (x, t|y, s)

],

where Wzx(t) > 0, representing the rate (probability perunit of time) for a transition to the state z given thatthe state is x at time t. An equation of the same struc-ture holds for the 1-point probability distribution of theprocess:

d

dtP1(x, t) =

∑z

[Wxz(t)P1(z, t)−Wzx(t)P1(x, t)

], (23)

which is known as Pauli master equation for a classicalMarkov process. The conditional transition probabilityof the process can be obtained solving Eq. (23) with theinitial condition P1(x, s) = δxy.

2. Non-Markovianity in the quantum regime

The above definition of a classical Markov process can-not be transferred immediately to the quantum regime(Vacchini et al., 2011). In order to illustrate the arisingdifficulties let us consider an open quantum system asdescribed in Sec. II.A. Suppose we carry out projectivemeasurements of an open system observable X at timestn > tn−1 > . . . > t1 > 0. For simplicity we assume thatthis observable is non-degenerate with spectral decom-position X =

∑x x|ϕx〉〈ϕx|. As in Eq. (5) we write the

unitary evolution superoperator as UtρSE = UtρSEU†t

and the quantum operation corresponding to the mea-surement outcome x as MxρSE = |ϕx〉〈ϕx|ρSE |ϕx〉〈ϕx|.Applying the Born rule and the projection postulate onecan then write a joint probability distribution

Pn(xn, tn;xn−1, tn−1; . . . ;x1, t1)

= trMxnUtn−tn−1

. . .Mx1Ut1ρSE(0)

(24)

which yields the probability of observing a certain se-quence xn, xn−1, . . . , x1 of measurement outcomes at therespective times tn, tn−1, . . . , t1. Thus, it is of course pos-sible in quantum mechanics to define a joint probabil-ity distribution for any sequence of measurement results.However, as is well known these distribution do in gen-eral not satisfy condition (15) since measurements carriedout at intermediate times in general destroy quantum in-terferences. The fact that the joint probability distribu-tions (24) violate in general the Kolmogorov condition(15) is even true for closed quantum system. Thus, thequantum joint probability distributions given by (24) donot represent a classical hierarchy of joint probabilitiessatisfying the Kolmogorov consistency conditions. Moregenerally, one can consider other joint probability distri-butions corresponding to different quantum operationsdescribing non-projective, generalized measurements.

For an open system coupled to some environment mea-surements performed on the open system not only influ-ence quantum interferences but also system-environmentcorrelations. For example, if the system-environmentstate prior to a projective measurement at time ti is givenby ρSE(ti), the state after the measurement conditionedon the outcome x is given by

ρ′SE(ti) =MxρSE(ti)

trMxρSE(ti)= |ϕx〉〈ϕx| ⊗ ρxE(ti), (25)

where ρxE is an environmental state which may dependon the measurement result x. Hence, projective mea-surements completely destroy system-environment corre-lations, leading to an uncorrelated tensor product stateof the total system, and, therefore, strongly influence thesubsequent dynamics.

We conclude that an intrinsic characterization andquantification of memory effects in the dynamics of openquantum systems, which is independent of any prescribedmeasurement scheme influencing the time evolution, hasto be based solely on the properties of the dynamics ofthe open system’s density matrix ρS(t).

C. Quantum non-Markovianity and information flow

The first approach to quantum non-Markovianity tobe discussed here is based on the idea that memory ef-fects in the dynamics of open systems are linked to theexchange of information between the open system andits environment: While in a Markovian process the opensystem continuously looses information to the environ-ment, a non-Markovian process is characterized by a flowof information from the environment back into the opensystem (Breuer et al., 2009; Laine et al., 2010a). In such away quantum non-Markovianity is associated to a notionof quantum memory, namely information which has beentransferred to the environment, in the form of system-environment correlations or changes in the environmen-

7

tal states, and is later retrieved by the system. To makethis idea more precise we employ an appropriate distancemeasure for quantum states.

1. Trace distance and distinguishability of quantum states

The trace norm of a trace class operator A is defined by||A|| = tr|A|, where the modulus of the operator is given

by |A| =√A†A. If A is selfadjoint the trace norm can be

expressed as the sum of the moduli of the eigenvalues aiof A counting multiplicities, ||A|| =

∑i |ai|. This norm

leads to a natural measure for the distance between twoquantum states ρ1 and ρ2 known as trace distance:

D(ρ1, ρ2) =1

2||ρ1 − ρ2||. (26)

The trace distance represents a metric on the state spaceS(H) of the underlying Hilbert space H. We have thebounds 0 6 D(ρ1, ρ2) 6 1, where D(ρ1, ρ2) = 0 if andonly if ρ1 = ρ2, and D(ρ1, ρ2) = 1 if and only if ρ1 andρ2 are orthogonal. Recall that two density matrices aresaid to be orthogonal if their supports, i.e. the subspacesspanned by their eigenstates with nonzero eigenvalue, areorthogonal. The trace distance has a series of interest-ing mathematical and physical features which make it auseful distance measure in quantum information theory(Nielsen and Chuang, 2000). There are two propertieswhich are the most relevant for our purposes.

The first property is that the trace distance betweentwo quantum states admits a clear physical interpreta-tion in terms of the distinguishability of these states. Toexplain this interpretation consider two parties, Alice andBob, and suppose that Alice prepares a quantum systemin one of two states, ρ1 or ρ2, with a probability of 1

2each, and sends the system to Bob. The task of Bobis to find out by means of a single quantum measure-ment whether the system is in the state ρ1 or ρ2. It canbe shown that the maximal success probability Bob canachieve through an optimal strategy is directly connectedto the trace distance:

Pmax =1

2

[1 +D(ρ1, ρ2)

]. (27)

Thus, we see that the trace distance represents the biasin favor of a correct state discrimination by Bob. For thisreason the trace distance D(ρ1, ρ2) can be interpreted asthe distinguishability of the quantum states ρ1 and ρ2.For example, suppose the states prepared by Alice areorthogonal such that we have D(ρ1, ρ2) = 1. In thiscase we get Pmax = 1, which is a well known fact sinceorthogonal states can be distinguished with certainty bya single measurement, an optimal strategy of Bob beingto measure the projection onto the support of ρ1 or ρ2.

The second important property of the trace distance isgiven by the fact that any completely positive and trace

D(1S(t), 2

S(t))D(1S , 2

S)

t

t

1S

2S 2

S(t)

1S(t)

Alice Bob

FIG. 2 (Color online) Illustration of the loss of distinguisha-bility quantified by the trace distance D between density ma-trices as a consequence of the action of a quantum dynamicalmap Φt. Alice prepares two distinct states, but a dynamicalmap acts as a noisy channel, thus generally reducing the in-formation available to Bob in order to distinguish among thestates by performing measurements on the system only.

preserving map Λ is a contraction for the trace distance,i.e. we have

D(Λρ1,Λρ2) 6 D(ρ1, ρ2) (28)

for all states ρ1,2. In view of the interpretation of thetrace distance we thus conclude that a trace preservingquantum operation can never increase the distinguisha-bility of any two quantum states. We remark that theequality sign in (28) holds if Λ is a unitary transforma-tion, and that (28) is also valid for trace preserving mapswhich are positive but not completely positive.

2. Definition and quantification of memory effects

In the preceding subsection we have interpreted thetrace distance of two quantum states as the distinguisha-bility of these states, where it has been assumed thatthe quantum state Bob receives is identical to the stateprepared by Alice. Suppose now that Alice preparesher states ρ1,2

S (0) as initial states of an open quantumsystem S coupled to some environment E. Bob willthen receive at time t the system in one of the statesρ1,2S (t) = Φtρ

1,2S (0), where Φt denotes the corresponding

quantum dynamical map. This construction is equiva-lent to Alice sending her states through a noisy quantumchannel described by the completely positive and tracepreserving map Φt. According to (28) the dynamics gen-erally diminishes the trace distance and, therefore, thedistinguishability of the states,

D(ρ1S(t), ρ2

S(t)) 6 D(ρ1S(0), ρ2

S(0)), (29)

such that it will in general be harder for Bob to discrimi-nate the states prepared by Alice. This fact is illustratedschematically in Fig. 2, in which the distinguishability ofstates prepared by Alice decreases due to the action of adynamical map, so that Bob is less able to discriminateamong the two. Thus, we can interpret any decrease ofthe trace distance D(ρ1

S(t), ρ2S(t)) as a loss of information

8

from the open system into the environment. Conversely,if the trace distanceD(ρ1

S(t), ρ2S(t)) increases, we say that

information flows back from the environment into theopen system.

This interpretation naturally leads to the following def-inition: A quantum process given by a family of quantumdynamical maps Φt is said to be Markovian if the tracedistance D(ρ1

S(t), ρ2S(t)) corresponding to all pair of ini-

tial states ρ1S(0) and ρ2

S(0) decreases monotonically for alltimes t > 0. Quantum Markovian behavior thus meansa continuous loss of information from the open systemto the environment. Conversely, a quantum process isnon-Markovian if there is an initial pair of states ρ1

S(0)and ρ2

S(0) such that the trace distance D(ρ1S(t), ρ2

S(t))is non-monotonic, i.e. starts to increase for some timet > 0. If this happens, information flows from the envi-ronment back to the open system, which clearly expressesthe presence of memory effects: Information containedin the open system is temporarily stored in the environ-ment and comes back at a later time to influence thesystem. A crucial feature of this definition of quantumMarkovian process is the fact that non-Markovianity canbe directly experimentally assessed, provided one is ableto perform tomographic measurement of different initialstates at different times during the evolution. No priorinformation on the dynamical map Φt is required, apartfrom its very existence.

environment!

system

environment!

systemd

dtIint > 0

d

dtIint < 0

FIG. 3 (Color online) Illustration of the information flowbetween an open system and its environment according toEq. (32). Left: The open system looses information to the en-vironment, corresponding to a decrease of Iint(t) and Marko-vian dynamics. Right: Non-Markovian dynamics is charac-terized by a backflow of information from the environment tothe system and a corresponding increase of Iint(t).

In order to explain in more detail this interpretationin terms of an information flow between system and en-vironment let us define the quantities

Iint(t) = D(ρ1S(t), ρ2

S(t)) (30)

and

Iext(t) = D(ρ1SE(t), ρ2

SE(t))−D(ρ1S(t), ρ2

S(t)). (31)

Here, Iint(t) is the distinguishability of the open systemstates at time t, while Iext(t) is the distinguishability of

the total system states minus the distinguishability of theopen system states at time t. In other words, Iext(t) canbe viewed as the gain in the state discrimination Bobcould achieve if he were able to carry out measurementson the total system instead of measurements on the opensystem only. We can therefore interpret Iint(t) as theinformation inside the open system and Iext(t) as theamount of information which lies outside the open sys-tem, i.e., as the information which is not accessible whenonly measurements on the open system can be performed.Obviously, we have Iint(t) > 0 and Iext(t) > 0. Since thetotal system dynamics is unitary the distinguishability ofthe total system states is constant in time. Moreover wehave D(ρ1

SE(0), ρ2SE(0)) = D(ρ1

S(0), ρ2S(0)) because, by

assumption, the initial total states are uncorrelated withthe same reduced environmental state. Hence, we obtain

Iint(t) + Iext(t) = Iint(0) = const. (32)

Thus, initially there is no information outside the opensystem, Iext(0) = 0. If Iint(t) decreases Iext(t) mustincrease and vice versa, which clearly expresses the ideaof the exchange of information between the open systemand the environment illustrated in Fig. 3. Employingthe properties of the trace distance one can derive thefollowing general inequality (Laine et al., 2010b) whichholds for all t > 0:

Iext(t)6 D(ρ1SE(t), ρ1

S(t)⊗ ρ1E(t)) (33)

+D(ρ2SE(t), ρ2

S(t)⊗ ρ2E(t)) +D(ρ1

E(t), ρ2E(t)).

The right-hand side of this inequality consists of threeterms: The first two terms provide a measure for the cor-relations in the total system states ρ1,2

SE(t), given by thetrace distance between these states and the product oftheir marginals, while the third term is the trace distancebetween the corresponding environmental states. Thus,when Iext(t) increases over the initial value Iext(0) = 0system-environment correlations are built up or the envi-ronmental states become different, implying an increaseof the distinguishability of the environmental states. Thisclearly demonstrates that the corresponding decrease ofthe distinguishability Iint(t) of the open system statesalways has an impact on degrees of freedom which areinaccessible by measurements on the open system. Con-versely, if Iint(t) starts to increase at some point of time t,the corresponding decrease of Iext(t) implies that system-environment correlations must be present already at timet, or that the environmental states are different at thispoint of time.

On the ground of the above definition for non-Markovian dynamics one is naturally led to introducethe following measure for the degree of memory effects

N (Φ) = maxρ1,2S

∫σ>0

dt σ(t), (34)

9

where

σ(t) ≡ d

dtD(Φtρ

1S ,Φtρ

2S

)(35)

denotes the time derivative of the trace distance of theevolved pair of states. In Eq. (34) the time integral isextended over all intervals in which σ(t) > 0, i.e., inwhich the trace distance increases with time, and themaximum is taken over all pairs of initial states ρ1,2

S ofthe open system’s state space S(HS). Thus, N (Φ) is apositive functional of the family of dynamical maps Φwhich represents a measure for the maximal total flowof information from the environment back to the opensystem. By construction we have N (Φ) = 0 if and onlyif the process is Markovian.

The maximization over all pairs of quantum states inEq. (34) can be simplified substantially employing sev-eral important properties of the functional N (Φ) andthe convex structure of the set of quantum states. Acertain pair of states ρ1,2

S is said to be an optimal statepair if the maximum in Eq. (34) is attained for this pairof states. Thus, optimal state pairs lead to the maxi-mal possible backflow of information during their timeevolution. One can show that optimal states pairs ρ1,2

S

lie on the boundary of the state space, and are in par-ticular always orthogonal (Wißmann et al., 2012). Thisis a quite natural result in view of the interpretation interms of an information flow since orthogonality impliesthat D(ρ1

S , ρ2S) = 1, which shows that optimal state pairs

have maximal initial distinguishability, corresponding toa maximal amount of initial information. An even moredrastic simplification is obtained by employing the fol-lowing equivalent representation (Liu et al., 2014):

N (Φ) = maxρ∈∂U(ρ0)

∫σ>0

dt σ(t), (36)

where

σ(t) ≡ddtD (Φtρ,Φtρ0)

D (ρ, ρ0)(37)

is the time derivative of the trace distance at time t di-vided by the initial trace distance. In Eq. (36) ρ0 is afixed point of the interior of the state space and ∂U(ρ0)an arbitrary surface in the state space enclosing, but notcontaining ρ0. The maximization is then taken over allpoints of such an enclosing surface. This representationoften significantly simplifies the analytical, numerical orexperimental determination of the measure since it onlyinvolves a maximization over a single input state ρ. Itis particularly advantageous if the open system dynam-ics has an invariant state and if this state is taken to beρ0, such that only one state of the pair evolves in time.Equation (36) may be called local representation for itshows that optimal state pairs can be found in any lo-cal neighborhood of any interior point of the state space.

Furthermore, the representation reveals the universalityof memory effects, namely that for a non-Markovian dy-namics memory effects can be observed everywhere instate space.

3. Generalizing the trace distance measure

There is an interesting generalization of the above defi-nition and quantification of non-Markovian quantum dy-namics first suggested by Chruscinski et al. (2011). Re-turning to the interpretation of the trace distance de-scribed in Sec. II.C.1, we may suppose that Alice pre-pares her quantum system in the states ρ1

S or ρ2S with

corresponding probabilities p1 and p2, where p1 +p2 = 1.Thus, we assume that Alice gives certain weights to herstates which need not be equal. In this case one can againderive an expression for the maximal probability for Bobto identify the state prepared by Alice:

Pmax =1

2

(1 + ‖p1ρ

1S − p2ρ

2S‖). (38)

As follows from (38) the quantity ‖p1ρ1S − p2ρ

2S‖ gives

the bias in favor of the correct identification of the stateprepared by Alice. We note that the operator ∆ = p1ρ

1S−

p2ρ2S is known as Helstrom matrix (Helstrom, 1967), and

that Eq. (38) reduces to Eq. (27) in the unbiased casep1 = p2 = 1

2 . Note that the interpretation in termsof an information flow between system and environmentstill holds for the biased trace distance, as can be seenreplacing in (30) and (31) the trace distance by the normof the Helstrom matrix.

Following this approach we can define a process to beMarkovian if the function ‖Φt(p1ρ

1S − p2ρ

2S)‖ decreases

monotonically with time for all p1,2 and all pairs of ini-

tial open system states ρ1,2S . Consequently, the extended

measure for non-Markovianity is then given by

N (Φ) = maxpi,ρiS

∫σ>0

dt σ(t), (39)

where

σ(t) ≡ d

dt‖Φt(p1ρ

1S − p2ρ

2S)‖. (40)

This generalized definition of quantum non-Markovianityleads to several important conclusions. First, we notethat also for the measure (39) optimal state pairs are or-thogonal, satisfying ‖p1ρ

1S − p2ρ

2S‖ = 1 and, therefore,

corresponding to maximal initial distinguishability. Themaximum in (39) can therefore be taken over all Hermi-tian (Helstrom) matrices ∆ with unit trace norm. Thisleads to a local representation analogous to Eq. (36):

N (Φ) = max∆∈∂U(0)

∫σ>0

dt σ(t), (41)

10

where σ(t) = ddt‖Φt∆‖/‖∆‖, and ∂U(0) is a closed sur-

face which encloses the point ∆ = 0 in the R-linear vectorspace of Hermitian matrices.

t not invertible

t invertible

P-divisible not P-divisible

CP-divisible

non-Markovian

N () > 0

Markovian

N () = 0

FIG. 4 (Color online) Schematic picture of the relations be-tween the concepts of quantum Markovianity and divisibilityof the family of quantum dynamical maps Φt. The blue areato the left of the thick black line represents the set of Marko-vian quantum processes for which the inverse of the dynam-ical map exists, which is identical to the set of P-divisibleprocesses. This set contains as subset the set of CP-divisibleprocesses, depicted in green. The blue area to the right repre-sents the non-Markovian processes which are therefore neitherCP nor P-divisible. The zig-zag line dividing the two regionspoints to the fact that neither of them is convex. The redareas mark the Markovian or non-Markovian processes forwhich the inverse of Φt does not exist.

Another important conclusion is obtained if we as-sume, as was done in Sec. II.A.2, that Φ−1

t exists. It canbe shown that under this condition the quantum processis Markovian if and only if Φt is P-divisible, which followsimmediately from a theorem by Kossakowski (1972a,b).The relationships between Markovianity and divisibilityof the dynamical map are illustrated in Fig. 4, where alsothe situation in which the inverse of Φt as a linear mapdoes not exist for all times has been considered. Thegeneralized characterization of non-Markovianity is sen-sitive to features of the open system dynamics neglectedby the trace distance measure (Liu et al., 2013b). It isworth stressing that in keeping with the spirit of the ap-proach, this generalized measure of non-Markovianity, inaddition to its clear connection with the mathematicalproperty of P-divisibility of the dynamical map, can stillbe directly evaluated by means of experiments. Morespecifically, both the quantification of non-Markovianityaccording to the measure (34), as well as its generaliza-tion (39) can be obtained from the very same data.

4. Connection between quantum and classical non-Markovianity

The definition of quantum non-Markovianity ofSec. II.C.3 allows to establish an immediate generalconnection to the classical definition. Suppose as inSec. II.A.2 that the inverse of the dynamical maps Φtexists such that we have the time-local quantum mas-ter equation (13), in which for the sake of simplicity weconsider only the rates to be time dependent. Supposefurther that Φt maps operators which are diagonal in afixed basis |n〉 to operators diagonal in the same ba-sis. If ρS(0) is an initial state diagonal in this basis, thenρS(t) can be expressed at any time in the form:

ρS(t) =∑n

Pn(t)|n〉〈n|, (42)

where Pn(t) denote the time-dependent eigenvalues. Thequantum master equation then leads to the followingequation of motion for the probabilities Pn(t):

d

dtPn(t) =

∑m

[Wnm(t)Pm(t)−Wmn(t)Pn(t)

], (43)

where

Wnm(t) =∑i

γi(t)|〈n|Ai|m〉|2. (44)

We observe that Eq. (43) has exactly the structure of aclassical Pauli master equation (23). Therefore, Eq. (43)can be interpreted as differential Chapman-Kolmogorovequation (22) for the conditional transition probability ofa classical Markov process if and only if the rates (44) arepositive. As we have seen a quantum process is Marko-vian if and only if the dynamics is P-divisible. Accord-ing to condition (14) P-divisibility implies that the rates(44) are indeed positive. Thus, we conclude that anyquantum Markovian process which preserves the diago-nal structure of quantum states in a fixed basis leads to aclassical Markovian jump process describing transitionsbetween the eigenstates of the density matrix.

D. Alternative approaches

In Sec. II.C we have considered an approach to quan-tum non-Markovianity based on the study of the dynam-ical behavior of the distinguishability of states preparedby two distinct parties, Alice and Bob. This viewpointdirectly addresses the issue of the experimental detectionof quantum non Markovian processes, which can be ob-tained by suitable quantum tomographic measurements.It further highlights a strong connection between quan-tum non-Markovianity and quantum information theory,or more specifically quantum information processing.

Alternative approaches to the study of quantum non-Markovian dynamics of open systems have also been re-cently proposed. We shall try to briefly discuss those

11

which are more closely related to the previous study,without any claim to properly represent the vast liter-ature on the subject. For a review on recent resultson quantum non-Markovianity see Rivas et al. (2014).The different approaches can to some extent be groupedaccording to whether they propose different divisibilityproperties of the quantum dynamical map (Chruscinskiand Maniscalco, 2014; Hall et al., 2014; Hou et al., 2011;Rivas et al., 2010; Wolf et al., 2008), different quantifiersof the distinguishability between states (Chruscinski andKossakowski, 2012; Dajka et al., 2011; Lu et al., 2010;Vasile et al., 2011a; Wißmann et al., 2013) or the study ofother quantities, be they related to quantum informationconcepts or not, which might exhibit both a monotonicand an oscillating behavior in time (Bylicka et al., 2014;Fanchini et al., 2014; Haseli et al., 2014; Lu et al., 2010;Luo et al., 2012).

1. Quantum non-Markovianity and CP-divisibility

Let us first come back to the notion of divisible mapsintroduced in Sec. II.A.2, which we express in the form

Φt,s = Φt,τΦτ,s, t > τ > s > 0, (45)

where the maps have been defined according to (11).In the classical case considering the matrices whose el-ements are given by the conditional transition prob-abilities (18) according to (Λt,s)xy = T (x, t|y, s) theChapman-Kolmogorov equation takes the form

Λt,s = Λt,τΛτ,s, t > τ > s > 0, (46)

where each Λt,s is a stochastic matrix, which obviouslybears a strict relationship with (45), once one consid-ers that positive maps are natural quantum counter-part of classical stochastic matrices. Given the fact thatin the classical case the Markov condition (17) entailsthe Chapman-Kolmogorov equation, and building on theconclusion drawn at the end of Sec. II.B.2, i.e. thatan intrinsic characterization of memory effects has to bebased on the system’s density operator only, whose dy-namics is fully described in terms of the time evolutionmaps, one is led to associate quantum Markovianity withP-divisibility. This observation reinforces the result ob-tained in Sec. II.C.3 relying on the study of the informa-tion flow as quantified by the generalized trace distancemeasure. Actually it can be shown that the P-divisibilitycondition (46) is equivalent to the monotonicity propertyof the function

K(P 11 (t), P 2

1 (t)) =∑x

|p1P11 (x, t)− p2P

21 (x, t)| (47)

which provides a generalization of the Kolmogorov dis-tance between two classical probability distributions

P 11 (t) and P 2

1 (t). Therefore these two equivalent prop-erties capture the feature of a classical Markovian pro-cess at the level of the one-point probability distribution,which is all we are looking for in order to obtain an in-trinsic definition of quantum non-Markovianity.

Relying on the fact that in the theory of open quantumsystems complete positivity suggests itself as a naturalcounterpart of positivity, as an alternative approach toquantum non-Markovianity it has been proposed to iden-tify quantum Markovian dynamics with those dynamicswhich are actually CP-divisible. This criterion was firstsuggested by Rivas et al. (2010). In order to check CP-divisibility of the time evolution one actually needs toknow the mathematical expression of the transition mapsΦt,s or of the infinitesimal generator Kt. Indeed given themaps Φt,s one can determine their complete positivitystudying the associated Choi matrices, while as discussedin Sec. II.A CP-divisibility corresponds to positivity atall times of the rates γi(t) appearing in (13). In order toquantify non-Markovianity as described by this criterion,different related measures have been introduced, relyingon an estimate of the violation of positivity of the Choimatrix (Rivas et al., 2010) or on various quantifiers ofthe negativity of the rates (Hall et al., 2014). In particu-lar, Rivas et al. (2010) suggest to consider the followingmeasure of non-Markovianity based on CP-divisibility

NRHP(Φ) =

∫ ∞0

dt g(t) (48)

with

g(t) = limε→0+

|| (Φt+ε,t ⊗ I) (|Ψ〉〈Ψ|) || − 1

ε, (49)

where |Ψ〉 is a maximally entangled state between opensystem and ancilla.

A further feature which can be considered in this con-text, and which has been proposed in Lorenzo et al.(2013) as another signature of non-Markovianity is thedynamical behavior of the volume of the accessible states,according to their parametrization in the Bloch repre-sentation, identifying non-Markovianity with the growthof this volume, which also allows for a geometrical de-scription. This study can be performed e.g. upon knowl-edge of the rates appearing in the time-local master equa-tion, and appear to be a strictly weaker requirement forMarkovianity with respect to P-divisibility (Chruscinskiand Wudarski, 2013).

To explain the relation between P-divisibility and CP-divisibility suppose that Alice prepares states ρ1

S or ρ2S on

an extended space HS⊗HS and sends the states throughthe channel Φt = Φt⊗I to Bob. The dynamical map Φt isthen Markovian, in the sense of P-divisibility, if and onlyif Φt is CP-divisible. Note however that this constructionrequires the use of a physically different system, describedby the tensor product space HS ⊗HS , which carries the

12

information transferred from Alice to Bob. More specifi-cally, the maximization over the initial states must allowfor entangled states ρ1,2

S which clearly explains the differ-ence between the definitions of non-Markovianity for Φtand Φt. This implies in particular that an experimentalassessment of non-Markovianity according to the crite-rion of CP-divisibility would in general imply the capabil-ity to prepare entangled states and perform tomographicmeasurements on the extended space. Alternatively oneshould be able to perform process tomography instead ofstate tomography, with an analogous scaling with respectto the dimensionality of the system.

2. Monotonicity of correlations and entropic quantities

Another line of thought about non-Markovianity of aquantum dynamics is based on the study of the behaviorin time of quantifiers of correlations between the opensystem of interest and an ancilla system. In this respectboth entanglement (Rivas et al., 2014) and quantum mu-tual information have been considered (Luo et al., 2012).In this setting one considers an ancilla system and stud-ies the behavior in time of the entanglement or the mu-tual information of an initially correlated joint system-ancilla state. Given the fact that both quantities arenon-increasing under the local action of a completely pos-itive trace preserving transformation, CP-divisibility ofthe time evolution would lead to a monotonic decreaseof these quantities. Their non monotonic behavior cantherefore be taken as a signature or a definition of quan-tum non-Markovianity. Actually, Rivas et al. (2014) haveconsidered CP-divisibility as the distinguishing featureof non-Markovianity, so that a revival of entanglementin the course of time is just interpreted as a witness ofnon-Markovianity. In (Luo et al., 2012) instead failureof monotonicity in the loss of quantum mutual informa-tion is proposed as a new definition of non-Markovianity.In both cases one considers as initial state a maximallyentangled state between system and ancilla. The quan-tification of the effect is obtained by summing up the in-creases over time of the considered quantity. Still otherapproaches have connected non-Markovianity with nonmonotonicity of the quantum Fisher information, consid-ered as a quantifier of the information flowing betweensystem and environment. In this case however the infor-mation flow is not directly traced back to a distance onthe space of states (Lu et al., 2010).

In the trace distance approach one studies the behav-ior of the statistical distinguishability between states,and relying on the introduction of the Helstrom matrix(Chruscinski et al., 2011) this notion can be directly re-lated to a divisibility property of the quantum dynamicalmap. The generalization of the trace distance approachalso allows for a stronger connection to a quantum infor-mation theory viewpoint. Indeed one can consider the

quantum dynamical maps as a collection of time depen-dent channels. In this respect the natural question ishow well Bob can recover information on the state pre-pared by Alice performing a measurement after a giventime t. A proposal related to this viewpoint has beenconsidered in (Bylicka et al., 2014), where two typesof capacity of a quantum channel, namely the so-calledentanglement-assisted classical capacity and the quan-tum capacity, have been considered. While the quantumdata processing inequality (Nielsen and Chuang, 2000)warrants monotonicity of these quantities provided thetime evolution is CP-divisible, lack of this monotonicityhas been taken as definition of non-Markovianity of thedynamics. The connection between non-Markovianityand the quantum data processing inequality has also beenrecently studied by Buscemi and Datta (2014).

III. MODELS AND APPLICATIONS OFNON-MARKOVIANITY

In this section we first address the description of simpleprototypical systems, for which one can exactly describethe non-Markovian features of the dynamics accordingto the concepts and measures introduced in Sec. II. Inparticular we will stress how a non-Markovian dynamicsleads to a recovery of quantum features according to thenotion of information backflow. In the second part wepresent applications of the introduced non-Markovianitymeasures to a number of more involved models illus-trating how memory effects allow to analyze their dy-namics and can be related to characteristic properties ofcomplex environments, thus suggesting to use the non-Markovianity of the open system dynamics as a quantumprobe for features of the environment.

A. Prototypical model systems

We consider first a two-level system undergoing puredecoherence. This situation can be described by a time-convolutionless master equation of the form (13) witha single decay channel and allows to demonstrate in asimple manner how the properties of the environmentand the system-environment coupling strength influencethe transition from a Markovian to a non-Markovian dy-namics for the open quantum system. A similar anal-ysis is then performed for a dynamics which also takesinto account dissipative effects, further pointing to thephenomenon of the failure of divisibility of the quantumdynamical map. We further discuss a dynamics drivenby different decoherence channels, which allows to bet-ter discriminate between different approaches to the de-scription of non-Markovian behavior. Finally, we presentsome results on the non-Markovian dynamics of the spin-boson model.

13

1. Pure decoherence model

Let us start considering a microscopic model for a puredecoherence dynamics which is amenable to an exact so-lution and describes a single qubit interacting with abosonic reservoir. The total Hamiltonian can be writ-ten as in Eq. (2) with the system Hamiltonian HS andthe environmental Hamiltonian HE given by

HS =1

2ω0σz, HE =

∑k

ωka†kak, (50)

where ω0 is the energy difference between ground state|0〉 and excited state |1〉 of the system, σz denotes a Pauli

matrix, ak and a†k are annihilation and creation opera-tors for the bosonic reservoir mode labelled by k withfrequency ωk, obeying the canonical commutation rela-tions [ak, a

†k′ ] = δkk′ . The interaction term is taken to

be

HI =∑k

σz

(gkak + g∗ka

†k

)(51)

with coupling constants gk. Considering a factorized ini-tial condition with the environment in a thermal state atinverse temperature β the model can be exactly solvedleading to a quantum dynamical map Φt which leavesthe populations invariant and modifies the off-diagonalmatrix elements according to

ρ11(t) = ρ11(0), ρ10(t) = G(t)ρ10(0), (52)

where ρij(t) = 〈i|ρS(t)|j〉 denote the elements of the in-teraction picture density matrix ρS(t). The function G(t)is often called decoherence function and in the presentcase it is real since we are working in the interaction pic-ture. It can be expressed in the form

G(t) = exp

[−∫ ∞

0

dω J(ω) coth

(βω

2

)1−cos (ωt)

ω2

], (53)

where we have introduced the spectral density J(ω) =∑k |gk|2δ(ω − ωk) which keeps track of the features of

the environment relevant for the reduced system descrip-tion, containing informations both on the environmentaldensity of the modes and on how strongly the systemcouples to each mode. It can be shown that the inter-action picture operator ρS(t) obeys a master equation ofthe form (13) with a single Lindblad operator,

d

dtρS(t) = γ(t) [σzρS(t)σz − ρS(t)] , (54)

where the time dependent decay rate γ(t) is related tothe decoherence function G(t) by

γ(t) = − 1

G(t)

d

dtG(t). (55)

Note that for a generic microscopic or phenomenologicaldecoherence model the decoherence function is in generala complex quantity even in the interaction picture, andin this case the modulus of the function should be consid-ered in formula (55). The map Φt is completely positivesince G(t) 6 1 which is equivalent to the positivity of the

time integral of the decay rate Γ(t) =∫ t

0dt′ γ(t′). In or-

der to discuss the non-Markovianity of the obtained timeevolution as discussed in Sec. II.C and Sec. II.D.1 weconsider the divisibility property of the time evolution.The master equation (54) has a single channel with decayrate γ(t) and thanks to the strict positivity of the deco-herence function Φ−1

t always exists. However, accordingto the criterion given by Eq. (14), P-divisibility fails attimes when the decoherence rate becomes negative, whichis exactly when CP-divisibility is lost. This is generallytrue when one has a master equation in the time-localform (13) with a single Lindblad operator, since in thiscase the condition for CP-divisibility, given by the re-quirement of positivity of the decay rate γ(t), coincideswith the condition (14) for P-divisibility.

The trace distance between time evolved states corre-sponding to the initial conditions ρ1,2

S is given accordingto (26) by

D(Φtρ

1S ,Φtρ

2S

)=√a2 +G2(t)|b|2, (56)

where a = ρ111−ρ2

11 and b = ρ110−ρ2

10 denote the differenceof the populations and of the coherences of the initialstates respectively. Its time derivative corresponding tothe quantity (35) in terms of which the non-Markovianitymeasure can be constructed then reads

σ(t) =G(t)|b|2√

a2 +G2(t)|b|2d

dtG(t). (57)

The non-monotonicity in the behavior of the trace dis-tance which is used as a criterion for non-Markovianityin Sec. II.C.2 is therefore determined by the non mono-tonic behavior of the decoherence function. In order toevaluate the measure Eq. (34) one has to maximize overall pairs of initial states. As shown by Wißmann et al.(2012) and discussed in Sec. II.C.2 the latter can be re-stricted to orthogonal state pairs. For the case at hand itfollows from Eq. (57) that the maximum is obtained forantipodal points on the equator of the Bloch sphere, sothat a = 0 and |b| = 1. The non-Markovianity measure(34) is then given by

N (Φ) =∑k

[G(tfk)−G

(tik)], (58)

where tik and tfk denote initial and final time point ofthe k-th interval in which G(t) increases. Since the in-crease of G(t) coincides with the negativity of the de-coherence rate according to Eq. (55), the growth of thetrace distance is in this case equivalent to breaking both

14

P-divisibility and CP-divisibility. Moreover, the gener-alized trace distance measure considered in Sec. II.C.3leads to the same expression (58) for non-Markovianity.

The measure for non-Markovianity of Eq. (48) basedon CP-divisibility takes in this case the simple expression

NRHP(Φ) = −2

∫γ<0

dt γ(t), (59)

which is the integral of the decoherence rate in the timeintervals in which it becomes negative. An example ofan experimental realization of a pure decoherence modelis considered in Sec. IV.C.1, where the decoherence func-tion is determined by the interaction between the polar-ization and the frequency degrees of freedom of a photon.In this case the decoherence function is, in general, com-plex valued, so that the previous formulae hold with G(t)replaced by |G(t)|. The geometric measure discussed inSec. II.D.1, which connects non-Markovian behavior withthe growth in time of the volume of accessible states,also leads to the same signature for non-Markovianity:The volume can again be identified with the decoherencefunction G(t) and therefore as soon as G(t) increases thevolume grows.

As we have discussed, in a simple decoherence modelnon-Markovianity can be identified with the revival ofcoherences of the open system, corresponding to a back-flow of information from the environment to the system.In this situation the time-local master equation (54) de-scribing the dynamics exhibits just a single channel, sothat all discussed criteria of non-Markovianity do coin-cide.

2. Two-level system in a dissipative environment

We now consider an example of dissipative dynamicsin which the system-environment interaction influencesboth coherences and populations of a two-state system.The environment is still taken to be a bosonic bath, sothat the free contributions to the Hamiltonian are againgiven by Eq. (50), but one considers an interaction termin rotating wave approximation given by

HI =∑k

(gkσ+ak + g∗kσ−a

†k

), (60)

where σ− = |0〉〈1| and σ+ = |1〉〈0| are the lowering andraising operators of the system. Such an interaction de-scribes, e.g., a two-level atom in a lossy cavity. For afactorized initial state with the environment in the vac-uum state, this model is again exactly solvable, thanksto the conservation of the number of excitations. Thequantum dynamical map Φt transforms populations andcoherences according to

ρ11(t) = |G(t)|2ρ11(0), ρ10(t) = G(t)ρ10(0), (61)

where the decoherence function is a complex function de-termined by the spectral density J(ω) of the model. Inparticular, denoting by f(t−t1) the two-point correlationfunction of the environment corresponding to the Fouriertransform of the spectral density, the function G(t) is de-termined as the solution of the integral equation

d

dtG(t) = −

∫dt1 f(t− t1)G(t1) (62)

with initial condition G(0) = 1. Also in this case it ispossible to write down the exact master equation obeyedby the density matrix ρS(t) of the system in the form ofEq. (13), which in the interaction picture reads (Breueret al., 1999)

d

dtρS(t) =− i

4S(t) [σz, ρS ] (63)

+ γ(t)

[σ−ρS(t)σ+ −

1

2σ+σ−, ρS(t)

],

where the time dependent Lamb shift is given by S(t) =−2=(G(t)/G(t)), and the decay rate can be written asγ(t) = −2<(G(t)/G(t)), or equivalently

γ(t) = − 2

|G(t)|d

dt|G(t)|. (64)

The analysis of non-Markovianity in the present modelclosely follows the discussion in III.A.1 due to the cru-cial fact that the master equation (63) still has a singleLindblad operator. The expression of the trace distanceis determined by the decoherence function and, with thesame notation as in (56), reads

D(Φtρ

1S ,Φtρ

2S

)= |G(t)|

√|G(t)|2a2 + |b|2, (65)

so that one has the time derivative

σ(t) =2|G(t)|2a2 + |b|2√|G(t)|2a2 + |b|2

d

dt|G(t)|. (66)

Complete positivity of the map is ensured by positivity ofthe integrated decay rate Γ(t), while the trace distanceshows a non monotonic behavior when the modulus ofthe decoherence function grows with time.

A typical expression for the spectral density is givenby a Lorentzian

J(ω) = γ0λ2/2π

[(ω0 + ∆− ω)2 + λ2

](67)

of width λ and centered at a frequency detuned from theatomic frequency ω0 by an amount ∆, while the rate γ0

quantifies the strength of the system-environment cou-pling. Let us first discuss the case when the qubit is inresonance with the central frequency of the spectral den-sity, so that one has a vanishing detuning, ∆ = 0. Thedecoherence function then takes the form

G(t) = e−λt/2[cosh

(dt

2

)+λ

dsinh

(dt

2

)], (68)

15

where d =√λ2 − 2γ0λ. For weak couplings, correspond-

ing to γ0 < λ/2, the decoherence function G(t) is a real,monotonically decreasing function so that (66) is alwaysnegative and the dynamics is Markovian. In this case, asdiscussed in Sec. II.A.2, the map is invertible and sincethe time-convolutionless form of the master equation hasjust a single channel, it is at the same time P-divisible andCP-divisible. Note that in the limit γ0 λ/2 the rateγ(t) becomes time-independent, γ = γ0, such that themaster equation (63) is of Lindblad form and leads to aMarkovian semigroup. However, for larger values of thecoupling between system and environment, namely forγ0 > λ/2, the function |G(t)| displays an oscillatory be-havior leading to a non-monotonic time evolution of thetrace distance and, hence, to non-Markovian dynamics.Thus, the trace distance measure of Eq. (34) is zero forγ0 < λ/2 and starts at the threshold γ0 = λ/2 to increasecontinuously to positive values (Laine et al., 2010a). It isinteresting to note that the transition point γ0 = λ/2 ex-actly coincides with the point where the perturbation ex-pansion of the time-convolutionless master equation (63)breaks down. We also remark that for this model opti-mal state pairs still correspond to antipodal points of theequator of the Bloch sphere (Xu et al., 2010), and thatthe criterion for non-Markovianity based on the Helstrommatrix considered in Sec. II.C.3 leads to the same results.

In the parameter regime γ0 > λ/2 the map Φt definedby (61) is no longer invertible for all times due to theexistence of zeros of the decoherence function. Thus, themodel provides an example for a family of maps lying inthe lower (red) region depicted in Fig. 4. Physically, thetwo-level system reaches the ground state before memoryeffects revive the coherences and the population of theupper state. Indeed, also in this case non-Markovianitycan be traced back to the revival of populations andcoherences, corresponding to a backflow of informationfrom the environment to the system. Since the inverseof Φt does not exist for all times, the criterion based onCP-divisibility cannot strictly speaking be applied. Thisis reflected by the fact that the corresponding measure,taking in this case again the expression (59), jumps fromzero to infinity.

In the non-resonant case, i.e., for nonzero detuning,one can observe a transition from Markovian to non-Markovian dynamics for a fixed value of the couplingin the weak coupling regime by increasing the detun-ing ∆. In this case the map is invertible for all times,since the decay rate γ(t) is finite for all times, so thatalso the measure (48) remains well defined. Again, theregions of non-Markovianity coincide for all approachessince the constraint in order to have CP-divisibility orP-divisibility is the same. The pair of states maximiz-ing the expression of the measure (34) are however nowgiven by the projections onto the excited and the groundstate. This shows how the optimal pair, which is alwaysgiven by orthogonal states, does depend on the actual

expression for the decoherence function. For this casethe derivative of the trace distance (66) takes the simple

form σ(t) = −γ(t) exp[−Γ(t)] with Γ(t) =∫ t

0γ(t′) dt′,

which puts into evidence the direct connection betweenthe direction of information flow and the sign of the decayrate.

3. Single qubit with multiple decoherence channels

In the preceding examples we have shown that for amaster equation in the time-convolutionless form (13)with a single Lindblad operator A(t), all the differentconsidered criteria for non-Markovianity coincide. It istherefore of interest to discuss an example in which onehas multiple decoherence channels with generally differ-ent rates. As we shall shortly see, such model allowsto discriminate among the different definitions of non-Markovian dynamics. To this end, we take a phenomeno-logical approach and following Chruscinski and Manis-calco (2014); Chruscinski and Wudarski (2013); and Vac-chini (2012) we consider for a two-level system the masterequation

d

dtρS(t) =

1

2

3∑i=1

γi(t) [σiρS(t)σi − ρS(t)] , (69)

where σi with i = x, y, z denote the three Pauli opera-tors. Considering, e.g., the decoherence dynamics of aspin- 1

2 particle in a complex environment the rates γi(t)would correspond to generally time dependent transver-sal and longitudinal decoherence rates. The dynamicalmap corresponding to (69) can be exactly worked out andis given by the random unitary dynamics

Φt(ρS) =

3∑i=0

pi(t)σiρSσi , (70)

where σ0 denotes the identity operator and the coeffi-cients pi(t) summing up to one are determined from thedecoherence rates. Introducing the quantities Aij(t) =exp[−(Γi(t)+Γj(t))], where as usual we have denoted byΓk(t) the time integral of the k-th decay rate, the coeffi-cients pi(t) take the explicit form p0,1 = (1/4)[1±A12(t)±A13(t) + A23(t)] and p2,3 = (1/4)[1 ∓ A12(t) ± A13(t) −A23(t)]. These equations show that the dynamical mapactually depends on the sum of the integrals of the de-cay rates. According to its explicit expression the mapΦt is completely positive provided the coefficients pi(t)remain positive, in which case they can be interpreted asa probability distribution and thus characterize the ran-dom unitary dynamics (70). Note that this can be thecase even if a decoherence rate stays negative at all timesas in Hall et al. (2014) and Vacchini et al. (2011).

As explained in Sec. II.A.2, the map is CP-divisiblewhen all of the decoherence rates remain positive for all

16

t > 0, so that as soon as at least one of the rates becomesnegative the measure (48) becomes positive indicating anon-Markovian behavior. However, the condition (14)for having P-divisibility is weaker. Indeed, in order tohave P-divisible dynamics only the sum of all pairs ofdistinct decoherence rates has to remain positive, i.e.,γi(t) + γj(t) > 0 for all j 6= i. The same constraint war-rants monotonicity in time of the behavior of the tracedistance, so that also in this case the measure (34) of non-Markovianity and its generalized version (39) based onthe Helstrom matrix and corresponding to P-divisibilitylead to the same result. In the characterization of non-Markovianity based on the backflow of information fromthe environment to the system memory effects thereforeonly appear whenever the sum of at least a pair of deco-herence rates becomes negative. To better understand,how this fact can be related to the dynamics of the sys-tem, one can notice (Chruscinski and Maniscalco, 2014)that the Bloch vector components 〈σi(t)〉, according toEq. (69), obey the equation

d

dt〈σi(t)〉 = − 1

Ti(t)〈σi(t)〉, (71)

where the relaxation times are given by Ti(t) = [γj(t) +γk(t)]−1 (with all three indices taken to be distinct) andcorrespond to experimentally measurable quantities. Ap-pearance of non-Markovianity and therefore failure of P-divisibility is thus connected to negative relaxation rates,corresponding to a rebuild of quantum coherences. Forthe present model one can also easily express the con-dition leading to a growth of the volume of accessiblestates within the Bloch sphere. In fact, this volume ofaccessible states is proportional to exp[−∑3

i=1 Γi(t)]. Asa consequence, the dynamics according to the geometriccriterion by Lorenzo et al. (2013) is Markovian if and only

if∑3i=1 γi(t) > 0 for t > 0, which is a strictly weaker cri-

terion with respect to either P or CP-divisibility.While in the examples considered above the non-

Markovianity measure (34) based on the trace distanceand its generalization (39) based on the Helstrom ma-trix agree in the indication of non-Markovian dynamics,a situation in which these criteria are actually differenthas been considered by Chruscinski et al. (2011).

4. Spin-boson model

Finally, we briefly discuss a further paradigmaticmodel for dissipative quantum dynamics with many ap-plications, namely the spin-boson model (Leggett et al.,1987). The system and the environmental Hamiltonianare again given by Eq. (50), while the interaction Hamil-tonian has the non-rotating-wave structure

HI =∑k

σx

(gkak + g∗ka

†k

). (72)

1 2 3 4 5 6 7 8 9 10

1

2

3

4

5

6

7

8

9

10

0.20.2

Ω/ω0

T/ω0

N (Φ)

< 10−6

10−5

10−4

10−3

10−2

10−1

> 1

FIG. 5 (Color online) The non-Markovianity measure N (Φ)defined by Eq. (34) for the spin-boson model as a function oftemperature T and cutoff frequency Ω in units of the transi-tion frequency ω0 (Clos and Breuer, 2012).

Figure 5 shows the non-Markovianity measure N (Φ) forthis model obtained from numerical simulations of thecorresponding second-order time-convolutionless masterequation, using an Ohmic spectral density of Lorentz-Drude shape with cutoff frequency Ω, reservoir tem-perature T and a fixed small coupling strength of sizeγ = 0.1ω0 (Clos and Breuer, 2012). As can be seen fromthe figure the dynamics is strongly non-Markovian bothfor small cutoff frequencies and for small temperatures(note the logarithmic scale of the color bar). The emer-gence of a Markovian region within the non-Markovianregime for small cutoffs and temperatures can be under-stood in terms of a resonance between the transition fre-quency and the maximum of an effective, temperature-dependent spectral density of the environmental modes.

B. Applications of quantum non-Markovianity

In Sec. III.A we have illustrated quantum non-Markovian behavior by means of simple model systems.The goal of the present section is to demonstrate thatmemory effects and their control also open new perspec-tives for applications. In fact, understanding various as-pects of non-Markovianity makes it possible to developmore refined tools for reservoir engineering and to usememory effects as an indicator for the presence of typi-cal quantum features. Moreover, it becomes possible todevelop schemes where a small open system is used asa quantum probe detecting characteristic properties ofthe complex environment it is interacting with. Here, wepresent a few examples from the recent literature illus-trating these points.

17

1. Analysis and control of non-Markovian dynamics

We first consider the model of a spin- 12 particle with

spin operator s0 coupled to a chain of N spin- 12 parti-

cles with spin operators sn (n = 1, 2, . . . , N), which hasbeen investigated by Apollaro et al. (2011). The systemHamiltonian is given by HS = −2h0s

z0, where h0 denotes

a local field acting on the system spin s0. The environ-mental Hamiltonian represents an XX-Heisenberg spinchain with nearest neighbor interactions of strength J ina transverse magnetic field h:

HE = −2J

N−1∑n=1

(sxns

xn+1 + syns

yn+1

)− 2h

N∑n=1

szn. (73)

The interaction Hamiltonian takes the form

HI = −2J0 (sx0sx1 + sy0s

y1) , (74)

describing an energy-exchange interaction between thesystem spin s0 and the first spin s1 of the chain withcoupling strength J0. This model leads to a rich dynami-cal behavior of the system spin and allows to demonstratenot only the impact of the system-environment interac-tion, but also how the open system dynamics changeswhen manipulating the interactions between the con-stituents of the environment and their local properties.

It can be shown analytically that optimal state pairsfor the trace distance measure correspond to antipodalpoints on the equator of the surface of the Bloch sphere.Moreover, when the local fields of the system and envi-ronmental sites are equal, h = h0, one can derive an an-alytic expression for the rate of change σ(t) of the tracedistance [see Eq. (35)] for optimal state pairs:

σ(t) = −(2/t) sgn[J1(2t)]J2(2t), (75)

where sgn is the sign function, and J1 and J2 denotethe Bessel functions of order 1 and 2, respectively. In thephysically more interesting case h 6= h0 the trace distancemeasureN (Φ) [see Eq. (34)] has to be determined numer-ically. Let us first consider the case, where the systemlocal field is zero, h0 = 0, the couplings between the spinsare equal, J = J0, and we tune the ratio h/J of the localfield with respect to the spin-spin interaction strength. Itturns out that the open system is then Markovian onlyat the point h/J = 1/2, i.e., when the strength of thelocal field is half of the coupling between the spins. Forall other values of h/J the open system dynamics dis-plays memory effects with N (Φ) > 0. Thus, the point ofMarkovianity separates two regions where memory effectsare present. This behavior persists for h0 6= 0: Denotingthe detuning between the local fields by δ = h − h0 andstill keeping J = J0, the point of Markovianity occurs atδh/J = 1/2. On both sides of this point the system againexhibits memory effects, but interestingly their proper-ties are different. For δh/J < 1/2 the system approaches

a steady state which is independent of the initial state.On the other hand, for δh/J > 1/2 the trace distancedoes not decay to zero asymptotically in time. This fea-ture can be interpreted as information trapping since itimplies that the distinguishability of states does not van-ish asymptotically. We also remark that the origin ofmemory effects and the differences in the dynamical be-havior can be understood in more detail by studying thespectrum of the total Hamiltonian (Apollaro et al., 2011).

As our second example we discuss applications to aphysical system which has been studied extensively dur-ing the last twenty years, namely Bose-Einstein conden-sates (BECs) which can be manipulated with high preci-sion by current technologies and thus provide means forenvironment engineering (Dalfovo et al., 1999; Pethickand Smith, 2008; Pitaevskii and Stringari, 2003). Fol-lowing Haikka et al. (2011) we consider an impurity-BECsystem, where the Markovian to non-Markovian transi-tion can be controlled by tuning the properties of theBEC. An impurity atom is trapped within a double-wellpotential, where the left (|L〉) and right (|R〉) states rep-resent the two qubit states. The BEC, which constitutesthe environment of the qubit, is trapped in a harmonicpotential. The Hamiltonian for the total impurity-BECsystem reads

H =∑k

Ekc†kck+

∑k

(ξkc†k+ξ∗kck)+

∑k

σz(gkc†k+g∗kck).

(76)Here, Ek denotes the energy of the Bogoliubov mode ckof the condensate, σz = |R〉〈R| − |L〉〈L|, while gk and ξkdescribe the impurity-BEC and the intra-BEC couplings,respectively.

For a background BEC at zero temperature the openqubit dynamics is described by a dephasing master equa-tion of the form of Eq. (54) and it turns out that thedecoherence rate γ(t) can be tuned by changing theinter-well distance, and by controlling the dimensional-ity and the interaction strength (scattering length) ofthe BEC. The detailed studies demonstrate that for a3D gas, an increase of the well-separation and of theintra-environment interaction leads to an increase of thenon-Markovianity measure for the impurity dynamics.Moreover, the dimensionality of the environment influ-ences the emergence of memory effects. Studying theMarkovian to non-Markovian transition as a function ofthe intra-environment interaction strength shows that a3D environment is the most sensitive. The Markovian– non-Markovian crossover occurs in 3D for weaker val-ues of the scattering length than in 1D and 2D environ-ments, and a 1D environment requires the largest scat-tering length for the crossover. This behavior can beultimately traced back to the question of how the di-mensionality of the environment influences the spectraldensity which governs the open system dynamics. In gen-eral, for small frequencies the spectral density shows a

18

power law behavior, J(ω) ∝ ωs. The spectral density iscalled sub-Ohmic, Ohmic, and super-Ohmic for s < 1,s = 1, and s > 1, respectively. For a 3D gas, even with-out interactions within the gas, the spectral density hassuper-Ohmic character. Introducing interactions withinthe environment then makes this character stronger andthereby the 3D gas is most sensitive to the Markovianto non-Markovian transition. In contrast, for the 1D gaswith increasing scattering length the spectral density firstchanges from sub-Ohmic to Ohmic and finally to super-Ohmic which then allows the appearance of memory ef-fects.

2. Open systems as non-Markovian quantum probes

In addition to detecting, quantifying and controllingmemory effects, it turns out to be fruitful to ask whetherthe presence or absence of such effects in the dynamicsof an open quantum system allows to obtain importantinformation about characteristic features of a complexenvironment the open system is interacting with. To il-lustrate this point we present a physical scenario, whereMarkovian behavior indicates the presence of a quan-tum phase transition in a spin environment (Sachdev,2011). Consider a system qubit, with states |g〉 and |e〉,which interacts with an environment described by a one-dimensional Ising model in a transverse field (Quan et al.,2006). The Hamiltonians of the environment and thequbit-environment interaction are given by

HE = −J∑j

(σzjσ

zj+1 + λσxj

), (77)

HI = −Jδ|e〉〈e|∑j

σxj , (78)

where J is a microscopic energy scale, λ a dimensionlesscoupling constant describing the strength of the trans-verse field, and σx,zj are Pauli spin operators. The sys-tem qubit is coupled with strength δ to all environmentalspins. The model yields pure dephasing dynamics de-scribed by a decoherence function G(t) which, for a pureenvironmental initial state |Φ〉, is given by

G(t) = 〈Φ|eiHgte−iHet|Φ〉, (79)

where the effective environmental Hamiltonians areHα = HE + 〈α|HI |α〉 with α = g, e. We note that ac-cording to Eq. (79) the decoherence function is directlylinked to the concept of the Loschmidt echo L(t) (Cuc-chietti et al., 2003) which characterizes how the environ-ment responds to perturbations by the system (Cucchi-etti et al., 2004; Gorin et al., 2006; Goussev et al., 2008;Peres, 1984). In fact, we have L(t) = |G(t)|2 and, hence,there is also a direct connection between the Loschmidtecho and the evolution of the trace distance for optimalstate pairs, i.e. D(ρ1

S(t), ρ2S(t)) =

√L(t).

N ()

FIG. 6 (Color online) The non-Markovianity measure N (Φ)of Eq. (34) for a qubit coupled to a one-dimensional Isingchain in a transverse field as a function of field strength λ∗

and particle number N . The measure vanishes along the blackline and is nonzero everywhere else (Haikka et al., 2012).

The system-environment interaction (78) leads to theeffective transverse field λ∗ = λ + δ when the system isin the state |e〉. At the critical point of the spin environ-ment (λ∗ = 1) the spin-spin interaction and the externalfield are of the same size and the environment is mostsensitive to external perturbations. This is reflected in aquite remarkable manner in the dynamics of the systemqubit as can be seen from Fig. 6 (Haikka et al., 2012).At the point λ∗ = 1 the system qubit experiences strongdecoherence. As a matter of fact, it turns out that the dy-namics of the system qubit always shows non-Markovianbehavior except at the critical point of the environmentwhere the dynamics is Markovian. Therefore, the sys-tem qubit acts as a probe for the spin environment andMarkovian dynamics represents a reliable indicator of en-vironment criticality. It is important to note that eventhough, strictly speaking, quantum phase transitions re-quire to take the thermodynamic limit, the above resultshold for any number of environmental spins.

IV. IMPACT OF CORRELATIONS AND EXPERIMENTALREALIZATIONS

In this section we study the effect of two fundamentalkinds of correlations on the non-Markovian behavior ofopen systems, namely correlations between the open sys-tem and its environment and correlations within a com-posite environment. We conclude the section by givingan overview of the experimental status of non-Markovianopen systems.

19

A. Initial system-environment correlations

1. Initial correlations and dynamical maps

Up to this point we have discussed the theory of openquantum systems via the concept of a dynamical map,which in Sec. II.A.1 we wrote down in terms of the initialstate of the environment and of the total Hamiltonian.In doing so we assumed that the open system and theenvironment are independent at the initial time of prepa-ration [see Eqs. (6) and (7)]. However, experimentalistsoften study fast processes in strongly coupled systems,where this independence is rarely the case, as originallypointed out by Pechukas (1994). Going beyond this ap-proximation brings us up against the problem of describ-ing the dynamical properties of open quantum systemswithout dynamical maps.

The theory of completely positive maps plays a crucialrole in many fields of quantum physics and thus a gen-eral map description for open systems, even in the pres-ence of initial correlations, would be very useful. Thus,naturally, a lot of effort has been put in trying to ex-tend the formalism along these lines (Alicki, 1995; Sha-bani and Lidar, 2009; Stelmachovic and Buzek, 2001).In this framework the aim has been to determine un-der which conditions the system dynamics can be de-scribed by means of completely positive maps if initialcorrelations are present. Unfortunately, no generally ac-knowledged answer to this question has been found andthe topic is still under discussion (Brodutch et al., 2013;Rodrıguez-Rosario et al., 2010; Shabani and Lidar, 2009).

As useful as a description in terms of completely pos-itive maps would be, one may as well ask the question:How do the initial correlations influence the dynamics?Especially, since an experimentalist is often restricted tolooking only at a subset of system states, he would wantto conclude something about the system based on thedynamics of this subset. It is not possible to construct amap, but perhaps something else could be concluded. Inthe following we present an approach, in which the dy-namics of information flow between the system and theenvironment is studied. We will see that the initial cor-relations modify the information flow and consequentlycan be witnessed from the dynamical features of the opensystem.

Let us have a closer look at the dynamics of the infor-mation inside the open system Iint(t), defined in Eq. (30),for a general open system, where initial correlations couldbe present. Suppose there are two possible preparationsgiving rise to distinct system-environment initial statesρ1SE(0) and ρ2

SE(0). Since the total system and the envi-ronment evolve under unitary dynamics, we have

Iint(t)− Iint(0) = Iext(0)− Iext(t) 6 Iext(0), (80)

where Iext(t) is the information outside the open systemdefined in Eq. (31). This inequality, as simple as it is,

reveals one very important point: The information in theopen system Iint(t) can increase above its initial valueIint(0) only if there is some information initially outsidethe system, i.e. Iext(0) > 0. Obviously, the contractionproperty (28) for completely positive maps is a specialcase of the inequality (80) which occurs when the systemand the environment are initially uncorrelated and theenvironmental state is not influenced by the system statepreparation, i.e. ρ1,2

SE = ρ1,2S ⊗ρE . This leads to Iext(0) =

0 and, thus, Eq. (80) reduces to Eq. (29).

2. Local detection of initial correlations

We found that initial information outside the open sys-tem can lead to an increase of trace distance. But howcan this be used to develop experimental methods to de-tect correlations in some unknown initial state ρ1

SE? Tothis end, let us combine (80) and the inequality (33) forthe initial time in order to reveal the role of initial cor-relations more explicitly (Laine et al., 2010b):

Iint(t)− Iint(0) 6 D(ρ1SE(0), ρ1

S(0)⊗ ρ1E(0))

+D(ρ2SE(0), ρ2

S(0)⊗ ρ2E(0))

+D(ρ1E(0), ρ2

E(0)). (81)

This inequality clearly shows that an increase of the tracedistance of the reduced states implies that there are ini-tial correlations in ρ1

SE(0) or ρ2SE(0), or that the initial

environmental states are different.

D(1S(t), 2

S(t))

tinitial!

correlations

no sign of!initial!

correlations

FIG. 7 (Color online) Schematic picture of the behavior ofthe trace distance with and without initial correlations. Ifthe trace distance between two states, prepared such thatρ2SE(0) = (Λ⊗I)ρ1SE(0), never exceeds its initial value (dashedblack line), the dynamics does not witness initial correlations.If, on the other hand, the trace distance for such pair of statesincreases above the initial value (solid red line), we can con-clude the presence of initial correlations.

Let us now assume that one can perform a state to-mography on the open system at the initial time zeroand at some later time t, to determine the reduced statesρ1S(0) and ρ1

S(t). In order to apply inequality (81) todetect initial correlations we need a second reference

20

state ρ2S(0), which has the same environmental state, i.e.

ρ2E(0) = ρ1

E(0). This can be achieved by performing alocal quantum operation on ρ1

SE(0) to obtain the state

ρ2SE(0) = (Λ⊗ I)ρ1

SE(0). (82)

The operation Λ acts locally on the variables of the opensystem, and may be realized, for instance, by the mea-surement of an observable of the open system, or by aunitary transformation induced, e.g., through an exter-nal control field. Now, if ρ1

SE(0) is uncorrelated then alsoρ2SE(0) is uncorrelated since it has been obtained through

a local operation. Therefore, for an initially uncorre-lated state the trace distance cannot increase accordingto inequality (81). Thus, any such increase represents awitness for correlations in the initial state ρ1

SE(0), as isillustrated in Fig. 7. We note that this method for thelocal detection of initial correlations requires only localcontrol and measurements of the open quantum system,which makes it feasible experimentally and thus attrac-tive for applications. In fact, experimental realizations ofthe scheme have been reported recently, see Sec. IV.C.2.Furthermore, Smirne et al. (2010) have studied how thescheme can be employed for detecting correlations inthermal equilibrium states.

I= 6=

2SE(0)

1SE(0)

2S(t)2

S(0)

1S(0) 1

S(t)trE U(t)

trE U(t)

trE

trE

FIG. 8 (Color online) Scheme for the local detection ofsystem-environment quantum correlations in some stateρ1SE(0), employing a local dephasing operation Λ to gener-ate a second reference state ρ2SE(0).

The above strategy can even be used in order to lo-cally detect a specific type of quantum correlations, i.e.system-environment states with nonzero quantum dis-cord (Modi et al., 2012). This is achieved by taking thelocal operation Λ in Eq. (82) to be a dephasing operationleading to complete decoherence in the eigenbasis of thestate ρ1

S(0) (Gessner and Breuer, 2011, 2013). The ap-plication of this dephasing operation destroys all quan-tum correlations of ρ1

SE(0), while leaving invariant itsmarginal states ρ1

S(0) and ρ1E(0) (see Fig. 8). Thus, one

can use the quantity C(ρ1SE(0)) = D(ρ1

SE(0), ρ2SE(0)) as a

measure for the quantum correlations in the initial stateρ1SE(0) (Luo, 2008). Since the trace distance is invariant

under unitary transformations and since the partial traceis a positive map, the corresponding local trace distanceD(ρ1

S(t), ρ2S(t)) provides a lower bound of C(ρ1

SE(0)) forall times t > 0. Hence, we have

maxt>0

D(ρ1S(t), ρ2

S(t)) 6 C(ρ1SE(0)). (83)

The left-hand side of this inequality yields a locally acces-sible lower bound for a measure of the quantum discord.Experimental realizations of the scheme are briefly de-scribed in Sec. IV.C.2. Most recent studies suggest thatthe method could also be applied for detecting a criti-cal point of a quantum phase transition via dynamicalmonitoring of a single spin alone (Gessner et al., 2014a).

B. Nonlocal memory effects

We now return to study the case of initially uncor-related system-environment states, for which the usualdescription in terms of dynamical maps can be used. Sofar, we have concentrated on examples with a single sys-tem embedded in an environment. However, many rel-evant examples in quantum information processing in-clude multipartite systems for which nonlocal proper-ties such as entanglement become important and it istherefore essential to study how non-Markovianity is in-fluenced by scaling up the number of particles.

The question of additivity of memory effects with re-spect to particle number was recently raised by Addiset al. (2013, 2014); and Fanchini et al. (2013) and it wasfound that the different measures for non-Markovianityhave very distinct additivity properties. Indeed, the re-search on multipartite open quantum systems is yet in itsinfancy and a conclusive analysis remains undone. In thefollowing, we will discuss a particular feature of bipartiteopen systems: the appearance of global memory effectsin the absence of local non-Markovian dynamics. Thisis at variance with the standard situation in which theenlargement of considered degrees of freedom leads froma non-Markovian to a Markovian dynamics (Martinazzoet al., 2011).

We will now take a closer look at nonlocal maps, forwhich memory effects may occur even in the absence oflocal non-Markovian effects. In Laine et al. (2012, 2013)it is shown that such maps may be generated from alocal interaction, when correlations between the environ-ments are present and that they may exhibit dynamicswith strong global memory effects although the local dy-namics is Markovian. The nonlocal memory effects arestudied in two dephasing models: in a generic model ofqubits interacting with correlated multimode fields (Wiß-mann and Breuer, 2013; Wißmann and Breuer, 2014) andin an experimentally realizable model of down convertedphotons traveling through quartz plates, which we willdiscuss later in detail.

Before we turn to discuss any experimental endeavor todetect non-Markovian dynamics, let us discuss in morerigor about the generation and possible applications ofnonlocal memory effects. Consider a generic scenario,where there are two systems, labeled with indices i = 1, 2,which interact locally with their respective environments.The dynamics of the two systems can be described via

21

the dynamical map

ρ12S (t) = Φ12

t ρ12S (0)

= trE

[U12SE(t)ρ12

S (0)⊗ ρ12E (0)U12†

SE (t)], (84)

where U12SE(t) = U1

SE(t)⊗U2SE(t) with U iSE(t) describing

the local interaction between the system i and its environ-ment. If the initial environment state ρ12

E (0) factorizes,i.e. ρ12

E (0) = ρ1E(0)⊗ ρ2

E(0), also the map Φ12t factorizes.

Thus, the dynamics of the two systems is given by a lo-cal map Φ12

t = Φ1t ⊗ Φ2

t . On the other hand, if ρ12E (0)

exhibits correlations, the map Φ12t cannot, in general, be

factorized. Consequently, the environmental correlationsmay give rise to a nonlocal process even though the in-teraction Hamiltonian is purely local (see Fig. 9).

open system 1

1t Markovian

open system

2t Markovian

2

open system 1+2

correlationsenvironment!1 environment!2

12t non-Markovian

interaction H1I interaction H2

I

FIG. 9 (Color online) Schematic picture of a system withnonlocal memory effects. The open systems 1 and 2 locallyinteract with their respective environments. Initial correla-tions between the local environments cause the occurrence ofnonlocal memory effects.

For a local dynamical process, all the dynamical prop-erties of the subsystems are inherited by the global sys-tem, but naturally for a nonlocal process the global dy-namics can display characteristics absent in the dynamicsof the local constituents. Especially, for a nonlocal pro-cess, even if the subsystems undergo a Markovian evo-lution, the global dynamics can nevertheless be highlynon-Markovian as was shown for the dephasing modelsby Laine et al. (2012, 2013).

Thus, a system can globally recover its earlier lostquantum properties although the constituent parts areundergoing decoherence and this way initial environmen-tal correlations can diminish the otherwise destructiveeffects of decoherence. This feature has been furtherdeployed in noisy quantum information protocols, suchas teleportation (Laine et al., 2014) and entanglementdistribution (Xiang et al., 2014), suggesting that non-Markovianity could be a resource for quantum informa-tion.

C. Experiments on non-Markovianity and correlations

Up to this point we have not yet discussed any ex-perimental aspects of the detection of memory effects.In the framework of open quantum systems the environ-ment is in general composed of many degrees of freedomand is therefore difficult to access or control. To performexperiments on systems where the environment induceddynamical features can be controlled is thus challenging.However, in the past years clever schemes for modifyingthe environment have been developed allowing the estab-lishment of robust designs for noise engineering. In thissection we will briefly review some of the experimentalplatforms where a high level of control over the envi-ronment degrees of freedom has been accomplished andnon-Markovian dynamics observed and quantified.

1. Control and quantification of memory effects in photonicsystems

Quantum optical experiments have for many decadesbeen the bedrock for testing fundamental paradigms ofquantum mechanics. The appeal for using photonic sys-tems arises from the extremely high level of control allow-ing, for example, controlled interactions between differentdegrees of freedom, preparation of arbitrary polarizationstates and a full state tomography. Needless to say, pho-tons thus offer an attractive experimental platform alsofor studying non-Markovian effects.

Liu et al. (2011) introduced an all-optical experimentwhich allows through careful manipulation of the initialenvironmental states to drive the open system dynam-ics from the Markovian to the non-Markovian regime,to control the information flow between the system andthe environment, and to determine the degree of non-Markovianity. In the experiment the photon polariza-tion degree of freedom (with basis states |H〉 and |V 〉 forhorizontal and vertical polarization, respectively) playsthe role of the open system. The environment is repre-sented by the frequency degree of freedom of the pho-ton (with basis |ω〉ω>0), which is coupled to the sys-tem via an interaction induced by a birefringent mate-rial (quartz plate). The interaction between the polar-ization and frequency degrees of freedom in the quartzplate of thickness L is described by the unitary operatorU(t) |λ〉 ⊗ |ω〉 = einλωt |λ〉 ⊗ |ω〉, where nλ is the refrac-tion index for a photon with polarization λ = H,V andt = L/c is the interaction time with the speed of light c.The photon is initially prepared in the state |ψ〉 ⊗ |χ〉,with |ψ〉 = α |H〉+ β |V 〉 and |χ〉 =

∫dωf(ω) |ω〉, where

f(ω) gives the amplitude for the photon to be in amode with frequency ω. The quartz plate leads to apure decoherence dynamics of superpositions of polariza-tion states (see Sec. III.A.1) described by the complexdecoherence function G(t) =

∫dω|f(ω)|2eiω∆nt, where

22

∆n = nV − nH . An optimal pair of states maximizingthe non-Markovianity measure (34) for this map is givenby |ψ1,2〉 = 1√

2(|H〉 ± |V 〉) and the corresponding trace

distance is D(ρ1S(t), ρ2

S(t)) = |G(t)|. Clearly, controllingthe frequency spectrum |f(ω)|2 changes the dynamicalfeatures of the map.

a)

b)

FIG. 10 (Color online) Non-Markovian dynamics arising froman engineered frequency spectrum. (a) The frequency spec-trum of the initial state for various values of the tilting angle θof the cavity. (b) The change of the trace distance as functionsof the tilting angle θ. The transition from the non-Markovianto the Markovian regime occurs at θ ≈ 4.1, and from theMarkovian to the non-Markovian regime at θ ≈ 8.0, corre-sponding to the occurrence of a double peak structure in thefrequency spectrum (a) (Liu et al., 2011).

The modification of the photon frequency spectrum isrealized by means of a Fabry-Perot cavity mounted on arotator which can be tilted in the horizontal plane. Theshape of the frequency spectrum is changed by chang-ing the tilting angle θ of the Fabry-Perot cavity, as canbe seen in Fig. 10a. Further, the frequency spectrumchanges the dephasing dynamics such that the open sys-tem exhibits a reversed flow of information and thusallows to tune the dynamics from Markovian to non-Markovian regime (see Fig. 10b). In the experiment, full

state tomography can further be performed thus allowinga rigorous quantification of the memory effects.

Also, a series of other experimental studies on non-Markovian dynamics in photonic systems have been per-formed recently. Tang et al. (2012) report a measure-ment of the non-Markovianity of a process with tunablesystem- environment interaction, Cialdi et al. (2011) ob-serve controllable entanglement oscillations in an effec-tive non-Markovian channel and in (Chiuri et al., 2012;Jin et al., 2014) simulation platforms for a wide class ofnon-Markovian channels are presented.

2. Experiments on the local detection of correlations

A number of experiments have been carried out in or-der to demonstrate the local scheme for the detectionof correlations between an open system and its environ-ment described in Sec. IV.A.2. Photonic realizations ofthis scheme have been reported in (Li et al., 2011; Smirneet al., 2011), where the presence of initial correlations be-tween the polarization and the spatial degrees of freedomof photons is shown by the observation of an increase ofthe local trace distance between a pair of initial state.

As explained in Sec. IV.A.2, it is also possible to reveallocally the presence of quantum correlations representedby system-environment states with nonzero discord if thelocal operation Λ in Eq. (82) is taken to induce completedecoherence in the eigenbasis of the open system state.The first photonic realizations of this strategy based onEq. (83) is described in (Tang et al., 2014). Moreover,Cialdi et al. (2014) have extended the method to enablethe discrimination between quantum and classical corre-lations, and applied this extension to a photonic experi-mental realization.

All experiments mentioned so far employ photonic de-grees of freedom to demonstrate non-Markovianity andsystem-environment correlations. The first experimentshowing these phenomena for matter degrees of freedomhas been described by Gessner et al. (2014b). In this ex-periment nonclassical correlations between the internalelectronic degrees of freedom and the external motionaldegrees of freedom of a trapped ion have been observedand quantified by use of Eq. (83). Important features ofthe experiment are that the lower bounds obtained fromthe experimental data are remarkably close to the truequantum correlations present in the initial state, and thatit also allows the study of the temperature dependenceof the effect.

3. Non-Markovian quantum probes detecting nonlocalcorrelations in composite environments

In Sec. IV.B we demonstrated that initial correlationsbetween local parts of the environment can lead to non-

23

local memory effects. In Liu et al. (2013a) such nonlocalmemory effects were experimentally realized in a pho-tonics system, where manipulating the correlations of thephotonic environments led to non-Markovian dynamics ofthe open system. The experimental scheme further pro-vided a controllable diagnostic tool for the quantificationof these correlations by repeated tomographic measure-ments of the polarization.

Let us take a closer look at the system under study inthe experiment. A general pure initial polarization stateof a photon pair can be written as |ψ12〉 = a |HH〉 +b |HV 〉 + c |V H〉 + d |V V 〉 and all initial states of thepolarization plus the frequency degrees of freedom areproduct states

|Ψ(0)〉 = |ψ12〉 ⊗∫dω1dω2 g(ω1, ω2) |ω1, ω2〉 , (85)

where g(ω1, ω2) is the probability amplitude for photon 1to have frequency ω1 and for photon 2 to have frequencyω2, with the corresponding joint probability distributionP (ω1, ω2) = |g(ω1, ω2)|2. If the photons pass throughquartz plates, the dynamics can be described by a gen-eral two-qubit dephasing map, where the different de-coherence functions can be expressed in terms of Fouriertransforms of the joint probability distribution P (ω1, ω2).For a Gaussian joint frequency distribution with identicalsingle frequency variances C and correlation coefficientK, the time evolution of the trace distance correspond-ing to the Bell-state pair |ψ±12〉 = 1√

2(|HH〉 ± |V V 〉) is

found to be

D(t) = exp

[−1

2∆n2C

(t21 + t22 − 2|K|t1t2

)], (86)

where ti denotes the time photon i has interacted withits quartz plate up to the actual observation time t. Foruncorrelated photon frequencies we have K = 0 and thetrace distance decreases monotonically, corresponding toMarkovian dynamics. However, as soon as the frequen-cies are anticorrelated, K < 0, the trace distance is non-monotonic which signifies quantum memory effects andnon-Markovian behavior. On the other hand, the localfrequency distributions are Gaussian and thus for the sin-gle photons the trace distance monotonically decreases.Therefore we can conclude, that the system is locallyMarkovian but globally displays nonlocal memory effects.

In the experiment two quartz plates act consecutivelyfor the photons, and the magnitude of the initial anti-correlations between the local reservoirs is tuned. Afterthe photon exits the quartz plates, full two-photon polar-ization state tomography is performed and by changingthe quartz plate thicknesses, the trace distance dynam-ics is recovered. From the dynamics it is evident thatinitial environmental correlations influence the quantumnon-Markovianity.

A further important aspect of the experimental schemeis that it enables to determine the frequency correlation

coefficient K of the photon pairs from measurements per-formed on the polarization degree of freedom. Thus byperforming tomography on a small system we can ob-tain information on frequency correlations, difficult tomeasure directly. Thus, in this context, the open system(polarization degrees of freedom) can serve as a quantumprobe which allows us to gain nontrivial information onthe correlations in the environment (frequency degrees offreedom).

V. SUMMARY AND OUTLOOK

In this Colloquium we have presented recent advancesin the definition and characterization of non-Markoviandynamics for an open quantum system. While in the clas-sical case the very definition of Markovian stochastic pro-cess can be explicitly given in terms of constraints on theconditional probabilities of the process, in the quantumrealm the peculiar role of measurements prevents a directformulation along the same path and new approaches arecalled for in order to describe memory effects. We havetherefore introduced a notion of non-Markovianity of aquantum dynamical map based on the behavior in time ofthe distinguishability of different system states, as quan-tified by their trace distance, which provides an intrinsiccharacterization of the dynamics. This approach allowsto connect non-Markovianity with the flow of informa-tion from the environment back to the system and nat-urally leads to the introduction of quantum informationconcepts. It is further shown to be connected to otherapproaches recently presented in the literature buildingon divisibility properties of the quantum dynamical map.In particular, a generalization of the trace distance cri-terion allows to identify Markovian time evolutions withquantum evolutions which are P-divisible, thus leadingto a clear-cut connection between memory effects in theclassical and quantum regimes. As a crucial feature thetrace distance approach to non-Markovianity can be ex-perimentally tested and allows for the study of system-environment correlations, as well as nonlocal memory ef-fects emerging from correlations within the environment.

We have illustrated the basic feature of the consideredtheoretical approaches to quantum non-Markovianity,typically leading to a recovery of quantum coherenceproperties, by analyzing in detail simple paradigmaticmodel systems, and further discussing more complexmodels which show important connections between non-Markovianity of the open system dynamics and featuresof the environment. It is indeed possible to obtain infor-mation on complex quantum systems via study of the dy-namics of a small quantum probe. The study of the timedevelopment of the trace distance between system statescan be shown to provide an indication of the presenceof initial correlations between system and environment,thus providing a powerful tool for the local detection of

24

correlations. Further considering suitable local opera-tions, such correlations can be further characterized and,in particular, one can distinguish quantum and classicalcorrelations.

Experimentally, it is in general obviously very difficultto control in detail the environmental degrees of freedomand therefore experiments on non-Markovian quantumsystems are still in a state of infancy. Still, some so-phisticated schemes for modifying the environment havebeen developed, thus leading to the realization of firstproof-of-principle experiments in which to test the studyof non-Markovianity, its connection with crucial featuresof the environment, as well as the capability to unveilsystem-environment correlations by means of local ob-servations on the system.

In recent years non-Markovian quantum systems havebeen enjoying much attention both due to fundamentalreasons and foreseeable applications, as shown by therapid growth in the related literature. Indeed, the po-tential relevance of memory effects in the field of com-plex quantum systems and quantum information has ledto an intense study, and we have pointed to some high-lights in this novel, so far fairly unexplored field of non-Markovianity. However, there are numerous open ques-tions yet to be studied. The latter include fundamentalquestions such as the mathematical structure of the spaceof non-Markovian quantum dynamical maps, the role ofcomplexity in the emergence of memory effects or therelevance of non-Markovianity in the study of the borderbetween classical and quantum aspects of nature, as wellas more applied issues like the identification of the en-vironmental features or system-environment correlationswhich can indeed be detected by means of local observa-tions on the system.

The theoretical and experimental investigations ofnon-Markovian quantum systems outlined in this Col-loquium pave the way for new lines of research by bothshedding light on fundamental questions of open quan-tum systems as well as by suggesting novel applicationsin quantum information and probing of complex systems.

ACKNOWLEDGMENTS

HPB, JP and BV acknowledge support from theEU Collaborative Project QuProCS (Grant Agreement641277). JP and BV acknowledge support by the COSTMP1006 Fundamental Problems in Quantum Physics,EML by the Academy of Finland through its Centresof Excellence Programme (2012-2017) under project No.251748, JP by the Jenny and Antti Wihuri and by theMagnus Ehrnrooth Foundation, and BV by the UnimiTRANSITION GRANT - HORIZON 2020.

REFERENCES

Accardi, L., A. Frigerio, and J. T. Lewis (1982), Publ. RIMSKyoto 18, 97.

Addis, C., P. , S. McEndoo, C. Macchiavello, and S. Manis-calco (2013), Phys. Rev. A 87, 052109.

Addis, C., B. Bylicka, D. Chruscinski, and S. Maniscalco(2014), Phys. Rev. A 90, 052103.

Alicki, R. (1995), Phys. Rev. Lett. 75, 3020.Alicki, R., and K. Lendi (1987), Quantum Dynamical Semi-

groups and Applications, Lecture Notes in Physics, Vol. 286(Springer, Berlin).

Apollaro, T. J. G., C. Di Franco, F. Plastina, and M. Pater-nostro (2011), Phys. Rev. A 83, 032103.

Borrelli, M., P. Haikka, G. deChiara, and S. Maniscalco(2013), Phys. Rev. A. 88, 010101(R).

Breuer, H.-P. (2012), J. Phys. B 45, 154001.Breuer, H.-P., B. Kappler, and F. Petruccione (1999), Phys.

Rev. A 59, 1633.Breuer, H.-P., E.-M. Laine, and J. Piilo (2009), Phys. Rev.

Lett. 103, 210401.Breuer, H.-P., and F. Petruccione (2002), The Theory of Open

Quantum Systems (Oxford University Press, Oxford).Brodutch, A., A. Datta, K. Modi, A. Rivas, and C. A.

Rodrıguez-Rosario (2013), Phys. Rev. A 87, 042301.Buscemi, F., and N. Datta (2014), e-print arXiv:1408.7062v1.Bylicka, B., D. Chruscinski, and S. Maniscalco (2014), Sci.

Rep. 4 (5720).Chancellor, N., C. Petri, and S. Haas (2013), Phys. Rev. B.

87, 184302.Chaturvedi, S., and F. Shibata (1979), Z. Phys. B 35, 297.Chin, A. W., S. F. Huelga, and M. B. Plenio (2012), Phys.

Rev. Lett. 109, 233601.Chiuri, A., C. Greganti, L. Mazzola, M. Paternostro, and

P. Mataloni (2012), Sci. Rep. 2.Chruscinski, D., and A. Kossakowski (2012), J. Phys. B 45,

154002.Chruscinski, D., A. Kossakowski, and A. Rivas (2011), Phys.

Rev. A 83, 052128.Chruscinski, D., and S. Maniscalco (2014), Phys. Rev. Lett.

112, 120404.Chruscinski, D., and F. A. Wudarski (2013), Phys. Lett. A

377, 1425 .Cialdi, S., D. Brivio, E. Tesio, and M. G. A. Paris (2011),

Phys. Rev. A 83, 042308.Cialdi, S., A. Smirne, M. G. A. Paris, S. Olivares, and B. Vac-

chini (2014), Phys. Rev. A 90, 050301.Clos, G., and H.-P. Breuer (2012), Phys. Rev. A 86, 012115.Cucchietti, F. M., D. A. R. Dalvit, J. P. Paz, and W. H.

Zurek (2003), Phys. Rev. Lett. 91, 210403.Cucchietti, F. M., H. M. Pastawski, and R. A. Jalabert

(2004), Phys. Rev. B 70, 035311.Dajka, J., J. Luczka, and P. Hanggi (2011), Phys. Rev. A 84,

032120.Dalfovo, F., S. Giorgini, L. P. Pitaevskii, and S. Stringari

(1999), Rev. Mod. Phys. 71, 463.Davies, E. B. (1976), Quantum Theory of Open Systems (Aca-

demic Press, London).Fanchini, F. F., G. Karpat, B. Cakmak, L. K. Castelano,

G. H. Aguilar, O. J. Farias, S. P. Walborn, P. H. S. Ribeiro,and M. C. de Oliveira (2014), Phys. Rev. Lett. 112, 210402.

Fanchini, F. F., G. Karpat, L. K. Castelano, and D. Z.Rossatto (2013), Phys. Rev. A 88, 012105.

25

Gardiner, C. W. (1985), Handbook of Stochastic Methods forPhysics, Chemistry and the Natural Sciences (Springer,Berlin).

Gessner, M., and H.-P. Breuer (2011), Phys. Rev. Lett. 107,180402.

Gessner, M., and H.-P. Breuer (2013), Phys. Rev. A 87,042107.

Gessner, M., M. Ramm, H. Haffner, A. Buchleitner, and H.-P. Breuer (2014a), EPL (Europhysics Letters) 107, 40005.

Gessner, M., M. Ramm, T. Pruttivarasin, A. Buchleitner, H.-P. Breuer, and H. Haffner (2014b), Nature Physics 10,105.

Gorin, T., T. Prosen, T. H. Seligman, and M. Znidaric (2006),Phys. Rep. 435, 33.

Gorini, V., A. Kossakowski, and E. C. G. Sudarshan (1976),J. Math. Phys. 17, 821.

Goussev, A., D. Waltner, K. Richter, and R. A. Jalabert(2008), New J. Phys. 10, 093010.

Haikka, P., J. Goold, S. McEndoo, F. Plastina, and S. Man-iscalco (2012), Phys. Rev. A 85, 060101.

Haikka, P., T. Johnson, and S. Maniscalco (2013), Phys. Rev.A. 87, 010103(R).

Haikka, P., S. McEndoo, G. De Chiara, G. M. Palma, andS. Maniscalco (2011), Phys. Rev. A 84, 031602.

Hall, M. J. W., J. D. Cresser, L. Li, and E. Andersson (2014),Phys. Rev. A 89, 042120.

Haseli, S., G. Karpat, S. Salimi, A. S. Khorashad, F. F. Fan-chini, B. Cakmak, G. H. Aguilar, S. P. Walborn, andP. H. S. Ribeiro (2014), Phys. Rev. A 90, 052118.

Helstrom, C. W. (1967), Information and Control 10 (3), 254.

Hou, S. C., X. X. Yi, S. X. Yu, and C. H. Oh (2011), Phys.Rev. A 83, 062115.

Huelga, S. F., A. Rivas, and M. B. Plenio (2012), Phys. Rev.Lett. 108, 160402.

Jin, J., V. Giovannetti, R. Fazio, F. Sciarrino, P. Mat-aloni, A. Crespi, and R. Osellame (2014), ArXiv e-printsarXiv:1411.6959 [quant-ph].

van Kampen, N. G. (1992), Stochastic Processes in Physicsand Chemistry (North-Holland, Amsterdam).

Kossakowski, A. (1972a), Bull. Acad. Polon. Sci. Math. 20,1021.

Kossakowski, A. (1972b), Rep. Math. Phys. 3, 247.Kraus, K. (1983), States, Effects, and Operations, Lecture

Notes in Physics, Vol. 190 (Springer, Berlin).Laine, E.-M., H.-P. Breuer, and J. Piilo (2014), Sci. Rep.

4 (4620).Laine, E.-M., H.-P. Breuer, J. Piilo, C.-F. Li, and G.-C. Guo

(2012), Phys. Rev. Lett. 108, 210402.Laine, E.-M., H.-P. Breuer, J. Piilo, C.-F. Li, and G.-C. Guo

(2013), Phys. Rev. Lett. 111, 229901.Laine, E.-M., J. Piilo, and H.-P. Breuer (2010a), Phys. Rev.

A 81, 062115.Laine, E.-M., J. Piilo, and H.-P. Breuer (2010b), EPL 92,

60010.Leggett, A. J., S. Chakravarty, A. T. Dorsey, M. P. A. Fisher,

A. Garg, and W. Zwerger (1987), Rev. Mod. Phys. 59, 1.Li, C.-F., J.-S. Tang, Y.-L. Li, and G.-C. Guo (2011), Phys.

Rev. A 83, 064102.Lindblad, G. (1976), Comm. Math. Phys. 48, 119.Lindblad, G. (1979), Comm. Math. Phys. 65, 281.Liu, B.-H., D.-Y. Cao, Y.-F. Huang, C.-F. Li, G.-C. Guo,

E.-M. Laine, H.-P. Breuer, and J. Piilo (2013a), Sci. Rep.3.

Liu, B.-H., L. Li, Y.-F. Huang, C.-F. Li, G.-C. Guo, E.-M.Laine, H.-P. Breuer, and J. Piilo (2011), Nature Physics7, 931.

Liu, B.-H., S. Wißmann, X.-M. Hu, C. Zhang, Y.-F. Huang,C.-F. Li, G.-C. Guo, A. Karlsson, J. Piilo, and H.-P. Breuer(2014), Sci. Rep. 4, 6327.

Liu, J., X.-M. Lu, and X. Wang (2013b), Phys. Rev. A 87,042103.

Lorenzo, S., F. Plastina, and M. Paternostro (2013), Phys.Rev. A 88, 020102.

Lu, X.-M., X. Wang, and C. P. Sun (2010), Phys. Rev. A 82,042103.

Luo, S. (2008), Phys. Rev. A 77, 022301.Luo, S., S. Fu, and H. Song (2012), Phys. Rev. A 86, 044101.Martinazzo, R., B. Vacchini, K. H. Hughes, and I. Burghardt

(2011), J. Chem. Phys. 134, 011101.Mazzola, L., E.-M. Laine, H.-P. Breuer, S. Maniscalco, and

J. Piilo (2010), Phys. Rev. A 81, 062120.Modi, K., A. Brodutch, H. Cable, T. Paterek, and V. Vedral

(2012), Rev. Mod. Phys. 84, 1655.Nakajima, S. (1958), Progr. Theor. Phys. 20, 948.Nielsen, M. A., and I. L. Chuang (2000), Quantum Compu-

tation and Quantum Information (Cambridge UniversityPress, Cambridge).

Pechukas, P. (1994), Phys. Rev. Lett. 73, 1060.Peres, A. (1984), Phys. Rev. A 30, 1610.Pethick, C. J., and H. Smith (2008), Bose–Einstein Con-

densation in Dilute Gases, 2nd ed. (Cambridge UniversityPress, Cambridge).

Pitaevskii, L., and S. Stringari (2003), Bose-Einstein con-densation (Oxford University Press, Oxford).

Quan, H. T., Z. Song, X. F. Liu, P. Zanardi, and C. P. Sun(2006), Phys. Rev. Lett. 96, 140604.

Rebentrost, P., and A. Aspuru-Guzik (2011), J. Chem. Phys.134 (10), 101103.

Rivas, A., S. F. Huelga, and M. B. Plenio (2010), Phys. Rev.Lett. 105, 050403.

Rivas, A., S. F. Huelga, and M. B. Plenio (2014), Rep. Prog.Phys. 77, 094001.

Rodrıguez-Rosario, C. A., K. Modi, and A. Aspuru-Guzik(2010), Phys. Rev. A 81, 012313.

Sachdev, S. (2011), Quantum Phase Transitions (CambridgeUniversity Press, Cambridge).

Shabani, A., and D. A. Lidar (2009), Phys. Rev. Lett. 102,100402.

Shibata, F., Y. Takahashi, and N. Hashitsume (1977), J. Stat.Phys. 17, 171.

Smirne, A., H.-P. Breuer, J. Piilo, and B. Vacchini (2010),Phys. Rev. A 82, 062114.

Smirne, A., D. Brivio, S. Cialdi, B. Vacchini, and M. G. A.Paris (2011), Phys. Rev. A 84, 032112.

Smirne, A., S. Cialdi, G. Anelli, M. G. A. Paris, and B. Vac-chini (2013), Phys. Rev. A 88, 012108.

Stelmachovic, P., and V. Buzek (2001), Phys. Rev. A 64,062106.

Tang, J.-S., C.-F. Li, Y.-L. Li, X.-B. Zou, G.-C. Guo, H.-P. Breuer, E.-M. Laine, and J. Piilo (2012), EPL (Euro-physics Letters) 97 (1), 10002.

Tang, J.-S., Y.-T. Wang, G. Chen, Y. Zou, C.-F. Li, G.-C. Guo, Y. Yu, M.-F. Li, G.-W. Zha, H.-Q. Ni, Z.-C.Niu, M. Gessner, and H.-P. Breuer (2014), “Experimen-tal detection of polarization-frequency quantum correla-tions in a photonic quantum channel by local operations,”arXiv:1311.5034v2 [quant-ph].

26

Vacchini, B. (2012), J. Phys. B 45, 154007.Vacchini, B., A. Smirne, E.-M. Laine, J. Piilo, and H.-P.

Breuer (2011), New J. Phys. 13, 093004.Vasile, R., S. Maniscalco, M. G. A. Paris, H.-P. Breuer, and

J. Piilo (2011a), Phys. Rev. A 84, 052118.Vasile, R., S. Olivares, M. Paris, and S. Maniscalco (2011b),

Phys. Rev. A. 83, 042321.Wißmann, S., and H.-P. Breuer (2013), “Nonlocal quan-

tum memory effects in a correlated multimode field,”arXiv:1310.7722 [quant-ph].

Wißmann, S., and H.-P. Breuer (2014), Phys. Rev. A 90,032117.

Wißmann, S., A. Karlsson, E.-M. Laine, J. Piilo, and H.-P.

Breuer (2012), Phys. Rev. A 86, 062108.Wißmann, S., B. Leggio, and H.-P. Breuer (2013), Phys. Rev.

A 88, 022108.Wolf, M. M., J. Eisert, T. S. Cubitt, and J. I. Cirac (2008),

Phys. Rev. Lett. 101, 150402.Xiang, G.-Y., Z.-B. Hou, C.-F. Li, G.-C. Guo, H.-P. Breuer,

E.-M. Laine, and J. Piilo (2014), EPL (Europhysics Let-ters) 107 (5), 54006.

Xu, Z. Y., W. L. Yang, and M. Feng (2010), Phys. Rev. A81, 044105.

Znidaric, M., C. Pineda, and I. Garcıa-Mata (2011), Phys.Rev. Lett. 107, 080404.

Zwanzig, R. (1960), J. Chem. Phys. 33, 1338.