# Quantum non-Markovian environment-to-system backflows of information: non-
operational vs. operational approaches

Adrián A. Budini  Consejo Nacional de Investigaciones Científicas y Técnicas
(CONICET), Centro Atómico Bariloche, Avenida E. Bustillo Km 9.5, (8400)
Bariloche, Argentina, and Universidad Tecnológica Nacional (UTN-FRBA), Fanny
Newbery 111, (8400) Bariloche, Argentina

###### Abstract

Quantum memory effects can be qualitatively understood as a consequence of an
environment-to-system backflow of information. Here, we analyze and compare
how this concept is interpreted and implemented in different approaches to
quantum non-Markovianity. We study a non-operational approach, defined by the
distinguishability between two system states characterized by different
initial conditions, and an operational approach, which is defined by the
correlation between different outcomes associated to successive measurement
processes performed over the system of interest. The differences, limitations,
and vantages of each approach are characterized in detail by considering
diverse system-environment models and dynamics. As a specific example, we
study a non-Markovian depolarizing map induced by the interaction of the
system of interest with an environment characterized by incoherent and
coherent self-dynamics.

##  I Introduction

The time-evolution of both classical and quantum systems may develop memory
effects vanKampen ; breuerbook ; vega ; wiseman . Nevertheless, the
characterization and definition of these effects is quite different in both
regimes BreuerReview ; plenioReview . As is well known, in a classical regime
memory effects can be rigorously defined in a probabilistic approach. The
independence or dependence of conditional probabilities on the previous system
history define respectively the (memoryless) Markovian and non-Markovian
regimes vanKampen .

In a quantum regime one is immediately confronted with an extra aspect. In
fact, the state of a quantum system (and consequently its history) can only be
determined by performing a measurement process, which intrinsically implies a
perturbation to its (originally unperturbed) dynamics. Therefore, the
definition of memory effects and quantum non-Markovianity can be tackled from
two intrinsically different approaches. In non-operational approaches, memory
effects are defined by taking solely into account the properties of the
unperturbed open system dynamics (its propagator). In operational approaches,
memory effects are defined by the statistical properties of different outcomes
associated to system measurement processes and transformations (like unitary
ones).

A wide variety of measures and memory witness has been utilized in the context
of non-operational approaches (see reviews BreuerReview ; plenioReview ). The
first proposals correspond to deviations of the system propagator from
divisibility divisible  and a non-monotonous behavior of the trace distance
(TD) between two distinct system states BreuerFirst . In this context, memory
effects were associated to an environment-to-system backflow of information:
information stored in the initial system state is transferred to the
environmental degrees of freedom; their influence on the system at later times
implies a backflow of information that leads to memory effects. In spite of
this clear and well-motivated interpretation EnergyBackFLow ; Energy ;
HeatBackFLow , the precise assessment of this concept is still under debate
megier ; maximal ; petruccione ; amato ; santis ; acin ; horo ; EntroBack .

The basic idea of operational approaches is to appeal to the standard
definition of memory effects in terms of probabilities vanKampen . Hence, the
(quantum) system must be subjected to a set of measurement processes such that
their statistical properties determine the presence or absence of memory
effects modi ; budiniCPF . The study and understanding of this approach has
been performed in the recent literature pollock ; pollockInfluence ; bonifacio
; han ; ban ; rio , including alternative definitions and analysis of
information flows goan ; BIF .

The main goal of this paper is to analyze and to compare how the concept of
environment-to-system backflow of information is interpreted and implemented
in operational and non-operational approaches. As a non-operational memory
witness we take the TD between two different system initial states BreuerFirst
, taking also into account the bounds on its revival behavior that has been
characterized recently EntroBack . As an operational memory witness we
consider a conditional past-future (CPF) correlation budiniCPF , both in a
deterministic and random schemes BIF . The comparison is performed by
considering different system-environment models and analyzing in each case the
information flows from the two perspectives. We consider statistical mixtures
of Markovian system evolutions, system coupled to incoherent maximal  and
coherent casual bystander environments casual , which are characterized by a
self-dynamics that is independent of the system degrees of freedom. In
addition, we consider (standard) unitary system-environment models breuerbook
. As a specific model, we study a depolarizing map induced by the interaction
of a system with a finite set of incoherent degrees of freedom. In this
regime, as well as in a quantum coherent one, we explain how and why both
approaches lead to different notions of quantum non-Markovianity and
environment-to-system backflows of information.

The paper is outlined as follows. In Sec. II we review the definition and main
properties of the considered non-operational BreuerFirst ; EntroBack  and
operational budiniCPF ; BIF  approaches. In Sec. III we study both approaches
by considering different system-environment models. In Sec. IV we study the
depolarizing map. In Sec. V we provide the conclusions.

##  II Quantum non-Markovianity

Here we briefly review the main characteristics of the different approaches to
quantum non-Markovianity.

###  II.1 Non-operational approach

If the open system is not affected or perturbed during its evolution, the
unique object that allows to defining the presence or absence of memory
effects is its (unperturbed) density matrix propagator. The rigorous theory of
quantum dynamical semigroups alicki  motivate to associating the (memoryless)
quantum Markovian regime with propagators whose time-evolution obey a Lindblad
equation (or Gorini-Kossakowski-Sudarshan-Lindblad equation). Consequently,
any (scalar) measure or property that quantifies departures of the system
propagator from a Lindblad equation can be taken as a witness of quantum
memory effects.

Lindblad equations lead to completely positive propagators between two
arbitrary times alicki . As is well known, completely positive transformations
lead to very specific contractive properties for different distance measures
and entropic quantities nielsen . For example, the TD between two arbitrary
density matrixes $\rho$ and $\sigma,$ defined as
$\mathbf{D}(\rho,\sigma)\equiv(1/2)\mathrm{Tr}|\rho-\sigma|,$ under a
completely positive transformation $\Phi$ fulfills the inequality
$\mathbf{D}(\Phi[\rho],\Phi[\sigma])\leq\mathbf{D}(\rho,\sigma).$
Consequently, it is possible to define quantum Markovianity by the condition
BreuerFirst

$\mathbf{D}(\rho_{t+\tau}^{s},\sigma_{t+\tau}^{s})\leq\mathbf{D}(\rho_{t}^{s},\sigma_{t}^{s}),$
(1)

where $\rho_{t}^{s}$ and $\sigma_{t}^{s}$ are two arbitrary evolved system
states that differ in their initial conditions,
$\rho_{0}^{s}\neq\sigma_{0}^{s}.$ Alternatively, one can interpret that
quantum memory effects are present whenever this inequality is not fulfilled
for a set of two arbitrary time intervals $t\geq 0$ and $\tau>0.$

In spite of the simplicity and efficacy of the previous theoretical frame, in
general it is not possible to know or infer which physical processes are
involved when the contractive condition (1) is not fulfilled. A remarkable
advance in this direction was recently obtained in Ref. EntroBack  by
establishing the inequality

$\displaystyle\mathbf{D}(\rho_{t+\tau}^{s},\sigma_{t+\tau}^{s})$
$\displaystyle\leq$
$\displaystyle\mathbf{D}(\rho_{t}^{s},\sigma_{t}^{s})+\mathbf{D}(\rho_{t}^{e},\sigma_{t}^{e})$
$\displaystyle\\!\\!\\!\\!\\!+\mathbf{D}(\rho_{t}^{se},\rho_{t}^{s}\otimes\rho_{t}^{e})+\mathbf{D}(\sigma_{t}^{se},\sigma_{t}^{s}\otimes\sigma_{t}^{e}).$

Here, $\rho_{t}^{se}$ and $\sigma_{t}^{se}$ are the evolved system-environment
states with initial conditions $\rho_{0}^{se}=\rho_{0}^{s}\otimes\rho_{0}^{e}$
and $\sigma_{0}^{se}=\sigma_{0}^{s}\otimes\sigma_{0}^{e}.$ As usual, the
system and bath states follows from a partial trace operation,
$\rho_{t}^{s}=\mathrm{Tr}_{e}[\rho_{t}^{se}]$ and
$\rho_{t}^{e}=\mathrm{Tr}_{s}[\rho_{t}^{se}].$ The asymmetry between system
and environment $(s\nleftrightarrow e)$ is introduced by taking in both cases
the same initial environmental state, $\rho_{0}^{e}=\sigma_{0}^{e}.$

The result (II.1) only relies on the triangle inequality fulfilled by the TD.
Thus, it is valid for arbitrary system-environment models. In addition, this
expression allows to bounding the environment-to-system backflow of
information defined by the “revivals”

$\mathbf{D}(\rho_{t+\tau}^{s},\sigma_{t+\tau}^{s})-\mathbf{D}(\rho_{t}^{s},\sigma_{t}^{s})>0.$
(3)

The remaining (bounding) contributions in the rhs of Eq. (II.1) have a clear
physical interpretation. One can relate the contribution
$\mathbf{D}(\rho_{t}^{e},\sigma_{t}^{e})$ to changes in the environmental
state, while the terms
$\mathbf{D}(\rho_{t}^{se},\rho_{t}^{s}\otimes\rho_{t}^{e})+\mathbf{D}(\sigma_{t}^{se},\sigma_{t}^{s}\otimes\sigma_{t}^{e})$
measure the correlations established between the system and the environment
EntroBack . Nevertheless, it is important to realize that these physical
processes do not guarantee the developing of revivals. The right conclusion is
that given that there exists revivals, their origin can related to changes in
the environmental state or to the establishing of system-environment
correlations.

It was also proven that the inequality (II.1) remains valid when the TD is
replaced by a telescopic relative entropy and the square root of a quantum
Jensen-Shannon divergence EntroBack . Thus, the interpretation of the bounds
remains the same when using these entropic quantities.

###  II.2 Operational approach

In a probabilistic frame, given a sequence of system states $x\rightarrow
y\rightarrow z$ with joint probability $P(z,y,x),$ Markovianity is defined by
the condition

$P(z,y,x)=P(z|y)P(y|x)P(x),$ (4)

where $P(b|a)$ denotes in general the conditional probability of $b$ given
$a.$ By Bayes rule, the equality (4) implies the (memoryless) condition
$P(z|y,x)=P(z|y).$ Similar constraints emerge when considering higher joint
probabilities involving an arbitrary number of events vanKampen .

For quantum systems, the definition of Markovianity in terms of probabilities
unavoidably implies performing a set of system measurement processes. In Ref.
modi , by means of a process tensor formalism, the Markovian condition is
taken into account for arbitrary (higher order) joint probabilities.
Nevertheless, for quantum systems coupled to standard environment models
(standard classical noises and/or unitary system-environment interaction
models), only three measurement events are enough for detecting departures
from a (probabilistic) Markovian regime budiniCPF . In such a case, the
condition (4) can be conveniently rewritten as a CPF independence,

$P(z,x|y)=P(z|y)P(x|y).$ (5)

This result follows straightforwardly by using that $P(z,x|y)=P(z,y,x)/P(y),$
where $P(y)=\sum_{z,x}P(z,y,x).$

The CPF independence (5) implies that any (conditional) correlation between
past and future events witnesses memory effects. Correspondingly, a CPF
correlation is defined as budiniCPF

$C_{pf}(t,\tau)|_{\breve{y}}\overset{d/r}{=}\sum_{z,x}zx[P(z,x|\breve{y})-P(z|\breve{y})P(x|\breve{y})],$
(6)

where $\\{x\\}$ and $\\{z\\}$ are the (past and future) measurement outcomes.
The time dependence $(t,\tau)$ emerges because the past, present, and future
measurements are performed at the initial time $t=0,$ at time $t,$ and
$t+\tau$ respectively. Evidently $C_{pf}(t,\tau)|_{\breve{y}}$ vanishes in a
(probabilistic) Markovian regime [Eq. (5)].

In Eq. (6) it was introduced the change $y\rightarrow\breve{y},$ which is
stretchy related with the definition of memory effects and information flows
in this approach. Two different measurement schemes are necessary BIF . In a
deterministic scheme (denoted with the supra $d),$ after the intermediate
measurement (whose outcome defines the conditional property) none change is
introduced. Hence, $\breve{y}=y.$ In a random scheme (denoted with the supra
$r),$ after the intermediate measurement the system state is randomly chosen
$(y\rightarrow\breve{y})$ over the set of possible states associated to the
outcomes $\\{y\\}.$ The CPF correlation is defined with this renewed
conditional state.

In the deterministic scheme, the CPF correlation
$[C_{pf}(t,\tau)|_{\breve{y}}\overset{d}{\neq}0]$ detects memory effects
[departures with respect to Eq. (4), or equivalently Eq. (5)] independently of
the specific system-environment model. In the random scheme, a non-vanishing
CPF correlation $[C_{pf}(t,\tau)|_{\breve{y}}\overset{r}{\neq}0],$ by
definition, detects the presence of environment-to-system backflows of
information (or bidirectional system-environment information flows). This
relation is motivated by the complementary case
$C_{pf}(t,\tau)|_{\breve{y}}\overset{r}{=}0$ that applies when the environment
(which induces the memory effects
$C_{pf}(t,\tau)|_{\breve{y}}\overset{d}{\neq}0)$ is unperturbed by its
coupling with the system BIF .

The previous characteristics of the deterministic and random schemes can be
easily understood from the properties of projective measurements performed
over bipartite systems casual . Interestingly, the formalism remains the same
and is also valid for purely (classically) incoherent system-environment
arrangements.

###  II.3 Bipartite propagator vs. single propagator

Before comparing both approaches (next section), here we clarify which
dynamical objects determine each one. In the non-operational approach, the
presence of memory effects [TD revivals defined by Eq. (3)] can be determined
after knowing solely the system (single) propagator. In contrast, for
determining the bound defined by Eq. (II.1) it is necessary to know the
bipartite system-environment propagator specified for a given initial bath
state.

In contrast, the operational approach can only be characterized by knowing
(exact or approximate) the bipartite propagator for different initial bath
states (the initial one and the bath state after the intermediate
measurement). As a matter of fact, the CPF correlation (6) can be written as a
function of the joint probability $P(z,\breve{y},x).$ Assuming that the three
measurements are projective ones, in the deterministic scheme it reads BIF

$\frac{P(z,\breve{y},x)}{P(x)}\overset{d}{=}\mathrm{Tr}_{se}(E_{z}\mathcal{G}_{t+\tau,t}^{se}[\rho_{\breve{y}}\otimes\mathrm{Tr}_{s}(E_{\breve{y}}\mathcal{G}_{t,0}^{se}[\rho_{x}^{se}])]),$
(7)

while in the random scheme it is BIF

$\frac{P(z,\breve{y},x)}{P(x)}\overset{r}{=}\mathrm{Tr}_{se}(E_{z}\mathcal{G}_{t+\tau,t}^{se}[\rho_{\breve{y}}\otimes\mathrm{Tr}_{s}(\mathcal{G}_{t,0}^{se}[\rho_{x}^{se}])])\wp(\breve{y}|x).$
(8)

In these expressions, $\mathcal{G}_{t+\tau,t}^{se}$ is the bipartite
propagator between $t$ and $t+\tau.$ In addition, $E_{m}\equiv|m\rangle\langle
m|$ and $\rho_{m}\equiv|m\rangle\langle m|$ $[m=z,\breve{y},x]$ represent the
(positive) effect measurement operators and post-measurement states
respectively. The set $\\{|m\rangle\\}$ $[m=z,\breve{y},x]$ are the
eigenstates of each measured observable. Furthermore,
$\rho_{x}^{se}\equiv\rho_{x}\otimes\rho_{0}^{e}$ and $P(x)=\langle
x|\rho_{0}^{s}|x\rangle.$ The random scheme is parametrized by an arbitrary
conditional probability $\wp(\breve{y}|x)$ that defines the change in the
system state $(y\rightarrow\breve{y})$ after the intermediate measurement.

The different dependence of both approaches on the bipartite propagator leads
to strong different conclusions about memory effects and information flows,
which are analyzed in the next section.

##  III Comparing both approaches

In order to perform a systematic comparison we consider different system-
environment models and approximations. In general, we assume that the
bipartite system-environment state $\rho_{t}^{se}$ evolves as

$\frac{d}{dt}\rho_{t}^{se}=(\mathcal{L}_{s}+\mathcal{L}_{e}+\mathcal{L}_{se})[\rho_{t}^{se}],$
(9)

where $\mathcal{L}_{s}$ and $\mathcal{L}_{e}$ define the self-dynamics of the
system and the environment respectively, while $\mathcal{L}_{se}$ defines
their mutual interaction. This interaction term may be unitary or includes
dissipative couplings.

###  III.1 Born-Markov approximation

For systems weakly coupled to their environments, the Born-Markov
approximation breuerbook  allows to write the bipartite state as

$\rho_{t}^{se}\simeq\rho_{t}^{s}\otimes\rho_{0}^{e},$ (10)

where $\rho_{t}^{s}$ is the system state, while $\rho_{0}^{e}$ is the (almost)
unperturbed environment state.

When this approximation is valid, in the non-operational approach, it is
simple to check that Eq. (II.1) reduces to Eq. (1). In fact,
$\mathbf{D}(\rho_{t}^{e},\sigma_{t}^{e})=\mathbf{D}(\rho_{t}^{se},\rho_{t}^{s}\otimes\rho_{t}^{e})=\mathbf{D}(\sigma_{t}^{se},\sigma_{t}^{s}\otimes\sigma_{t}^{e})=0.$
Furthermore, $\rho_{t}^{s}$ can be well approximated by a Lindblad equation,
which guarantees the absence of any revival in
$\mathbf{D}(\rho_{t}^{s},\sigma_{t}^{s}).$ Thus, the dynamics is Markovian.

In the operational approach, by introducing the approximation (10) into Eqs.
(7) and (8) straightforwardly it follows that
$C_{pf}(t,\tau)|_{\breve{y}}\overset{d/r}{=}0$ [Eq. (6)]. These results are
independent of which observables are measured. Thus, the dynamics is
Markovian.

In this case [Eq. (10)] both approaches coincides. Strong differences appear
in the cases studied below.

###  III.2 Casual bystander environments

A wide class of “non-Markovian” dynamics can be derived by assuming that the
system interacts with a “casual bystander” environment. These baths are
defined by the independence of their marginal states
$\rho_{t}^{e}=\mathrm{Tr}_{s}[\rho_{t}^{se}]$ of any degree of freedom of the
system. Alternatively, the time evolution of $\rho_{t}^{e}$ can be written in
the environment Hilbert space without involving any operator or state of the
system. These properties must be valid for arbitrary system and environment
(separable) initial conditions.

For fulfilling the previous properties, the interaction term
$\mathcal{L}_{se}$ in the general evolution (9) must be restricted such that

$\mathrm{Tr}_{s}(\mathcal{L}_{se}[\rho_{t}^{se}])=\mathcal{A}[\rho_{t}^{e}],$
(11)

where $\mathcal{A}$ is an arbitrary superoperator acting on $\rho_{t}^{e}$
that does not have any dependence on the system degrees of freedom. In
general, this constraint can only be satisfied by dissipative (non-unitary)
system-environment couplings. On the other hand, the bath dynamics can be
quantum casual  or a classical (incoherent) one maximal .

In the non-operational approach, the independence of the environment state on
the system degrees of freedom cannot be translated to any restriction on the
inequality defined by Eq. (II.1). In fact, under the constraint (11), the TD
may or not present revivals, property that can only be cheeked for each
specific model. Thus, some dynamics are classified as Markovian and other as
non-Markovian. The unique simplification that can be introduced is to assume
that the environment state does not evolve in time,
$\rho_{t}^{e}=\rho_{0}^{e},$ that is, the environment begins in its stationary
state. In this case, Eq. (II.1) reduces to

$\displaystyle\mathbf{D}(\rho_{t+\tau}^{s},\sigma_{t+\tau}^{s})-\mathbf{D}(\rho_{t}^{s},\sigma_{t}^{s})$
$\displaystyle\leq$
$\displaystyle\mathbf{D}(\rho_{t}^{se},\rho_{t}^{s}\otimes\rho_{0}^{e})$ (12)
$\displaystyle\\!\\!\\!\\!\\!+\mathbf{D}(\sigma_{t}^{se},\sigma_{t}^{s}\otimes\rho_{0}^{e}).$

Even in this case $(\rho_{t}^{e}=\rho_{0}^{e}),$ the TD may or not present
revivals, that is, depending on the model, the system may be classified as
Markovian or non-Markovian.

In Eq. (12), any environment-to-system backflow of information can be related
to the establishing of the correlations
$\mathbf{D}(\rho_{t}^{se},\rho_{t}^{s}\otimes\rho_{0}^{e})+\mathbf{D}(\sigma_{t}^{se},\sigma_{t}^{s}\otimes\rho_{0}^{e}).$
Certainly, the system-environment correlations (always) changes in time.
Nevertheless, even when there is not revivals in the TD system-environment
correlations are established. This feature represent a central problem for the
interpretation of this approach. In addition, here the environment state is
completely independent of the system (and even of time). Thus, the revivals of
the TD must be taken as a (mathematical) model-dependent property whose origin
cannot be related to any physical process that implies a physical transfer of
information from the environment to the system.

A different perspective emerges in the operational approach. By using the
independence of the environment state
$[\rho_{t}^{e}=\mathrm{Tr}_{s}(\rho_{t}^{se})]$ of any degree of freedom of
the system, it is possible to check that the joint probability (7) of the
deterministic scheme does not fulfill the Markov property (4). In contrast, it
is simple to check that the joint probability (8) of the random scheme fulfill
the Markov property (4). Consequently, a casual bystander environment leads to
the CPF correlations [Eq. (6)]

$C_{pf}(t,\tau)|_{\breve{y}}\overset{d}{\neq}0,\ \ \ \ \ \ \ \ \ \
C_{pf}(t,\tau)|_{\breve{y}}\overset{r}{=}0.$ (13)

In this approach, the property
$C_{pf}(t,\tau)|_{\breve{y}}\overset{d}{\neq}0,$ valid for any model under the
constraint (11), implies that the system dynamics is non-Markovian. Its origin
can be related to the establishing of (arbitrary) system-environment
correlations. On the other hand, the property
$C_{pf}(t,\tau)|_{\breve{y}}\overset{r}{=}0,$ which is valid for arbitrary
measurement processes and specific models, is read as the absence of
bidirectional system-environment information flows. In fact, given that the
environment is characterized by a self-dynamics that is completely independent
of the system, any environment-to-system backflow of information (as detected
in the non-operational approach) does not rely on any physical process that
affect the environment state neither its dynamics.

The meaning of the previous analysis is clarified by specifying different
bipartite models that fulfill the evolution (9) and the constraint (11).

####  III.2.1 Classical mixture of quantum Markovian dynamics

Given a set of different system Lindblad superoperators
$\\{\mathcal{L}_{s}^{c}\\},$ which may include both unitary and dissipative
contributions, and given a set of normalized positive weights $\\{p_{c}\\},$
$\sum_{c}p_{c}=1,$ a classical statistical mixture of Markovian dynamics is
defined by the bipartite state

$\rho_{t}^{se}=\sum_{c}\exp(t\mathcal{L}_{s}^{c})[\rho_{0}^{s}]\otimes
p_{c}|c\rangle\langle c|.$ (14)

Here, $\\{|c\rangle\langle c|\\}$ is a set of projectors associated to the
environment space. The marginal system and environment states read

$\rho_{t}^{s}=\sum_{c}p_{c}\exp(t\mathcal{L}_{s}^{c})[\rho_{0}^{s}],\ \ \ \ \
\ \rho_{t}^{e}=\sum_{c}p_{c}|c\rangle\langle c|.$ (15)

Memory effects in this kind of non-Markovian system dynamics has been explored
in the literature poland . Notice that the environment does not have any
dynamics. Even more, the system dynamics can be performed by mixing in a
random way (with weight $p_{c})$ each of the evolved Markovian system states
$\exp(t\mathcal{L}_{s}^{c})[\rho_{0}^{s}].$ Thus, the detection of an
environment-to-system backflow of information via Eq. (3) seems to have a
formal mathematical interpretation rather than a physical one. On the other
hand, in the operational approach this case is characterized by Eq. (13),
which guaranty the presence of memory effects
$C_{pf}(t,\tau)|_{\breve{y}}\overset{d}{\neq}0,$ but not any bidirectional
information flow, $C_{pf}(t,\tau)|_{\breve{y}}\overset{r}{=}0.$

####  III.2.2 Interaction with stochastic classical degrees of freedom

When the system interacts with stochastic classical degrees of freedom, the
bipartite state can be written as

$\rho_{t}^{se}=\sum_{c}\rho_{t}^{c}\otimes p_{c}(t)|c\rangle\langle c|.$ (16)

In contrast to the previous case [Eq. (14)], the weights $\\{p_{c}(t)\\}$ are
time-dependent and the evolution of the states $\\{\rho_{t}^{c}\\}$ may
involve coupling between all of them. In fact, under the constraint (11), the
more general evolution can be written as maximal

$\frac{d\tilde{\rho}_{t}^{c}}{dt}=\mathcal{L}_{s}^{c}[\tilde{\rho}_{t}^{c}]-\sum_{c^{\prime}}\gamma_{c^{\prime}c}\tilde{\rho}_{t}^{c}+\sum_{c^{\prime}}\gamma_{cc^{\prime}}\mathbb{S}_{cc^{\prime}}[\tilde{\rho}_{t}^{c^{\prime}}].$
(17)

Here, $\tilde{\rho}_{t}^{c}\equiv p_{c}(t)\rho_{t}^{c}.$ Thus,
$p_{c}(t)=\mathrm{Tr}_{s}(\tilde{\rho}_{t}^{c}).$ Furthermore,
$\\{\mathbb{S}_{cc^{\prime}}\\}$ are arbitrary completely positive system
transformations, which are trace preserving,
$\mathrm{Tr}_{s}(\mathbb{S}_{cc^{\prime}}[\rho])=\mathrm{Tr}_{s}(\rho).$
Consequently, the environment probabilities $\\{p_{c}(t)\\}$ obey a classical
master equation

$\frac{dp_{c}(t)}{dt}=-\sum_{c^{\prime}}\gamma_{c^{\prime}c}p_{c}(t)+\sum_{c^{\prime}}\gamma_{cc^{\prime}}p_{c^{\prime}}(t),$
(18)

which in turn shows the role played by the coupling rates
$\\{\gamma_{c^{\prime}c}\\}.$ In contrast, the system dynamics depart from a
Markovian (Lindblad) evolution. From some specific models, it is possible to
recover some phenomenological non-Markovian master equations (see for example
lidar ).

In the non-operational approach it is very difficult to predict if a given
dynamics [Eq. (16)] lead or not to revivals in the TD. If the incoherent
degrees of freedom begin in their stationary state,
$p_{c}(0)=\lim_{t\rightarrow\infty}p_{c}(t),$ one is confronted with the
bounds defined by Eq. (12). Even in this case, one cannot predict when there
exist or not an environment-to-system backflow of information.

Interestingly, the origin of the contributions
$\mathbf{D}(\rho_{t}^{se},\rho_{t}^{s}\otimes\rho_{0}^{e})+\mathbf{D}(\sigma_{t}^{se},\sigma_{t}^{s}\otimes\rho_{0}^{e})$
in Eq. (12) [or in general in Eq. (II.1)] can be easily read from the
evolution (17). In fact, this equation shows that the system evolution is
totally conditioned to the environment dynamics. The contributions
$\mathcal{L}_{s}^{c}$ are “active” whenever the environment is in the state
$|c\rangle\langle c|.$ Furthermore, the system suffers the transformation
$\rho\rightarrow\mathbb{S}_{c^{\prime}c}[\rho]$ whenever the environment
“jumps” between the states $c\rightarrow c^{\prime}.$ This is the physical
mechanism that leads to the system-environment correlations, which in turn
does not imply any system-dependent change in the environment state or
dynamics. Thus, the interpretation of revivals in the TD as environment-to-
system backflow of information is again controversial.

Independently of the Lindblad contributions $\\{\mathcal{L}_{s}^{c}\\},$ the
superoperators $\\{\mathbb{S}_{cc^{\prime}}\\},$ and rates
$\\{\gamma_{c^{\prime}c}\\},$ the operational approach is characterized by Eq.
(13), that is, the dynamics is non-Markovian
$[C_{pf}(t,\tau)|_{\breve{y}}\overset{d}{\neq}0]$ without the developing of
any bidirectional system-environment information flow
$[C_{pf}(t,\tau)|_{\breve{y}}\overset{r}{=}0].$

####  III.2.3 Environmental quantum degrees of freedom

The condition Eq. (11) can be satisfied even when the environment is a quantum
one, that is, it develops coherent behaviors. In this case, the bipartite
state can be written as

$\rho_{t}^{se}=\sum\rho_{t}^{c}\otimes p_{c}(t)|c_{t}\rangle\langle c_{t}|.$
(19)

In contrast to Eq. (16), due to the quantum nature of the environment, the
projectors $\\{|c_{t}\rangle\langle c_{t}|\\}$ are time-dependent. In fact,
they define the base in which the environment density matrix $\rho_{t}^{e}$ is
diagonal. The more general bipartite evolution (9) under the constraint (11),
in its diagonal representation, is given by casual

$\displaystyle\frac{d}{dt}\rho_{t}^{se}$ $\displaystyle=$
$\displaystyle(\mathcal{L}_{s}+\mathcal{L}_{e})[\rho_{t}^{se}]+\sum_{\alpha}\Gamma_{\alpha}\
B_{\alpha}\mathbb{S}_{\alpha}[\rho_{t}^{se}]B_{\alpha}^{{\dagger}}$ (20)
$\displaystyle-\frac{1}{2}\sum_{\alpha}\Gamma_{\alpha}\\{B_{\alpha}^{{\dagger}}B_{\alpha},\rho_{t}^{se}\\}_{+},$

where $\\{\cdot,\cdot\\}_{+}$ is an anticommutator operation. Furthermore,
$\\{B_{\alpha}\\}$ are arbitrary environment operators, while
$\mathbb{S}_{\alpha}$ are completely positive trace-preserving system
superoperators. The rates $\\{\Gamma_{\alpha}\\}$ set the environment
dynamics. In fact,

$\frac{d}{dt}\rho_{t}^{e}=\mathcal{L}_{e}[\rho_{t}^{e}]+\sum_{\alpha}\Gamma_{\alpha}\
(B_{\alpha}\rho_{t}^{e}B_{\alpha}^{{\dagger}}-\frac{1}{2}\\{B_{\alpha}^{{\dagger}}B_{\alpha},\rho_{t}^{e}\\}_{+}),$
(21)

which is a Lindblad dynamics completely independent of the system degrees of
freedom. These evolutions recover, as particular cases, some phenomenological
collisional models introduced in the literature (see for example collision ).

The physical interpretation of the evolution (20) is quite similar to that of
Eq. (17). In fact, here the application of the system superoperators
$\mathbb{S}_{\alpha}$ occur whenever the environment suffer a transition
associated to the operators $B_{\alpha}.$ This (unidirectional) mechanism
defines how the system-environment correlations are build up.

In the non-operational approach, even when the environment begins in its
stationary state $\rho_{0}^{e}=\lim_{t\rightarrow\infty}\rho_{t}^{e}$ [where
$\rho_{t}^{e}$ obeys Eq. (21)], it is not possible to infer for an arbitrary
model the presence or absence of revivals in the TD [Eq. (3)]. In contrast,
the operational approach is still characterized by Eq. (13).

###  III.3 Unitary system-environment interactions

Independently of the specific models, the correlation between the system and
the casual bystander environments introduced previously does not involve
quantum entanglement entanglement  [see the separable states Eqs. (14), (16),
and (19)]. In contrast, quantum entanglement may emerges when considering
Hamiltonian (time-reversible) system-environment interactions. In fact, solely
for special system-environment initial conditions a bipartite unitary dynamics
does not induce quantum entanglement katarzyna .

The total Hamiltonian is written as

$H_{T}=H_{s}+H_{e}+H_{I}.$ (22)

Each contribution correspond respectively to the system, environment, and
interaction Hamiltonians respectively. The bipartite propagator is

$\mathcal{G}_{t,t_{0}}^{se}[\bullet]=\exp[-i(t-t_{0})H_{T}]\bullet\exp[+i(t-t_{0})H_{T}].$
(23)

In the non-operational approach, each contribution in the rhs of Eq. (II.1)
make complete sense in this context. In fact, almost all unitary interactions
lead to a change in the environment state and also induce the developing of
(arbitrary) system-environment correlations. When revivals in the TD develops,
Eq. (II.1) defines a bound with a clear physical meaning. Nevertheless, in
general it is not possible to infer which kind of dynamics develop or not
revivals in the TD. Even for a given (Hamiltonian) model, depending on the
underlying parameters the system dynamics may be Markovian or not.
Consequently, it is not clear which physical property defines the boundary
between Markovian and non-Markovian dynamics.

In the operational approach, given that the state and dynamics of the
environment are in general modified by a unitary interaction, instead of Eq.
(13), here it follows

$C_{pf}(t,\tau)|_{\breve{y}}\overset{d}{\neq}0,\ \ \ \ \ \ \ \ \ \
C_{pf}(t,\tau)|_{\breve{y}}\overset{r}{\neq}0.$ (24)

Both inequalities can be supported by performing a perturbation theory based
on projectors techniques bonifacio . Consistently, it has been shown that even
close to the validity of a Born-Markov approximation the operational approach
can detect memory effects rio .

The inequality $C_{pf}(t,\tau)|_{\breve{y}}\overset{d}{\neq}0$ implies that
the system dynamics is non-Markovian (system-environment correlations are
developed during the evolution), while
$C_{pf}(t,\tau)|_{\breve{y}}\overset{r}{\neq}0$ detects the presence of
bidirectional information flows. In fact, here the environment state and
evolution always depend on the system degrees of freedom.

There exist a unique exception to Eq. (24), which reduces to Eq. (13). Hence,
even when the environment state is modified, for any system observables one
obtain $C_{pf}(t,\tau)|_{\breve{y}}\overset{r}{=}0.$ While this property is
certainly undesirable, this case has a clear physical interpretation. It
emerges when, in a given environmental base $\\{|e\rangle\\},$ the diagonal
part of the bipartite propagator (23) can be written as

$\langle
e|\mathcal{G}_{t,0}^{se}[\bullet]|e\rangle=\mathcal{T}_{t,0}^{(e)}\langle
e|\bullet|e\rangle,$ (25)

where $\mathcal{T}_{t,0}^{(e)}$ is a system (density matrix) propagator that
parametrically depends on each environmental state $\\{|e\rangle\\}.$ The
condition (25) is fulfilled, for example, when the environment and interaction
Hamiltonians commutate

$[H_{e},H_{I}]=0.$ (26)

Introducing the condition (25) into Eqs. (7) and (8) it is possible to check
that $C_{pf}(t,\tau)|_{\breve{y}}\overset{d}{\neq}0,$ and
$C_{pf}(t,\tau)|_{\breve{y}}\overset{r}{=}0.$ This last equality does not
imply that the environment in not affected. It emerges because the system
state assumes the structure

$\rho_{t}^{s}=\mathrm{Tr}_{e}(\mathcal{G}_{t,0}^{se}[\rho_{0}^{s}\otimes\rho_{0}^{e}])=\sum_{e}\langle
e|\rho_{0}^{e}|e\rangle\mathcal{T}_{t,0}^{(e)}[\rho_{0}^{s}].$ (27)

Therefore, the system evolution can be written as a statistical superposition
of unitary maps, quite similar to Eq. (15). In consequence, for unitary
system-environment models the condition
$C_{pf}(t,\tau)|_{\breve{y}}\overset{r}{=}0$ allows to detect when the system
dynamics (even between measurements) can be represented by a Hamiltonian
ensemble, property that has been of interest in the recent literature nori .

##  IV Example

In this section we consider an explicit example of the dynamics discussed
previously. The quantum system $(s),$ taken for simplicity as a two-level
system, interacts with an incoherent environment $(e)$ (see Sect. 3.2.2),
which here is defined by four discrete states denoted as
$\\{|1\rangle,|2\rangle,|3\rangle,|4\rangle\\}.$ Correspondingly, the
bipartite system-environment state is written as

$\rho_{t}^{se}=\sum_{k=1,2,3,4}\tilde{\rho}_{k}(t)\otimes|k\rangle\langle k|.$
(28)

The system and environment states then read

$\rho_{t}^{s}=\sum_{k=1,2,3,4}\tilde{\rho}_{k}(t),\ \ \ \ \ \ \
\rho_{t}^{e}=\sum_{k=1,2,3,4}p_{k}(t)|k\rangle\langle k|,$ (29)

where $p_{k}(t)=\mathrm{Tr}_{s}[\tilde{\rho}_{k}(t)].$ The evolution of the
unnormalized system states $\\{\tilde{\rho}_{k}(t)\\}_{k=1}^{k=4}$ is taken as

$\displaystyle\frac{d\tilde{\rho}_{4}(t)}{dt}$ $\displaystyle=$
$\displaystyle-\gamma\tilde{\rho}_{4}(t)+\phi\sum_{k=1,2,3}\sigma_{k}\tilde{\rho}_{k}(t)\sigma_{k},$
(30a) $\displaystyle\frac{d\tilde{\rho}_{k}(t)}{dt}$ $\displaystyle=$
$\displaystyle-\phi\tilde{\rho}_{k}(t)+\left(\frac{\gamma}{3}\right)\sigma_{k}\tilde{\rho}_{4}(t)\sigma_{k},\
\ \ k=1,2,3.\ \ \ \ \ \ \ \ $ (30b) In this expression, $\gamma$ and $\phi$
are characteristic coupling rates. Furthermore, the set of Pauli matrixes is
denoted as
$(\sigma_{x},\sigma_{y},\sigma_{z},\mathrm{I})\leftrightarrow(\sigma_{1},\sigma_{2},\sigma_{3},\sigma_{4}),$
where $\mathrm{I}$ is the identity matrix in the two-dimensional system
Hilbert space. From Eq. (30), the evolution of the environment populations is
defined by the following classical master equation
$\displaystyle\frac{dp_{4}(t)}{dt}$ $\displaystyle=$ $\displaystyle-\gamma
p_{4}(t)+\phi\sum_{k=1,2,3}p_{k}(t),$ (31a)
$\displaystyle\frac{dp_{k}(t)}{dt}$ $\displaystyle=$ $\displaystyle-\phi
p_{k}(t)+\left(\frac{\gamma}{3}\right)p_{4}(t),\ \ \ k=1,2,3.\ $ (31b) This
equation is completely independent of the system degrees of freedom. Thus, the
evolution (30) has a simple interpretation. When the environment suffer the
transition $|4\rangle\overset{\gamma/3}{\rightarrow}|k\rangle$ or the
transition $|k\rangle\overset{\phi}{\rightarrow}|4\rangle$ $(k=1,2,3),$ the
transformation $\sigma_{k}\bullet\sigma_{k}$ is conditionally applied over the
open quantum system.

Eq. (30) can be solved after specifying the bipartite initial conditions. We
consider a separable state, $\rho_{0}^{se}=\rho_{0}^{s}\otimes\rho_{0}^{e},$
which implies $\tilde{\rho}_{k}(0)=\rho_{0}^{s}p_{k}(0).$ In general, each
auxiliary state $\tilde{\rho}_{k}(t)$ can be written as a superposition of the
Pauli channels acting on the initial system state $\rho_{0}^{s},$ that is,

$\tilde{\rho}_{k}(t)=\sum_{j=1,2,3,4}g_{k}^{j}(t)\sigma_{j}\rho_{0}^{s}\sigma_{j},$
(32)

where $\\{g_{k}^{j}(t)\\}$ are (sixteen) scalar functions that depend on time.
Their initial conditions are $g_{k}^{4}(0)=p_{k}(0)$ and $g_{k}^{j}(0)=0,$
with $j=1,2,3,$ and $k=1,2,3,4.$ The evolution of the set $\\{g_{k}^{j}(t)\\}$
follows after inserting the previous expression for $\tilde{\rho}_{k}(t)$ into
Eq. (30). Consistently with their definition,
$p_{k}(t)=\mathrm{Tr}_{s}[\tilde{\rho}_{k}(t)],$ the environment populations
are recovered as

$p_{k}(t)=\sum_{j=1,2,3,4}g_{k}^{j}(t).$ (33)

###  IV.1 Depolarizing dynamics

The evolution of the auxiliary states Eq. (30) is (structurally) the same for
the states
$\\{\tilde{\rho}_{1}(t),\tilde{\rho}_{2}(t),\tilde{\rho}_{3}(t)\\}.$ Thus, if
we consider environment initial conditions where $p_{1}(0)=p_{2}(0)=p_{3}(0),$
from Eqs. (29) and (32) it follows that the solution map
$\rho_{0}^{s}\rightarrow\rho_{t}^{s}$ must be a depolarizing channel nielsen ,
that is,

$\rho_{t}^{s}=w(t)\rho_{0}^{s}+\frac{1-w(t)}{3}\sum_{k=1,2,3}\sigma_{k}\rho_{0}^{s}\sigma_{k},$
(34)

where the positive weight $w(t),$ from Eq. (32), follows as

$w(t)=\sum_{k=1,2,3,4}g_{k}^{4}(t).$ (35)

Consistently, $[1-w(t)]/3=\sum_{k=1,2,3,4}g_{k}^{j}(t),$ with $j=1,$ $2,$ $3.$

The more natural initial conditions for the environment are their stationary
populations $p_{k}^{\infty}\equiv\lim_{t\rightarrow\infty}p_{k}(t),$ where
$p_{k}(t)$ are defined by Eq. (31). Straightforwardly, we get

$p_{4}^{\infty}=\frac{\phi}{\gamma+\phi},\ \ \ \ \ \ \ \
p_{k}^{\infty}=\frac{1}{3}\frac{\gamma}{\gamma+\phi}\ \ \ \ \ (k=1,2,3).$ (36)

Under the assumption $p_{k}(0)=p_{k}^{\infty},$ after getting the set
$\\{g_{k}^{j}(t)\\}$ in an explicit analytical way, the function $w(t)$ that
characterizes the depolarizing channel Eq. (34) can be written as

$w(t)=\frac{(\gamma^{2}+3\phi^{2})}{3(\gamma+\phi)^{2}}+\frac{4\gamma\phi}{3(\gamma+\phi)^{2}}e^{-(\gamma+\phi)t}+\frac{2\gamma}{3(\gamma+\phi)}e^{-\phi
t},$ (37)

which consistently satisfies $w(0)=1.$ Furthermore,
$\lim{}_{t\rightarrow\infty}w(t)\neq 0.$ On the hand, the environment dynamics
is stationary, that is, $p_{k}(t)=p_{k}(0)=p_{k}^{\infty}$ [Eq. (36)].

###  IV.2 Operational vs. non-operational quantum non-Markovianity

In the non-operational approach, quantum non-Markovianity is defined by the
revivals in the trace distance between two different initial states, Eq. (3).
By using that $(\mathrm{I}/2)=(\rho+\sum_{k=1,2,3}\sigma_{k}\rho\sigma_{k})/4$
nielsen , the depolarizing map (34) can be rewritten as
$\rho_{t}^{s}=w(t)\rho_{0}^{s}+(1/3)[1-w(t)](2\mathrm{I}-\rho_{0}^{s}).$ Thus,
the trace distance straightforwardly can be written as

$D[\rho_{t}^{s},\sigma_{t}^{s}]=\left|\frac{4w(t)-1}{3}\right|D[\rho_{0}^{s},\sigma_{0}^{s}]\equiv
d(t)D[\rho_{0}^{s},\sigma_{0}^{s}]$ (38)

where $D[\rho_{0}^{s},\sigma_{0}^{s}]$ is the trace distance between the two
initial states $\rho_{0}^{s}$ and $\sigma_{0}^{s}.$ Notice that the decay of
the trace distance does not depends on the initial states, being dictated by
the function $d(t).$

In Fig. 1(a) we plot the function $d(t)$ for different values of the
characteristic parameter $\phi/\gamma.$ As expected from Eq. (37),
$D[\rho_{t}^{s},\sigma_{t}^{s}]$ decay in a monotonous way without developing
any revival. Thus, under the trace distance criteria, the dynamics is
Markovian and there is not any environment-to-system backflow of information.
Nevertheless, notice that for any value of $\phi/\gamma$ system-environment
correlations are built up during the dynamics [see Eq. (28)]. This feature,
which is irrelevant for the TD decay behavior, is relevant for the CPF
correlation.

Figure 1: (a) Decay of the trace distance $d(t)$ [Eq. (38)] corresponding to
the model (30). (b) Time dependence of the CPF correlation
$C_{pf}(t,t)|_{\breve{y}}$ in the deterministic scheme CPF  corresponding to
the same model. The value of the quotient $\phi/\gamma$ is indicated in each
plot.

In the operational approach, the presence of memory effects is witnessed by
the CPF correlation [Eq. (6)] in the deterministic scheme. We assume that the
three measurements are projective ones, all of them being performed in the
$z$-direction of the Bloch sphere. Furthermore the initial condition of the
system is taken as $\rho_{0}^{s}=|\psi\rangle\langle\psi|,$ where
$|\psi\rangle$ is an eigenstate of the $x$-Pauli matrix. Explicit general
expression for $C_{pf}(t,\tau)|_{\breve{y}}$ in terms of the coefficients
$\\{g_{k}^{j}(t)\\}$ can be found in Ref. multi  (see corresponding Appendix
D). Under the previous assumptions, the CPF correlation can be obtained in an
analytical way, which is written in CPF . Simple expressions are obtained for
specific values of the decay rates. For example, for $\phi=\gamma$ it follows

$\displaystyle
C_{pf}(t,\tau)|_{\breve{y}}\overset{d}{=}\frac{4}{81}(1-e^{-\gamma
t})(1-e^{-\gamma\tau})$ (39) $\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\times(2+e^{-\gamma t}+e^{-\gamma\tau}+5e^{-\gamma(t+\tau)}).$

Due to the symmetry of the problem, in all cases $C_{pf}(t,\tau)|_{\breve{y}}$
does not depend on the value of the conditional $\breve{y}=\pm 1.$ In Fig.
1(b) we plot the CPF correlation at equal times, $C_{pf}(t,t)|_{\breve{y}},$
for different values of $\phi/\gamma.$ In contrast to the non-operational
approach, here for all possible values of the characteristic parameter
$\phi/\gamma$ it is fulfilled $C_{pf}(t,\tau)|_{\breve{y}}\overset{d}{\neq}0,$
which indicates a non-Markovian regime. In fact, the system is strongly
correlated with the environment [Eq. (28)].

The system-environment correlations emerges due to a unidirectional dependence
of the system dynamics on the environment transitions [Eq. (30)]. In fact, the
environment populations do not depend on the system degrees of freedom [see
Eq. (31)]. These properties are relevant in the random scheme and imply that
$C_{pf}(t,\tau)|_{\breve{y}}\overset{r}{=}0$ [Eq. (13)]. This result is valid
for arbitrary measurement processes, indicating in the operational approach,
the absence of any environment-to-system backflow of information.

###  IV.3 Environment-to-system backflow of information

In the previous section we concluded that both approaches differ in the
classification of the dynamics (Markovian vs. non-Markovian), but (due to
different reasons) agree in the absence of any environment-to-system backflow
of information. Here, we show that in general both approaches also differ in
this last aspect. Different mechanisms can be proposed for getting a revival
in the trace distance Eq. (38).

####  IV.3.1 Slow modulation of the stationary environment state

First, we consider the same model [Eq. (30)], but in addition it is assumed
that the characteristic rates are time dependent,
$\gamma\rightarrow\gamma(t),$ $\phi\rightarrow\phi(t),$ with

$\gamma(t)=\gamma[1+b(t)]>0,\ \ \ \ \ \ \phi(t)=\phi[1-b(t)]>0.$ (40)

Here, $b(t)$ is an arbitrary function of time that fulfill the constraint
$-1<b(t)<1.$ The previous structure is chosen for simplifying the argument and
calculus. Nevertheless, we remark that similar dependences can be implemented
in different experimental situations (see for example Ref. Udo ). The more
relevant aspect is that the assumption (40) can be implemented by affecting
solely the environmental degrees of freedom [see Eq. (31)].

In addition, in Eq. (40) it is assumed that

$\left|\frac{d}{dt}b(t)\right|\ll\gamma,\ \ \ \
\left|\frac{d}{dt}b(t)\right|\ll\phi.$ (41)

Hence, the time dependence of $b(t)$ can be considered slow with respect to
the decay times $(1/\gamma)$ and $(1/\phi).$ Consequently, the full dynamics
can be described in an adiabatic approximation, where the full bipartite
system in the long time regime $(\gamma t\gg 1,\ \phi t\gg 1)$ rapidly adjust
to the instantaneous values of $\gamma(t)$ and $\phi(t).$ In particular, in
this regime, the environment populations, from Eq. (36), can be written as

$p_{4}^{\infty}(t)\simeq\frac{\phi}{\gamma+\phi}[1-b(t)],\ \ \ \
p_{k}^{\infty}(t)\simeq\frac{1}{3}\frac{\gamma}{\gamma+\phi}[1+b(t)],$ (42)

where $k=1,2,3.$ For simplicity, we assumed that
$(\gamma-\phi)\ll(\gamma+\phi),$ which allows to approximate
$\gamma(t)+\phi(t)=(\gamma+\phi)+b(t)(\gamma-\phi)\simeq(\gamma+\phi).$

In the long time regime, the non-operational approach is characterized by the
value $\lim_{t\rightarrow\infty}w(t)\neq 0$ [see Eqs. (37) and (38)]. For
time-independent rates this quantity can be written in terms of the stationary
populations $\\{p_{k}^{\infty}\\}_{k=1}^{k=4}$ [Eq. (36)] as
$\lim_{t\rightarrow\infty}w(t)=[p_{4}^{\infty}]^{2}+\sum_{k=1,2,3}[p_{k}^{\infty}]^{2}.$
Given that in the slow modulation regime [Eq. (41)] these values become time
dependent, $p_{k}^{\infty}\longrightarrow p_{k}^{\infty}(t)$ [Eq. (42)], it
follows that

$w(t)\overset{slow}{\simeq}[p_{4}^{\infty}(t)]^{2}+\sum_{k=1,2,3}[p_{k}^{\infty}(t)]^{2},\
\ \ \ \gamma t\gg 1,\ \phi t\gg 1.$ (43)

Therefore, under the previous hypothesis, the stationary values of the TD in
Fig. 1(a) $[d(t)=\left|4w(t)-1\right|/3]$ become proportional to the arbitrary
function $b(t).$ This result implies that one can get arbitrary revivals in
the trace distance [Eq. (38)] by choosing different time-dependences of the
function $b(t).$ Alternatively, an arbitrary environment-to-system backflow of
information can be produced by changing solely in a slow way the
(“stationary”) environment populations. Nevertheless, we remark that the full
dynamics is essentially the same than in the static-rate case. While one can
associate the revivals in the TD to the system-environment correlations, these
correlations have the same origin and structure than in absence of revivals,
that is Fig. 1(a) (static case) and when $b(t)$ does not lead to revivals.

In the deterministic scheme, the operational approach is characterized by the
stationary value CPF

$\lim_{\begin{subarray}{c}t\rightarrow\infty\\\
\tau\rightarrow\infty\end{subarray}}C_{pf}(t,\tau)|_{\breve{y}}\overset{d}{=}\frac{8\gamma(\gamma-3\phi)^{2}(\gamma+3\phi)}{81(\gamma+\phi)^{4}},$
(44)

which can also be written in terms of $\\{p_{k}^{\infty}\\}_{k=1}^{k=4}$ [Eq.
(36)]. Thus, under the same conditions that guarantee the slow modulation
regime [Eqs. (41) and (42)], the stationary values of
$C_{pf}(t,t)|_{\breve{y}}$ plotted in Fig. 1(b) also become proportional to
the function $b(t).$ Nevertheless, in this approach this property does not
implies the presence of any backflow of information. In fact, given that the
environment state does not depend at all on the system degrees of freedom,
even in the slow modulation regime, it follows that
$C_{pf}(t,\tau)|_{\breve{y}}\overset{r}{=}0$ [Eq. (13)]. In this way, it is
clear that both the non-operational and operational approaches also strongly
disagree in this aspect.

####  IV.3.2 Quantum coherent contributions in the environment dynamics

The system-environment dynamics associated to the depolarizing channel [Eq.
(30)] can alternatively be represented through a Lindblad equation. In fact,
the evolution of the bipartite state $\rho_{t}^{se}$ can be written as

$\displaystyle\frac{d\rho_{t}^{se}}{dt}$ $\displaystyle=$
$\displaystyle+\frac{\gamma}{3}\sum_{k=1,2,3}(B_{k}\sigma_{k}[\rho_{t}^{se}]\sigma_{k}B_{k}^{\dagger}-\frac{1}{2}\\{B_{k}^{\dagger}B_{k},\rho_{t}^{se}\\}_{+})$
(45)
$\displaystyle+\phi\sum_{k=1,2,3}(B_{k}^{\dagger}\sigma_{k}[\rho_{t}^{se}]\sigma_{k}B_{k}-\frac{1}{2}\\{B_{k}B_{k}^{\dagger},\rho_{t}^{se}\\}_{+})$
$\displaystyle-i[H_{e},\rho_{t}^{se}],$

where the bath operators are $B_{k}\equiv|k\rangle\langle 4|,$ $k=1,2,3.$ As
before, $\\{|1\rangle,|2\rangle,|3\rangle,|4\rangle\\}$ are the environment
base. Defining the states $\tilde{\rho}_{k}\equiv\langle
k|\rho_{t}^{se}|k\rangle,$ it is simple to check that the first two lines of
the previous Lindblad dynamics recover the time evolution introduced in Eq.
(30).

From Eq. (45) it is simple to check that the bath state
$(\rho_{t}^{e}=\mathrm{Tr}_{s}[\rho_{t}^{se}])$ obeys a Lindblad equation
that, even with the extra contribution $-i[H_{e},\rho_{t}^{se}],$ is
independent of the system degrees of freedom. Thus, the environment is still a
casual bystander one [see Eqs. (20) and (21)]. In order to obtain a (system)
depolarizing channel [Eq. (34)] the symmetry between the bath states
$\\{|1\rangle,|2\rangle,|3\rangle\\}$ must be granted. For example, the
Hamiltonian

$H_{e}=\frac{\Omega}{2}\sum_{k=1,2,3}(|k\rangle\langle 4|+|4\rangle\langle
k|),$ (46)

fulfill this property.

Figure 2: Decay of the trace distance $d(t)$ [Eqs. (38) and (49)]
corresponding to the model (45) with $\phi=\gamma$ for different values of the
Hamiltonian frequency $\Omega/\gamma.$

In consistence with the solution defined by Eqs. (28) and (32), here the
bipartite state is written as

$\rho_{t}^{se}=\sum_{k=1,2,3,4}(\sigma_{k}\rho_{0}^{s}\sigma_{k})\otimes\varrho_{t}^{k},$
(47)

where $\\{\varrho_{t}^{k}\\}$ are states in the environment Hilbert space. In
order to obtain analytical treatable solutions we assume the bipartite initial
condition

$\rho_{0}^{se}=\rho_{0}^{s}\otimes\rho_{0}^{e}=\rho_{0}^{s}\otimes|4\rangle\langle
4|.$ (48)

Under this assumption $(\rho_{0}^{e}=|4\rangle\langle 4|),$ given that the
underlying system stochastic dynamics associated to Eq. (45) is the same than
in the incoherent case [Eq. (30)], it follows that the system state goes back
to the initial condition $\rho_{0}^{s}$ whenever the environment goes back to
the state $|4\rangle.$ This property straightforwardly follows from
$\sigma_{k}^{2}=\mathrm{I}.$ Therefore, under the assumption (48), here the
depolarizing map Eq. (34) is defined with the function

$w(t)=\mathrm{Tr}_{e}[\varrho_{t}^{4}]=\langle 4|\rho_{t}^{e}|4\rangle,$ (49)

where $\rho_{t}^{e}$ is the density matrix of the environment. Consistently
$[1-w(t)]/3=\mathrm{Tr}_{e}[\varrho_{t}^{k}]=\langle k|\rho_{t}^{e}|k\rangle,$
with $k=1,$ $2,$ $3.$ In consequence, the decay of the trace distance is
proportional to the bath population $\langle 4|\rho_{t}^{e}|4\rangle.$ Its
explicit analytical expression is rather complex and non-informative p4 .

In this alternative situation, it is clear that $H_{e}$ induces intrinsic
quantum coherent oscillations in the environment dynamics, which in turn may
lead to oscillations in the trace distance [Eq. (38)]. In Fig. 2 we plot the
TD decay $d(t)=\left|4w(t)-1\right|/3$ taking $\phi=\gamma$ and for different
values of $\Omega/\gamma.$ When $\Omega/\gamma<1,$ a monotonous decay is
observed. Nevertheless, for $\Omega/\gamma>1$ revivals in the TD are observed.

The CPF correlation in the deterministic scheme cannot be calculated in an
analytical way. Nevertheless, given that the system dynamics is still
controlled by the environment (self) transitions, it follows that
$C_{pf}(t,\tau)|_{\breve{y}}\overset{d}{\neq}0.$ Thus, the dynamics becomes
non-Markovian in both approaches $(\Omega/\gamma>1).$ Nevertheless, given that
the environment is a casual bystander one, in the random scheme it is valid
that $C_{pf}(t,\tau)|_{\breve{y}}\overset{r}{=}0$ [Eq. (13)] for any value of
$\Omega/\gamma.$ Consequently, in the same way as in the previous model [Eq.
(40)] the non-operational and operational approaches gives different results
about the presence of environment-to-system backflows of information.

##  V Summary and conclusions

The interpretation of quantum memory effects in terms of an environment-to-
system backflow of information is still under a vivid debate. In this
contribution we presented a partial view of this problem by comparing how this
concept is introduced an interpreted in non-operational and operational
approaches to quantum non-Markovianity.

Our main contribution is a comparison between both formalisms for different
environment models. We considered casual bystander environments, which are
characterized by a density matrix that does not depend on the system degrees
of freedom. This class cover classical statistical mixtures of Markovian
dynamics [Eq. (14)], interaction with stochastic classical degrees of freedom
[Eq. (16)], and also environmental quantum degrees of freedom [Eq. (19)]. In
addition, we considered unitary system-environment models [Eq. (22)].

As a non-operational approach we used the TD between two system states with
different initial conditions. This formalism is characterized by the bound Eq.
(II.1). We have argued that, in general, it is not possible to predict if for
a given model the TD presents or not revivals in its time behavior. This
property is valid for all environmental models. In the case of casual
bystander ones, the previous feature represent an obstacle for giving a
consistent physical interpretation of any environment-to-system backflow of
information defined as revivals in the TD [Eq. (3)]. In fact, for these
dynamics the system-environment correlations emerges due to a unidirectional
dependence of the system dynamics in the state of the environment and its
transitions. In particular, for stationary environments it is not possible to
know when the system-environment correlations lead to the presence or absence
of backflows of information. The possibility of getting monotonous decay
behaviors of the TD for unitary interaction models also represents an
undesirable property because in general the environment state is modified by
its interaction with the system.

As an operational approach we used a CPF correlation [Eq. (6)], which is
defined by three consecutive system measurement processes. Both a
deterministic and random schemes were considered [with associated joint
probabilities Eqs. (7) and (8)]. In the case of casual bystander environments,
the CPF correlation in the deterministic scheme does not vanish, while in the
random scheme it vanishes identically for any chosen measurement observables
[Eq. (13)]. Thus, in this approach, any casual bystander environment leads to
a non-Markovian system dynamics but not any bidirectional information flow is
detected. In the case of Hamiltonian models, in general, in both schemes the
CPF correlation does not vanish indicating a non-Markovian system dynamics and
the presence of bidirectional information flows [Eq. (24)]. An undesirable
exception to this last property emerges when the system dynamics can
equivalently be represented by a random unitary map [Eqs. (25) and (27)].

As an specific example we considered a system coupled to an environment able
to induce a depolarizing dynamics [Eqs. (30), (40) and (45)]. We have found
that both approaches differ in the Markovian and non-Markovian regimes as well
in the presence or absence of environment-to-system backflows of information.

In general, both operational and non-operational approaches to quantum non-
Markovianity provide necessary and complementary points of view for defining
and understanding memory effects in open quantum systems. The present results
shed light on some conceptual differences and properties of these approaches.
They may be useful for extending the application of these formalisms for the
understanding of memory effects induced by structured or spatially extended
environments.

## Acknowledgments

This paper was supported by Consejo Nacional de Investigaciones Científicas y
Técnicas (CONICET), Argentina.

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