Quantum Non-Markovian Environment-to-System Backflows of Information: Nonoperational vs. Operational Approaches

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 nonoperational 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 [1][2][3][4].Nevertheless, the characterization and definition of these effects is quite different in both regimes [5,6].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 [1].
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 nonoperational 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 [5,6]).The first proposals correspond to deviations of the system propagator from divisibility [7] and a non-monotonous behavior of the trace distance (TD) between two distinct system states [8].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 [9][10][11], the precise assessment of this concept is still under debate [12][13][14][15][16][17][18][19].
The basic idea of operational approaches is to appeal to the standard definition of memory effects in terms of probabilities [1].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 [20,21].The study and understanding of this approach has been performed in the recent literature [22][23][24][25][26][27], including alternative definitions and analysis of information flows [28,29].
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 nonoperational memory witness we take the TD between two different system initial states [8], taking also into account the bounds on its revival behavior that has been characterized recently [19].As an operational memory witness we consider a conditional past-future (CPF) correlation [21], both in a deterministic and random schemes [29].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 [13] and coherent casual bystander environments [30], 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 [2].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 environmentto-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 [8,19] and operational [21,29] 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.

A. 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 [31] 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 [31].As is well known, completely positive transformations lead to very specific contractive properties for different distance measures and entropic quantities [32].For example, the TD between two arbitrary density matrixes ρ and σ, defined as D(ρ, σ) ≡ (1/2)Tr|ρ − σ|, under a completely positive transformation Φ fulfills the inequality D(Φ[ρ], Φ[σ]) ≤ D(ρ, σ).Consequently, it is possible to define quantum Markovianity by the condition [8] where ρ s t and σ s t are two arbitrary evolved system states that differ in their initial conditions, ρ s 0 = σ s 0 .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 ≥ 0 and τ > 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. [19] by establishing the inequality Here, ρ se t and σ se t are the evolved system-environment states with initial conditions ρ se 0 = ρ s 0 ⊗ ρ e 0 and σ se 0 = σ s 0 ⊗σ e 0 .As usual, the system and bath states follows from a partial trace operation, ρ s t = Tr e [ρ se t ] and ρ e t = Tr s [ρ se t ].
The asymmetry between system and environment (s e) is introduced by taking in both cases the same initial environmental state, ρ e 0 = σ e 0 .The result (2) only relies on the triangle inequality fulfilled by the TD.Thus, it is valid for arbitrary systemenvironment models.In addition, this expression allows to bounding the environment-to-system backflow of information defined by the "revivals" The remaining (bounding) contributions in the rhs of Eq. ( 2) have a clear physical interpretation.One can relate the contribution D(ρ e t , σ e t ) to changes in the environmental state, while the terms D(ρ se t , ρ s t ⊗ρ e t )+D(σ se t , σ s t ⊗ σ e t ) measure the correlations established between the system and the environment [19].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 (2) remains valid when the TD is replaced by a telescopic relative entropy and the square root of a quantum Jensen-Shannon divergence [19].Thus, the interpretation of the bounds remains the same when using these entropic quantities.

B. Operational approach
In a probabilistic frame, given a sequence of system states x → y → 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 [1].
For quantum systems, the definition of Markovianity in terms of probabilities unavoidably implies performing a set of system measurement processes.In Ref. [20], 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 [21].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). ( This result follows straightforwardly by using that P (z, x|y) = P (z, y, x)/P (y), where P (y) = 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 [21] C pf (t, τ )| y d/r = z,x zx[P (z, x|y) − P (z|y)P (x|y)], (6) where {x} and {z} are the (past and future) measurement outcomes.The time dependence (t, τ ) emerges because the past, present, and future measurements are performed at the initial time t = 0, at time t, and t + τ respectively.Evidently C pf (t, τ )| y vanishes in a (probabilistic) Markovian regime [Eq.( 5)].
In Eq. ( 6) it was introduced the change y → y, which is stretchy related with the definition of memory effects and information flows in this approach.Two different measurement schemes are necessary [29].In a deterministic scheme (denoted with the supra d), after the intermediate measurement (whose outcome defines the conditional property) none change is introduced.Hence, y = y.In a random scheme (denoted with the supra r), after the intermediate measurement the system state is randomly chosen (y → 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 ] 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, τ )| y r = 0], by definition, detects the presence of environment-tosystem backflows of information (or bidirectional systemenvironment information flows).This relation is motivated by the complementary case C pf (t, τ )| y r = 0 that applies when the environment (which induces the memory effects C pf (t, τ )| y d = 0) is unperturbed by its coupling with the system [29].
The previous characteristics of the deterministic and random schemes can be easily understood from the properties of projective measurements performed over bipartite systems [30].Interestingly, the formalism remains the same and is also valid for purely (classically) incoherent system-environment arrangements.

C. 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. ( 2) 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, y, x).Assuming that the three measurements are projective ones, in the deterministic scheme it reads [29] P (z, y, x) P (x) ), (7) while in the random scheme it is [29] P (z, y, x) P (x) In these expressions, G se t+τ,t is the bipartite propagator between t and t + τ.In addition, E m ≡ |m m| and ρ m ≡ |m m| [m = z, y, x] represent the (positive) effect measurement operators and post-measurement states respectively.The set {|m } [m = z, y, x] are the eigenstates of each measured observable.Furthermore, ρ se x ≡ ρ x ⊗ ρ e 0 and P (x) = x|ρ s 0 |x .The random scheme is parametrized by an arbitrary conditional probability ℘(y|x) that defines the change in the system state (y → 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 ρ se t evolves as where L s and L e define the self-dynamics of the system and the environment respectively, while L se defines their mutual interaction.This interaction term may be unitary or includes dissipative couplings.

A. Born-Markov approximation
For systems weakly coupled to their environments, the Born-Markov approximation [2] allows to write the bipartite state as where ρ s t is the system state, while ρ e 0 is the (almost) unperturbed environment state.
When this approximation is valid, in the nonoperational approach, it is simple to check that Eq. ( 2) reduces to Eq. ( 1).In fact, D(ρ e t , σ e t ) = D(ρ se t , ρ s t ⊗ ρ e t ) = D(σ se t , σ s t ⊗ σ e t ) = 0. Furthermore, ρ s t can be well approximated by a Lindblad equation, which guarantees the absence of any revival in D(ρ s t , σ s t ).Thus, the dynamics is Markovian.
In this case [Eq.( 10)] both approaches coincides.Strong differences appear in the cases studied below.

B. 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 ρ e t = Tr s [ρ se t ] of any degree of freedom of the system.Alternatively, the time evolution of ρ e t 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 L se in the general evolution ( 9) must be restricted such that where A is an arbitrary superoperator acting on ρ e t 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 [30] or a classical (incoherent) one [13].
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. ( 2).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, ρ e t = ρ e 0 , that is, the environment begins in its stationary state.In this case, Eq. ( 2) reduces to Even in this case (ρ e t = ρ e 0 ), 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 D(ρ se t , ρ s t ⊗ ρ e 0 ) + D(σ se t , σ s t ⊗ ρ e 0 ).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 [ρ e t = Tr s (ρ se t )] 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)] In this approach, the property C pf (t, τ )| y d = 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) systemenvironment correlations.On the other hand, the property C pf (t, τ )| y 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).

Classical mixture of quantum Markovian dynamics
Given a set of different system Lindblad superoperators {L c s }, which may include both unitary and dissipative contributions, and given a set of normalized positive weights {p c }, c p c = 1, a classical statistical mixture of Markovian dynamics is defined by the bipartite state Here, {|c c|} is a set of projectors associated to the environment space.The marginal system and environment states read Memory effects in this kind of non-Markovian system dynamics has been explored in the literature [33].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(tL c s )[ρ s 0 ].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 , but not any bidirectional information flow, C pf (t, τ )| y r = 0.

Interaction with stochastic classical degrees of freedom
When the system interacts with stochastic classical degrees of freedom, the bipartite state can be written as In contrast to the previous case [Eq.( 14)], the weights {p c (t)} are time-dependent and the evolution of the states {ρ c t } may involve coupling between all of them.In fact, under the constraint (11), the more general evolution can be written as [13] Here, ρc t ≡ p c (t)ρ c t .Thus, p c (t) = Tr s (ρ c t ).Furthermore, {S cc ′ } are arbitrary completely positive system transformations, which are trace preserving, Tr s (S cc ′ [ρ]) = Tr s (ρ).Consequently, the environment probabilities {p c (t)} obey a classical master equation which in turn shows the role played by the coupling rates {γ c ′ 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 [34]).
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→∞ 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 D(ρ se t , ρ s t ⊗ ρ e 0 ) + D(σ se t , σ s t ⊗ ρ e 0 ) in Eq. (12) [or in general in Eq. ( 2)] 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 L c s are "active" whenever the environment is in the state |c c|.Furthermore, the system suffers the transformation ρ → S c ′ c [ρ] whenever the environment "jumps" between the states c → c ′ .This is the physical mechanism that leads to the systemenvironment 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 {L c s }, the superoperators {S cc ′ }, and rates {γ c ′ c }, the operational approach is characterized by Eq. ( 13), that is, the dy- = 0] without the developing of any bidirectional system-environment information flow [C pf (t, τ )| y r = 0].

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 In contrast to Eq. ( 16), due to the quantum nature of the environment, the projectors {|c t c t |} are timedependent.In fact, they define the base in which the environment density matrix ρ e t is diagonal.The more general bipartite evolution (9) under the constraint (11), in its diagonal representation, is given by [30] where {•, •} + is an anticommutator operation.Furthermore, {B α } are arbitrary environment operators, while S α are completely positive trace-preserving system superoperators.The rates {Γ α } set the environment dynamics.
In fact, 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 [35]).
The physical interpretation of the evolution ( 20) is quite similar to that of Eq. ( 17).In fact, here the application of the system superoperators S α occur whenever the environment suffer a transition associated to the operators B α .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 ρ e 0 = lim t→∞ ρ e t [where ρ e t 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).

C. 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 [36] [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 [37].
The total Hamiltonian is written as Each contribution correspond respectively to the system, environment, and interaction Hamiltonians respectively.The bipartite propagator is In the non-operational approach, each contribution in the rhs of Eq. (2) 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. ( 2) 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 Both inequalities can be supported by performing a perturbation theory based on projectors techniques [24].Consistently, it has been shown that even close to the validity of a Born-Markov approximation the operational approach can detect memory effects [27].
The inequality C pf (t, τ )| y d = 0 implies that the system dynamics is non-Markovian (system-environment correlations are developed during the evolution), while C pf (t, τ )| y r = 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, τ )| y r = 0.While this property is certainly undesirable, this case has a clear physical interpretation.It emerges when, in a given environmental base {|e }, the diagonal part of the bipartite propagator ( 23) can be written as where t,0 is a system (density matrix) propagator that parametrically depends on each environmental state {|e }.The condition ( 25) is fulfilled, for example, when the environment and interaction Hamiltonians commutate Introducing the condition (25) into Eqs.( 7) and ( 8) it is possible to check that C pf (t, τ )| y d = 0, and C pf (t, τ )| y r = 0.This last equality does not imply that the environment in not affected.It emerges because the system state assumes the structure Therefore, the system evolution can be written as a statistical superposition of unitary maps, quite similar to Eq. ( 15).In consequence, for unitary systemenvironment models the condition C pf (t, τ )| y 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 [38].

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 , |2 , |3 , |4 }.Correspondingly, the bipartite systemenvironment state is written as The system and environment states then read In this expression, γ and φ are characteristic coupling rates.Furthermore, the set of Pauli matrixes is denoted as (σ x , σ y , σ z , I) ↔ (σ 1 , σ 2 , σ 3 , σ 4 ), where 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 This equation is completely independent of the system degrees of freedom.Thus, the evolution ( 30 Eq. ( 30) can be solved after specifying the bipartite initial conditions.We consider a separable state, ρ se 0 = ρ s 0 ⊗ ρ e 0 , which implies ρk (0) = ρ s 0 p k (0).In general, each auxiliary state ρk (t) can be written as a superposition of the Pauli channels acting on the initial system state ρ s 0 , that is, where {g j k (t)} are (sixteen) scalar functions that depend on time.Their initial conditions are g 4 k (0) = p k (0) and g j k (0) = 0, with j = 1, 2, 3, and k = 1, 2, 3, 4. The evolution of the set {g j k (t)} follows after inserting the previous expression for ρk (t) into Eq.( 30).Consistently with their definition, p k (t) = Tr s [ρ k (t)], the environment populations are recovered as A. Depolarizing dynamics The evolution of the auxiliary states Eq. ( 30) is (structurally) the same for the states { ρ1 (t), ρ2 (t), ρ3 (t)}.
The more natural initial conditions for the environment are their stationary populations p ∞ k ≡ lim t→∞ p k (t), where p k (t) are defined by Eq. (31).Straightforwardly, we get Under the assumption p k (0) = p ∞ k , after getting the set {g j k (t)} in an explicit analytical way, the function w(t) that characterizes the depolarizing channel Eq. ( 34) can be written as which consistently satisfies w(0) = 1.Furthermore, lim t→∞ w(t) = 0. On the hand, the environment dynamics is stationary, that is, p k (t) = p k (0) = p ∞ k [Eq.(36)].

B. 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 (I/2) = (ρ + k=1,2,3 σ k ρσ k )/4 [32], the depolarizing map (34) can be rewritten as . Thus, the trace distance straightforwardly can be written as where D[ρ s 0 , σ s 0 ] is the trace distance between the two initial states ρ s 0 and σ s 0 .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 φ/γ.As expected from Eq. ( 37), D[ρ s t , σ s t ] decay in a monotonous way without developing any revival.Thus, under the trace distance criteria, the dynamics is Markovian and there is  38)] corresponding to the model (30).(b) Time dependence of the CPF correlation C pf (t, t)| y in the deterministic scheme [40] corresponding to the same model.The value of the quotient φ/γ is indicated in each plot.
not any environment-to-system backflow of information.Nevertheless, notice that for any value of φ/γ systemenvironment 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.
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 ρ s 0 = |ψ ψ|, where |ψ is an eigenstate of the x-Pauli matrix.Explicit general expression for C pf (t, τ )| y in terms of the coefficients {g j k (t)} can be found in Ref. [39] (see corresponding Appendix D).Under the previous assumptions, the CPF correlation can be obtained in an analytical way, which is written in [40].Simple expressions are obtained for specific values of the decay rates.For example, for φ = γ it follows ×(2 + e −γt + e −γτ + 5e −γ(t+τ ) ).
Due to the symmetry of the problem, in all cases C pf (t, τ )| y does not depend on the value of the conditional y = ±1.In Fig. 1(b) we plot the CPF correlation at equal times, C pf (t, t)| y , for different values of φ/γ.In contrast to the non-operational approach, here for all possible values of the characteristic parameter φ/γ it is fulfilled C pf (t, τ )| y d = 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, τ )| y 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.

C. 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-tosystem 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).

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, γ → γ(t), φ → φ(t), with 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. [41]).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 Hence, the time dependence of b(t) can be considered slow with respect to the decay times (1/γ) and (1/φ).Consequently, the full dynamics can be described in an adiabatic approximation, where the full bipartite system in the long time regime (γt ≫ 1, φt ≫ 1) rapidly adjust to the instantaneous values of γ(t) and φ(t).In particular, in this regime, the environment populations, from Eq. ( 36), can be written as where k = 1, 2, 3.For simplicity, we assumed that (γ − φ) ≪ (γ + φ), which allows to approximate γ(t) In the long time regime, the non-operational approach is characterized by the value lim t→∞ w(t) = 0 [see Eqs.(37) and (38)].For time-independent rates this quantity can be written in terms of the stationary populations {p ∞ k } k=4 k=1 [Eq.(36)] as lim t→∞ w Given that in the slow modulation regime [Eq.(41)] these values become time dependent, p ∞ k −→ p ∞ k (t) [Eq.( 42)], it follows that (43) Therefore, under the previous hypothesis, the stationary values of the TD in Fig. 1(a) [d(t) = |4w(t) − 1| /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 timedependences 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 [40] lim which can also be written in terms of {p ∞ k } k=4 k=1 [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)| 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, τ )| y r = 0 [Eq.( 13)].In this way, it is clear that both the non-operational and operational approaches also strongly disagree in this aspect.

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 ρ se t can be written as  20) and ( 21)].In order to obtain a (system) depolarizing channel [Eq.(34)] the symmetry between the bath states {|1 , |2 , |3 } must be granted.For example, the Hamiltonian fulfill this property.
In consistence with the solution defined by Eqs. ( 28) and (32), here the bipartite state is written as where {̺ k t } are states in the environment Hilbert space.In order to obtain analytical treatable solutions we assume the bipartite initial condition Under this assumption (ρ e 0 = |4 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 ρ s 0 whenever the environment goes back to the state |4 .This property straightforwardly follows from σ 2 k = I.Therefore, under the assumption (48), here the depolarizing map Eq. ( 34) is defined with the function where ρ e t is the density matrix of the environment.Consistently [1 − w(t)]/3 = Tr e [̺ k t ] = k|ρ e t |k , with k = 1, 2, 3.In consequence, the decay of the trace distance is proportional to the bath population 4|ρ e t |4 .Its explicit analytical expression is rather complex and non-informative [42].
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) = |4w(t) − 1| /3 taking φ = γ and for different values of Ω/γ.When Ω/γ < 1, a monotonous decay is observed.Nevertheless, for Ω/γ > 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, τ )| y d = 0. Thus, the dynamics becomes non-Markovian in both approaches (Ω/γ > 1).Nevertheless, given that the environment is a casual bystander one, in the random scheme it is valid that C pf (t, τ )| y r = 0 [Eq.( 13)] for any value of Ω/γ.Consequently, in the same way as in the previous model [Eq.(40)] the nonoperational 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 nonoperational 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. ( 2).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-tosystem backflow of information defined as revivals in the TD [Eq.( 3)].In fact, for these dynamics the systemenvironment 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 environmentto-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.

FIG. 1 :
FIG.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)| y in the deterministic scheme[40] corresponding to the same model.The value of the quotient φ/γ is indicated in each plot.
Lindblad equation that, even with the extra contribution −i[H e , ρ se t ], is independent of the system degrees of freedom.Thus, the environment is still a casual bystander one [see Eqs. ( where the bath operators are B k ≡ |k 4|, k = 1, 2, 3.As before, {|1 , |2 , |3 , |4 } are the environment base.Defining the states ρk ≡ k|ρ se t |k , it is simple to check that the