Maximally non-Markovian quantum dynamics without environment backflow of information

The degree of non-Markovianity allows to characterizing quantum evolutions that depart from a Markovian regime in a similar way as Schmidt number measures the degree of entanglement of pure states. Maximally non-Markovian dynamics are the analogous of maximally entangled states [D. Chru\'sci\'nski and S. Maniscalco, Phys. Rev. Lett. 112, 120404 (2014)]. Here, we demonstrate that there exists a class of maximally non-Markovian quantum evolutions where the associated environment (degrees of freedom not belonging to the system) obeys a Markovian (memoryless) dynamics, which in turn is unperturbed by the system state or dynamics. These properties imply the absence of any"physical environment-to-system backflow of information."Non-Markovian features (as usual in quantum systems coupled to dissipative classical degrees of freedom) arise from a unidirectional dependence of the system dynamics on the reservoir states.


I. INTRODUCTION
In contrast to classical stochastic systems, for open quantum systems [1] the definition of memory, or equivalently departure from a memoryless (Markovian) regime [1,2], is much more subtle. The theory of quantum dynamical semigroups [3] gives a rigorous basis for defining Markovian dynamics, where the time evolution of the system density matrix is given by the so called Lindblad equations (or Gorini-Kossakowski-Sudarshan-Lindblad equations). Assuming that all available information is encoded in the system density matrix, any departure in the properties of its propagator with respect to that of a quantum dynamical semigroup may be utilized as an indicator of the presence of memory, or equivalently, for defining a quantum non-Markovian regime [4,5].
The previous point of view was introduced in the seminal contribution of Breuer, Laine, and Piilo [6], where the memory indicator is given by the (non-monotonous) behavior of the distinguishability between two initial states. Since then, many other indicators and measures were introduced, such as for example based on the divisibility of the propagator [7][8][9][10], geometry of the set of accessible states [11], negativity of the dissipative rates in a canonical form of the quantum master equation [12], non-Markovianity degree [13,14], spectra of the dynamical map [15], quantum regression theorem [16], power spectrum [17], Fisher information flow [18], mutual information [19], fidelity [20], and accessible information [21] just to name a few [4,5]. In addition, many aspects were analyzed such as for example the relation with the definition of non-Markovian classical stochastic processes [22,23], and the correspondence between the different (inequivalent) non-Markovian indicators and measures [24][25][26][27][28][29].
The authors of Ref. [6] also introduced a new perspective for understanding non-Markovian or memory effects: one can read any departure from a Markovian regime (measured, in general, with any of the previous indica-tors) in terms of a backflow of information from the reservoir to the system. Instead, the memoryless (Markovian) case corresponds to a unidirectional loss of information from the system to the reservoir. This conceptual frame was criticized in Ref. [30]. On the other hand, there are physical situations where it has a clear meaning. For example, in a Markovian regime, an excitation of a quantum optical transition is transferred unidirectionally from the system to the environment [1]. In the other extreme, for a two-level system interacting with a quantized harmonic oscillator (Jaynes-Cumming model [1]) an excitation is continuously transferred back and forth between the (two-level) system and the environment (quantum harmonic oscillator). Spin-boson models [1] interpolate between both extreme behaviors. This kind of bidirectional physical backflow of information, which is mediated between the system and the environment by a physical variable such as energy or heat, was analyzed in Refs. [31][32][33] and recently studied experimentally in Ref. [34].
It is important to notice that memory effects (as detected by any of the proposed non-Markovian indicators based on the system dynamics) may happen without a "physical environment-to-system backflow of information." For example, the environment, defined by all degrees of freedom not belonging to the system, may has its own Markovian (time-memoryless) dynamics which in turn is unperturbed by the system dynamics or state. Hence, even when the system dynamics is non-Markovian, the environment (its partial dynamics) is completely unaware of its coupling with the system.
A class of dynamics with the previous property is given by quantum systems coupled to environments modeled through stochastic Hamiltonians whose fluctuations are written in terms of classical non-white (Gaussian) noises. Even when the system may develops strong non-Markovian features [35], the environment (the noise) is unaffected by the system. In addition, quantum systems coupled to dissipative classical degrees of freedom [36] may lead to the same situation, that is, the system dynamics is non-Markovian while the environment follows a Markovian evolution that is unaffected by the system state. For example, in Ref. [37] a class of time-convoluted non-Markovian master equations were derived over the previous basis, where the non-Markovian indicator is the relative entropy with respect to the stationary state. In addition, a class of non-Markovian collisional models [38] can be recovered in a similar way, where a quantum ancilla system (part of the environment) in a bipartite embedding follows its own independent quantum Markovian Lindblad evolution [39]. More recently, master equations leading to eternal non-Markovianity, a Lindbladlike equation with a time-dependent rate that is negative at all times [12], were derived on similar grounds [40]. The underlying (Markovian) environment dynamics can be read in alternative ways such as random dephasing channels and Markov chains [36].
The main goal of this paper is to extend the previous results [37,39,40] to the case of maximally non-Markovian quantum evolutions. This kind of dynamics was introduced by Chruściński and Maniscalco in Ref. [13]. Studying the positivity of the propagator between two arbitrary times in an extended Hilbert space (divisibility), the authors defined a non-Markovianity degree. This parameter is the analog of Schmidt number in entanglement theory, and allows to compare (to rank) different non-Markovian evolutions. Maximally non-Markovian quantum evolutions are the analog of maximally entangled states. Here, we demonstrate that there exists of class of such kind of extreme non-Markovian dynamics that are induced by environments whose own dynamics is unperturbed by the system state, that is, without existing a physical environment-to-system backflow of information. The evolutions rely on simple quantumclassical hybrid arranges (collisional models [38] with a finite number of events), where the classical degrees of freedom (the environment) follow their own local in time Markovian evolution [36]. Similarly to the results of Refs. [37,39,40], non-Markovian effects arise due to a unidirectional dependence of the system evolution on the environment states.
The paper is organized as follows. In Sec. II, different examples of maximally non-Markovian evolutions are introduced. In Sec. III, the examples are derived from hybrid quantum-classical dynamics. In Sec. IV, the more general underlying quantum-classical dynamics that lead to non-Markovian effects without a physical environment backflow of information are characterized. Sec. V is devoted to the Conclusions. Information that supports the main obtained results is provided in the Appendixes.

II. MAXIMALLY NON-MARKOVIAN EVOLUTIONS
Maximally non-Markovian evolutions saturate the degree of non-Markovianity. Explicit examples that satisfy the required conditions found in [13] are provided. The underlying formalism is briefly reviewed in Appendix A. Both dephasing and random unitary maps are given.
A. Dephasing channel Consider a qubit system whose density matrix ρ t follows the evolution where γ(t) is a time-dependent rate and σ z is the z-Pauli matrix. The solution can be written as where Γ(t) ≡ t 0 γ(τ )dτ. Eq. (2) shows the dephasing property of the solution map, that is, only coherences are affected.
The evolution is completely positive if and only if Γ(t) ≥ 0. Furthermore, the evolution is maximally non-Markovian if [13] lim t→∞ This simple condition implies the extreme non-Markovian property lim t→∞ ρ t = ρ 0 [see Eq. (2)]. Hence, in the long time limit (stationary state) the initial density matrix is recovered.
In order to fulfill the previous condition, we propose the following time dependent rate where the free scaling parameter γ > 0 determines the initial rate value, lim t→0 γ(t) = 2γ. Its time integral is given by which is positive, Γ(t) ≥ 0, and also satisfies condition (3). Hence, the corresponding evolution [Eq. (1)] is completely positive and maximally non-Markovian. In Fig. 1 we plot both the rate γ(t) and its time integral Γ(t). The time dependent rate is a smooth function that does not present any divergence and consistently assumes positive and negative values. The coherences behavior in Eq.
(2) follows straightforwardly as e −Γ(t) = [1 − 2γte −γt ]. From this solution, it is possible to rewriting the evolution (1) with a timeconvoluted structure, with memory kernel where δ(t) is the delta-Dirac function. This kind of kernel, with both local and nonlocal in time contributions, were also found in Refs. [37,40].

B. Random unitary dynamics
In this case, the qubit dynamics is governed by the time-dependent generator (σ k is the k-Pauli matrix) where two decoherence channels were added. The solution is written as a random unitary map, where The set of probabilities{p α (t)} and the set of time dependent rates {γ k (t)} depend each of the other. Given the probabilities, the rates can be expressed as [41] which implies the relation γ 0 (t) = − 3 k=1 γ k (t). The coefficients {H αβ } correspond to a square four dimensional Hadamard matrix [41]. On the other hand, in Appendix B we find a time-convoluted master equation that is equivalent to Eq. (8) (see also Ref. [42]). Defining Γ k (t) = t 0 γ k (τ )dτ, the solution map (9) is maximally non-Markovian, for example, if γ 2 (t)+γ 3 (t) 0 and [13] As an explicit example we take the positive and normalized probabilities where γ > 0. From Eq. (10), the time dependent rates can be written as where the auxiliary functions are In Fig. 2 we plot the set {γ k (t)}. All functions are smooth, do not diverge, assume positive and negative values, and vanish asymptotically. From the expressions for the rates [Eq. (13)] it is possible to obtain lim t→∞ Γ 1 (t) = ln(2), while lim t→∞ Γ 2 (t) = lim t→∞ Γ 3 (t) = 0. Furthermore, γ 2 (t) + γ 3 (t) = γg a (t) 0. Hence, from Eq. (11) we conclude that the evolution is maximally non-Markovian.

III. QUANTUM-CLASSICAL REPRESENTATION OF DYNAMICS
In the previous section, two maximally non-Markovian evolutions were explicitly defined. Here, we demonstrate that both examples can be recovered from simple quantum-classical dynamics, which admit different representations such as in terms of Markov chains, Lindblad rate equations, and bipartite evolutions. In all cases, the environment follows its own unaffected Markovian dynamics. These examples give a basis for constructing a full class of dynamics with the same memory properties.

Markov chain representation
We introduce a classical stochastic system endowed with three different states (Fig. 3). Their probabilities {P i (t)}, i = 0, 1, 2, obey by the classical master equation with initial condition P 0 (0) = 1. Thus, at random times the classical system undergoes the successive transitions 0 → 1 → 2. In addition, in each transition the quantum system suffers the disruptive transformation Therefore, the density matrix of the quantum system is This solution can be read as a quantum collisional model [38] with a finite number of events. Eq. (19) can trivially be written with the structure given by Eq. (9), with p 0 (t) = P 0 (t) + P 2 (t), p 1 (t) = p 2 (t) = 0, and p 3 (t) = P 1 (t). By solving Eq. (17) with P 0 (0) = 1, we get  1) and (4)], the environment consists in three (unidirectionally) coupled states, which obey the classical master equation (17). In each transition, the transformation (18) (8) and (13)], whose solution is the mixed state (28).
In the previous stochastic representation, the property lim t→∞ ρ t = ρ 0 can be understood straightforwardly from the occurrence of two successive flips in the sign of the coherences, which are induced by the transformation ρ → σ z ρσ z . Furthermore, the environment, represented here by the classical system, is completely unaware of the (unidirectional) dependence of the system dynamics on its states.

Lindblad rate equation
As is well known, the class of stochastic dynamics described previously can be formulated in terms of Lindblad rate equations [36]. These generalized Lindblad equations give the more general evolution for a quantumclassical (hybrid) arrange where the behavior of the classical part is inherently irreversible in time.
The system density matrix is written as where the auxiliary states {ρ i (t)} evolve as Lindblad rate equations also arise from a generalized Born-Markov approximation. Hence, the previous dynamics [Eq. (22)] can alternatively be read as the result of the interaction with a complex structured reservoir [36].

Bipartite Lindblad representation and measurement trajectories
Lindblad rate equations can be embedded in a bipartite quantum dynamics [36]. In the present case, the "ancilla system" has three states, |0 , |1 , and |2 . Denoting with ρ sa t the system and ancilla density matrix, its evolution can be written as where the initial state is ρ sa 0 = ρ 0 ⊗|0 0|. The dissipative channels are defined by the operators The dynamics given by the Lindblad rate equation (22) is recovered from ρ i (t) = i|ρ sa t |i , i = 0, 1, 2. The bipartite representation [Eq. (23)], through the standard quantum jump approach [1], allows to unravelling the evolution in measurement trajectories. Assuming that the measurement device detects the transitions |0 → |1 and |1 → |2 , the bipartite measurement operators are V 1 and V 2 . Similarly to Ref. [39], the "collisional trajectories," defined by Eq. (18), can be recovered from the bipartite measurement realizations. The main difference is that here only a finite number of transitions occur.

Decoherence and recoherence
The underlying quantum-classical dynamics can be generalized without affecting the non-Markovianity degree of the solution map. For example, the introduction of extra classical states also leads to the property lim t→∞ ρ t = ρ 0 if an even number of transformations (18) happen (Fig. 3). By increasing the number of collisions the phenomenon of recoherence occurs, that is, after an exponential decay the system coherence (almost) vanishes and emerges at a later time, recovering its initial value in the (stationary) long time regime.
Denoting the qubit coherence as ρ 12 (t) = c n (t)ρ 12 (0), where the function c n (t) gives its characteristic time behavior and the subindex n denotes the (even) number of collisions, in Appendix C we obtain This expression follows by conditioning the quantum dynamics to a classical system with n + 1 states, where each transition implies the transformation (18). The (unidirectional) coupling rate between successive states is γ. For the model with three states (Fig. 1), it follows c 2 (t) = 1 − 2γte −γt = e −Γ(t) , where consistently Γ(t) is given by Eq. (5). In all cases the time dependent rate [Eq. (

1)] follows as γ(t) = (d/dt) ln[1/c n (t)], with Γ(t) = ln[1/c n (t)], which fulfills Eq. (3).
In Fig. 4 it is plotted c n (t) as a function of time for n = 2, 10, and 20. For n ≫ 1, the coherence vanishes following the decay behavior c n (t) ≃ exp(−2γt), while recoherence always occurs in the long time regime, lim t→∞ c n (t) = 1. On the other hand, a Markovian limit is recovered as lim n→∞ c n (t) = exp(−2γt). In this case, a standard collisional dynamics is recovered [38], which in a given realization contains an infinite number of disruptive transformations.

B. Random unitary dynamics
The Maximally non-Markovian random unitary dynamics defined by Eq. (9) and the probabilities (12) can also be derived taking into account an environment that is unaffected by the system dynamics. With this goal, in addition to the three auxiliary states defined by Eq. (22), an extra state ρ M (t) is introduced (Fig. 3). It is completely uncoupled of the other states and follows the Markovian evolution Here, σ x is the x−Pauli matrix. The solution is given by The system density matrix is written as where 0 ≤ r ≤ 1. This expression corresponds to a statistical mixture (or convex combination [43]) of a Markovian and a maximally non-Markovian dephasing evolution, Eqs. (26) and (1) respectively. Taking r = 1/2, from the solutions (19) and (27) it follows the probabilities (12), recovering in consequence the maximally non-Markovian random unitary evolution presented in the previous section. Notice that also in this case the environment is unaware of the dependence of the system dynamics on its states, implying in consequence, the absence of any physical backflow of information. On the other hand, the state (28) can also be characterized in terms of Lindblad rate equations and bipartite evolutions. It is interesting to note that Eq. (28) leads to a Markovian solution map for r = 1. By an explicit calculation, it is possible to demonstrate that this is the unique value at which the map departs from a maximally non-Markovian dynamics. In fact, for 0 ≤ r < 1 we get Hence, condition Eq. (11) is fulfilled. In contrast, a statistical mixture of a maximally entangled state and the identity matrix state (Werner state) departs from an entangled state at r < 1 [44]. This difference is expectable because the non-Markovianity degree relies on a formal analogy with entanglement [13].

IV. NON-MARKOVIAN QUANTUM DYNAMICS WITHOUT ENVIRONMENT BACKFLOW OF INFORMATION
The previous examples, as well as the results of Refs. [37,39,40], demonstrate that strong non-Markovian effects may arise from simple quantumclassical hybrid dynamics. The environment consists in a set of classical degrees of freedom that follows their own dynamics. Here, we write the more general underlying equations that lead to these properties and analyze how memory effects may emerge.

A. Discrete dissipative classical degrees of freedom
The system density matrix is written in terms of a set of auxiliary states, ρ t = R ρ R (t), whose evolution is given by a Lindblad rate equation, with initial conditions ρ R (0) = P R (0)ρ 0 . In the first line, H R and L R are the (conditional) unitary and dissipative Lindblad dynamics of the system given that the environment is in the state R. The second line introduces a coupling between the auxiliary states with rates {γ R ′ R }. In each transition R ′ → R, the state ρ R ′ suffers the disrup- } are trace preserving completely positive superoperators, The probability that the environment is in the state R at time t is given by P R (t) = Tr[ρ R (t)] [36]. From Eq. (29), it follows the classical master equation This result implies that, in fact, the environment is completely unperturbed by the system state or dynamics. Hence, the environment is only a "spectator" or "casual bystander" during the whole evolution [40]. Eq. (29) is the more general quantum-classical evolution that leads to these properties [45]. The underlying equations (29) may induce strong non-Markovian quantum effects, which in turn may depend on the initials conditions {P R (0)}. For example, the particular case (22) The possibility of observing memory or non-Markovian effects without a physical environment-to-system backflow of information follows from the unidirectional dependence of the system evolution on the environment states [Eq. (29)]. This dependence is present even when E RR ′ = I or when {γ R ′ R } = 0. In this last extreme case, {P R (t)} = {P R (0)}. By writing the system density matrix as ρ t = Λ t ρ 0 = V t,s Λ t ρ 0 , we realize that the intermediate propagator V t,s (which is used for defining the non-Markovianity degree [13]) may develops strong departures from a completely positive "Markovian" propagator. In fact, V t,s must to takes into account the previous dependence of the system dynamics on the environment states. This fact explain why, even when the environment is frozen {P R (t)} = {P R (0)} [40], the system dynamics may develops memory effects without happening a physical environment-to-system backflow of information.

B. Non-white Gaussian stochastic Hamiltonian
Non-Markovian effects induced by non-white Gaussian stochastic Hamiltonians [35] also falls in the previous category. In order to show this fact, we write the density matrix as where the auxiliary state ρ η (t) depends on the real parameter η, and obeys the equation Both H and ∆H are Hamiltonian operators. The environment dynamics, defined by the variable η, follows from P η (t) ≡ Tr[ρ η (t)], which gives This Fokker-Planck equation [47] is independent of the system state. Furthermore, the quantum state parametrically depends on the bath state η. Therefore, non-Markovian effects obtained from Eq. (33) occur without a physical environment-to-system backflow of information. On the other hand, this equation can be mapped to a stochastic Hamiltonian evolution. In fact, Eq. (34) is equivalent to the Langevin equation [47] d dt where the white Gaussian noise ξ(t) has correlation ξ(t)ξ(t ′ ) = Dδ(t − t ′ ). In contrast, the non-white Gaussian process η(t) has an exponential correlation, which in the stationary regime reads In addition, the system density matrix ρ t [Eq. (32)] can be read as an average over realizations of η(t), ρ t = ρ st (t) , where ρ st (t) follows the stochastic evolution Therefore, stochastic non-white Hamiltonians also lead to non-Markovian effects [35] without happening a physical environment-to-system backflow of information.

V. SUMMARY AND CONCLUSIONS
Similarly to Schmidt number in entanglement theory, the degree of non-Markovianity allows to ranking memory effects developed by quantum evolutions. Maximally non-Markovian evolutions are the analog of maximally entangled states. Here, we showed that exist a class of such kind of extreme non-Markovian dynamics that can be modeled from simple quantum-classical hybrid dynamics. The environment, defined by the classical degrees of freedom, follows its own Markovian evolution, which in turn is unaffected by the system state or dynamics. Therefore, memory effects arise without the occurrence of any physical environment-to-system backflow of information. Departure with respect to a quantum Markovian regime follows from the unidirectional dependence of the system dynamics on the environment states.
Both dephasing and random unitary maximally non-Markovian maps were modeled. In the former case, the dynamics can be read in terms of a finite (even) number of collisional events that change the sign of the qubit coherence, each event occurring at random times. The phenomena of decoherence and recoherence are captured in this frame. On the other hand, in the last case (random unitary maps), the system dynamics follows from a statistical mixture (convex combination) of the former non-Markovian evolution and a Markovian dephasing channel in a different direction. In both cases, the classical degrees of freedom are unaffected by the system state. The underlying dynamics admit different representations such as in terms of quantum Markov chains, Lindblad rate equations, and measurement trajectories.
General Lindblad rate equations that lead to memory effects without a physical backflow of information were also characterized. The proposed models can be generalized in different ways (dimensionality, collision transformations, quantum ancillas, etc.), providing a basis for the study of a full family of maximally non-Markovian dynamics. This approach may be useful for modeling decoherence in quantum information channels where the environment is coupled during a set of finite intervals of time (collisional models with a finite number of events).
In accordance with previous results [37,39,40], the present analyses confirm that, in general, non-Markovian indicators or measures based solely on the system dynamics are unable to distinguish between memory (non-Markovian) effects developed with or without a physical environment-to-system backflow of information. and ancilla of dimension k. This result allows to defining a series of non-Markovianity measures. Departure from k divisibility is measured with where and, similarly for N + k [X] (where the integral is over intervals such that λ k (X; t) < 0), and The supremum is taken over all Hermitian X. Using this expression it follows This expression is the analog of a similar relation between the coefficients of a Schmidt decomposition of an entangled pure quantum state. Maximally non-Markovian dynamics corresponds to the case M 1 [Λ t ] = 1, which immediately leads to in perfect analogy with maximally quantum entangled states [44].
Appendix B: Time-convoluted evolution for random unitary maps Given the evolution defined by the general random unitary map Eq. (9), the goal here is to obtain a timeconvoluted evolution where L t is a memory superoperator. The derivation is similar to the time-convolutionless case [41]. Expressing ρ t in terms of its propagator, ρ t = Λ t ρ, and working the previous time evolution in the Laplace domain [f (z) = ∞ 0 dte −zt f (t)], it follows the relation Writing Eq. (9) in the Laplace domain, similarly to Ref. [41], let us observe that where the eigenvalues are given by where Assuming that L z can be written as where {k α (z)} are memory functions, using the property (B4) with p α (z) → k α (z), it follows that L z (σ α ) = (1/2) 3 α=0 H αβ k β (z)σ α . From this relation and Eq. (B6), and by using that H −1 = (1/4)H, it follows a close expression for the memory functions The memory functions {k α (z)} only depend on the set of probabilities {p α (z)}. On the other hand, general kernels {k α (z)} that guarantee a completely positive solution map were characterized in Ref. [42]. For the dephasing map defined by Eqs. (1) and (4), the probabilities are p 0 (t) = 1 − γt exp(−γt), p 1 (t) = p 2 (t) = 0, and p 3 (t) = γt exp(−γt), Eq. (20). From Eq. (B9) it follows k 1 (z) = k 2 (z) = 0, and k 3 (z) = 2z 2 γ/(z 2 + γ 2 ), which consistently is the Laplace transform of Eq. (7). For the probabilities Eq. (12), the kernels have the same structure, where their specific time dependence can be written in terms of exponential and trigonometric functions.