The interplay between local and non-local master equations: exact and approximated dynamics

Master equations are a useful tool to describe the evolution of open quantum systems. In order to characterize the mathematical features and the physical origin of the dynamics, it is often useful to consider different kinds of master equations for the same system. Here, we derive an exact connection between the time-local and the integro-differential descriptions, focusing on the class of commutative dynamics. The use of the damping-basis formalism allows us to devise a general procedure to go from one master equation to the other and vice-versa, by working with functions of time and their Laplace transforms only. We further analyze the Lindbladian form of the time-local and the integro-differential master equations, where we account for the appearance of different sets of Lindbladian operators. In addition, we investigate a Redfield-like approximation, that transforms the exact integro-differential equation into a time-local one by means of a coarse graining in time. Besides relating the structure of the resulting master equation to those associated with the exact dynamics, we study the effects of the approximation on Markovianity. In particular, we show that, against expectation, the coarse graining in time can possibly introduce memory effects, leading to a violation of a divisibility property of the dynamics.


Introduction
Any realistic physical system is unavoidably coupled to some external degrees of freedom and should then be treated as an open system. This is especially relevant in the realm of quantum physics, which describes phenomena on small scales, typically fragile under the interaction with the environment. As a consequence, the theory of open quantum systems [1,2] has a wide application area, e.g., in quantum chemistry [3], quantum information [4] and even biophysics [5,6].
However, accounting for the interaction of the system of interest with the external degrees of freedom has its price. In particular, the equations of motion describing the evolution of open quantum systems are either fully characterized, but derived under very restrictive assumptions, or in principle appropriate for a wide range of dynamical evolutions, but computationally demanding and at least partially unexplored. The former situation refers to the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation [7,8], which is associated to quantum dynamical semigroups; the latter is based on more general time-local master equations or on integro-differential master equations, fixed by a memory kernel. The focus of this paper is precisely on the connection between the latter types of description.
Both the time-local and the integro-differential descriptions of the open-system dynamics are highly relevant. The local one is better suited to access some of the properties of the evolution and to conduct calculations (e.g. numerically), while the approach based on the use of memory kernels often gives a better insight of the physical processes underlying the dynamics. This is the case, for example, for quantum semi-Markov processes, where continuous in time quantum evolutions are randomly interrupted by jumps [9][10][11][12][13][14]. The structure of the memory kernels for these processes clearly reflects this origin and additionally guarantees the property of complete positivity (CP) of the corresponding dynamical maps. For a given system it is in general of advantage to know both representations of its evolution, as each can extend one's knowledge and it is rarely possible to know a-priori which is the most suitable one.
The first goal of this paper is to shed some light on the connection between the descriptions given by the time-local and the integro-differential approach, both treated in an exact way. We use a damping-basis representation of the generator and memory kernel, which proves to be a powerful tool to characterize their structural properties, as well as the connection between them [15]. Focusing on commutative dynamics, i.e., such that the dynamical maps at different times commute [16,17], we define a general procedure to go directly from one form of the master equation to the other, and viceversa.
The second part of the paper concerns the study of a possible simplified treatment of the integro-differential master equation, which enforces a time-local structure, and can be seen as an approximated link between the two different descriptions of the opensystem dynamics. Relying on the idea that the memory kernel is generally localized around the origin, with a width which can be interpreted as the memory time of the dynamics, one can define a general coarse-graining operation leading to a time-local master equation, along the lines of the seminal work of Redfield [18]. Here, we first show that the damping-basis description also enables us to connect this approximated time-local description and the two original, time-local and integro-differential, ones. Furthermore, we study how moving to a Redfield-like master equation modifies the (non-)Markovian nature of the dynamics.
In recent years, the investigation of memory effects in the open quantum system dynamics, summarized under the term of non-Markovianity", has attracted a great interest, among others, in connection with quantum thermodynamics [19][20][21], quantumcontrol theory [22] and quantum metrology [23][24][25][26]. Till now, a wide range of non-equivalent characterizations of quantum non-Markovianity was introduced in the

Structure of the master equations
The properties of the open quantum system alone are fixed by its density operator ρ(t), also referred to as reduced state, at a generic time t. The dynamics of the open system is thus fully characterized by a family of dynamical maps {Λ t } t≥0 , which map the initial reduced state ρ(0) to the reduced state at later times, according to ρ(t) = Λ t [ρ(0)] (Λ 0 = 1). In the time-local description of the dynamics, the family of dynamical maps satisfies the following equation where the superscript TCL stands for time-convolutionless [1]; instead, the integrodifferential approach builds on the equation where the symbol * denotes the convolution in time, K NZ t is the memory kernel and NZ stands for Nakajima-Zwanzig [1]. The conditions on the time-local generator K TCL t and on the memory kernel K NZ t guarantying that Λ t is a proper dynamical map, i.e. completely positive ‡ and trace preserving (CPTP), are in general unknown. Only in some limited cases definite statements in this respect have been obtained, see e.g. [9,14,[36][37][38][39][40][41][42][43][44][45].

Damping bases
For the sake of simplicity, we consider that the reduced system is associated with a Hilbert space H of finite dimension N . The set of linear operators on H is denoted as B(H) and it is an N 2 dimensional Hilbert space. Hence, given a linear map Ξ acting on B(H), often referred to as super-operator, and a basis {σ α } α=1,...,N 2 of B(H) orthonormal with respect to the Hilbert-Schmidt scalar product, we can write the action of Ξ on a generic element ω ∈ B(H) as [16,46] Now, the map Ξ is said to be diagonalizable iff the corresponding matrix M Ξ is. In this case, there exist two families of operators {τ α } α=1,...,N 2 and {ς α } α=1,...,N 2 , such that and The two bases of B(H), {τ α } α=1,...,N 2 and {ς α } α=1,...,N 2 , which are not necessarily orthogonal bases, are sometimes referred to as bi-orthogonal bases. Within the context of open quantum systems, these bases were introduced in [47] and named damping bases; in particular, there they were associated with a GKSL generator, i.e. Ξ = L.
The damping bases are also strictly connected to the relation between the map Ξ and its dual Ξ , where the latter is defined by Using the damping bases, one finds where c * is the complex conjugate of c. From Eqs. (5), (6) and (8) one can then see that the operators {τ α } α=1,...,N 2 and {ς α } α=1,...,N 2 are the eigenvectors, respectively, of the linear map Ξ and of its dual Ξ with respect to complex conjugates eigenvalues, i.e.
Indeed, if Ξ is a normal operator, i.e., [Ξ, Ξ ] = 0, then the damping bases both coincide with a single orthonormal basis; if Ξ is a Hermitian map, in addition the eigenvalues are real, λ α = λ * α . Let us now move to one-parameter families of maps, {Ξ t } t≥0 , which are used to describe the dynamics of open quantum systems. In general, both the coefficients and the operators in Eq. (5) will depend on time. However, it can well happen that the time dependence is enclosed in the eigenvalues only, while the corresponding damping bases are time-independent, i.e., that one has Eq. (10) implies that the maps at different times commute, the converse implication holds, if we further assume that the maps Ξ s and Ξ t are diagonalizable. In particular, one can consider a one-parameter family of CPTP dynamical maps {Λ t } t≥0 which commute at different times (a situation which has been thoroughly investigated in [16,17]), i.e., such that If the family {Λ t } t≥0 satisfies a time-local master equation of the form Eq. (1) then the dynamical maps satisfy Eq. (12) if and only if the time-local generator satisfies the analogous commutation relation If we further assume diagonalizability of the dynamical maps and the generator, they will share the same time-independent damping bases, see Eq. (10), according to: Let us stress that the dynamics satisfying Eq. (12) include, but are not restricted to, the case where Eq. (1) holds with K TCL t = γ(t)L, which has been widely studied in the literature [36,37,[48][49][50].

Novel connection between time-local and integro-differential representation
Given a diagonalizable commutative dynamics, its damping-basis decomposition sets a representation of both the dynamical map and the time-local generator in which the time dependence is fully enclosed in the eigenvalues -i.e., in complex functions of time -but not in the operatorial structure, see Eqs. (14) and (15). Here, we show that this feature is also shared by the memory kernel, which will allow us to derive some novel connections between the time-local and the integro-differential master equations associated to a given dynamics. Proposition 1. Consider a family of dynamical maps {Λ t } t≥0 with time-local generator K TCL t and memory kernel K NZ t . Moreover, for any couple of operators τ α and ς α let M α be the linear map acting on B(H) defined as The following propositions are equivalent: i) the time-local generator K TCL t has the damping-basis diagonalization ii) the memory kernel K NZ t has the damping-basis diagonalization moreover, the corresponding eigenvalues are related by where we introduced the function f t (u) ≡ f u is the Laplace transform of f t and I (g(u)) (t) the inverse Laplace transform of g(u).
Proof. The proof is given in Appendix A.
Proposition 1 is the central result of this paper. According to it, the time-local generator and the memory kernel have the same time-independent damping bases, while the time dependence appears in the eigenvalues only: to go from one master equation to the other, only functions of time (and their Laplace transforms) are involved, while the operatorial structure is unchanged; see the left part of Fig. 1.
As a consequence, Proposition 1 is particularly useful when we want to compare the structures of the time-local generator and the memory kernel. As a general relevant example, consider the following corollary, which directly follows from the proposition above.
Corollary 1. Under the same assumptions of Proposition 1, consider a diagonalizable GKSL generator L with only one-nonzero eigenvalue , possibly degenerate with degeneracy d, i.e., then the following identities are equivalent where m NZ (t) and G(t) are related as in Eqs. (19) and (21) where now Proof. Simply note that m TCL (t) = γ(t) and d α=1 M α = L/ , see Eq. (23); then apply Proposition 1.
In other terms, if L has only one non-zero eigenvalue, the super-operatorial part of the time-local generator and the memory kernel are exactly the same; the difference between the master equations is enclosed in one overall time-dependent factor.

Lindbladian form
The time-local generator and the memory kernel in Corollary 1 are directly proportional to a GKSL generator of quantum dynamical semigroups. Here we show that, starting from the damping-basis decomposition in Eqs. (17) and (18), it is always possible to write the time-local generator and the memory kernel in a way which is directly related to the GKSL generator; we will refer to such form as Lindbladian. To do so, we essentially apply the general prescription given by Lemma 2.3 in [7] to the situation of interest for us. Besides providing us with a canonical reference structure which eases the comparison between super-operators, as we show by different examples, the Lindbladian form allows us to infer the (C)P and the (C)P-divisibility of the dynamics in a more direct way.
Any linear map acting on B(H) can be represented in several ways. A relevant example is given by the matrix representation in Eq. (4), which is at the basis of the damping-basis decomposition, see Eq. (5). Alternatively, given an orthonormal basis N , any linear map Ξ acting on B(H) can be uniquely written as Note that such a representation is strictly related to the CP of the map Ξ: in fact, the matrix of coefficients with elements c αβ is positive semidefinite if and only if Ξ is CP, in which case the decomposition in Eq. (27) directly leads to the Kraus decomposition of CP maps. Most importantly for us, the representation in Eq. (27) gives a general characterization also of the time-local and the integro-differential master equations associated with open-system dynamics. In fact, let us consider a time-local generator or a memory kernel K t . The dynamics {Λ t } t≥0 resulting from Eqs. (1) and (2) is trace and Hermiticity preserving (where the latter means that any Hermitian operator ω = ω † is mapped at any time t into an Hermitian operator Λ t (ω) = (Λ t (ω)) † ) if and only if the corresponding time-local generator and memory kernel satisfy § Restricting for the sake of convenience to diagonalizable super-operators, in the form where the coefficients κ αβ (t) = κ βα (t) * are given by while the Hamiltonian H(t) = H † (t) is given by In addition, since the matrix with elements κ αβ (t) is Hermitian, it can be diagonalized by a unitary matrix V (t) with elements V αβ (t), so that Eq. (30) can be rewritten as with Eqs. (31)-(34) provide us with the wanted recipe to get the Lindbladian form, starting from the damping-basis representation, Eqs. (17) and (18). The main difficulty is that the different non-zero eigenvalues m α (t) will mix" in a non-trivial way, so that there is not a direct connection between them and the coefficients r α (t) in the Lindbladian structure. As a first consequence, one looses the correspondence between the superoperatorial structures of, respectively, the time-local generator and the memory kernel, which is guaranteed by Corollary 1 in the case of one non-zero eigenvalue; this is shown by the examples below.
Example 1 Consider the time-local generator which describes, for example, the reduced dynamics of a two-level system interacting with a zero-temperature bosonic bath via a Jaynes-Cummings interaction term [46,51], neglecting for the sake of simplicity the free Hamiltonian term. This is a special case of the generator treated (for constant coefficients) in [47]. The dual generator is {ς α } α=1,...,4 Indeed, the relations in Eq. (6) are satisfied; moreover, we note that we have now two eigenvalues different from 0 (one two-fold degenerate) and the damping bases are not made of self-adjoint operators. The time dependence is enclosed in the eigenvalues only, so that we are in the case of commuting dynamics treated in the previous sections. In particular, by applying Proposition 1, we find that the integro-differential generator K NZ t has the same damping bases, Eqs. (38) and (39), with eigenvalues given by Eqs. (19) and (22) with respect to the functions in Eq. (37). In other terms, both the time-local generator and the memory kernel can be written in the form [see Eq. (5)] with m X i (t) the corresponding eigenvalues. But, by using Eqs. (31)- (34), one can see that the generator as in Eq. (40) corresponds to a Lindbladian form Crucially, while for K TCL t one has m TCL 2 (t) = 2m TCL 3 (t) [see Eq. (37)], so that the pure dephasing term cancels out, this is generally not the case for K NZ t , which will then present a pure-dephasing term, in addition to the spontaneous-emission term; an explicit example of this is given in [46,51]. This implies in particular that at variance with the case of the standard GKSL generator, a direct physical interpretation of the operatorial contribution is not available. As anticipated, even though the time-local master equation we started from, Eq. (35), is in the form given by Eq. (24), now the corresponding GKSL generator has more than one eigenvalue different from 0. As a consequence, the transformation to the memory kernel no longer preserves the super-operatorial part of the Lindbladian structure [compare with Corollary 1], but rather generates one more term.

Example 2 A somehow opposite example is obtained by starting with a memory kernel of the form
with k(t) such that the described dynamics is CPTP, as e.g. in [31]. First, we note that such a generator can be written in a "manifest" Lindbladian structure as so that this time we start from a non-local generator in the form K NZ t = k(t)L [compare with Eq. (24)]. Such a generator is self-adjoint, K NZ t = K NZ t , so that the eigenvalues are real, and the damping bases coincide, yielding an orthonormal basis: Also in this case, since we have two eigenvalues different from 0 we cannot apply Corollary 1, which would guarantee that the time-local equation would have the same Lindbladian structure. Instead, we can apply Proposition 1, so that both the time-local generator and the memory kernel have the form K X t (ω) = m X 2 (t) (Tr [σ x ω] σ x + Tr [σ y ω] σ y )+m X 4 (t)Tr [σ z ω] σ z , X = TCL, NZ (46) where for the memory kernel the eigenvalues are given by Eq. (44), while for the timelocal generator they are obtained from the latter via Eqs. (21) and (22) For the memory kernel m NZ 4 (t) = 2m NZ 2 (t), so that the pure-dephasing term cancels out, while this will not generally be the case for the time-local generator. An example is given in [31]. Once again, the multiple eigenvalues in the damping-basis decomposition generate further terms, this time when going from the integro-differential to the timelocal master equation. Of course, the situation is symmetrical, so that we could obtain further examples by simply inverting the starting points in the examples above; the only difference when going, respectively, from the time-local to the integro-differential master equation or viceversa is the connection between the eigenvalues of the damping-basis decomposition, i.e., whether one should use Eq. (19) or Eq. (21).
We conclude that Propositions 1 and Eqs. (31)-(34) yield a systematic procedure to obtain the integro-differential master equation from the time-local one and viceversa, whenever we are able to diagonalize them and the resulting damping bases are timeindependent. An example of application for a higher dimensional system is given in Appendix C.

Redfield-like master equation
Until now, we compared the time-local and integro-differential master equations in an exact way, i.e., without introducing any approximation. On the other hand, as recalled in the Introduction, it is useful to consider situations in which an integro-differential master equation is transformed into a time-local one by means of some approximations. Here, we focus on a master equation which is obtained via a coarse graining in time analogous to the one introduced by Redfield in [18].
More precisely, if τ R is the relaxation time of the open-system dynamics and K NZ t is appreciably different from zero only on a time scale much shorter than τ R , one might approximate the dynamical maps Λ t with Λ Red t , where the latter is obtained by replacing Eq.
When this approximation is used in the presence of a weak-coupling interaction between the open system and its environment, the resulting equation is often called Redfield equation [1,18]. We will refer to Eq. (48) as Redfield-like master equation, in order to emphasize that we take it into account without necessarily restricting to the weakcoupling regime. The Redfield equation is commonly exploited in several different contexts, such as the study of transport processes in condensed-matter or biophysical systems [52][53][54][55][56]. Importantly, the Redfield equation might lead to a not well-defined evolution, as studied by now extensively in the literature [57][58][59]. We will show how the damping-basis representation enables us to determine the structural properties of the resulting time-local generator K Red t , as well as to investigate the Markovian nature of the dynamics Λ Red t t≥0 .

Structure of the time-local generator
We first consider the following simple connection between the time-local generator, the memory kernel and the Redfield-like generator, in the case of commutative dynamics.
with m Red Moreover, under the same assumptions of Corollary 1, the Redfield-like generator K Red Accordingly, the Redfield-like dynamics has the same operational structure as the exact one, with time-dependent eigenvalues obtained from the corresponding original ones. This situation is depicted in Fig. 1. What is more, the eigenvalues m Red α (t) can be written in a compact form in terms of the functions G α (t) defined in Eq. (22).
can be written as Proof. The proof is given in Appendix B.

Markovianity of the dynamics
As mentioned in the Introduction, a highly relevant property of the dynamics of an open quantum system is its (non-)Markovianity. Quantum non-Markovianity is often defined in terms of the divisibility property, positive or completely positive, of the corresponding dynamical maps. As we will see below, the damping-basis representation enables us to make some statements about the preservation of (C)P-divisibility under the approximation leading to the Redfield-like master equation. First, we note that the relation in Eq. (54) directly allows us to infer that (C)Pdivisibility is preserved in the Redfield-like master equation, whenever L has only one non-zero eigenvalue.
) is (C)P-divisible if and only if γ(t) ≥ 0 (γ Red (t) ≥ 0). Thus, since we assume {Λ t } t≥0 (C)P-divisible, we have γ(t) ≥ 0. Moreover, has to be real and negative, since it is the only non-zero eigenvalue of L. It follows that the function G(t), see Eq. (26), is negative. But then, from Eq. (54) we have that m Red (t)/ is positive and thus Λ Red t t≥0 is (C)P-divisible. Note that the reverse statement, that is that (C)P-divisibility of the Redfield-like dynamics originates from (C)P-divisibility of the original dynamics, is in general not true (even in the case of a single non-zero eigenvalue). Indeed, in the next example we will consider two dynamics, which both have a (C)P-divisible approximated evolution, but only one of them has this property originally.
Example 3 Let us consider the case of a pure dephasing dynamics for a twolevel system, characterized by a monotonic reduction of coherences described by the decoherence funtion ϕ(t). Starting from the expression of the dynamics we can easily work out the corresponding time-local generator, which takes the form with and It can be easily checked that L has only one non-zero eigenvalue, so that relying on Corollary 1 we get the corresponding memory kernel, Since Eq. (58) exactly corresponds to Eq. (55), this memory kernel describes the very same CPTP dynamics. In particular, it describes a CP-divisible dynamics since ϕ(t) is monotonically decreasing. According to Proposition 2, the Redfield-like generator is given by where the function K(t) = t 0 dτ k(τ ) is indeed positive sinceK(u) = − φ(u)/(1 + φ(u)) and ϕ(t) is monotonically decreasing. The Redfield-like generator therefore also describes a CP-divisible dynamics. Let us now start from Eq. (58) changing the operatorial structure in the memory kernel via the replacement L → L with The associated Redfield-like generator reads and again corresponds to a CP-divisible dynamics. Making reference to Corollary 1 (also L has only one non-zero eigenvalue), we can obtain via the damping-basis approach the time-local generator exactly corresponding to Eq. (62), namely where nowγ (t) = − 1 2 The sign ofγ(t) is now not necessarily fixed, so that the dynamics is in general not C(P)-divisible. The evolution is however always well defined, that is CPTP, since t 0 dτγ(τ ) 0 [28]. The same therefore holds for the evolution described by K NZ t , due to the exact correspondence between Eq. (64) and Eq. (62). We have therefore shown an example of two dynamics whose associated Redfield-like master equations both describe a CP-divisible collection of maps, despite the fact that only one of the two was originally CP-divisible, while the other can even break P-divisibility. If we go beyond the case of only one non-zero eigenvalue, the situation gets more involved. Focusing in particular on Pauli maps, one can see that P-divisibility is more robust under the Redfield approximation than CP-divisibility.

Corollary 3. Consider a Pauli dynamical map in the form
Assume that the dynamical map is P-divisibile and Eq. (53) holds. Then the associated Redfield-like map is also P-divisible.
Proof. For Pauli dynamical maps the eigenvalues of the time-local generator read: A map in such a form is P-divisible iff [60,61] Accordingly, G α (t) ≤ 0, ∀α. Using this result, from Eq. (54) we conclude that m Red α (t) ≤ 0, ∀α, and the Redfield-like map is then also P-divisible.
On the other hand, CP-divisibility can be lost in the approximation leading to the Redfield-like master equation, as shown in the following example; indeed, this is a consequence of the presence of more than one non-zero eigenvalues.

Example 4
We analyze the dephasing dynamics in random directions first introduced in [62]: where L i [ρ] = σ i ρσ i − ρ, describing dephasing along the i-th direction, and the x i 's are the corresponding probabilities constrained by x 1 + x 2 + x 3 = 1. The time evolution obeys the following time-local master equation where the time dependent decoherence rates can be expressed as (t) (panel (b) the approximated dynamics) are positive, that is the dynamics is CP-divisible, up to the given time; different colours of the areas correspond to different times (in arbitrary units). Initially (t = 0, yellow triangle) the only constraint is in both cases the condition x 1 + x 2 + x 3 = 1. For all later times from γ Red k (t) > 0 it follows γ k (t) > 0, but not the other way around, so that green/blue/black areas in panel (a) are larger than the corresponding areas in panel (b). The black color corresponds to t → ∞, so that for these choices of x 1 , x 2 , x 3 the corresponding dynamical map always remains CP-divisible. Only the choices x i = x j ≤ x k , with i = j = k lead to CP-divisible Redfield-like dynamics ((b) tripod configuration consisting of three lines), in contrast to the exact dynamics ((a) black star shape). The four black dots denote choices of parameters x 1 , x 2 , x 3 for which both the original and the approximated generator have only one non-zero eigenvalue and the corresponding dynamical map is CP-divisible (in panel (a) the dot in the middle is not visible). with The dynamics given by Eq. (70) is always P-divisible, since Eqs. (68) are satisfied. However, depending on the choice of the x i parameters one of the decay rates can become negative so that the evolution looses its CP-divisibility, see Fig. 2 (a). In particular, the well-known eternally non-Markovian master equation belongs to this family [32]. It corresponds to the choice x 1 = x 2 = 1 2 , x 3 = 0 and generates an evolution which for all t > 0 is non-CP-divisible, as in this case γ 3 (t) = − tanh(t) < 0, ∀t > 0. The evolution resulting from the Redfield-like master equation Eq. (48) has the form d dt with γ Red γ Red γ Red and Y k (t) = exp(−2x k t)(x k − 1). This is still a proper CPTP quantum dynamics, as it fulfils the relevant constraints given in [61]. The dynamics (72) is P-divisible iff the condition (68) for γ Red k (t) is satisfied. As is always positive, by conducting our approximation the dynamics keeps its P-divisibility (in accordance with Corollary 3). However, almost all of the dynamics which were CPdivisible for the original map, are no more CP-divisible for the Redfield-like master equation, see Fig. 2 (a), for the original dynamics and Fig. 2 (b), for the approximated evolution. Initially, for t = 0, all γ(t)'s and γ Red (t)'s are positive. For later times from γ Red (t) > 0 it follows γ(t) > 0, but not the other way round. The Redfield-like dynamical maps are CP-divisible only for the choices x i = x j ≤ x k , with i = j = k (all γ Red (t)'s are positive for all times). This is in contrast to the original evolution, where a significant fraction of the dynamics are CP-divisible [62]. Note that there are only four choices of parameters x 1 , x 2 , x 3 (black dots in Fig. 2), for which the resulting original generator has only one eigenvalue and the corresponding dynamical map is CP-divisible.
In accordance with Corollary 1 and Corollary 2 these properties are also present in the approximated dynamics. The results of this Section concerning preservation or occurrence of (C)P-divisibility under the approximation leading to the Redfield-like master equation are summarized in Table 1. When going to higher dimensional systems the situation gets more involved and no easy connections between the (C)P-divisibility of the exact and approximated dynamics were found. However, in a case of the generalized Pauli channels [63] some statements are still possible, as elaborated in Appendix D.

Summary
In this paper, we investigated the connection between the time-local and the memorykernel master equations associated with the dynamics of an open quantum system. Focusing on the class of commutative dynamics and making use of the damping-basis approach, we formulated a general strategy to obtain each kind of master equation from the other one. Only transformations among functions of time and their Laplace transforms are involved, as the operational structure of the two master equations exact dynamics Redfield-like single eigenvalue C(P)-div ⇒ C(P)-div Pauli channel P-div ⇒ P-div CP-div CP-div Table 1. Connections between C(P)-divisibility property for exact and Redfield-like approximated dynamics. Arrows ⇒ and barred arrows express the implications shown by means of proofs and counterexamples (in the case of a single non-zero eigenvalue by Corollary 2 and Example 3, and in the case of a Pauli channel by Corollary 3 and Example 4, respectively). coincide in the damping-basis picture. In the presence of a single non-zero eigenvalue also the Lindbladian structure of the time-local generator and the memory kernel is exactly the same. Instead, when more eigenvalues are present, new Lindbladian operators can be generated in going from the time-local master equation to the integro-differential one, or viceversa.
Furthermore, we analyzed the impact of the approximation leading to the Redfieldlike master equation on both the operatorial structure of the master equation and the Markovianity of the dynamics. In the case of a single non-zero eigenvalue both CP-divisibility and P-divisibility are preserved, but this is no longer guaranteed for more general dynamics. In particular, restricting to Pauli dynamical maps, we showed that, while P-divisibility is still preserved, CP-divisibility can be lost as a result of the Redfield-like coarse graining in time.
Our results highlight the relevance of describing open system dynamics with different representations -such as the time-local or the integro-differential master equations, as well as the damping-basis or the Lindbladian picture -and the possibility to find direct connections among them. Indeed, it will be of interest to push forward such an analysis and deal with more general types of evolution in order to shed light on their relevance and usefulness for the description and characterization of non-Markovianity.