Adding Dynamical Generators in Quantum Master Equations

Adding dynamical generators in quantum master equations

Jan Kolodyński, 1 Jonatan Bohr Brask,2 Mart´ıPerarnau-Llobet,1, 3 and Bogna Bylicka1 1ICFO–Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain 2Group of Applied Physics, University of Geneva, 1211 Geneva, Switzerland 3Max-Planck-Institut fürQuantenoptik, Hans-Kopfermann-Str. 1, D-85748 Garching, Germany The quantum master equation is a widespread approach to describing open quantum system dynamics. In this approach, the effect of the environment on the system evolution is entirely captured by the dynamical generator, providing a compact and versatile description. However, care needs to be taken when several noise processes act simultaneously or the Hamiltonian evolution of the system is modified. Here, we show that generators can be added at the master equation level without compromising physicality only under restrictive conditions. Moreover, even when adding generators results in legitimate dynamics, this does not generally correspond to the true evolution of the system. We establish a general condition under which direct addition of dynamical generators is justified, showing that it is ensured under weak coupling and for settings where the free system Hamiltonian and all system-environment interactions commute. In all other cases, we demonstrate by counterexamples that the exact evolution derived microscopically cannot be guaranteed to coincide with the dynamics naively obtained by adding the generators.

I. INTRODUCTION In this work, we address the questions of when: (i) The naive addition of generators yields physically It is generally impossible to completely isolate a small valid dynamics. system of interest from the surrounding environment. (ii) The corresponding evolution coincides with the true Thus, dissipative effects caused by the environment are system dynamics derived from the underlying micro- important in almost every quantum experiment, ranging scopic model. from highly controlled settings, where much effort is in- First, we show that (i) is satisfied for generators which vested in minimising them, to areas where the dissipation are commutative, semigroup-simulable (can be inter- is the key object of interest. In many cases, exact mod- preted as a fictitious semigroup at each time instance), elling of the environment is not practical and its effect and preserve commutativity of the dynamics under addi- is instead accounted for by employing effective models tion. These reach beyond the case of Markovian genera- describing the induced noise. Different approaches ex- tors for which (i) naturally holds. Outside of this class, ist, e.g., quantum Langevin and stochastic Schrödinger we find examples of simple qubit QMEs which lead to equations [1,2], quantum jump and state-diffusion mod- unphysical dynamics. We observe that (ii) holds if and els [3,4], or Hilbert-space averaging methods [5]. only if the cross-correlations between distinct environ- Arguably, the most widely applied approach is to use ments can be ignored within a QME. We show this to be the quantum master equation (QME) description [1,2]. the case in the weak-coupling regime, extending previous In this approach, the system evolution is given by a time- results in this direction [7, 12, 13]. We also provide a suf- local differential equation, where the effect of the envi- ficient condition for (ii) dictated by the commutativity of ronment is captured by the dynamical generator. A mas- Hamiltonians at the microscopic level. We combine these ter equation can be derived from a microscopic model of generic considerations with a detailed study of a specific the system and environment, and their interaction, by open system, namely a qubit interacting simultaneously tracing over the environment and applying appropriate with multiple spin baths, for which we provide examples approximations [1,2]. However, QMEs are also often ap- where (ii) is not satisfied, while choosing the microscopic

arXiv:1704.08702v3 [quant-ph] 19 May 2018 plied directly, without explicit reference to an underlying Hamiltonians to fulfil particular commutation relations. model. In that case, care needs to be taken when several Our results are of relevance to areas of quantum noise processes act in parallel, as simultaneous coupling physics where careful description of dissipative dynam- to multiple baths in a microscopic model does not gen- ics plays a key role, e.g., in dissipative quantum state erally correspond to simple addition of noise generators. engineering [14–17], dissipative coupling in optomechan- Moreover, when the Hamiltonian evolution of the sys- ics [18], or in dissipation-enhanced quantum transport tem is modified, e.g., when controlling system dynamics scenarios [11, 19], including biological processes [20]. In by coherent driving [6], the form of noise generators in a particular, they are of importance to situations in which QME may significantly change. Additivity of noise at the QMEs are routinely employed to account for multiple QME level has been discussed recently for qubits when sources of dissipation, e.g., in quantum thermodynamics analysing dynamical effects of interference between dif- [21–24] when dealing with multiple heat baths [8, 25–27] ferent baths [7,8], non-additivity of relaxation rates in or in quantum metrology [28–30] where the relation be- multipartite systems [9, 10], as well as in the context of tween dissipation and Hamiltonian dynamics, encoding charge (excitation) transport [11]. the estimated parameter, is crucial [31–34]. 2

The manuscript is structured as follows. In Sec.II, we A. Physicality of dynamical generators discuss QMEs at an abstract level—as defined by families of dynamical generators whose important properties we For the QME (1) to be physically valid, it must yield summarise in Sec.IIA. We specify in Sec.IIB conditions dynamics that is consistent with quantum theory. In par- under which the addition of physically valid generators ticular, upon integration the QME must lead to a family is guaranteed to yield legitimate dynamics. We demon- of (dynamical) maps Λt (parametrised by t) that satisfy strate by explicit examples that even mild violation of ρS(t) = Λt[ρS(0)] for any t ≥ 0 and initial ρS(0), with these conditions may lead to unphysical evolutions. each Λt being completely positive and trace preserving In Sec.III, we view the validity of QMEs from the mi- (CPTP) [38, 39]. croscopic perspective. In particular, we briefly review in On the other hand, any QME (1) is unambiguously Sec.IIIA the canonical derivation of a QME based on specified by the family of (dynamical) generators Lt ap- an underlying microscopic model, in order to discuss the pearing in Eq. (1). However, as discussed in App.A, effect of changing the system Hamiltonian on the QME, although the CPTP condition can be straightforwardly as well as the generalisation to interactions with multi- checked for maps Λt, it does not directly translate onto ple environments. We then formulate a general criterion the generators Lt. As a result, for a generic QME for the validity of generator addition in Sec.IIIB, which its physicality cannot be easily inferred at the level of we explicitly show to be ensured in the weak coupling Eq. (1), unless its explicit integration is possible. Never- regime, or when particular commutation relations of the theless, we formally call a family of dynamical generators microscopic Hamiltonians are fulfilled. Lt physical if the family of maps it generates consists In Sec.IV, we develop an exactly solvable model of a only of CPTP transformations. In what follows (see also qubit interacting with multiple spin baths, which allows App.A1), we describe properties of dynamical genera- us to explicitly construct counterexamples that disprove tors that ensure their physicality. the microscopic validity of generator addition in all the Any family of dynamical generators, whether physical regimes in which the aforementioned commutation rela- or not, can be uniquely decomposed as [37] tions do not hold. Finally, we conclude in Sec.V. d2−1 X 1 L [ρ] = −i[H(t), ρ] + D (t) F ρF † − {F †F , ρ} , t ij i j 2 j i II. TIME-LOCAL QUANTUM MASTER i,j=1 EQUATIONS (2) d2 where d is the Hilbert space dimension and {Fi}i=1 is any QMEs constitute a standard tool to describe reduced † orthonormal operator basis with Tr{Fi Fj} = δij and all dynamics of open quantum systems. They provide a com- √ F traceless except F 2 = 11/ d. The Hamiltonian part, pact way of defining the effective system evolution at the i d Ht, of the generator in Eq. (1) is then determined by level of its density matrix, ρS(t), without need for ex- H(t) of Eq. (2), while the dissipative part Dt is defined plicit specification neither of environmental interactions by the Hermitian matrix D(t). Although general criteria nor the nature of the noise. Although a QME may be for physicality of dynamical-generator families are not expressed in a generalised form as an integro-differential known, two natural classes of physical dynamics can be equation involving time-convolution [35], its equivalent identified based on the above decomposition. (c.f. [36]) and more transparent time-local formulation is In particular, when D(t) is positive semidefinite, Lt typically favoured, providing a more direct connection to is said to be of Gorini-Kossakowski-Sudarshan-Lindblad the underlying physical mechanisms responsible for the (GKSL) form [37, 40]. If this is the case for all t ≥ 0, dissipation [1,2]. Given a time-local QME: then the corresponding evolution is not only physical but d also CP-divisible, i.e., the corresponding family of maps ρS(t) = Lt[ρS(t)] = Ht[ρS(t)] + Dt[ρS(t)], (1) can be decomposed as Λt = Λ˜ t,sΛs, where Λ˜ t,s is CPTP dt for all 0 ≤ s ≤ t. This property is typically associated all the information about the system evolution is con- with Markovianity of the evolution [41–43]. tained within the dynamical generator, Lt, that is Furthermore, when, in addition, H and D in Eq. (2) are uniquely defined at each moment of time t. Moreover, time-independent, the dynamics forms a semigroup, such Lt can always be decomposed into its Hamiltonian and that the generator and map families are directly related purely dissipative parts, i.e., Lt = Ht +Dt in Eq. (1) with via Λt = exp[tL] with all Lt = L [44]. See App.A1 for a Ht[ρ] = −i[H(t), ρ] and some Hermitian H(t)[37]. more detailed discussion of different types of evolutions. Although the QME (1) constitutes an ordinary differ- For the purpose of this work, we also identify another ential equation, the system evolution may exhibit highly important class of physical dynamics: non-trivial memory features thanks to the arbitrary de- Definition 1. A given dynamical family Λt is pendence of Lt on the local time-instance t, but also on semigroup-simulable (SS) if for any t ≥ 0 the map the (fixed) initial time t0 at which the evolution com- Z = log Λ is of the GKSL form (2) with some D(t) ≥ 0. mences [36]—which, without loss of generality, we choose t t to be zero (t0 = 0) and drop throughout this work. Formally, Zt constitutes the instantaneous exponent of 3

Zt the dynamics, satisfying Λt = e (see Ref. [45] and App.A1). Physicality of the evolution is then guaranteed by the GKLS form of Zt, because at any t the dynamical map Λt can be interpreted as a fictitious semi- Ztτ group Λt = e τ=1 generated by Zt (at this particular time instance). Λt must therefore be CPTP at t. In general, it is not straightforward to verify whether a given QME (1) yields SS dynamics [45], even after de- composing its dynamical generators according to Eq. (2). However, in the special case of commutative dynamics, for which [Ls, Lt] = 0 (or equivalently [Λs, Λt] = 0) for all s, t ≥ 0, one may directly identify the SS subclass, because (see App.A1): Lemma 1. Any commutative dynamics is SS iff (if and only if) for any t ≥ 0 the decomposition of its dynamical FIG. 1. Cross-section of the vector space defined by generators (2) fulfills the families of dynamical generators. The non-convex set (orange) describes a cut through the set of all physical Z t dτ D(τ) ≥ 0. (3) families, while the inner convex sets (blue) correspond to cuts 0 through convex cones of various dynamical subclasses possessing additive generators. Sets containing CP-divisible and In short, we term any such semigroup-simulable and com- semigroup evolutions are indicated, as well as two exemplary mutative dynamics SSC. SSC classes of dynamics. Physicality can be broken by adding (1) to a family Lt , which is SSC but non-CP-divisible, another Note that the condition (3) is clearly weaker than pos- (2) itive semi-definiteness, D(t) ≥ 0, at all times. Hence, family Lt that lies outside of the particular SSC class, even there exist commutative dynamics which are SS but not a semigroup. CP-divisible. However, let us emphasise that there also exist commutative dynamics which are physical but not hand, means that all the elements of the convex cone, (1) (2) even SS. An explicit example is provided by the eternally {αLt + βLt }α,β≥0, are physical. non-Markovian model of Ref. [46], as well as by other in- For CP-divisible dynamics, we observe that when both stances of random unitary [47, 48] and phase covariant (1) (2) Lt and Lt are of the GKLS form or even form a semi- [33] qubit dynamics, which we discuss in detail in App.B. group, so must any non-negative linear combination of them. Hence, it naturally follows that generator families describing CP-divisible evolutions constitute a convex cone contained in the physical set, with semigroups B. Additivity of dynamical generators forming a subcone. Furthermore, as we demonstrate in App.A3: We define the notion of additivity for families of dy- (1) (2) namical generators as follows: Lemma 2. Any SSC generator families Lt and Lt are (1) (2) (1) additive if upon addition, αLt +βLt with any α, β ≥ 0, Definition 2. Two physical families of generators Lt (2) they yield commutative dynamics. and Lt are additive if all their non-negative linear com- (1) (2) Hence, the non-negative linear span of any such SSC pair binations, αLt + βLt with α, β ≥ 0, are also physical. forms a convex cone contained in the physical set. More- over, one may then naturally expand such a cone by con- Note that according to Def.2 a pair of generator fami- sidering more than two, in particular, a complete set of lies can be additive only if each of them is individually SSC generator families whose non-negative linear combi- rescalable—remains physical when multiplied by an non- nations are all commutative. We term the convex cone negative scalar, i.e., Lt → αLt remains physical for any so-constructed a particular SSC class. α ≥ 0. However, as such a multiplication does not in- In Fig.1, we schematically depict the cross-section of validate the GKSL form of the decomposition (2) or the the set of physical generator families, which then also cuts condition (3), it follows that any generator family which through the convex cones containing generator families of is CP-divisible or SSC must be rescalable. the aforementioned dynamical subclasses. Importantly, From the linear algebra perspective [49], one may for- as all physical dynamics do not form a convex cone in the mally define the vector space containing families of dy- vector space, the ones lying within such a hyperplane are namical generators. Physical generators then form its described by a non-convex set that, in turn, contains the particular subset. Rescalability of a given Lt states convex sets of: CP-divisible dynamics, its semigroup sub- then that the whole ray {αLt}α≥0 lies within the phys- set, as well as ones representing particular SSC classes. (1) (2) ical set. Additivity of Lt and Lt , on the other Now, as indicated in Fig.1 by the dashed line, by 4 adding a generator family that is SSC but not CP- divisible (i.e., non-Markovian [41–43]) and another physical family, even a semigroup, which does not commute with the first—i.e., is not contained within the corresponding SSC class—one may obtain unphysical dynamics. Consider an example of two purely dissipative (Lt = Dt in Eq. (1)) qubit generators:

(1) Lt [ρ] = γ1(t)(σxρσx − ρ), (4a) 1 L(2)[ρ] = γ (t)(σ ρσ − {σ σ , ρ}), (4b) t 2 − + 2 + − where σ± = (σx ±iσy)/2 and σx, σy, σz are the Pauli operators, and γ1(t), γ2(t) are chosen such that the genera- (1) (2) tors are physical. Importantly, the families Lt and Lt despite being commutative, do not commute between one another. They belong to different SSC classes of qubit dynamics (see App.B), namely, random-unitary [47, 48] and phase-covariant [33] evolutions, respectively. In order to prove the situation indicated in Fig.1, we (1) (2) FIG. 2. Eigenvalues of the CJ matrix as a function construct examples in which both Lt and Lt are phys- of time, whose negativity demonstrates non-physicality of ical, but their sum is not. We take instances of γ (t) (1) (2) (•) 1 the dynamical maps generated by Lt + Lt with Lt as and γ2(t) with one rate being constant (semigroup), and defined in Eq. (4). We choose in (a): γ1(t) = sin(2t) and the other taking negative values for some times (non- γ2(t) = 1; while in (b): γ1(t) = 1/2 and γ2(t) = sin(t). R t Markovian) while fulfilling 0 dτγ(τ) ≥ 0 of Eq. (3) (SSC). Two simple examples are provided by choosing III. MICROSCOPIC APPROACH TO QMEs γ1(t) = sin(ωt) and γ2(t) = γ and vice versa, with γ and ω being positive constants. We consider then the (1) (2) Let us recall that the QME (1) constitutes an effective generator family Lt + Lt and solve analytically in description of the reduced dynamics, whose form must App.B2b for the families of maps Λ t that arise in both always originate from an underlying physical mechanism cases. For each Λt, we compute the eigenvalues of its responsible for both free (noiseless) and dissipative parts Choi-Jamiolkowski (CJ) matrix—all of which must be of the system evolution. In particular, given a micro- non-negative at all times for the map to be CPTP (see scopic model one should arrive at Eq. (1) starting from a App.A1). We depict them as a function of time in Fig.2 closed dynamics describing the evolution of: the system, for a choice of parameters which clearly demonstrates its environment, as well as their interaction; after tracing that the physicality is, indeed, invalidated at finite times. out the environmental degrees of freedom [1,2]. Their negativity, as demonstrated in App.B2b, can also be verified analytically. In App. B2b, we also consider additional choices of A. Microscopic derivation of a QME γ1(t) and γ2(t) for the generators (4), in order to show that the same conclusion holds when both semigroup and In a microscopic model of an evolving open quan- non-Markovian contributions come from explicit micro- tum system, as illustrated in Fig.3(a), one considers scopic derivations. In particular, as the generators de- a system of interest S, coupled to an environment E scribe dephasing (4a) and spontaneous-emission (4b) pro- that is taken sufficiently large for the total system to be cesses, we consider their non-Markovian forms derived closed. The global evolution is then unitary, U (t) = (see App.B1) from spin-boson and Jaynes-Cummings SE exp[−i(H + H + H )t], being determined by the free models, respectively [2]. S E I Hamiltonians HS and HE, and the system-environment Note that it follows from the above observations that interaction HI . In the Schrödingerpicture, the reduced physicality of a non-Markovian QME can be easily bro- state of the system, ρS(t) = TrE ρSE(t), evolves as ken by addition of even a time-invariant (semigroup) dissipative term. Moreover, one should be extremely careful d ρ (t) = −i Tr [H + H + H , ρ (t)] , (5) when dealing with dynamics described by generator fam- dt S E S E I SE ilies that are not even rescalable, e.g., see App.A2: ones that exhibit singularities at finite times [47], are derived where ρSE is the total system-environment state. assuming weak-coupling interactions [50], or lead to phys- If the environment and the system are initially uncorre- ical (even commutative) but non-SS dynamics [46]. lated, so that ρSE(0) = ρS(0) ⊗ ρE, and ρE is stationary, 5

in Eq. (7). However, under certain circumstances this can be justiﬁed. In App.C6, we discuss in detail the natural cases ˜ when variations of HS do not aﬀect the form of Lt in Eq. (7), yet we summarise them here by the following lemma:

0 Lemma 3. Consider a change HS → HS(t) = HS +V (t) in Eq. (5). The corresponding time-local QME can be 0 FIG. 3. Microscopic description of an open quan- obtained by just replacing HS with HS(t) in Eq. (7) while tum system S, (a): interacting with a single environment keeping L˜ unchanged, if at all times [V (t),HI ] = 0 and E; (b): simultaneously interacting with multiple, independent t either [HS,HI ] = 0 or [V (t),HS] = 0 (or both). environments E1, E2, E3, ... . Unfortunately, if the above suﬃcient condition cannot i.e., [HE, ρE] = 0, Eq. (5) can be conveniently rewritten be met, one must, in principle, rederive the QME (7) as (see also App.C2)[2, 51]: ˜ 0 and the corresponding generator Lt for HS(t). Moreover, such treatment is required independently of the interac- d Z t ρ¯S(t) = − ds TrE[H¯I (t), [H¯I (s), ρ¯SE(s)]], (6) tion strength, i.e., also in the weak-coupling regime dis- dt 0 cussed below. A prominent physical example is provided

i(H +H )t −i(H +H )t by coherently driven systems, for which V (t) represents where by the bar, •¯ := e S E • e S E , we the externally applied force. In their case, it is common denote the interaction picture with respect to the free that the time-dependence of V (t) is naturally carried over system-environment Hamiltonian HS + HE. Eq. (6) con- onto, and significantly amends, the dynamical generator stitutes the integro-differential QME discussed at the be- irrespectively of the coupling strength [51, 52]. ginning of Sec.II that, in practice, is typically recast into the time-local form (1), which after returning to the Schrödingerpicture (see App.C3 andC4) reads: 2. Generalisation to multiple environments d ρ (t) = −i[H , ρ (t)] + L˜ [ρ (t)] . (7) dt S S S t S Another important question one should pose is under ˜ what conditions the full derivation of the QME (7) can Importantly, Lt above can be unambiguously identified as the dynamical generator—containing both Hamilto- also be bypassed when dealing with a system that simul- nian and dissipative parts as in Eq. (1)—that arises taneously interacts with multiple environments—as de- purely due to the interaction with the environment; with picted in Fig.3(b). Motivated by the analysis of Sec.IIB, the system free evolution (dictated by the system Hamil- one may then naively expect that, given multiple additive generator families describing each separate interac- tonian HS) being explicitly separated. ˜(i) tion, Lt , they should be simply added to construct the ˜ P ˜(i) overall QME of the form (7) with Lt = i Lt [53]. 1. Dependence on the system Hamiltonian Such a procedure may, however, lead to incorrect dynamics, as may be demonstrated by considering explic- However, as detailed in App.C5, despite the separa- itly the microscopic model that incorporates interactions tion of terms in Eq. (7) the form of the dynamical gener- P with multiple environments—with now HE = i HEi ˜ P ator, Lt, may in general strongly depend on the system and HI = i HIi in Eq. (5). Following the derivation Hamiltonian HS. Crucially, this means that the evolu- steps of the time-local QME (7), while assuming its ex- tion of systems with different HS, which interact with istence both in the presence of each single environment the same type of environment, cannot generally be mod- and all of them, one arrives at a generalised QME (see elled with the same QME after simply changing the HS also App.D):

Z t d X (i) X ρ (t) = −i [H , ρ (t)] + L˜ [ρ (t)] − ds e−iHS (t−s) Tr H¯ (t − s), H , ρ (s) eiHS (t−s), (8) dt S S S t S Eij Ii Ij SEij i i6=j 0

i H +H τ −i H +H τ (i) ¯ ( S Ei ) ( S Ei ) ˜ the generator arising when only the ith environment is where HIi (τ) = e HIi e , Lt is 6

present, ρSEij denotes the joint-reduced state of the system and environments i and j, while TrEij stands for the trace over these environments. Crucially, the naive addition of generators would lead to a QME that contains only the ﬁrst two terms in Eq. (8). In particular, it would completely ignore the last term, which we here name the cross-term, as it accounts for the cross-correlations that may emerge between each two environments due to their indirect interaction being mediated by the system.

B. Microscopic validity of generator addition

The generalisation of the QME to multiple environ- FIG. 4. Validity of generator addition as assured ments (8) allows one to unambiguously identify when the by the commutativity of microscopic Hamiltonians. true dynamics derived microscopically coincides with the The QME (8) describing the multiple-environment scenario evolution obtained by naively adding the generators. of Fig.3(b) is considered. Each set (circle) above indicates that commutativity of the interaction Hamiltonians with: II Observation 1. A dynamical generator corresponding – each other, IS – the system Hamiltonian, IE – all the free to a system simultaneously interacting with multiple en- Hamiltonians of environments; can be assumed. vironments can be constructed by simple addition of the generators associated with each individual environment derivation stage, so that the QME so-obtained, despite iff the cross-term in Eq. (8) identically vanishes. correctly reproducing the reduced dynamics of the sys- In what follows, we show that Obs.1 allows one to prove tem under weak coupling, may yield upon integration the validity of generator addition in the weak-coupling closed dynamics with overall system-environment states regime. However, as the cross-term in Eq. (8) involves a that significantly deviate from the ansatz (9) and, in par- time-convolution integral, in order to prove that it identi- ticular, its tensor-product structure [52]. cally vanishes given the microscopic Hamiltonians satisfy In App.D1, we first employ operator Schmidt decom- particular commutation relations, we must consider the position [54] to reexpress each of the interaction Hamil- P Ei dynamics in its integrated form—at the level of the cor- tonians in Eq. (8) as HIi = k Ai;k ⊗ Bk , i.e, as a sum responding dynamical (CPTP) map. of operators that act separately on the system and corresponding environment. This decomposition, together with the ansatz (9) allows us to rewrite the overall QME 1. Weak-coupling regime (8) in terms of correlation functions involving only pairs of baths. Furthermore, the tensor-product structure of Lemma 4. The dynamical generator of the evolution of Eq. (9) ensures that each of these reduces to a product of a system simultaneously interacting with multiple envi- single-bath correlation functions. Hence, as any single- ronments in the weak coupling regime can be constructed bath (one-time) correlation function can always be as- by simple addition of the generators associated with each sumed to be zero, every summand in the cross-term of individual environment. the QME (8) must independently vanish. Note that, in particular, this holds for all QMEs de- Here, we summarise the proof of Lemma4 that can be rived using the time-convolutionless approach [55] up to found in App.D1, and generalises the argumentation of second order in all the interaction parameters represent- Ref. [12, 13] applicable to the more restricted regime in ing coupling strengths for each environment. which the Born-Markov approximation is valid. In particular, it applies to any QME valid in the weak- coupling regime (c.f. [2, 43, 51]) which is derived using 2. Commutativity of microscopic Hamiltonians the ansatz: O Next, we investigate the implications that commutativ- ρSE(t) ≈ ρS(t) ⊗ %Ei (t), (9) i ity of the system, environment, and interaction Hamilto- nians has on the validity of generator addition. We con- where ρS(t) is the reduced system state at t, while %E (t) i sider the cases when all HIi commute with each other can be arbitrarily chosen for t > 0—it does not need (II), with HS (IS), or with all the HEi (IE), and sum- to represent the reduced state of the ith environment, marise the results in Fig.4. We find that:

ρEi (t) = Tr¬Ei ρSE(t), as long as it initially coincides with its stationary state, i.e., %Ei (0) = ρEi . Let us em- Lemma 5. Only when the interaction Hamiltonians phasise that the assumption (9) is employed only at the commute among themselves and with the system free 7

Hamiltonian, i.e., [HIi ,HIj ] = 0 and [HIi ,HS] = 0 for consisting of many spin-1/2 systems. Within this model, all i, j; can the overall QME be constructed by adding the closed dynamics of the global qubit-magnets system dynamical generators associated individually with each can be solved and, after tracing out the magnets’ degrees environment—ignoring the cross-term in Eq. (8). of freedom, the exact open dynamics of the qubit can be obtained. As a result, we can determine the QME gener- Again, we summarise here the proof of Lemma5 that ators describing dynamics of the qubit when coupled to can be found in App.D2. However, in contrast to the one or more magnets, so that comparison with evolutions discussion of the weak-coupling regime, we are required obtained by adding the corresponding generators can be to return to the microscopic derivation of the QME (8). explicitly made. Crucially, the commutativity of interaction Hamilto- nians with one another as well as with HS—the region II ∩ IS marked ‘Yes’ in Fig.4—assures that the unitary of the global von Neumann equation (5) factorises, i.e.: A. Magnet as an environment

P −i(HS + HE +HI )t −iHS t Y −i(HI +HE )t USE(t)=e i i i =e e i i . Within our model, we allow the system free Hamilto- i nian HS to be chosen arbitrarily, yet, for simplicity, we (10) take the free Hamiltonian of the environment to vanish, As a result, the system dynamics is described by a prod- HE = 0. As a result, any initial environment state is (i) Q ˜ stationary, with [ρE,HE] = 0 trivially for any ρE. uct of commuting CPTP maps,ρ ¯S(t) = i Λt [¯ρS(0)], associated with each individual environment and given The environment is represented by a magnet that con- (i) ˜ −i(HIi +HEi )t i(HIi +HEi )t sists of N spin-1/2 particles, for which we introduce the by Λt [•] = TrEi{e (• ⊗ ρEi ) e }. By differentiating the dynamics with respect to t, it is magnetisation operator: then evident that the QME takes the form (8) with each N N (i) ˙ (i) (i) L˜ = Λ˜ ◦ (Λ˜ )−1 and the cross-term being, indeed, X (n) X t t t mˆ = σz = mk Πk, (11) ˜(i) absent. Note that, as all Lt must then represent gener- n=1 k=0 ator families belonging to a common commutative class, if each of them yields dynamics that is also SS, they all where Πk is the projector onto the subspaces with mag- must belong to the same SSC class in Fig.1. netisation mk (i.e., with k spins pointing up). The mag- In all other cases marked ‘No’ in Fig.4, the commu- netisation mk takes N + 1 equally spaced values between tativity does not ensure the generators to simply add. −N and N: We demonstrate this by providing explicit counterexamples based on a concrete microscopic model, for which mk = −N + 2k for k = 0, ..., N. (12) the evolution of the system interacting with each environment separately, as well as all simultaneously, can be Consistently with Sec.III, the initial state of the spin- explicitly solved. It is sufficient to do so for the settings magnet system reads ρSE(0) = ρS(0) ⊗ ρE, where we take the initial magnet state to be a classical mixture of in which either all HIi commute with all HEi and HS different magnetisations, i.e., (intersection IS ∩ IE in Fig.4), or all HIi commute with each other and all HEi (intersection II∩IE), since it then N follows that neither II, IS, nor IE alone can ensure the X validity of generator addition. ρE = qkΠk. (13) k=0 Note that it is known that [HI ,HS] = 0 implies the evolution to be CP-divisible [56]. Thus, as families of CP- The initial probability for an observation of the magneti- divisible generators are additive (c.f. Fig.1), our coun- sation to yield m is then terexample for IS ∩ IE below corresponds to a case where k generator addition results in dynamics which is physical p(m ) = Tr[ρ Π ] = q Tr Π . (14) but does not agree with the microscopic derivation. For k E k k k a setting in which H do not commute neither among Ii In the limit of large N, p(m ) approaches a continuous each other nor with H , validity of adding generators k S distribution p(m), whose moments can then be computed has been discussed in Ref. [7]. as follows:

N Z ∞ IV. SPIN-MAGNET MODEL X s s (mk) p(mk) −→ dm m p(m). (15) N→∞ k=0 −∞ In this section, we construct counterexamples to the validity of adding generators at the QME level for We consider only interaction Hamiltonians which cou- the relevant cases summarised in Fig.4. Inspired by ple the system qubit to the magnet’s magnetisation, i.e, Ref. [57, 58], we consider a single qubit (spin-1/2 par- ticle) in contact with multiple magnets—environments HI = A ⊗ mˆ (16) 8 with A being an arbitrary qubit observable. However, in As no coupling between different magnetisation sub- order to be consistent with Sec.III, we must impose that spaces (labelled by k) is present, after rewriting the l.h.s. above using Eq. (20), one obtains a set of uncoupled N X differential equations for each conditional state: TrE{HI ρE} = A Tr{mρˆ E} = A mk p(mk) = 0, k=0 ρ˙(k)(t) = −i[H + m A, ρ(k)(t)], (23) (17) S S k S which implies that we must restrict to distributions with ρ(k)(t) = ρ (0) for each k. p(mk) (and p(m) in the N → ∞ limit) with zero mean. S S (k) In the examples discussed below, we consider two ini- Hence, every ρS evolves unitarily within our model tial magnetisation distributions for the magnet in the (k) with US (t) := exp[−i(HS + mkA)t], while the overall asymptotic N limit. In particular, we consider a Gaus- evolution of the qubit (21) is given by the dynamical map, sian distribution: Λt, corresponding to a mixture of such (conditional) uni- 2 tary transformations distributed according to the initial 1 − m p(m) = √ e 2σ2 , (18) magnetisation distribution of the magnet, p(m ), i.e.: 2πσ k X (k) (k)† which formally corresponds to the asymptotic limit of ρS(t) = Λt[ρS(0)] = p(mk) US (t) ρS(0) US (t). a magnet being described by a microcanonical ensemble k [57]—its every spin configuration being equally probable, (24) N with qk = 1/2 in Eq. (14), yielding a binomial distri- Furthermore, as Λt constitutes a mixture of unitaries in bution of magnetisation with variance equal to the num- the model, it must be unital, i.e., for all t ≥ 0: Λt[11] = 11. ber of spins (σ2 = N). We also consider the case when the magnetisation follows a Lorentzian distribution in the N → ∞ limit, i.e.: 1. Bloch ball representation λ p(m) = , (19) We rewrite the above qubit dynamics employing the π(λ2 + m2) 1 Bloch ball representation [54], i.e., ρ ≡ 2 (11 + r · σ) with parametrised by the scale parameter λ (specifying the the Bloch vector r unambiguously specifying a qubit state half width at half maximum). ρ. Then, Eqs. (23) and (24) read, respectively: Given the above initial magnet state (13) and the inter- (k) (k) action Hamiltonian (16), the global system-magnet state r˙ (t) · σ = −i[HS + mkA, r (t) · σ] (25) constitutes at all times a mixture of states with different and magnet magnetisation. In particular, it can be decomposed at any t ≥ 0 as " # X (k) r(t) = Dt r(0) = p(mk) R (t) r(0). (26) X (k) ρSE(t) = qkρS (t) ⊗ Πk, (20) k k The rotation matrices above, R(k), constitute the SO(3) (k) (k) where every ρS can be understood as the (normalised) representations of the unitaries U ∈ SU(2) in Eq. (24), state of the system conditioned on the magnet possessing S and are thus similarly mixed according to p(mk). the magnetisation m . Consequently, the full reduced k The qubit dynamical map, Λt of Eq. (24), is repre- system state at time t reads sented by an affine transformation of the Bloch vector: X (k) Z ∞ ρS(t) = TrE ρSE(t) = p(mk)ρS (t). (21) X (k) Dt := p(mk) R (t) −→ dm p(m) R(m, t), k N→∞ k −∞ Crucially, within the model each of the conditional (27) (k) which is linear due to Λ being unital within the magnet states ρS in Eq. (21) evolves independently. In order to t show this, we substitute the system-environment state model, i.e., does not contain a translation. (20) and the microscopic Hamiltonians into the global Now, as the spaces of physical Λt (dynamical maps) von Neumann equation (5) to obtain and Dt (affine transformations) are isomorphic [54], the ˙ −1 dynamical generators of the former Lt := Λt ◦ Λt (see ρ˙SE(t) = −i[HS + HI , ρSE(t)] ˙ −1 App.A1) directly translate onto Lt := Dt Dt of the lat- = −i[HS ⊗ 11 + A ⊗ m,ˆ ρSE(t)] ter, with the map composition and inversion replaced by X X (l) matrix multiplication and inversion, respectively. More- = −i[H ⊗ 11 + m A ⊗ Π , q ρ (t) ⊗ Π ] S k k l S l over, as the vector spaces containing families of genera- k l tors defined in this manner must also be isomorphic, all X (k) = −i qk[HS + mkA, ρS (t)] ⊗ Πk. (22) the notions described in Sec.IIB—in particular, rescal- k ability and additivity—naturally carry over. 9

However, in order to define the Bloch ball represen- to asymptotically read tation of the environment-induced generator L˜ in the t   QME (7), we must correctly relate it to the interaction e−f(t) 0 0 −f(t) and Schrödingerpictures of the dynamics, summarised Dt =  0 e 0 , (32) in App.C1. In general (see App.C4 for the derivation 0 0 1 from the dynamical maps perspective), the Bloch ball 1 2 2 2 representation of L˜ reads with f(t) = 2 σ g t and f(t) = γgt in case of the Gaus- t sian (18) and Lorentzian (19) distributions p(m), respec- −1 −1 tively. Hence, the corresponding generators (28) in Bloch L˜ := L − R˙ S(t) RS(t) = RS(t) L¯ RS(t) , (28) t t t ball representation take a simple form: where R (t) ∈ SO(3) is the rotation matrix of the Bloch S −γ(t) 0 0 vector that represents the qubit unitary map, US(t) := exp[−iH t] ∈ SU(2), induced by the system free Hamil- Lt =  0 −γ(t) 0 , (33) S 0 0 0 tonian HS. ˜ As stated in Eq. (28), Lt may be equivalently specified which corresponds to the standard dephasing generator ¯ ¯˙ ¯ −1 with help of Lt := Dt Dt , i.e, the Bloch ball repre- with a time-dependent rate (as defined in Eq. (B6) of sentation of the dynamical generator defined in the in- App.B), i.e.: ¯ ¯˙ ¯ −1 teraction picture, Lt := Λt ◦ Λt —see also App.C3 ¯ L [•] = γ(t)(σ • σ − •) (34) for its formal microscopic definition. Importantly, Lt t z z may be directly computed for a given Dt of Eq. (27) by 2 2 first transforming it to the interaction picture, i.e., de- with γ(t) = σ g t in the Gaussian case, and constant ¯ −1 (semigroup) γ(t) = λg in the Lorentzian case. termining Dt := RS (t) Dt that is the Bloch-equivalent ¯ † of Λt[•] := US(t)Λt[•] US(t) discussed in App.C1. ˜ ¯ On the other hand, as Lt and Lt are linearly related B. Counterexamples to sufficiency of the via Eq. (28), their vector spaces must be isomorphic. commutativity assumptions Hence, in what follows, we may equivalently stick to the ¯ interaction picture and consider Lt instead, in particular, We now prove the regions marked ‘No’ in Fig.4. In ¯ (1) ¯ (2) ˜ (1) ˜ (2) Lt + Lt ⇔ Lt + Lt , when verifying the validity of particular, we provide explicit counterexamples which generator addition. assure that the commutativity assumption—associated with the particular region of the Venn diagram—is generally not sufficient for the system dynamics to be recov- 2. Example: magnet-induced dephasing erable by simple addition of the generators attributed to each of the environments. In order to do so, it is enough To illustrate the model, we first consider the case when to consider the scenario in which the qubit is indepen- it is employed to provide a simple microscopic derivation dently coupled to just two magnets via the mechanism of the qubit dephasing dynamics. We take: described above.

1 H = 0,H = 0,H = g σ ⊗ m,ˆ (29) E S I 2 z 1. IS ∩ IE commutativity assumption with the system Hamiltonian being absent, so that all the We start with an example of dynamics, in which the in- ¯ ˜ generators, Lt = Lt = Lt in Eq. (28), become equivalent. teraction Hamiltonians (trivially) commute with the free From Eq. (25) we get system and all the environmental Hamiltonians, but not among each other. In particular, we simply set: (k) i (k) r˙ (t) · σ = − gmk[σz , r (t) · σ], (30) 2 HS = HE1 = HE2 = 0, (35) 1 1 which just yields rotations of the Bloch ball around the z HI1 = g1 σz ⊗ mˆ 1,HI2 = g2 σx ⊗ mˆ 2 axis with angular speed depending on the magnetisation 2 2 mk, i.e., Eq. (27) with (in Cartesian coordinates) with subscripts {1, 2} labelling to the first and the second magnet. As in the case of the dephasing-noise derivation cos (gmt) − sin (gmt) 0 above, the system Hamiltonian is absent, so the genera- ¯ ˜ R(m, t) = sin (gmt) cos (gmt) 0 (31) tors in Eq. (28) coincide with Lt = Lt = Lt. 0 0 1 In the case of simultaneous coupling to two magnets, Eq. (20) naturally generalises to for the N → ∞ limit. X (k,k0) Integrating Eq. (31) over the initial magnetisation dis- 0 0 ρSE1E2 (t) = qk,k ρS (t) ⊗ Πk ⊗ Πk , (36) tribution p(m), we obtain the affine transformation (27) k,k0 10 where qk,k0 now represents the joint probability of find- Hence, we conclude that the commutativity of the in- ing the first and the second magnet in magnetisations teraction Hamiltonians with both system and environ- (k,k0) ment Hamiltonians, but not with each other, cannot as- mk and mk0 , respectively, while ρS (t) stands for the corresponding conditional reduced state of the system. sure the generators to simply add at the level of the Consequently, the (conditional) von Neumann equa- QME—as denoted by the ‘No’ label in the region of Fig.4 representing the IS ∩ IE commutativity assumption. tion (25), which now must be derived for HI = HI1 +HI2 , describes the dynamics of Bloch-vectors that represent 0 each conditional state, ρ(k,k )(t), being also parametrised S 2. II ∩ IE commutativity assumption by the two indices k and k0, i.e.:

(k,k0) i (k,k0) r˙ (t) · σ = − [g1m1,kσz + g2m2,k0 σx , r (t) · σ]. In order to construct an example of reduced dynamics 2 in which, at the microscopic level, the interaction Hamil- (37) tonians commute with each other and all free environ- Eq. (37) leads to coupled equations in the Cartesian ba- mental Hamiltonians but not the system Hamiltonian, sis, i.e. (dropping the indices k, k0 and the explicit time- we consider again a two-magnet model but this time set: dependence for simplicity): 1 r˙ = −g m r , (38a) H = ωσ ,H = H = 0, (40) x 1 1 y S 2 x E1 E2 r˙y = g1m1rx − g2m2rz, (38b) 1 1 HI = g1 σz ⊗ mˆ 1,HI = g2 σz ⊗ mˆ 2. r˙z = g2m2ry, (38c) 1 2 2 2 which can be analytically solved to obtain the R-matrix In contrast to the previous example, since HS 6= 0, in in Eq. (27)—labelled R12 to indicate that both magnets order to investigate the validity of generator addition we are involved. R12(m1, m2, t) possesses now two magneti- must either consider the environment-induced generators ˜ ¯ sation parameters associated with each of the magnets, Lt defined in Eq. (28) or the generators Lt directly com- and we state its explicit form in App.E1. puted in the interaction picture. Furthermore, we can straightforwardly obtain the so- We do the latter and compute both HI in the interac- ¯ i lution of the equations of motion when only one of the tion picture, i.e., HIi (t), which are obtained by replacing magnets is present by simply setting either g1 = 0 or σz Pauli operators in Eq. (40) with g2 = 0 in R12. In presence of only the first mag- iHS t −iHS t net (g2 = 0), we recover the magnet-induced dephasing σ¯z(t) := e σze = cos(ωt)σz + sin(ωt)σy. (41) noise described above—with R12(m1, m2, t) simplifying to R1(m1, t) that takes exactly the form (31). On the Importantly,σ ¯z(t) should be interpreted as a (time- other hand, when only the second magnet (g1 = 0) is dependent) operator A in the general expression (16) for present, which couples to the system via σx rather than HI that, in contrast to the previous case, is now identical σz, see Eq. (35), we obtain R2(m2, t) as in Eq. (31) but for both magnets. Thus, inspecting the general expres- with coordinates cyclically exchanged—see App.E1 for sion for the dynamics (25), we obtain the equation of explicit expressions. motion for the Bloch vector in the interaction picture, (k,k0) −1 (k,k0) We then average each Rx, where x = {12, 1, 2} denotes ¯r (t) := RS (t) r (t), that represents the qubit the magnet(s) being present, over the initial magnetisa- state conditioned on the first and second magnet pos- tions in the N →∞ limit. This way, we obtain the affine sessing magnetisations mk and mk0 , respectively, as maps (27) representing the corresponding qubit dynam- (k,k0) i ics for all the three cases in the asymptotic N limit as: ¯r˙ (t) · σ = − (g m + g m 0 )× (42) 2 1 1,k 2 2,k Z ∞ (x) (k,k0) Dt = dmx p(mx) Rx(mx, t), (39) × [cos(ωt)σz + sin(ωt)σy , ¯r (t) · σ], −∞ which leads to coupled equations (again, dropping the where in presence of both magnets m12 ≡ (m1, m2) and indices k, k0 and the explicit time-dependence): p(m12) ≡ p(m1) p(m2); and we take each p(mi) to follow a Gaussian distribution (18) with variance σ . Similarly, i r¯˙ = (g m + g m )(sin(ωt)¯r − cos(ωt)¯r ), (43a) we obtain the integral expressions for the time-derivatives x 1 1 2 2 z y ˙ (x) R ˙ r¯˙y = (g1m1 + g2m2) cos(ωt)¯rx, (43b) of the affine maps, Dt = dmx p(mx) Rx(mx, t) after also computing analytically all the corresponding R˙ x. r¯˙z = −(g1m1 + g2m2) sin(ωt)¯rx. (43c) Finally, we choose particular values of g1, g2, σ1, σ2 and time t, in order to numerically compute the integrals As before, see App.E2, we solve the above equations (x) of motion in order to obtain the R¯ -matrix of Eq. (27) in over the magnetisation parameters and obtain all Dt ¯ ˙ (x) the interaction picture, i.e., R12(m1, m2, t). Again, by and Dt . Our choice, allows us then to explicitly con- setting either g2 = 0 or g1 = 0, we obtain expressions (x) ˙ (x) (x) −1 struct dynamical generators Lt = Dt (Dt ) , which for R¯ 1 and R¯ 2, respectively, corresponding to the cases (12) (1) (2) importantly exhibit Lt 6= Lt + Lt , see App.E1. when only first or second magnet is present. We then 11

˙ also compute all R¯ x with x = {12, 1, 2}, in order to ar- dition does not generally correspond to the real evolu- rive at integral expressions for both the affine maps and tion derived from a microscopic model describing inter- ¯ (x) ¯˙ (x) actions with multiple environments. We have formulated their time-derivatives, i.e., Dt and Dt , respectively, computed now in the interaction picture. a general criterion under which the addition of genera- As in the previous example, we take initial magneti- tors associated with each individual environment yields sation distributions of both magnets to be Gaussian and the correct dynamics. We have then shown that this condition is generally fix all the model parameters (i.e., ω and g1, g2 for the satisfied in the weak-coupling regime, whenever it is cor- system and interaction, σ1, σ2 for the magnets, as well as the time t) in order to numerically perform the in- rect to use a master equation derived assuming a tensor- product ansatz for the global state describing the sys- tegration over magnetisations mx. We then find, see App.E2, a choice of parameters for which it is clear tem and environments. Finally, we have demonstrated (x) (x) (x) that, at the microscopic level, the commutativity of in- that the dynamical generators, L¯ = D¯˙ (D¯ )−1, ful- t t t teraction Hamiltonians among each other and with the ¯ (12) ¯ (1) ¯ (2) fil Lt 6= Lt + Lt . system Hamiltonian also ensures addition of dynamical Hence, we similarly conclude that the commutativity of generators to give the correct dynamics. all the interaction Hamiltonians with each other, and all We believe that our results may prove useful in ar- the free Hamiltonians of environments also cannot assure eas where the master equation description of open quan- the generators to simply add at the level of the QME— tum systems is a common workhorse, including quantum proving the ‘No’ label in the region of Fig.4 representing metrology, thermodynamics, transport, and engineered the II ∩ IE commutativity assumption. dissipation. ACKNOWLEDGMENTS

V. CONCLUSIONS We would like to thank L. Aolita and N. Bernades for interesting discussions on non-Markovianity that sparked We have investigated under what circumstances modi- this work, as well as A. Smirne, M. Lostaglio, S. Huelga, fications to open system dynamics can be effectively dealt L. Correa and A. S. Sørensen for helpful exchanges. with at the master equation level by adding dynamical J.K. and B.B. acknowledge support from the Spanish generators. We have identified a condition—semigroup MINECO (Grant QIBEQI FIS2016-80773-P and Severo simulability and commutativity preservation—applicable Ochoa SEV-2015-0522), FundacióPrivada Cellex, Gen- beyond Markovian (CP-divisible) dynamics which guar- eralitat de Catalunya (SGR875 and CERCA Program). antees generator addition to yield physical evolutions. J.K. is also supported by the EU Horizon 2020 pro- We have also demonstrated by considering simple qubit gramme under the MSCA Fellowship Q-METAPP (no. generators that even mild violation of this condition may 655161), while B.B. by the ICFO-MPQ Fellowship. M.P.- yield unphysical dynamics under generator addition. L. acknowledges also support from the Alexander von Moreover, even when physically valid, generator ad- Humboldt Foundation.

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tum systems evolving in a Markovian regime,” (2014), matrices {ρt ∈ B(Hd)}t≥0 that describe the system state arXiv:1408.6352 [quant-ph]. at each time t ≥ 0. The system evolution is then defined [57] A. E. Allahverdyan, R. Balian, and T. M. Nieuwen- by a family of dynamical maps (quantum channels [54]), huizen, “Understanding quantum measurement from the {Λt} with Λ0 = I being the identity map, such that solution of dynamical models,” Phys. Rep. 525, 1 – 166 t≥0 for any initial system state, ρ0, the state at time t ≥ 0 is (2013). given by [58] Mart´ıPerarnau-Llobet and Theodorus Maria Nieuwen- huizen, “Simultaneous measurement of two noncommut- ρ = Λ [ρ ] . (A1) ing quantum variables: Solution of a dynamical model,” t t 0 Phys. Rev. A 95, 052129 (2017). [59] E. C. G. Sudarshan, P. M. Mathews, and Jayaseetha Importantly, for a given dynamics to be physical the Rau, “Stochastic dynamics of quantum-mechanical sys- family {Λt}t≥0 must consist of completely-positive trace tems,” Phys. Rev. 121, 920–924 (1961). preserving (CPTP) maps. Only then, for any given 0 [60] M. M. Wolf, J. Eisert, T. S. Cubitt, and J. I. Cirac, “As- enlarged initial state %(0) ∈ B(Hd ⊗ Hd) with arbi- 0 0 sessing non-markovian quantum dynamics,” Phys. Rev. trary d = dim Hd, the state at every time t ≥ 0, i.e., Lett. 101, 150402 (2008). %(t) = Λt ⊗ I[%(0)], is guaranteed to be correctly de- [61] T. S. Cubitt, J. Eisert, and M. M. Wolf, “The complex- scribed by a positive semidefinite matrix. ity of relating quantum channels to master equations,” In practice, any linear map Λ : B(Hd) → B(Hd0 ) may Commun. Math. Phys. 310, 383–418 (2012). be verified to be CPTP by constructing its correspond- [62] A. G. Redfield, “On the theory of relaxation processes,” ing Choi-Jamiolkowski (CJ) matrix Ω ∈ B(H 0 ⊗ H ) IBM J. Res. Dev. 1, 19–31 (1957). Λ d d [63] R. Dümcke and H. Spohn, “The proper form of the gener- defined as [38, 39]: ator in the weak coupling limit,” Z. Phys. B 34, 419–422 (1979). ΩΛ := Λ ⊗ I [|ψihψ|] , (A2) [64] F. Benatti, D. Chru´sciński, and S. Filippov, “Tensor Pd d power of dynamical maps and positive versus completely with |ψi = i=1 |ii|ii and {|ii}i=1 being some orthonor- positive divisibility,” Phys. Rev. A 95, 012112 (2017). mal basis spanning Hd. In particular, a map Λ is CP and [65] Bassano Vacchini, “A classical appraisal of quantum def- TP iff its CJ matrix is positive semi-definite, i.e., ΩΛ ≥ 0, initions of non-markovian dynamics,” J. Phys. B: At., and satisfies TrH 0 {ΩΛ} = 11d, respectively. Mol. Opt. Phys. 45, 154007 (2012). d [66] D. Chru´scińskiand F. A. Wudarski, “Non-markovianity degree for random unitary evolution,” Phys. Rev. A 91, b. Dynamical generators 012104 (2015). [67] A. J. Leggett, S. Chakravarty, A. T. Dorsey, Matthew P. A. Fisher, Anupam Garg, and W. Zwerger, “Dynamics One may associate with any given dynamics (A1) the of the dissipative two-state system,” Rev. Mod. Phys. 59, family of dynamical generators, {Lt}t≥0, that specify for 1–85 (1987). each t ≥ 0 the time-local QME stated in Eq. (1) of the [68] P. Haikka, T. H. Johnson, and S. Maniscalco, “Non- main text [45–47], i.e.: markovianity of local dephasing channels and time- invariant discord,” Phys. Rev. A 87, 010103 (2013). ˙ [69] J.-G. Li, J. Zou, and B. Shao, “Non-markovianity of the ρ˙t = Lt[ρt] ⇐⇒ Λt = Lt ◦ Λt (A3) damped jaynes-cummings model with detuning,” Phys. Rev. A 81, 062124 (2010). d with •˙ ≡ dt • and the dynamical generator being then [70] A. S. Holevo, “A note on covariant dynamical semi- formally defined at each t ≥ 0 as groups,” Rep. Math. Phys 32, 211–216 (1993). [71] Bassano Vacchini, “Covariant Mappings for the De- ˙ −1 Lt := Λt ◦ Λt , (A4) scription of Measurement, Dissipation and Decoher- ence in Quantum Mechanics,” in Theoretical Founda- −1 −1 where Λt is the inverse (Λt ◦ Λt = I) of the dynam- tions of Quantum Information Processing and Commu- ical map at time t, and is not necessarily CPTP. How- nication: Selected Topics, edited by Erwin Brüningand ever, Λ−1 in general may cease to exist at certain time- Francesco Petruccione (Springer, 2010) pp. 39–77. t instances, at which the corresponding generators Lt then become singular, even though the family of maps is per- fectly smooth. Nevertheless, under particular conditions Appendix A: QMEs as families of dynamical [47], the resulting QME (A3) can still be integrated and generators yield correctly the original dynamics (A1). In the other direction, given a family of dynamical gen- 1. Describing open system dynamics erators {Lt}t≥0, one may write the corresponding dynamical map at any time t with help of a time-ordered a. Physically valid quantum dynamics exponential, expressible in the Dyson-series form [45, 47]:

Z t ∞ A particular evolution of an open quantum system is X (i) Λt = T← exp Lτ dτ = St [L•] , (A5) formally represented by a continuous family of density 0 i=0 14

(0) where St [•] = I and for all i ≥ 1: so that Eqs. (A5) and (A8) both, respectively, read t t 1 Z t Z t Z t Z Z (i) Λ = exp dτL = exp dτX . (A13) St [L•] := dt1 dt2 ... dti T← Lt1 ◦Lt2 ◦...◦Lti t τ τ i! 0 0 0 0 0 Z t Z t1 Z ti−1

= dt1 dt2 ... dti Lt1 ◦Lt2 ◦...◦Lti . 0 0 0 e. CP-divisible dynamics (A6) The dynamics is said to be divisible into CPTP maps— CP-divisible (or Markovian [41–43])—if its corresponding c. Instantaneous generators family of maps, {Λt}t≥0, satisﬁes for all 0 ≤ s ≤ t:

Given a family of dynamical maps {Λt}t≥0 defining the Λt = Λ˜ t,s ◦ Λs, (A14) evolution (A1), one can also construct the corresponding where Λ˜ t,s is a CPTP map itself. family of instantaneous generators {Xt}t≥0, defined for each t ≥ 0 as [45]: At the level of dynamical generators, this is equivalent to the statement that all {Lt}t≥0 are of the Gorini- d X := log Λ = Z˙ with Z := log Λ (A7) Kosakowski-Sudarshan-Linblad (GKSL) form [37, 40]: t dt t t t t 1 L [•] = −i [H , •] + Φ [•] − {Φ?[11] , •} , (A15) being the so-called instantaneous exponent, such that: t t t 2 t t Z where Ht is a time-dependent Hermitian operator, Φt is Λt = exp Zt = exp dτXτ , (A8) a completely-positive (CP) map, and they, respectively, 0 represent the Hamiltonian, Ht, and dissipative, Dt, parts ? where the above expression, in contrast to Eq. (A5), does in the QME (1) of the main text. Φt is the dual map of not involve the time-ordering operator T←. Φt that—given a Kraus representation of the CP map Φt, P † Although the instantaneous generators Xt cannot be i.e., a set of operators {Vi(t)}i satisfying i Vi(t) Vi(t) = P † directly used to construct the QME (A3), the family of 0 for all t ≥ 0 such that Φt[•] = i Vi(t) • Vi(t) [59]—is dynamical generators {L } can be formally related to ? P † t t≥0 defined as Φt [•] := i Vi (t) • Vi(t). the family of instantaneous ones. In particular, by sub- Thus, one may rewrite Eq. (A15) also as [2]: stituting into Eq. (A4) X 1 L [•] = −i [H , •] + V (t) • V (t)† − V (t)†V (t), • , d Z 1 t t i i 2 i i sZt (1−s)Zt Λ˙ t = exp Zt = ds e ◦ Xt ◦ e (A9) i dt 0 (A16) which—after fixing a particular orthonormal basis of ma- −1 −Zt and Λt = e , one observes that n † o trices {Fj}j satisfying Tr Fi Fj = δij in which each Z 1 P sZt −sZt Vi(t) = j Vij(t)Fj—can be further rewritten as in Lt = ds e ◦ Xt ◦ e . (A10) 0 Eq. (2) of the main text: X † 1 n † o Lt[•] = −i [Ht, •] + Dij(t) Fj • F − F Fj, • d. Commutative dynamics i 2 i i,j (A17) A dynamics is defined to be commutative if the maps with the time dependence of the dissipative part being describing the evolution in Eq. (A1), or equivalently— now fully contained within the matrix D(t). as follows from Eq. (A5)—all the dynamical generators Although any dynamical generator Lt, constituting defining the QME (A3) commute between one another, a traceless and Hermiticity-preserving operator, can be i.e., for all s, t ≥ 0: decomposed as above, the GKSL form (A15) ensures that for all t ≥ 0 there exists a matrix V(t) such that [Λs, Λt] = 0 ⇐⇒ [Ls, Lt] = 0. (A11) D(t) = V(t)†V(t). Hence, it follows that any dynamics is Moreover, Eq. (A8) implies then that also all instan- CP-divisible iff one may at all times decompose its cor- taneous exponents must commute with each another, responding dynamical generators according to Eq. (A17) with some positive semi-definite D(t) ≥ 0. [Zs, Zt] = 0, but also with instantaneous generators, 0 = ∂s [Zs, Zt] = [Xs, Zt]. Hence, in case of commutative dynamics the dynamical and instantaneous generators must coincide at all times, as f. Semigroup dynamics Z 1 Z 1 sZt −sZt An important subclass of commutative and CP- Lt = ds e ◦ Xt ◦ e = ds Xt = Xt, (A12) 0 0 divisible dynamics are semigroups, for which the whole 15 evolution is determined by a single fixed generator L, Crucially, the decomposition (A21) proves Lemma1 of the main text, as it becomes clear that the GKSL form {Lt ≡ L}t≥0 =⇒ {Λt = exp[t L]}t≥0 , (A18) of Zt is then fully ensured by the condition which in order to describe physical dynamics (so that all Z t Γ(t) := dτ D(τ) ≥ 0, (A22) Λt are CPTP) must be of the GKSL form (A15) with 0 both the Hamiltonian H and the positive semi-definite matrix D ≥ 0 in Eq. (A17) being now time-independent. stated in Eq. (3) of the main text. Hence, given a commutative dynamics for which condition (A22) holds, it must also be SS—constitute a semigroup-simulable and g. Semigroup-simulable (SS) dynamics commutative (SSC) evolution. In some previous works [60, 61], the SS property has been identified as Markovianity of the dynamics. Let us Definition 3. We define a map Λt to be (instanta- neously) semigroup-simulable (SS) at time t if its corre- emphasise that such a notion is non-trivially related to the concept of CP-divisibility introduced in App.A1e, sponding instantaneous exponent Zt = log Λt in Eq. (A7) is of the GKSL form, i.e.: which is more commonly associated with Markovian- ity [41–43]. The CP-divisibility ensures D(t) ≥ 0 in h i 1 n o Eq. (A17) at all times, so that (in case of commutative Z [•] = −i H˜ , • + Φ˜ [•] − Φ˜ ?[11] , • , (A19) t t t 2 t dynamics) the SSC condition (A22) is trivially fulfilled. However, as D(t) ≥ 0 is a stronger requirement, there where, similarly to Eq. (A15), H˜t and Φ˜ t are some Her- must exist (also commutative) evolutions that are SS mitian operator and CP map, respectively. If all instan- but not CP-divisible, e.g., instances of qubit dynamics taneous exponents, {Zt}t≥0, can be decomposed according discussed below in App.B. This fact can also be un- to Eq. (A19), we term the whole dynamics to be SS. derstood by inspecting Eq. (A10), from which it is clear that the GKSL form (A19) of the instantaneous exponent Importantly, given that Λt is SS at time t, a semigroup Z (and, hence, of X = Z˙ ) does not generally ensure parametrised by τ ≥ 0 with physical (of GKSL form) t t t the corresponding dynamical generator Lt to also be of generator L = Zt may be defined: GKSL form (A15). n o Ztτ Λ(˜ t)τ with Λ(˜ t)τ = e , (A20) τ≥0 2. Rescalability of dynamical generators so that it coincides with the original map at τ = 1, Λ(˜ t)τ=1 = Λt, or, in other words, “simulates” its action Definition 4. We define a physical family of dynamical at this particular instance of “fictitious time” τ. generators {Lt}t≥0 to be rescalable if by multiplying all its elements by any non-negative constant, α ≥ 0, one Observation 2. The SS property provides a sufficient obtains a generator family, but not necessary condition for physicality of dynamics. {L0 := αL } , (A23) Zt t t t≥0 If for a dynamical family Λt = e t≥0 all its instantaneous exponents are of the GKSL form (A19), it must that also yields physical dynamics. consist of maps which coincide with semigroups at all Given a family of dynamical maps {Λ } , by rescal- t ≥ 0 and, hence, all must be CPTP. In the other di- t t≥0 rection, however, there exist dynamics that are not SS ing its corresponding dynamic generators {Lt}t≥0, as in 0 but nonetheless physical. Examples may be found by Eq. (A23), we obtain a family of maps, {Λt}t≥0, that considering instances of, e.g.,random unitary and phase according to Eq. (A5) reads covariant, qubit evolutions, as shown below in App.B1. Z t ∞ ∞ X (i) X (i) In case of commutative dynamics, it follows from Λ0 = T exp L0 dτ = S [L0 ] = αiS [L ] . R t t ← τ t • t • Eq. (A13) that Zt = 0 dτLτ , so that one may explic- 0 i=0 i=0 itly connect the decomposition (A17) of the dynamical (A24) generator at time t with the one of the instantaneous Crucially, as the above Dyson series includes now the exponent in Eq. (A19), as follows factor α ≥ 0, it is non-trivial to determine whether the resulting map Λ0 is CPTP; even in case of commutative Z t t dynamics for which time-ordering, T←, can be dropped. Zt[•] = − i dτHτ , • (A21) 0 The rescalability, however, is naturally ensured in case Z t of CP-divisible evolutions (and, hence, semigroups), as X † 1 n † o + dτ Dij(τ) Fj • Fi − Fi Fj, • , the GKSL form (A15) of any dynamical generator is then 0 2 0 i,j trivially carried over to Lt in Eq. (A23) for any α ≥ 0. On the other hand, any family of dynamical generators R t ˜ where 0 dτHτ constitutes then Ht in Eq. (A19). yielding SSC dynamics must also be rescalable. As the 16 instantaneous exponents are then related to the dynami- with α, β ≥ 0, yield families of dynamical generators, R t {L0 } , that are physical. cal generators via Zt = 0 dτLτ , they transform similarly t t≥0 0 to Eq. (A23) with Zt := αZt. Thus, the condition (A22) ensuring their GKSL form (A19) is fulfilled for any α ≥ 0. Firstly, we realise that (as for rescalability) all pairs Nevertheless, non-rescalability of generators also nat- of generator families describing CP-divisible evolutions urally emerges in some particular situations, e.g., when must be additive, as by adding families of CP-divisible dealing with: dynamics according to Eq. (A25) one obtains generators a. Dynamical generators with singularities, which that are also of the GKSL form (A15). emerge in case of evolutions whose family of dynami- On the other hand, by considering generator families cal maps, {Λt}t≥0, contains non-invertible CPTP maps. describing SSC dynamics, we observe that: In this case, the dynamics can be unambiguously recov- ered from the dynamical generators only for times smaller Lemma 6. Any pair of SSC dynamics with generator n (1)o n (2)o than T , denoting the occurrence of the (first) singularity families Lt and Lt whose addition (A25) t≥0 t≥0 [47]. As a result, even though the dynamics is physical 0 yields commutative dynamics {Lt}t≥0 for any α, β ≥ 0 despite {Lt}t≥0 containing singular generators, as soon must be additive. as α 6= 1 in Eq. (A23) the integrability of the corresponding QME (A3)—and, hence, the physicality—is lost for n (1)o n (2)o Proof. As all the families Lt , Lt and times t ≥ T . We provide an explicit example of such a t≥0 t≥0 0 phenomenon below in App.B2a, where we discuss the {Lt}t≥0 are commutative, their instantaneous exponents Jaynes-Cummings model describing a qubit that under- 0 (1) (2) also add according to Eq. (A25), i.e., Zt = αZt +βZt . goes spontaneous emission [2]. n (1)o n (2)o (1) Moreover, as Lt , Lt are SS, both Zt b. Weak-coupling-based generators, which are ap- t≥0 t≥0 proximate and only valid for a particular timescale T and Z(2) must satisfy Eq. (A22) with Γ(1)(t) ≥ 0 and 2 t (0 ≤ t ≤ T ). Consider a family λ Lt of gen- (2) 0 0≤t≤T Γ (t) ≥ 0. Hence, any family {Lt}t≥0 must also be SS erators derived by employing a microscopic model and (and, hence, physical), as recomputing condition (A22) assuming the system-environment coupling constant, λ, 0 for Zt with help of Eq. (A21) it reads small enough, so that the weak-coupling approximation 2 real 2 0 (1) (2) to O(λ ) holds and Lt ≈ λ Lt [50] (e.g., by assum- Γ (t) = α Γ (t) + β Γ (t) ≥ 0, (A26) ing the Redfield form of the QME [62]). One may then simply interpret the rescaling factor as the square of the and is trivially fulfilled for any α, β ≥ 0. coupling constant, α = λ2. Importantly, such a generator family is guaranteed to yield physical dynamics—a family −2 of CPTP maps—only on timescales with T λ [63]. Appendix B: Rescalability and additivity of qubit Hence, by rescaling the generators with large enough dynamical generators α = λ2 or, in other words, by choosing strong enough coupling, one must at some point invalidate the weak- 1. RU and PC classes of qubit dynamics coupling approximation and, eventually, the physicality. c. Commutative but not SS dynamics. Although all families of dynamical generators that lead to SSC dy- We consider two important classes of commutative namics must be rescalable, the commutativity property qubit dynamics, namely, random unitary (RU) [47, 48] alone is not enough. A direct example is provided by the and phase-covariant (PC) [33] evolutions. In order to eternally non-Markovian model introduced in Ref. [46] provide their physically motivated instances, we explic- and discussed below in App.B1. In particular, when itly discuss exemplary microscopic derivations for the rescaling its generators according to Eq. (A23), one ob- (generalised) dephasing and amplitude damping models tains dynamics that is not physical for any 0 ≤ α < 1 that fall into the RU and PC classes, respectively. [64].

a. Random unitary (RU) dynamics 3. Additivity of dynamical generators Random unitary (RU) qubit dynamics are formed by The Def.2 of the main text may be restated in a more considering smooth families of Pauli channels, which up detailed form as: to unitary transformations represent the most general qubit unital (Λ[11] = 11) maps [65]. In particular, any Deﬁnition 5. Two families of physical and rescalable RU evolution is described by a qubit QME (A3) with n (1)o n (2)o dynamical generators Lt and Lt are ad- [47, 48]: t≥0 t≥0 ditive, if all their non-negative linear combinations, X Lt[•] = γk(t)(σk • σk − •) , (B1) 0 (1) (2) Lt := αLt + βLt (A25) k={x,y,z} 17

so that the RU generator family {Lt}t≥0 is fully speci- 0, with the notions of physicality and SS then trivially fied by the three rates γk(t) defining a diagonal form of coinciding. the general D-matrix in Eq. (A17), with Pauli operators The dephasing dynamics (B6) can be explicitly ob- {11, σx, σy, σz} constituting a basis for two-dimensional tained by considering various microscopic derivations, in Hermitian matrices. Hence, it directly follows that the which a qubit is coupled to a large environment via some GKSL form (A15) of RU generators, and hence the CP- Hint ∝ σn ⊗ Oenv. In Sec. IV A 2 of the main text, divisibility (A14) of the dynamics, is ensured iff at all we provide a compact example by using a toy-model times all γk(t) ≥ 0 are non-negative in Eq. (B1). of a qubit coupled to a large magnet. The most com- One may straightforwardly verify that any RU dynam- mon microscopic derivation, however, is constructed by ics (B1) is commutative, so that by Eq. (A12) dynamical considering a qubit coupled to a large, thermal bosonic and instantenuous generators coincide, and the instanta- bath [2]. The interaction is then modelled by Hint ∝ neous exponents according to Eq. (A21) read: P ? † σn ⊗ k gkaˆk + gkaˆk which couples the qubit to a X bosonic reservoir of an Ohmic-like spectral density [67]: Zt[•] = Γk(t)(σk • σk − •) (B2) s k={x,y,z} X 2 ω − ω J(ω) := g δ(ω − ωk) = e ωc , (B7) k ωs−1 R t k c with Γk(t) := 0 dτγk(τ). Hence, it directly follows from the condition (A22) that any RU dynamics is SS iff where ωc represents the reservoir cutoff frequency, while s ≥ 0 is the so-called Ohmicity parameter. ∀t≥0,k={x,y,z} :Γk(t) ≥ 0. (B3) Assuming further the reservoir to be at zero tempera- Note that, as one may easily construct families of dy- ture, the dynamical generators describing the qubit evo- namical generators (B1) that satisfy Eq. (B3) without lution take then exactly the form (B6) with the dephasing rate reading [68]: requiring γk(t) ≥ 0 for all t, there exist RU dynamics that are SS but not CP-divisible. − s h 2i 2 On the other hand, an explicit condition for the phys- γ(t) = ωc 1 − (ωct) Γ[s] sin[s arctan(ωct)] , (B8) icality of RU dynamics is known [47, 66]. In particular, RU dynamics is physical iff for all the cyclic permutations where Γ[s] above represents the Euler gamma function. of i, j, k ∈ {x, y, z} (i.e., such that ijk = 1): Moreover, one may show that the dephasing rate tem- porarily takes negative values iff s > 2, so that the dy- µi(t) + µj(t) ≤ 1 + µk(t), (B4) namics ceases then to be CP-divisible [68]. where each µi(t) := exp[−2 (Γj(t) + Γk(t))]. It is easy to verify that the physicality condition (B4) is less re- b. Phase-covariant (PC) dynamics strictive than the SS condition (B3). Hence, there exist RU dynamics that are physical but not SS, despite being A phase-covariant (PC) qubit evolution corresponds to commutative. a family of dynamical maps that possess azimuthal sym- a. Eternally non-Markovian model. An example of metry with respect to rotations about the z axis in the RU dynamics that is physical but not SS is also pro- Bloch-ball representation. The most general PC dynam- vided by the eternally non-Markovian model introduced ics is described by a qubit QME (A3) with [33]: in Ref. [46], which corresponds to the following choice of rates in Eq. (B1): 1 Lt[•] = γ−(t) σ− • σ+ − {σ+σ−, •} 1 1 2 γx(t) = γy(t) = , γz(t) = − tanh(t) , (B5) 1 2 2 + γ (t) σ • σ − {σ σ , •} + + − 2 + − for which the physicality condition (B4) holds, even though γ (t) < 0 (and hence Γ (t) < 0) for all t ≥ 0. z z + γz(t) σz • σz − • , (B9) b. Dephasing dynamics. The simplest example of RU dynamics (B1) is provided by the dephasing model: which represents a combination of relaxation, excitation and dephasing processes occurring with rates: γ (t), Lt[•] = γ(t)(σn • σn − •) , (B6) − γ (t) and γ (t), respectively; while σ := √1 (σ ± iσ ) + z ± 2 x y P 2 where σn = n·σ = i niσi, and σn = 11 implies knk = 1. are the transition operators. The unit vector n should be interpreted as a choice (a Although dynamical generators commute within each passive rotation in the Bloch-ball picture) of the Pauli- of the RU (B1) and PC (B9) classes of dynamics, they do operator basis, in which then Eq. (B6) corresponds to not generally commute in between the two. In fact, their (rank-one Pauli) RU dynamics (B1) with only a single common commutative subset corresponds to all unital term present in the sum. One may easily verify that for PC evolutions for which Lt in Eqs. (B1) and (B9) co- R t 1 1 the dephasing model to be physical Γ(t) = 0 dτγ(τ) ≥ incide with γx(t) = γy(t) = 2 γ+(t) = 2 γ−(t). Note 18

p 2 that, the eternally non-Markovian model with decay purely imaginary, d = i |d| with |d| = 2γ0λ − λ , so rates specified in Eq. (B5) is, in fact, both RU and PC, that the relaxation rate (B13) simplifies to while the dephasing (RU) dynamics belongs to the PC 2γ0λ class only when aligned along the z direction, i.e., when γ−(t) = , (B14) |d|t n = {0, 0, 1} in Eq. (B6). λ + |d| cot 2 As {11, σ+, σ−, σz} equivalently constitute a basis for 2 −λ two-dimensional Hermitian matrices, the properties of and diverges at every t = |d| arccot |d| + nπ with PC dynamics can be determined analogously to the RU + n ∈ N [2]. case. In particular, considering now k = {+, −, z}, a given PC evolution is CP-divisible iff all γk(t) ≥ 0 at all R t times, while it is SS iff all Γk(t) = 0 γk(τ)dτ ≥ 0 for any 2. Counterexamples to rescalability and additivity t ≥ 0. Moreover, it is not hard to verify that the family of PC generators (B9) is physical iff for all t ≥ 0 [33]: a. Non-rescalable qubit dynamics

2 2 2 η||(t) ± κ(t) ≤ 1 and 1 + η||(t) ≥ 4η⊥(t) + κ(t) , An explicit example of non-rescalable qubit dynamics (B10) is provided by the Jaynes-Cummings model of sponta- 1 −δ(t) − (δ(t)−4Γz (t)) where η||(t) := e , η⊥(t) := e 2 , κ(t) := neous emission described just above, considered in the −δ(t) R t δ(τ) on-resonance, strong-coupling regime. It leads to an ex- e 0 dτ e [γ+(τ) − γ−(τ)] and δ(t):=Γ+(t)+Γ−(t). Hence, similarly to the case of RU dynamics, as the ample of dynamics described in App.A2a with dynam- physicality condition (B10) is less restrictive than the SS ical generators being singular due to the damping rate requirement (Γk(t) ≥ 0), there exist (commutative) PC (B14) being divergent periodically in t. dynamics that are not SS but still physically legitimate. Considering then the rescaled version of the amplitude- 0 The eternally non-Markovian model [46], being both RU damping generator (B11), i.e., Lt of Eq. (A23), one may and PC, provides again an appropriate example. simply integrate the resulting QME (A3) for any t in a. Amplitude damping dynamics. The most com- order to explicitly determine the form of the rescaled dy- 0 mon example of PC dynamics, which is not RU, is the namical map Λt in Eq. (A24). For instance, when set- amplitude damping evolution that represents the pure ting γ0 = 3/2 and λ = 1 for simplicity in Eq. (B14), the corresponding CJ-matrix (A2), Ω 0 , may be explicitly relaxation process, i.e., spontaneous emission of a two- Λt level (qubit) system [2]; and corresponds to the choice computed and its non-zero eigenvalues read γ+(t) = γz(t) = 0 in Eq. (B9), i.e., √ 2α vals −α −αt t t λ± = 1 ± 2 e 2 cos √ + sin √ . 1 2 2 Lt[•] = γ−(t) σ− • σ+ − {σ+σ−, •} . (B11) (B15) 2 Crucially, although in the case of original dynamics vals Its canonical microscopic derivation stems from the (when α = 1) both λ± ≥ 0 for any t ≥ 0, only for times before the occurrence of the first singularity, i.e., when Jaynes-Cummings interaction model [2], Hint ∝ σ+ ⊗ √ √ † 2 −λ P ? t < T = |d| arccot |d| + π = 2 π − arctan( 2) , k gkaˆk + gkaˆk , in which the qubit is coupled to a cavity possessing Lorentzian spectral density [69]: the eigenvalues are guaranteed to be real and non- negative independently of α. In particular, for any t ≥ T 2 one may easily find α ≥ 0 (α 6= 1) such that the eigen- 1 γ0λ J(ω) = , (B12) values (B15) take complex values with the QME being, 2π (ω − ω − ∆)2 + λ2 0 in fact, not even integrable. where ∆ describes the difference between qubit transition, ω , and cavity central frequencies, while λ repre- 0 b. Non-additive qubit dynamics sents the cavity spectral width. Crucially, such a model— after tracing out degrees of freedom of the cavity—leads to a qubit QME (A3) with the dynamical generator In order to construct counterexamples to additivity, (B11), whose relaxation rate reads [69]: we consider dynamical generators given in Eq. (4) of the main text and make specific choices for their dissipation ( ) rates γ1(t) and γ2(t). We then solve the QME obtained 2γ0λ γ−(t) = Re dt , (B13) after adding the generators as in Eq. (A25) with some λ − i∆ + d coth 2 α, β ≥ 0, i.e., and does not exhibit a singular behaviour as long d 0 (1) (2) ρ(t) = Lt[ρ(t)] = αLt [ρ(t)] + βLt [ρ(t)], (B16) as the real part of the complex parameter d := dt p 2 (λ − i∆) − 2γ0λ is positive. in order to explicitly compute the corresponding family 0 However, this is not the case in the on-resonance (∆ = of maps {Λt}t≥0. Crucially, we find in this way fami- λ 0), strong-coupling (γ0 > 2 ) regime, in which d becomes lies containing maps that cease to be CPTP—with their 19

CJ matrices, {Ω 0 } as defined in Eq. (A2), exhibiting and similarly in the case of semigroup γ (t), both gener- Λt t≥0 2 negative eigenvalues at some time instances. ator families are SSC and hence rescalable, so that their In order to solve the QME (B16), we choose a qubit additivity may be unambiguously considered. operator basis: Considering their non-negative linear combinations, √ √ √ √ L0 = αL(1) + βL(2), one obtains generator families with µˆ0 = 11/ 2, µˆ1 = σx/ 2, µˆ2 = σy/ 2, µˆ3 = σz/ 2, t t t (B17) which allows us to use the matrix and vector repre-  0 0 0 0  sentations for generators and states, respectively. As βγ 0  0 − 2 0 0  Tr[ˆµiµˆj] = δij, any generator L may then be represented M =   . t  0 0 − βγ − 2α sin(ωt) 0  by a matrix M with entries Mij = Tr[ˆµiL[ˆµj]], while 2 any state ρ by a vector x with components x = Tr[ρµˆ ] βγ 0 0 −βγ − 2α sin(ωt) √ √ i i (B21) (x = Tr[ρ]/ 2 = 2/2 by definition). The QME (B16) 0 Solving Eq. (B18), one finds is then equivalent to the set of linear, coupled differential equations:   d x0(0) 0 − 1 βγt x(t) = Mt x(t), (B18)  e 2 x1(0)  dt  2α cos(ωt) βγt  x(t) = − 2α + − ,  e ω ω 2 x2(0)  0 0   where M is the matrix representation of L . − 2α + 2α cos(ωt) −βγt 2α t t e ω ω [x (0) + βγe ω I(t) x (0)] For our first example, we take dephasing and ampli- 3 0 (B22) tude damping generators of Eq. (4) in the main text to √ be non-Markovian and semigroup, respectively, with where x0(0) = 2/2 and

γ1(t) = sin(ωt) and γ2(t) = γ, (B19) Z t βγs− 2α cos(ωs) I(t) = ds e ω . (B23) where ω, γ > 0 are some ﬁxed constants. Crucially, since 0 for all t ≥ 0:

Z t The four eigenvalues of the CJ matrices, ΩΛ0 , for the 1 − cos(ωt) 0 t ds γ1(s) = ≥ 0, (B20) corresponding family of maps {Λt}t≥0 read 0 ω

r 2! vals 1 − 2α −βγt 2α +βγt 2α cos(ωt) 4α(cos(ωt)+1) 2α 2α cos(ωt) λ = e ω e ω ∓ e ω ± β2γ2e ω I(t)2 ∓ eβγt e ω + e ω , (B24) ∓,± 2

and are plotted in Fig.5(a) for α = β = 1, ω = 2, and SSC (and thus rescalable) and upon addition yield γ = 1. For t = π, the integral in Eq. (B23) evaluates to I ≈ 23.36, and it is easy to check that two of the 0 0 0 0  0 − β 0 0 eigenvalues are negative. Hence, the evolution is clearly 0  2  Mt = sin(ωt)  2αγ β  , unphysical. 0 0 − sin(ωt) − 2 β 0  2αγ β 0 0 − sin(ωt) − β (B26) For the second example, we consider the symmetric which after solving Eq. (B18) leads to case with the dissipation rates exchanged, i.e.:   x0(0) β cos(ωt) − β  e 2ω 2ω x1(0)  x(t) =  β β cos(ωt)   − −2αγt+   e 2ω 2ω x2(0)  − β −2αγt+ β cos(ωt) β/ω e ω ω [x3(0) + βe I(t) x0(0)] γ (t) = γ and γ (t) = sin(ωt). (B25) 1 2 √ (B27) with again x0(0) = 2/2 and now Z t 2αγs− β cos(ωs) I(t) = ds sin(ωs) e ω . (B28) 0 By the same argumentation as before both generators are The four CJ eigenvalues this time read: 20

(1) (2) (1) (2) FIG. 5. CJ eigenvalues as functions of time after adding generators, αLt + βLt . Lt and Lt describe qubit dephasing along x and amplitude damping in z, respectively, as in Eq. (4) of the main text. In all plots α = β = 1, while the rate functions are chosen so that: (a) γ1(t) = sin(2t), while γ2(t) = 1; (b) γ1(t) = 1/2, while γ2(t) = sin(t); (c) γ1(t) is set according to Eq. (B8) (super-Ohmic regime) with cut-off frequency ωc = 1 and Ohmicity parameter s = 4.5, while γ2(t) = 1; (d) γ1(t) = 1, while γ2(t) is fixed according to Eq. (B13) (off-resonant regime) with detuning ∆ = 3, spectral width λ = 0.05 and excited-state decay rate γ0 = 150. Note that in all cases negative eigenvalues occur, indicating that each evolution ceases to be physical at some point in time.

r ! vals 1 − β −2αγt β +2αγt β cos(ωt) β(cos(ωt)+1) β(cos(ωt)+1) 2 λ = e ω e ω ∓ e ω ± e ω β2e ω I(t)2 + (e2αγt + 1) , (B29) ∓,± 2

and are plotted in Fig.5(b) for α = β = 1, ω = 1, be constant, while the damping rate to the one of and γ = 1/2. For t = 2π the integral (B28) yields I ≈ Eq. (B13)—derived basing on the Jaynes-Cummings mi- −204.81, and again two of the eigenvalues are negative, croscopic model in which the qubit is coupled to a cav- proving the evolution to be unphysical. ity with a Lorentzian frequency spectrum—in the off- We repeat the above analysis, but considering this time resonant (∆ 6= 0) regime. Again, we find the CJ eigen- the dissipation rates of either dephasing or amplitude- values by solving Eq. (B18) numerically for fixed param- damping in Eq. (4) of the main text to have a functional eter values. These are plotted in Fig.5(d) for α = β = 1, form derived explicitly from an underlying microscopic ∆ = 3, λ = 0.05, γ0 = 150, and γ = 1. We observe again model yielding non-Markovian dynamics. that the evolution becomes unphysical, this time a bit Firstly, in an analogy to Eq. (B19), we consider the de- before t = π/2. phasing rate to be specified by Eq. (B8)—as if the qubit were coupled to a bosonic reservoir with an Ohmic-like spectrum—while the damping rate to be constant. In this case, we can solve Eq. (B18) numerically at each t Appendix C: Microscopic derivations of QMEs for given parameter settings, in order to compute the corresponding CJ eigenvalues. These are plotted in Fig.5(c) for α = β = 1, ωc = 1, s = 4.5, which corresponds to a We consider the situation depicted in Fig.3(a) of the super-Ohmic spectrum [68], and γ = 1. We observe that main text, in which a system of interest and its environ- the evolution becomes unphysical around t = π/2. ment evolve under closed dynamics determined by a time- Secondly, we consider the symmetric case in an anal- invariant total (T) Hamiltonian—consisting of Hamilto- ogy to Eq. (B25), this time setting the dephasing to nians associated with the system (S), the environment 21

(E) and their interaction (I): 2. QME in the integro-diﬀerential form

HT = HS + HE + HI . (C1) The von Neumann equation describing the unitary evolution of the closed system-enviroment (SE) system, i.e., Eq. (5) of the main text, in the IP reads:

1. Interaction and Schr¨odingerpictures dρ¯ (t) SE = −i H¯ (t), ρ¯ (t) . (C11) dt I SE The interaction picture (IP), which we denote here with an over-bar, is then deﬁned in the same man- Assuming the SE to initially be in a product state, ner for all operators and states acting on the system- ρ (0) = ρ (0) ⊗ ρ (C12) environment Hilbert space, i.e., as SE S E

with ρE being a stationary state of the environment that O¯ := ei(HS +HE )t O e−i(HS +HE )t (C2) satisfies H¯E(t), ρE = [HE, ρE] = 0, one may write the integral of Eq. (C11) as: for any given O ∈ B(HS ⊗ HE) that is specified in the Z t Schrödingerpicture (SP). ρ¯SE(t) = ρS(0) ⊗ ρE − i ds H¯I (s), ρ¯SE(s) . (C13) In contrast, a general dynamical map, Λt,t0 , that de- 0 scribes the evolution of solely the system between the initial time t0 and some later t transforms from SP to IP Tracing out the environment in Eq. (C11), so that (and vice versa) as: its l.h.s. reduces to dρ¯S(t)/dt and substituting into its r.h.s. forρ ¯SE(t) according to Eq. (C13), one arrives at the S† S S S† integro-differential equation describing the system den- Λ¯ t,t = U ◦ Λt,t ◦ U ⇐⇒ Λt,t = U ◦ Λ¯ t,t ◦ U , 0 t 0 t0 0 t 0 t0 sity matrix in the IP at time t: (C3) dρ¯ (t) where by S = −i Tr H¯ (t), ρ (0) ⊗ ρ (C14) dt E I S E t S † −iHS t iHS t Z U [ • ] := US(t) • U (t) = e • e (C4) t S − ds TrE H¯I (t), H¯I (s), ρ¯SE(s) . 0 we denote the unitary transformation induced by the The first term in Eq. (C14) may be dropped, as without system free Hamiltonian, H . However, as we consider S loss of generality one may impose throughout this work dynamical maps that commence at zero time (t = 0), see Eq. (A1), Eq. (C3) simplifies to 0 TrE H¯I (t)ρE = 0. (C15)

¯ S† S ¯ by shifting the zero point energy of Hamiltonians, i.e., by Λt = Ut ◦ Λt ⇐⇒ Λt = Ut ◦ Λt , (C5) changing HI and HS as follows which allows us to explicitly compute how the corre- H0 = H − Tr {H ρ } ⊗ 11 ,H0 = H + Tr {H ρ } , ¯ I I E I E E S S E I E sponding dynamical generators of Λt and Λt transform (C16) between the SP and IP. so that condition (C15) is ensured, given [HE, ρE] = 0, In particular, defining the IP-based dynamical genera- without affecting the total Hamiltonian HT in Eq. (C1). tor in accordance with Eq. (A4) as As a result, we obtain the QME in its integro- differential form that does not involve any approxima- ¯ ¯˙ ¯ −1 tions, but only assumes Eq. (C12) with [HE, ρE] = 0, Lt := Λt ◦ Λt , (C6) Z t ¯ dρ¯S(t) and substituting for Λt according to (C5), we obtain = − ds TrE H¯I (t), H¯I (s), ρ¯SE(s) , dt 0 (C17) ¯ ˙ S† S† ˙ −1 S Lt = (Ut ◦ Λt + Ut ◦ Λt) ◦ Λt ◦ Ut (C7) and constitutes Eq. (6) of the main text. ˙ S† S S† ˙ −1 S = Ut ◦ Ut + Ut ◦ Λt ◦ Λt ◦ Ut (C8) S† S = i[HS, •] + Ut ◦ Lt ◦ Ut , (C9) 3. QME in the time-local form

S where in the last line we have used the deﬁnition of Ut The QME (C17) despite being compact and exact is (C4), and accordingly deﬁned the SP-based dynamical typically not of much use, as it involves the full system- generator, i.e., as in Eq. (A4): environment state and a time-convoluted integral. Nev- ertheless, one may always formally rewrite it as a function ˙ −1 Lt := Λt ◦ Λt , (C10) of the system state at a given time. 22

After integrating the closed von Neumann dynamics as the environment-induced dynamical generator, which (C11), one should arrive at can be then associated solely with the impact of the environment on the system. ¯ ¯ † ˜ ρ¯SE(t) = USE(t)(ρS(0) ⊗ ρE) USE(t) (C18) Still, as Lt generally contains both Hamiltonian and ˜ dissipative parts, i.e., Lt [•] = −i [H(t), •] + Dt [•], one with the unitary rotation being formally defined as a may conveniently rewrite the QME (C23) as [2, 51]: time-ordered exponential: dρ (t)  t  S = −i [H + H(t), ρ (t)] + D [ρ (t)] , (C27)  Z  dt S S t S U¯SE(t) := T← exp −i ds H¯I (s) . (C19)   0 which allows to explicitly identify H(t) as the environment-induced Hamiltonian correction to the sys- Now, as the reduced state of the system is obtained at tem free evolution, e.g., representing the Lamb shifts any time by tracing out the environment, the dynamical when describing atom-light interactions [2]. Whereas, map, Λ¯ , associated solely with the system evolution in t Dt in Eq. (C27) may then be entirely associated with the the IP may be identified as dissipative impact of the environment. Lastly, let us emphasise that when investigating ¯ n ¯ ¯ † o ρ¯S(t) = Λt[ρS(0)] := TrE USE(t)(ρS(0) ⊗ ρE) USE(t) . whether by adding generators associated with different ˜ (C20) environments—i.e., the generators Lt in Eq. (C23) ob- Hence, if at a given t one can compute the inverse of the tained by considering the impact of each environment ¯ −1 ¯ −1 dynamical map, i.e., Λt such that ρS(0) = Λt [¯ρS(t)], separately—one reproduces the correct dynamics, it is as well as its the time-differential Λ¯˙ , which is now for- equivalent to consider all the generators in the IP. t ¯ mally determined by Eq. (C17) as As the IP-based generators, Lt, are linearly related to ˜ the environment-induced ones, Lt, by Eq. (C26), the vec- ˙ Λ¯ t [•] = (C21) tor spaces formed by their families must be isomorphic. Z t Hence, the notions of rescalability and additivity of gen- h ¯ h ¯ ¯ ¯ † ii erator families, discussed in Sec.IIB of the main text − ds TrE HI (t), HI (s), USE(s)(• ⊗ ρE) USE(s) , 0 and App.A above, are naturally carried over between the two, e.g., for any α, β, t ≥ 0: one may equivalently rewrite the exact QME (C17) into its time-local form (in the IP): ˜0 ˜(1) ˜(2) ¯0 ¯(1) ¯(1) Lt = αLt + βLt ⇐⇒ Lt = αLt + βLt (C28) dρ¯S(t) = L¯ [¯ρ (t)] , (C22) ˜x S ¯x S† dt t S with Eq. (C26) relating all Lt = Ut ◦ Lt ◦ Ut for each x = {0, (1), (2)}. ¯ ¯˙ ¯ −1 where Lt = Λt ◦Λt is the IP-based dynamical generator defined in Eq. (C6). 5. HS -covariant dynamics

˜ 4. QME in the Schr¨odingerpicture and the Although the dynamical generators Lt deﬁned in environment-induced generator Eq. (C26) arise due to the presence of the environment, their form may still strongly depend on the system free ˜ We rewrite the QME (C22) in the SP as Hamiltonian HS. Thus, Lt and, in particular, both its Hamiltonian H(t) and dissipative parts Dt in Eq. (C27) dρ (t) S = L [ρ (t)] , (C23) cannot be generally associated with the properties of just dt t S the environment and the interactions. In fact, only in very special cases the form of L can be derived from H , where in accordance with Eq. (C9) the IP-based dynam- t I H , and ρ . ical generator must be transformed to E E An important example is provided when the system S ¯ S† and interaction Hamiltonians alone commute: Lt[•] = −i[HS, •] + Ut ◦ Lt ◦ Ut [•]. (C24) As a result, we arrive at the time-local QME as stated in [HS,HI] = 0. (C29) Eq. (7) of the main text: As the global unitary U¯SE in Eq. (C19) then also com- ¯ ¯ dρ (t) mutes with HS, USE,HS = 0, the IP-based map Λt in S = −i [H , ρ (t)] + L˜ [ρ (t)] , (C25) dt S S t S Eq. (C20) is assured to be HS-covariant, i.e., to commute with any HS-induced unitary (C4) (and so must trivially and identify with help of Eq. (C24): the SP-based Λt), so that for any s ≥ 0 [70]: ˜ ˙ S S† S ¯ S† S ¯ ¯ S S ¯ ¯ S Lt := Lt − Ut ◦ Ut = Ut ◦ Lt ◦ Ut (C26) Us ◦ Λt = Λt ◦ Us ⇐⇒ Us ◦ Lt = Lt ◦ Us . (C30) 23

As noted above, the HS-covariance must be naturally in- alently obtained by simply adding V (t) to the Hamiltoni- ¯ herited by the IP-based dynamical generators Lt [70, 71], ans in Eq. (C27)—even though H(t) and Dt non-trivially which, in turn, must then coincide with the environment- depend on the original HS (but not on V (t)). ˜ ¯ induced ones, with Lt = Lt in Eq. (C26). As a result, Lastly, let us note that in case of HS-covariant dy- ˜ the form of Lt must then, indeed, be independent of HS. namics and [HS,HI ] = 0, one may play a similar trick in order to deal with the case when [V (t),HS] = 0 but [V (t),HI ] 6= 0, so that the modified dynamics is no longer 0 guaranteed to be HS-covariant. By redefining the inter- 6. Externally modifying the system Hamiltonian 0 action Hamiltonian this time as HI := V (t) + HI , which importantly commutes with HS, it becomes clear that In general, a modification of the system free Hamilto- the HS-covariance must be preserved. Nonetheless, al- nian H may affect both the Hamiltonian and the dissi- ˜ S though the dynamical generator Lt in Eq. (C25) remains pative parts of the generator L˜ in Eq. (C27). Neverthe- ˜ t then independent of HS, the form of Lt may depend on less, let us consider a transformation: 0 V (t) and must thus be rederived, i.e., based now on HI . 0 HS → HS(t) := HS + V (t) (C31) Appendix D: Microscopic validity of generator with V (t) being an arbitrary (potentially time- addition dependent) Hermitian operator. Crucially, by considering particular commutation relations satisfied by the 1. Weak-coupling regime microscopic Hamiltonians of Eq. (C1) and V (t), one may identify two important cases for which the microscopic Below, we prove Lemma4 stated in the main text, in rederivation of the QME (C27) can be bypassed—with particular, we show that under weak coupling the cross- the impact of V (t) being directly accountable for at the term in Eq. (8) can always be assumed to vanish. Hence, level of the QME: in accordance with Obs.1, it is then valid to add dynam- a. [HS,HI ] = 0, ∀t≥0 :[V (t),HI ] = 0. If the ical generators corresponding to each individual environ- system Hamiltonian commutes with the interaction ment, without necessity to rederive the overall QME. Hamiltonian—so that the dynamics is HS-covariant— The following proof can be regarded as an extension of and so does the perturbation V (t) for all t, then the argumentation found in Ref. [12, 13], which applies 0 the modified dynamics must also be HS-covariant, as to the more stringent regime in which the Born-Markov 0 ˜ [HS(t),HI ] = 0 at all times. Hence, the form of Lt in approximation holds. Eq. (C25) is unaffected by the modification of HS, re- Firstly, we perform the operator Schmidt decomposi- maining fully determined by HE, HI and ρE. Moreover, tion of each interaction Hamiltonian (indexed by i)[54]: the new dynamics is then correctly described by simply 0 X X replacing HS with H (t) in Eq. (C25) (or Eq. (C27)). Ei ¯ ¯ ¯Ei S HIi = Ai;k ⊗ Bk ⇔ HIi (t)= Ai;k(t) ⊗ Bk (t). b. [HS,HI ] 6= 0, ∀t≥0 :[V (t),HI ] = [V (t),HS] = 0. k k The above conclusion also holds when dealing with non- (D1) H -covariant dynamics, given that V (t) commutes with Ei S where {Ai;k}k and {Bk }k form then sets of Hermitian both the interaction and the system Hamiltonian. As operators that act separately on the system and corre- then [H ,H ] 6= 0, the generator L˜ in principle depends S I t sponding environment subspaces, i.e., HS and HEi , re- on HS. However, without affecting the total Hamiltonian spectively. HT in Eq. (C1) and hence the dynamics, we may redefine As noted above, this decomposition preserves its 0 the interaction Hamiltonian as HI := HI + HS, pretend- tensor-product structure in the IP, which is now defined ing the system Hamiltonian to be absent. In such a ficti- according to Eq. (C2) with the free system-environment tious picture, the QME (C25) possesses just the second Hamiltonian incorporating multiple environments, HS + ˜0 0 P term with Lt now being derived based on HI . As impor- i HEi . Hence, carrying out here the analysis in the IP 0 tantly [V (t),HI ] = 0 is fulfilled at all times, it becomes for compactness of the expressions, we rewrite the gen- clear that the dynamics must be V (t)-covariant. Hence, eral and exact QME (8) of the main text, which describes the perturbation must lead to a QME that may be equiv- a system interacting with multiple environments, as

Z t d X (i) X ρ¯ (t) = L¯ [¯ρ (t)] + ds Tr H¯ (t), H¯ (s), ρ¯ (s) (D2) dt S t S Eij Ii Ij SEij i i6=j 0 Z t nh h E iio X ¯(i) X X ¯ ¯Ei ¯ ¯ j = Lt [¯ρS(t)] + ds TrEij Ai;k(t) ⊗ Bk (t), Aj;l(s) ⊗ Bl (s), ρ¯SEij (s) , (D3) i i6=j k,l 0 24 where the second term above is the crucial, inter- reducing to a product of single-bath correlation functions: environment cross-term whose absence assures the va- n E o iHEi (t−s) i −iHEi (t−s) lidity of generator addition. Ci;k(t, s) := Tr e Bk e %Ei (s) . There exist various approaches to obtain simplified (D9) forms of QMEs for the weak-coupling regime [2, 13, Furthermore, as the two-bath correlation function in 43, 51]. Here, in order keep the derivation general Eq. (D5) factorises to Eq. (D8) with s = s0, the whole and emphasise necessary requirements for our arguments cross-term (D5) is guaranteed to vanish whenever at all to apply, we assume that the appropriate QME under times t ≥ 0 for each environment (labelled by i) and its weak coupling is derived after approximating the global Ei each operator Bk (labelled by k): system-environments state at every time t ≥ 0 as Ei Ci;k(t) := Ci;k(t, t) = Tr{Bk %Ei (t)} = 0. (D10) O ? ρSE(t) ≈ ρS(t) ⊗ %Ei (t), (D4) i Crucially, the condition Eq. (D10) can always be ensured by shifting adequately the interaction and the sys- where ρS(t) = TrE ρSE(t) is the reduced state of the system Hamiltonians without affecting the total Hamilto- tem at time t. Although in the weak-coupling approxi- nian (C1) (similarly to Eq. (C16) of App.C2). In par- mations [2, 51] the separable state of each environment ticular, one can redefine the system and each interaction in Eq. (D4) is frequently taken to be its reduced state Hamiltonian to be generally time-dependent and read: at time t, i.e., %Ei (t) ≡ ρEi (t) := Tr¬Ei ρSE(t), in what 0 X follows it can be chosen arbitrarily—as long as for all HS(t) = HS + Ci;k(t) Ai;k, (D11) environments (labelled by i) %Ei (0) = ρEi to maintain i,k consistency with the derivation in App.C2. 0 X ∀i : H (t) = HI − Ci;k(t)(Ai;k ⊗ 11E ), (D12) Let us stress that the tensor-product ansatz of Eq. (D4) Ii i i for the system-environments state is employed only to ob- k tain the form of the QME valid under the weak coupling, so that the decomposition (D1) of the interaction Hamil- and does not force the solutions of the QME to actually tonian for each environment becomes be separable states. In particular, the resulting QME, 0 X 0Ei X Ei H (t) = Ai;k⊗B (t) = Ai;k⊗[B −Ci;k(t)11E ], before tracing out environmental degrees of freedom, can Ii k k i k k yield upon integration statesρ ¯SE(t) that strongly de- (D13) viate from the form (D4) already at moderate times t, with the new correlation function (D10) identically van- even though the validity of the dynamics—and, hence, ishing by construction, as for any t ≥ 0: the QME employed—is still assured by weak coupling n o 0 0Ei [52]. Ci;k(t) = TrEi Bk %Ei (t) (D14) Thanks to the condition (D4), the crucial cross-term n o Ei within the exact QME (D3) can be reexpressed as = TrEi (Bk − Ci;k(t))%Ei (t) = 0. X X Z t For consistency, le us also note that the necessary re- ds C[i;k][j;l](t, s; s)× (D5) 0 quirement TrE H¯ (t)ρE = 0, introduced in App.C2, 0 Ii i i6=j k,l is then trivially fulfilled for every environment. Decom- × A¯ (t)A¯ (s)¯ρ (s) − A¯ (t)¯ρ (s)A¯ (s) posing H0 (t) according to Eq. (D13) and remembering i;k j;l S i;k S j;l Ii that [HE , ρE ] = 0 for each i, one gets (in the IP): −A¯j;l(s)¯ρS(s)A¯i;k(t) +ρ ¯S(s)A¯j;l(s)A¯i;k(t) , i i n o X ¯ 0Ei X ¯ where now Ai;k(t) TrEi Bk ρEi = Ci;k(0) Ai;k(t) = 0, k k n E o 0 ¯Ei ¯ j 0 (D15) C[i;k][j;l](t, s; s ) := Tr Bk (t) ⊗ Bl (s) %¯Eij (s ) , as each 0 (0) = 0 is zero by Eq. (D14). (D6) Ci;k Note that, in particular, the above argumentation is the two-bath correlation function that is indepen- holds for all QMEs derived using the time-convolutionless dent of the reduced system state, being evaluated only approach [55] up to the second order in all the interac- on% ¯Eij (t) :=% ¯Ei (t) ⊗ %¯Ej (t). Note that, as i and tion parameters—in which case the QME (D2) is from j are just labels of distinct environments, the correla- the start assumed to exhibit a time-local form, rather 0 tion function (D6) is symmetric with C[i;k][j;l](t, s; s ) = than involve a time-convolution integral. 0 C[j;l][i;k](s, t; s ). On the other hand, the most conservative Born- Moreover, Eq. (D4) assures all the correlation func- Markov approximation discussed in Ref. [12, 13] enforces 0 tions (D6) to factorise, so that for any s, s , t ≥ 0: every %Ei (t) in Eq. (D4) to be at all times the initial, stationary state ρE of each environment. As a result, all the n E o i 0 ¯Ei 0 ¯ j 0 C[i;k][j;l](t, s; s ) ≈ Tr Bk (t)¯%Ei (s ) ⊗ Bl (s)¯%Ej (s ) single-bath correlation functions, Ci;k(t, s) in Eq. (D9), become then t- and s-independent due to [H , ρ ] = 0, (D7) Ei Ei 0 0 and identically vanish by Eq. (D14). Hence, then trivially = Ci;k(t, s ) Cj;l(s, s ), (D8) C[i;k][j;l] = Ci;kCj;l = 0 at all times. 25

2. Commutativity of microscopic Hamiltonians be decomposed in the SP, as follows

−i(HS +HE +HI )t −iHS t Y −i(HE +HI )t USE(t) = e = e e i i , Here, we provide the proof of Lemma5 stated in the i main text, which assures that dynamical generators as- (D17) sociated with each individual environment can be simply so that at the level of the corresponding unitary maps: added at the QME level, if the interaction Hamiltonians SE S Y IEi commute between each other and with the system Hamil- Ut = Ut ◦ Ut , (D18) tonian. This condition corresponds to the II ∩ IS region i in the Venn diagram of Fig.4—marked ‘Yes’ to indicate IEi −i(HE +HI )t the validity of generator addition. where we have deﬁned Ut [•] := e i i • i(H +H )t e Ei Ii , and by Q we denote also the conjugation Let us note that whenever for all i and j: of multiple maps, i.e., for a given set of maps {Λi}i:

n Y Λi := Λn ◦ Λn−1 ◦ · · · ◦ Λ2 ◦ Λ1. (D19) i=1 HIi ,HIj = 0 and [HI ,HS] = 0, (D16) As a result, after straightforwardly generalising Eq. (C20) to multiple environments and transforming it P with HI := i HIi being the full interaction Hamilto- to the SP, we can generally write the system reduced nian, the global unitary dynamical operator (C19) can state at a given time t as

( " #) SE S Y SEi O ρS(t) = TrE Ut [ρS(0) ⊗ ρE] = TrE Ut ◦ Ut ρS(0) ⊗ ρEk (D20) i k      n o  S Y SEi SE1 O = Ut  TrEi6=1 Ut  TrE1 Ut [ρS(0) ⊗ ρE1 ] ⊗ ρEk   (D21)  i6=1 k6=1        S Y SEi ˜ (1) O = Ut  TrEi6=1 Ut  Λt [ρS(0)] ⊗ ρEk   = ··· = (D22)  i6=1 k6=1        S Y SEi ˜ (2) ˜ (1) O = Ut  TrEi6=1,2 Ut  Λt ◦ Λt [ρS(0)] ⊗ ρEk   = ··· = (D23)  i6=1,2 k6=1,2  " # S Y ˜ (i) S ˜ = Ut Λt [ρS(0)] =: Ut ◦ Λt[ρS(0)], (D24) i

where in (D22) by ··· we mean repeating the procedure in Eq. (D16), the overall map Λ˜ t in Eq. (D24) could have for the 2nd environment, and similarly in (D23) for all the ˜ (i) been constructed by composing the maps Λt in any or- other environments. The overall dynamical (SP-based) der. Hence, all maps originating from interactions with ˜ Q ˜ (i) map Λt = Λt is HS-independent and constitutes diﬀerent reservoirs commute, i.e., for all i and j: i n o ˜ (i) SEi a composition of maps Λt [•] := TrEi Ut [• ⊗ ρEi ] , ˜ (i) ˜ (j) ˜ (j) ˜ (i) each of which describing the impact of the ith environ- Λt ◦ Λt = Λt ◦ Λt . (D25) ment.

As a consequence, while due to the HS-covariance all As discussed in App. C5 above, the condition the dynamical generators induced by separate environ- [HI ,HS] = 0 in Eq. (D16) ensures the dynamics to be ˜(i) ments, i.e., Lt in Eq. (C26) indexed now by i, coincide HS-covariant—as explicitly manifested in Eq. (D24) in with their corresponding IP-based generators, they also S SEi which Ut , thanks to commuting with all Ut , commutes add at the level of the QME (C25) thanks to Eq. (D25). ˜ (i) also with all the Λt maps. On the other hand, as all Computing explicitly the dynamical generator induced SEi ˜ Ut commute between one another due to [HIi ,HIj ] = 0 by all the environments together, i.e., Lt associated with 26 the overall map Λ˜ t in (D24), we have of the main text, discussing the counterexamples to the   commutativity assumptions based on the spin-magnet −1 model. ˜ ˜˙ ˜ −1 X ˜˙ (i) Y ˜ (j) Y ˜ (k) Lt = Λt ◦ Λt =  Λt ◦ Λt  ◦ Λt i j6=i k −1 X ˜˙ (i) ˜ (i) X ˜(i) = Λt ◦ Λt = Lt , (D26) i i ˜(i) where Lt is the generator corresponding to the interaction with the ith environment alone, and we have used 1. IS ∩ IE commutativity assumption the commutativity of the maps (D25).

We solve the equations of motion (38) that describe the Appendix E: Spin-magnet model Bloch vector dynamics in order to obtain an explicit form of the R-matrix in Eq. (27), which is then parametrised We provide here the details and explicit form of rele- by magnetisations of the two magnets, m1 and m2, and vant quantities for the calculations presented in Sec.IVB reads:

√ √ √  2 2 2 2 2 2 2 2 2 2 2 2 2 1 2 2 2 2  cos t g m +g m g m +g m sin t g m +g m g1m1 2 sin t g m +g m g1g2m1m2 1 1 2 2 1 1 2 2 − √ 1 1 2 2 2 1 1 2 2 g2m2+g2m2 2 2 2 2 g2m2+g2m2  1 1 2 2 g m +g m 1 1 2 2  √ 1 1 2 2 √  sin t g2m2+g2m2 g m sin t g2m2+g2m2 g m   1 1 2 2 1 1 p 2 2 2 2 1 1 2 2 2 2  R12(m1, m2, t) =  √ cos t g m + g m − √  . (E1)  g2m2+g2m2 1 1 2 2 g2m2+g2m2   √ 1 1 2 2 √ √1 1 2 2  2 sin2 1 t g2m2+g2m2 g g m m sin t g2m2+g2m2 g m g2m2+cos t g2m2+g2m2 g2m2  2 1 1 2 2 1 2 1 2 √ 1 1 2 2 2 2 1 1 1 1 2 2 2 2  g2m2+g2m2 2 2 2 2 g2m2+g2m2 1 1 2 2 g1 m1+g2 m2 1 1 2 2

(x) ˙ (x) (x) −1 When only the ﬁrst magnet is present (g2 = 0), the above Lt = Dt (Dt ) the relevant aﬃne maps and their expression reduces to time-derivatives. For instance, when taking g =g = 2 and σ =σ = 1,   1 2 1 2 cos (tg1m1) − sin (tg1m1) 0 we obtain at t = 0.5: sin (tg m ) cos (tg m ) 0 R1(m1, t) =  1 1 1 1  , (E2)     0 0 1 −2 0 0 0 0 0 (1) (2) Lt=0.5 =  0 −2 0 , Lt=0.5 = 0 −2 0  , 0 0 0 0 0 −2 while, when in contact with only the second magnet (g1 = 0), it becomes (E4) and 1 0 0  −1.56835 0 0  R (m , t) = 0 cos (tg m ) − sin (tg m ) . (E3) 2 2  2 2 2 2  L(12) = 0 −7.26687 0 , (E5) 0 sin (tg m ) cos (tg m ) t=0.5   2 2 2 2 0 0 −1.56835

We also analytically compute the time-derivative R˙ 12 (as (12) (1) (2) which provides the desired example of Lt 6= Lt +Lt . well as R˙ 1 and R˙ 2, after setting g2 = 0 and g1 = 0, respectively) which we, however, do not include here due its cumbersome form. 2. II ∩ IE commutativity assumption With the exact expressions (E1-E3) at hand, we can explicitly write each affine map D(x) with x = {12, 1, 2} t We observe that, in order to solve the equations of according to Eq. (39) of the main text, i.e., as an average motion (43) stated in the main text, which describe the of the corresponding R over the Gaussian distributions x dynamics of the Bloch vector in the IP, i.e., ¯r(t) = p(m ) of fixed variance σ in Eq. (18); and similarly in i i R−1(t) r(t), it is convenient to move to a rotating frame ˙ (x) ˙ S case of the time-derivatives Dt by averaging Rx. defined as ˇr(t) := V(t) ¯r(t), where We perform the averaging integrals numerically af-   ter fixing the parameters g1, g2, σ1, σ2, and the time 1 0 0 t. As a result, we obtain the expressions of dynami- V(t) := 0 − cos(ωt) sin(ωt) (E6) (1) (2) (12) cal generators Lt , Lt , and Lt by substituting into 0 sin(ωt) cos(ωt) 27 is an orthogonal matrix such that V(t) = PRS(t) with the SP with the y → −y coordinate inverted. P = diag{1, −1, 1} and RS(t) being the SO(3) represen- Defining also γ = g1m1 + g2m2 for compactness, we −iHS t tation of qubit unitary US(t) = e induced by the obtain a simpler set of equations of motion: 1 system free Hamiltonian HS = 2 ωσx of Eq. (40). Hence, ˇr(t) = P r(t) can be interpreted as the Bloch vector in rˇ˙x = γrˇy, rˇ˙y = ωrˇz − γrˇx, rˇ˙z = −ωrˇy, (E7) which can be explicitly solved, yielding

√ √ √  cos γ2+ω2 t γ2+ω2 γ sin γ2+ω2 t γω cos γ2+ω2 t −1  2 2 √ − 2 2  γ +ω γ2+ω2 γ +ω   √ √  ˇ  γ sin γ2+ω2 t p ω sin γ2+ω2 t  R(m1, m2, t) =  − √ cos γ2 + ω2 t √  . (E8)  γ2+ω2 γ2+ω2   √ √ √   γω cos γ2+ω2 t −1 ω sin γ2+ω2 t γ2+ω2 cos γ2+ω2 t  − 2 2 − √ 2 2 γ +ω γ2+ω2 γ +ω

We then construct the R-matrix determining the affine For example, when choosing g1 = g2 = 2, σ1 = σ2 = 1 map Dt in Eq. (27) by transforming back the above Rˇ - and ω = 2, we obtain at t = 0.5: matrix to the IP, so that 0.379798 0. 0.  ¯ (1) ¯ (2) ¯ −1 ˇ Lt=0.5 = Lt=0.5 =  0. 0.779093 −1.40007 , R(m1, m2, t) = V (t) R(m1, m2, t) V(0). (E9) 0. 1.70235 −3.05922 (E10) We do not enclose here the explicit forms, but, as in giving the previous example, we also compute all the relevant   ¯ ¯˙ 0.759597 0. 0. Rx and Rx in the IP, with x = {12, 1, 2}, which allow ¯ (1) ¯ (2) Lt=0.5 + Lt=0.5 =  0. 1.55819 −2.80015 . us to obtain the integral expressions for the correspond- 0. 3.4047 −6.11843 ¯ (x) ¯˙ (x) ing affine maps, Dt , and their time-derivatives, Dt . (E11) Again, we choose the initial magnetisations of both mag- On the other hand, we find in presence of both magnets: nets to be Gaussian distributed with both p(mi) as in Eq. (18) of fixed variance σi. 1.28248 0. 0.  ¯ (12) As before, we perform the averaging integrals numer- Lt=0.5 =  0. 13.0326 −16.6865 , (E12) ically, after fixing the model parameters—now: ω, g1, 0. 28.4767 −36.4608 g2, σ1, σ2, and the time t—in order to obtain numerical ¯ (x) ˙ (x) (x) −1 ¯ (12) ¯ (1) ¯ (2) expressions for all Lt = Dt (Dt ) . what, thus, provides an instance of Lt 6= Lt + Lt .