Application of Quantum Darwinism to a Structured Environment

Application of quantum Darwinism to a structured environment

Graeme Pleasance∗ and Barry M. Garraway Department of Physics and Astronomy, University of Sussex, Falmer, Brighton, BN1 9QH, United Kingdom (Dated: April 9, 2020) Quantum Darwinism extends the traditional formalism of decoherence to explain the emergence of classicality in a quantum universe. A classical description emerges when the environment tends to redundantly acquire information about the pointer states of an open system. In light of recent interest, we apply the theoretical tools of the framework to a qubit coupled with many bosonic subenvironments. We examine the degree to which the same classical information is encoded across collections of: (i) complete subenvironments and (ii) residual “pseudomode” components of each sub- environment, the conception of which provides a dynamic representation of the reservoir memory. Overall, significant redundancy of information is found as a typical result of the decoherence process. However, by examining its decomposition in terms of classical and quantum correlations, we discover classical information to be non-redundant in both cases (i) and (ii). Moreover, with the full collection of pseudomodes, certain dynamical regimes realize opposite effects, where either the total classical or quantum correlations predominantly decay over time. Finally, when the dynamics are non- Markovian, we find that redundant information is suppressed in line with information backflow to the qubit. By quantifying redundancy, we concretely show it to act as a witness to non-Markovianity in the same way as the trace distance does for nondivisible dynamical maps.

I. INTRODUCTION The QMI defines a measure of correlations which act as a communication channel between the subsystems. The theory of open quantum systems formulates the The central thesis of quantum Darwinism is that the emergence of classical-like behavior in quantum objects class of system-environment states produced by deco- through the action of environment induced decoherence herence is unique, in that the states contain correla- [1–5]. This is a phenomenon where superpositions are tions encoding many local copies of classical data about removed over time, leaving certain mixtures of stable the open system. This is done under a selection pro- macroscopic states, known as pointer states [6–8]. In- cess during which only records specific to the pointer evitably, mitigating this process by, for example, utiliz- states can create lasting copies of themselves, and so end ing reservoir engineering techniques [9, 10] in an effort to up passing their information into many different frag- successfully realize quantum technological devices relies ments. The redundancy Rδ [13] quantifies the average on further understanding the fragility of quantum states number of fragments that record up to classical information I(ρ ) = (1 δ)S(ρ ), with a small deficit δ. To and their information. SXf − S During the decoherence process, the issue of how in- this effect, a large Rδ implies widely deposited classical formation about the pointer states is communicated to data in the environment on the pointer observable—this the external observer is usually forgone by “tracing out” being, in practice, the only knowledge of the quantum and ignoring the active role of the environment. Quan- system accessible to measurements. tum Darwinism instead proposes that a new step can In this paper we join this framework to a qubit sys- be made [11]. When an open system S interacts with tem coupled to an ensemble of bosonic subenvironments. its surroundings, correlations invariably develop between The motivation is two-fold. Firstly, it is of fundamental the two. Thus, to understand the nature of the informa- interest to understand whether the success of quantum tion passed to the environment, it is useful to consider its Darwinism applies to an as of yet unexplored regime. It role explicitly in the formalism rather than as a passive has in fact been shown that classical features are inher- ent to generic models of quantum Darwinism [14], but

arXiv:1711.03732v2 [quant-ph] 8 Apr 2020 sink of coherence. In our paper we focus on some particular region of interest within the environment, labeled it is unclear under what conditions these features mani- X, which we will contextualise shortly. Given that the fest beyond the scope of this work. Secondly, we aim to system interacts with the degrees of freedom of X, the consolidate recent investigations which found quantum quantum mutual information (QMI) [12] can be used to Darwinism to be inhibited when memory effects arise in ascertain what is known about the system by a fragment the dynamics [15, 16]. of this part of the environment, Our approach enacts a partitioning of the environment into its memory and non-memory parts, where X is as- I(ρ ) = S(ρ ) + S(ρ ) S(ρ ), (1) SXf S Xf − SXf signed to either one of two cases: (i) the full environment modes or (ii) the memory region (see Fig. 1). The in- where Xf makes up some fraction (in size) of X and S(ρ) = Tr [ρ ln ρ] is the von Neumann entropy of ρ. tent is to specifically look at where information is shared − redundantly: also, in view of Refs. [17, 18], we aim to examine how quantum Darwinism is affected for state (ii) which mixes over time by evolving through a noisy ∗ [email protected] 2 quantum channel. As we will show, in case (i) redundant Non-Markovian environment e information tends to always emerge at long times, which | i is also found in case (ii) though this can only be quan- ω0 Sub-environments tified within certain dynamical limits. However, judging g . . whether true effects of quantum Darwinism appear from | i . . E a large redundancy of the total information does not give ⊗ ⊗ 1 Ω1 an accurate picture: writing the QMI in terms of the ac- . . S . . cessible information and the quantum discord shows most E2 Ω2 ⊗ ⊗ of the environment has to be measured to gain close to full classical data on the qubit. Further, the redundancy Ef measure Rδ is used to explore how non-Markovian behav- (i) ior of the open system affects its shared correlations with the environment. To gain a consistent interpretation of Pseudomodes Markovian information backflow using Eq. (1), we check the time environment . evolution of the redundancy against temporal changes in . the trace distance for an arbitrary pair of input states to P1 Γ Ω1 . the channel. . This paper is organised as follows. In Sec. II we intro- S Ω duce the dynamical solutions of the model via a master 2 P2 equation that describes the dynamics of the qubit and Pf R pseudomodes (memory). In Sec. III we address the ap- (ii) plication of the quantum Darwinism to our model. In Sec. IV we compute the average of the QMI for different fragments and examine the multipartite structure of FIG. 1. Schematic showing two dynamical representations of the total system-environment correlations, as quantified the model. (i) The original picture provided by Eqs. (2) and (3), where the qubit system S couples to many subenviron- by Eq. (1), in terms of its classical and quantum compo- ments E ,E ,... with strengths Ω , Ω ,... . (ii) An equiv- nents. Finally, we analyze the non-Markovian behavior of 1 2 1 2 alent picture in terms of pseudomodes, labeled P1,P2,... , the model with regard to its affect on information redun- which are each damped by independent Markovian reservoirs dancy. The paper is then summarized in Sec. V, while R at a rate Γ. The environment is sampled by constructing details of all methods are outlined in the Appendices. fragments out of the bare subenvironments or the pseudomodes, indicated by Ef and Pf , respectively.

II. DYNAMICAL MODEL A. Solutions to the model We start by considering a qubit S interacting with an environment E of bosons. The environment is arranged The pseudomode method [19, 20] provides complete #E solutions to the model by replacing the environment as E = k=1Ek and initially in a vacuum state, where the index⊗k = 1, 2,..., #E labels individual subenviron- with a set of damped harmonic oscillators—the pseudomodes—which are identified through evaluating the poles ments Ek. Each are composed of harmonic oscillators of the spectral distribution ρ (g )2 when analytically with frequencies ωλ. The Hamiltonian of the qubit and λ k,λ environment are thus given by (~ = 1) continued to the complex plane. For a single shared excitation between the system and environment it is possi- X ble to derive a master equation describing the combined HS = ω0σ+σ ,HE = ωλa† ak,λ. (2) − k,λ time evolution of the qubit and pseudomodes. Impor- k,λ tantly, since the resulting equation is of Lindblad form, the original non-Markovian system is effectively mapped Here, the operator σ+ (σ ) raises (lowers) the qubit en- − onto an enlarged qubit-plus-pseudomode system, the dy- ergy by an amount ω , while a (a† ) is the bosonic 0 k,λ k,λ namics of which are Markovian. annihilation (creation) operator of the λ-mode in the k To retrieve the solutions using such an approach, it sub-environment. Note variables between subenviron- first proves useful to extract the frequency dependence ments commute as [ak,λ, ak† 0,λ0 ] = δk,k0 δλ,λ0 . Within the of the system-environment coupling constants onto the rotating wave approximation and interaction picture, the structure function D(ωλ), interaction Hamiltonian HI reads 2 X 2 Ωk H (t) = (σ B (t) + h.c.) , (3) ρλ(gk,λ) = D(ωλ), (4) I + ⊗ k 2π k where ρλdωλ gives the number of modes in the bandwidth P where Bk(t) = λ gk,λak,λexp[ i(ωλ ω0)t] and gk,λ is dωλ, and Ωk is the coupling parameter of the qubit and the coupling to the qubit. − − k sub-environment. The above is normalized according 3 to 1

Z ∞ dωD(ω) = 2π, (5) 0.8 (a) −∞ which introduces the total coupling strength Ω0 via the 0.6 2 P 2 P 2 relation Ω0 = k Ωk = k,λ(gk,λ) . In the following we consider structure functions of the form 0.4 Populations X wkΓ D(ω) = 2 2 , (6) 0.2 (ξk ω) + (Γ/2) k − 0 where Γ is the width and ξk is the peak frequency of an 0 2 4 6 8 10 12 individual Lorentzian, each weighted by real positive con- Ω0t stants w satisfying P w = 1. For simplicity, we have k k k 1 imposed that the Lorentzian widths—associated with the coupling of the system to each sub-environment—are equal. 0.8 (b) Suppose the qubit is initially prepared in the state 0.6 ψ S = cg g + ce(0) e , with g ( e ) its ground (excited)| i state.| i As the number| i of excitations| i | i are conserved in this model, the total state is restricted to the single 0.4 excitation manifold, admitting the closed form Populations X 0.2 ψ(t) = c g, 0 + c (t) e, 0 + c (t) g, 1 , (7) | i g | i e | i k,λ | k,λi k,λ 0 0 2 4 6 8 10 12 where e, 0 and g, 1k,λ indicate the excitation in the Ω0t qubit and| thei λ-mode| ofi the k sub-environment, respectively. By substituting Eq. (7) into the Schr¨odinger FIG. 2. Time evolution of the qubit Green’s function 2 2 2 equation, we obtain a dynamical equation for ce(t) by (blue solid curve) from ce(t) = G(t) ce(0) and ex- | | | | | | eliminating the variables ck,λ(t). This leads toc ˙e(t) = cited pseudomode population for ce(0) = 1. Parameters are R t 2 dsf(t s)c (s), where the memory kernel f(t s) is Γ = 10− , 0.1, 0.5 Γ+ (red long-dashed, violet dashed, black − 0 − e − { } obtained by taking the continuum limit over all subenvi- dotted curves) for ∆ = 0. (a) Weak coupling case, Γ+ = 10Ω0. ronments as follows: (b) Strong-coupling (to moderate-coupling) case, Γ+ = Ω0. X h i f(t s) = B (t),B†0 (s) − k k k,k0 The dynamics of the joint qubit-pseudomode degrees of 2 Z freedom are formulated in terms of a Markovian master Ω0 ∞ i(ω ω )(t s) = dωD(ω)e− − 0 − . (8) equation [21] 2π −∞ X ρ˙ = i[H (t), ρ ] + Γ [a ]ρ , (12) Because Eq. (6) is meromorphic and contains simple SP − 0 SP D k SP poles in the lower half complex plane, we can rewrite k the preceding equation of motion as 1 where [a ] a a† a† a , and a (a† ) is the an- D k · ≡ k · k − 2 { k k ·} k k X i∆kt nihilation (creation) operator of the k-pseudomode. This c˙e(t) = i Ωke− bk(t), (9) − master equation is exact and describes the joint unitary k Γ time evolution of the qubit and pseudomodes, generated b˙ (t) = b (t) iΩ ei∆ktc (t), (10) by the Hamiltonian k − 2 k − k e

X i∆kt which are defined in a new rotating frame with respect H0(t) = Ωk e− σ+ ak + h.c. . (13) P ⊗ to ∆ a† a , and where ∆ = ξ ω . The coefficients k k k k k k k − 0 Z t Hence, the original environment is equally represented Γt/2 (i∆k Γ/2)s bk(t) = iΩke− ds e − ce(s) (11) in terms of a new structured one with a bipartite inner − 0 structure, composed of a set of uncoupled pseudomodes #E are interpreted as those of pseudomodes. We note the P = k=1Pk and Markovian reservoirs R. The qubit one-to-one correspondence between the number of poles interacts⊗ directly with the pseudomodes, which each in contained in Eq. (6) when extended to the complex ω- turn leak into R at a rate Γ (see Fig. 1). plane and the resulting number of pseudomodes. As Eqs. (9) and (10) are linear, exact solutions to 4 the coefficients may be found directly through numeri- further, first we set ∆ = 0, this being the case we shall cal inversion of the equations. However, we consider the focus on from now on. In the strong-coupling regime continuum limit of pseudomodes, and, more generally, of [23], defined by 4Ω0 > Γ+, it is noticed that the qubit the subenvironments by taking #E . This allows undergoes oscillatory dynamics—the excited population analytical solutions to be obtained provided→ ∞ a suitable increases in time during intervals when γ(t) < 0, which, distribution for the weights wk = w(ξk) is assumed in from Eq. (18), gives a simultaneous and equal decrease in the conversion P R dξ ρ (see appendix A). A key the pseudomode population, while, in the weak-coupling k → k k result is that by choosing a single Lorentzian distribution regime 4Ω0 < Γ+, the time evolution of the qubit tends of width ΓW Eq. (6) can be written as towards exponential decay at the Markov emission rate γ = 4Ω2/Γ . In this instance γ(t) is positive at all Γ 0 0 + D(ω) = + . (14) times and the qubit-pseudomode populations show no (ω ∆ ω)2 + (Γ /2)2 0 − − + oscillations. We therefore interpret the non-Markovian behavior as being causal to the backflow of population The parameter Γ+ = Γ + ΓW is the resulting increased width compared to the single pseudomode case [where and energy between the two. As discussed in Ref. [24], Eq. (6) is a single Lorentzian of width Γ], while ∆ is the this indicates that the pseudomode region P in Fig. 1 detuning from the qubit frequency. acts as a memory for the qubit in the presence of strong interactions. At this point it is also worth elaborating on the pseu- B. Qubit and pseudomode dynamics domode population dynamics, obtained via the density matrix ρP = TrS[ρSP ]. Before we go into this, we first highlight the fact that the time evolution of G(t) 2 de- The master equation (12) provides fully amendable pends only on the memory kernel (8). In turn,| this means| solutions for strong system-environment interactions, the qubit dynamics is solely determined by the structure which are shown in Fig. 2 to illustrate their behav- function (6). One might intuitively expect something ior (matching solutions are in appendix A). We ini- similar for the dynamics of the pseudomode coefficients. tially check the response of the qubit by tracing out However, we actually discover two damping time scales the pseudomodes from Eq. (12). This yields the time- that affect bk(t). Its solution, provided in appendix A, convolutionless master equationρ ˙S = Γ(t)[σ ρS, σ+] + − separates into two parts, each with different exponen- Γ∗(t)[σ , ρSσ+] [22], where Γ(t) = G˙ (t)/G(t) and − − tial prefactors, causing one part containing functional G(t) is the Green’s function of the qubit, i.e., ce(t) = terms to decay at a rate Γ+/4, and another part with G(t)ce(0). By defining static terms to decay at a rate Γ/2. Because of “mix- P 2 i∆kt ing” between terms in the population bk(t) , it is X Im e ce(t)b∗(t) k k difficult to single out their individual effect| in a| typical γ(t) = Γ(t) + Γ∗(t) = 2Ωk 2 , ce(t) k | | time evolution, which generally shows complex behavior. (15) It becomes apparent, though, when we introduce a large

i∆kt separation of time scales through X Re e ce(t)bk∗(t) s(t) = i [Γ∗(t) Γ(t)] = 2Ωk 2 , − ce(t) k | | (16) 1 1 t . (19) the master equation of the qubit can be expressed in Γ+ Γ canonical form: s(t) ρ˙S = i [σ+σ , ρS] + γ(t) [σ ]ρS, (17) − 2 − D − In Fig. 2, the effect of the fast and slow terms becomes where γ(t) is a time-dependent decay rate and s(t) is a increasingly noticeable towards the regime Γ Γ+. We Lamb shift. The dynamics associated with Eq. (17) are see the fast terms decay quickly and predominantly in- known to be nondivisible and non-Markovian if γ(t) takes fluence the short-time evolution, while the slow terms on negative values. It is instructive to consider this as- decline exponentially and thus survive into the long-time pect with regard to the behavior of the qubit-pseudomode limit. populations, as described by the following relation: Let us now comment further on the dynamics in such a case where Eq. (19) is valid. Within the strong- X ∂ + Γ b (t) 2 = γ(t) c (t) 2. (18) coupling regime, there is a distinct cross-over owing to ∂t | k | | e | the fact that the fast oscillatory terms decay on the fixed k timescale t O(1/Γ+). The dynamics are then cate- Note this has been derived using Eqs. (9), (10) and (15). gorised into∼ two phases. As we have seen, the short- The lefthand side of the above shows the rate of change time evolution is characterized by memory effects where the pseudomode population compensated against irre- the qubit and pseudomode populations oscillate in time. versible losses, which occur at a rate Γ. To examine this When t 1/Γ , the pseudomode population instead + 5 decays monotonically as [ce(0) = 1] additionally construct fragments out of the memory part of the environment, using the pseudomodes. 2 2 2 X 2 Γt X 16Ωk ∆k + (ΓW /2) Let us recall X, which was originally used to indi- bk(t) e− 2 | | ≈ 1 2 cate some region of interest in the environment [see Eq. k k 2i∆ + (Γ Γ ) + Ω2 k 2 − W (1)]. In view of the bipartite structure of the envi- (20) ronment, we will consider two cases where a fragment p 2 2 m where Ω = 4Ω0 (Γ+/2) [see Eq. (A6) in appendix X = { }X is made up using a random combination − f ⊗k=1 k A]. At this point the qubit has essentially relaxed and m = k1, . . . , km of either (i): the bare subenviron- thus decoupled from the memory, i.e., G( ) 2 0. The ments{ } E{ , or (ii): the} pseudomodes P . Both schemes | ∞ | ≈ k k density matrix ρP then obeys the master equation are depicted in Fig. 1. We emphasise that the idea of X using a two-pronged approach to check the locality of re- ρ˙ = Γ [a ]ρ , t 1/Γ . (21) P D k P + dundant information is entirely based on the fact that k the two representations of the environment coincide. Similar phases also exist in the pseudomode dynamics Successful quantum Darwinism is characterized by a large redundancy, R 1, needing the full mutually when the qubit undergoes a Markovian evolution. Notice δ here, however, that the cross-over is not as distinct since induced decoherence of many system-fragment combi- the fast terms do not decay on the same fixed time scale. nations for the environment to learn anything about Nevertheless, there is still a transition to slow exponen- the qubit. Our objective is to then use the QMI to tial decay close to when the qubit has fully dissipated its gauge the extent to which system-environment correla- energy. tions produced under the time evolution of the pure state When t 1/Γ, Eq. (20) predicts that the pseudo- (7), and/or the density matrix ρSP (12), are universally modes tend to form a quasi-bound state at long times shared between fragments: or, more simply put, how many fragments communicate roughly the same infor- Ω0t 1 as a result of the cross-over, i.e., Γ Γ+. Al- though this occurs generally with respect to the coupling mation about the qubit, in each case. To this end, we quantify the size of a given fragment using Ω0, the excitation is most efficiently “trapped” by the m pseudomodes in the strong-coupling limit since increas- f = , (22) ing Γ+ (with the ratio Γ/Γ+ fixed) also increases the rate #E at which population leaks to the Markovian reservoir. where f is the fraction, 0 f 1, and m = 1, 2,..., #E Overall, we find the validity of Eq. (20) in describing is the number of objects in≤ the≤ fragment: e.g., for f = 1, the long-time dynamics to only really be affected by the the full environment is sampled and so Xf=1 = X. The degree of separation between Γ and Γ+. The trapping ef- computation of the redundancy follows directly from the fect then appears to be a feature of the narrow-Lorentzian “partial information” I(ρSXf ) = I(f) [11], where structure of D(ω) [Eq. (6)]. Indeed, by taking the broad denotes the average overh all fragmentsi h ofi size f. In termsh·i Lorentzian limit Γ Γ+ one recovers the usual single of Eq. (22), the redundancy is given by pseudomode dynamics≈ from Ref. [20], which does not display any of the trapping features seen here. 1 Rδ = , (23) The presence of a large pseudomode population well fδ into the long-time limit suggests that a significant pro- in which portion of the total correlations of S+E develops between the qubit and memory region of the environment. Since I(fδ) = (1 δ)S(ρS). (24) h i − we are working within the context of quantum Darwin- ism, it seems justified to ask if such correlations translate into redundant information. This is part of what we con- IV. RESULTS sider in forthcoming sections. In order to examine the redundant recording of information in cases (i) and (ii), we employ a Monte Carlo III. APPLICATION OF QUANTUM procedure to randomly sample fragments of size f for ev- DARWINISM ery m = 1, 2,..., #E. The partial information is then computed by averaging the QMI over the ensemble. In this section we move onto investigate emergent features of quantum Darwinism. Here, the central quantity under study is the QMI (1), which is computed for a A. Quantum Darwinism given choice of fragment. As we are working in a typical decoherence setting [4] the most obvious choice is to Numerical results are obtained for both Markovian and construct fragments out of the subenvironments. In this non-Markovian dynamics of the qubit, assuming an ini- paper we would also like to go further to address where tial system-environment state ρSE(0) = e e 0 0 . information regarding the pointer states manifests within We proceed by discussing each of the cases| i (i) h and| ⊗ | (ii)i h in| the environment. A natural way to go about this is to turn. 6

2 2 (a) (b)

) 1.5 ) 1.5 S S ρ ρ ( (

/S 1 /S 1 i i ) ) f f ( ( I I h 0.5 1 δ h 0.5 −

0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 f f

FIG. 3. Partial information I(f) /S(ρS ) between the qubit and subenvironments shown as a function of fraction size f at h i times Ω0t = 10, 15, 25, 40, 50 (black dotted, violet short-dashed, orange dashed, red long-dashed, and blue solid curves), for { } parameters (a) Γ+ = Ω0 and (b) Γ+ = 10Ω0. In the shown examples, the spreading of correlations over time leads to emergent redundant information, indicated by the presence of the ﬂat plateau. The intercept between the black dashed line (shown for δ = 0.15) and each curve highlights the average size of a fraction fδ = 1/Rδ [Eq. (23)] containing up to (1 δ)S(ρS ) information on the qubit pointer states. −

1. Case (i) until S(ρ ) S(ρ ), so that I(ρ ) S(ρ ). If Ef ≈ SEf SEf ≈ S we increase Ω0 enough, the same entropy oscillates over Figure 3 shows the partial information plots of the time and for very strong coupling these oscillations con- tinue in the limit t . This disrupts the process by qubit and subenvironments. At short times, the qubit → ∞ quickly develops correlations with the environment un- which the fragment acquires information due to periodic til its state is maximally mixed, from which point the intervals of decorrelation. We see more precisely how the average entropy of the system decreases as it relaxes to exchange of population between the qubit and memory the ground state. For a pure global state, as in Eq. (7), affects the plateau in Sec. IV D. the plots are antisymmetric about f = 1/2: mathemat- Before moving on to the next case we examine a reduc- ically, the sum of the (average) QMI between the qubit tion of Eq. (1) to a much simpler analytical form, which and any two complimentary fragments Ef and E1 f of we can use to check the accuracy of our numerical results. the environment satisfies [13] − This is first achieved by mapping the density matrix of a fragment state onto a single qubit [25]. Note the mapping

I(ρSEf ) + I(ρSE1−f ) = 2S(ρS), (25) is not specific to either case (i) or (ii), and, accordingly, we shall use it to approximate the QMI in both such which stems from the fact that the marginal entropies of cases. The ground state of the collective qubit is univer- SEf E1 f are equal, i.e., S(ρSE ) = S(ρE − ). The ⊗ − f 1 f sally defined as e0 = 0 , while here (Xf = Ef ) bipartite S + E also means the sum of the information | iXf |{ }iXf from complimentary fragments provides the total infor- the excited state is formed using mation I(ρ ) = 2S(ρ ). As decoherence sets in, we 1 X SE S 1 = c (t) 1 , (26) e Ef k,λ k,λ see that the partial information increases more quickly | i ηE (t) | i f k E ,λ to the maximum value around f 1, in turn matched ∈ f ≈ by a gradually steeper gradient close to the origin. This where k Xf denotes summation over objects in the channels the middle region of the plot into a flat plateau fragment,∈ and shape, where its length indicates the availability of clas- s X 2 sical information (1 δ)S(ρ ) from fractions of the envi- ηE (t) = ck,λ(t) . (27) − S f | | ronment. The redundancy typically measures the length k Ef ,λ ∈ of this plateau. Snapshots are displayed at increasing By approximating times where the plateau begins to level out, indicating 2 2 X 2 that eventually many fractions gain access to the same η (t) fη (t) = f ck,λ(t) , (28) Ef ≈ E | | information on the qubit for both the strong (moderate) k,λ and weak system-environment coupling. However, when the dynamics is strongly non- for all fraction sizes, the joint system-fragment state can be written as Markovian, i.e. for Ω0 Γ+, we find the partial informa-  2  tion oscillates about f = 1/2 and hence no stable plateau (1 f)ηE 0 0 0 − 2 develops (not shown). The origin of such behavior is ex-  0 fηE √fce∗ηE 0 ρSEf =  2  , (29) posed by examining the dynamics of S(ρEf ) for a typical  0 √fceηE ce 0 fragment, which for weak coupling, grows monotonically 0 0| 0| 0 7 being taken in the basis Γ+ = Ω0 Γ+ = 10Ω0 2 g, e0Ef , g, e1Ef , e, e0Ef , e, e1Ef . The eigenval- {| i | i | i | i} (a) (b) ues of ρSEf provide the following expression for the ) S 1.5 ρ

partial information of the qubit and fragment state, (

/S 1 i I(ρ ) = h c (t) 2 + h (χ (f)) h (χ (1 f)) , ) SEf e E E f | | − − (

I 0.5

(30) h

0 where 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 (c) (d) )

2 S 1.5 f smallf ρ

χE(f) = fηE(t), (31) (

/S 1 i and h(x) = x ln x (1 x)ln(1 x). In Figs. 4(a- ) f

− − − − ( b) we see Eq. (30) reproduces the numerical results re- I 0.5 h markably well, with only small discrepancies appearing 0 at limiting values of the fraction size for Γ+ = Ω0, close 0 0.2 0.4 0.6 0.8 10 0.2 0.4 0.6 0.8 1 to the boundaries of the plots at f = 0 and 1. Thus, f f our simple analytical model manages to predict the key features of the partial information plots to a good degree FIG. 4. Analytical approximations of the partial informa- of accuracy. tion (solid curves) plotted against the corresponding numerical results at times Ω0t = 10, 15, 40 (black circles, red squares and blue triangles), for{ ∆ = 0.} (a), (b) Average 2. Case (ii) of I(ρSEf ) [Eq. (30)]. (c), (d) Average of I(ρSPf ) [Eq. (37)] 3 for Γ = 10− Γ+. Note the left-hand column is for parameters The global state becomes mixed beyond t = 0 and Γ+ = Ω0, and the right-hand column is for Γ+ = 10Ω0. It can so the partial information plots do not acquire the same be seen that the analytical results ﬁt the numerics accurately form. The QMI instead satisﬁes the inequality within the plateau region at longer times.

I(ρ ) I(ρ ), t 0, f, (32) SPf ≤ SEf ≥ ∀ where the upper bound is set by the strong sub-additivity of the von Neumann entropy [12]. As the equality only minimum value on the time scale t O(1/Γ) from the strictly holds at t = 0, Eq. (32) is understood from the fact that the gradient of P (t) is approximately∼ ten times 3 idea that the state ρSP evolves under a noisy quantum larger between Γ+ = Ω0 and 10Ω0 when Γ = 10− Γ+. channel where information irreversibly leaks out to the For Γ Γ , we can then expect the information content + Markovian reservoir. The rate at which the vacuum state of ρSP to stay closer to the equality of Eq. (32) than the population increases signifies the noisiness of the channel. examples seen in Figs. 5(a) and (b) for f = 1. This is In view of this aspect, the top row of Fig. 5 shows the because the purity declines more slowly when there is a partial information plots of the qubit and pseudomodes large separation of time scales (assuming the same val- within the lossy regime Γ+ Γ, where the vacuum pop- ues of the coupling Ω0 are used from before). As such, ulation increases significantly≈ at short times. Here, cor- the partial information plots should retain some of the relations between the qubit and pseudomodes typically antisymmetry features expressed by Eq. (25). decay quickly, though with strong system-environment Figures 5(c) and 5(d) show the development of a flat interactions the QMI dissipates more slowly and redun- classical plateau over time for parameters Γ Γ+. Small dant correlations have time to develop. In this instance differences in the partial information plots are apparent we notice the appearance of a similar plateau feature between the strong- and weak-coupling limits as the rate from before. at which purity decays slightly increases with higher val- If we were to find the redundancy, however, it gener- ues of Γ+ [see Fig. 6]. While Figs. 5(c) and 5(d) tend ally proves troublesome to compute using a fixed fraction to deviate from a complete antisymmetric form at longer size fδ due to lack of antisymmetry of the partial infor- times, they maintain a similar shape to those in Fig. 3 mation—that is, the plateau drops below the threshold for almost all fraction sizes below f = 1. Crucially then, (1 δ)S(ρS) for δ 1. This issue raises the question: because the partial information saturates to the limit in are− there circumstances where the application of the re- Eq. (24) (δ 1) once there is sufficient decoherence of dundancy measure Rδ is possible? To answer this, we the state, the redundancy measure Rδ can be used even briefly look at how the time-dependent behavior of the without the qubit-pseudomode state being pure. This is 2 purity P (t) = Tr[ρSP (t)] changes with respect to the pa- clear from comparing these plots between the two cases rameters Γ, Γ+, and Ω0. Figure 6 depicts P (t) for differ- (i) and (ii) at equal times, where both exhibit a similar ent values of the spectral widths and coupling strength. plateau. First, when Γ Ω , we notice the purity decays to its We can map the state of a fragment P to that of a 0 f 8

Γ+ = Ω0 Γ+ = 10Ω0 1 1 (a) (b) 0.8 0.8 ) ) S S ρ ρ

( 0.6 ( 0.6 /S /S i i ) )

f 0.4 f 0.4 ( ( I I h h 0.2 0.2

0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 f f 2 2 (c) (d)

) 1.5 ) 1.5 S S ρ ρ ( (

/S 1 /S 1 i i ) ) f f ( 1 (

I I(ρ ) 1 I < SPf = I(ρ 1 ) 1 h 2 ' h SPf = 0.5 0.5 2 ∼

0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 f f

FIG. 5. Partial information I(f) /S(ρS ) between the qubit and pseudomodes shown as a function of fraction size f at times h i (a) Ω0t = 5, 15, 25, 40 , (b) Ω0t = 0.5, 1, 5, 10 and (c),(d) Ω0t = 10, 15, 25, 40, 50 (t increases in the same order as each { } { } { } curve in Fig. 3 from dotted to solid). The left-hand column is for strong (moderate) coupling Γ+ = Ω0 while the right-hand column is for weak coupling Γ+ = 10Ω0 (all ∆ = 0). Unlike case (i), total correlations are erased over time as the state ρSPf evolves through a noisy quantum channel. (a), (b) Γ = 0.4Γ+: partial information decays quickly though for strong coupling 3 redundancy features are qualitatively noticeable. (c), (d) Γ = 10− Γ+, a classical plateau is present in the long-time limit since the partial information retains approximate asymmetry about f = 1/2 (indicated by the arrows), except at the boundary f = 1.

collective qubit, the excited state of which is defined by using the basis states g, e0Pf , g, e1Pf , e, e0Pf , e, e1Pf . {| i | i | i | i} 1 X It turns out the partial information is then e1 = bk(t) 1k , (33) Pf 2 1 2 | i ηPf (t) | i k Pf I(ρSPf ) = h ce(t) +h χP (f) h χP (1 f) , (37) ∈ | | − − with normalization where the coefficients are s X η (t) = b (t) 2. (34) 1 2 Pf | k | χP (f) = fηP (t), k P ∈ f 2 2 χP (f) = fηP (t) + Πp(t), (38) Here, we look to follow a similar method that lead to a simple analytical expression for the partial information and Πp(t) is the vacuum population of the pseudomodes [see Eqs. (28) and (29)], provided in Eq. (30), with the (see definition in Appendix A). In Figs. 4(c) and 4(d), purpose of reproducing the results shown in Fig. 5. If we results obtained from the approximate form of the partial again assume on average that information (37) are presented against the previously discussed numerical results at various times, with Γ Γ+ 2 2 X 2 η (t) fη (t) = f bk(t) , (35) and Γ = Ω . Our analytical formula shows remarkable Pf ≈ P | | + 0 k agreement with the numerics, though small differences are noticeable: in particular, the partial information is then the density matrix ρSP is given by f slightly overestimated for small values of f. Regardless,  2  Πp + (1 f)ηP 0 0 0 the main features of these plots are captured, the most − 2  0 fηP √fce∗ηP 0 important being the increasing flattening of the plateau ρSPf =  2  , (36)  0 √fceηP ce 0 over time and subsequent emergence of redundant infor- 0 0| 0| 0 mation. 9

1 X = E 2 (a) 0.9 1.5

0.8 1 ) t (

P 0.5 0.7 3 Γ = 10− Γ+, Γ+ = Ω0 Ave. correlations 3 Γ = 10− Γ+, Γ+ = 10Ω0 0 0.6 Γ = 0.4Γ+, Γ+ = Ω0 f Γ = 0.4Γ+, Γ+ = 10Ω0 (b) 1.5 0.5 0 5 10 15 20 1 Ω0t

0.5

FIG. 6. Time evolution of the purity P (t) = Tr[ρSP ] Ave. correlations shown for parameters ∆ = 0, Γ+ = Ω0 (blue solid, violet 0 dashed curves), and Γ+ = 10Ω0 (red long-dashed, black dot- 0 0.2 0.4 0.6 0.8 1 ted curves). f

FIG. 7. Total partial information (violet solid curve), partial accessible information (blue long-dashed curve), and partial Just as in case (i), a large redundancy in this case quantum discord (red dashed curve) between the qubit and indicates widely accessible classical information. How- subenvironments [see Eqs. (39)-(41)], plotted over S(ρS ) at 3 ever, because this information is located in the pseudo- time Ω0t = 50, with Γ = 10− Γ+ (∆ = 0). Snapshots are modes it reveals more about the interaction: specifically, shown for parameters (a) Γ+ = Ω0 and (b) Γ+ = 10Ω0. The that records containing up to classical information on sum of the classical and quantum correlations (violet open the qubit are held redundantly within the memory region points) are also shown, indicating the validity of Eq. (44). of the environment. This differs from lossy interactions (Γ Γ+), where the damping noise (coming from the increase≈ in vacuum population) severely restricts the time where h i window in which redundant information can form before Xf J(ρSXf ) = max S(ρS) S(ρS M ) , (40) X j being lost completely (see Fig. 5). M f − |{ } { j } On the same point, it is noteworthy that the QMI h i ¯ Xf δ(ρSXf ) = min S(ρXf ) S(ρSXf ) + S(ρS M ) . of the full fraction of pseudomodes decays on a much X j M f − |{ } faster time scale than the redundant information (in the { j } plateau region). For a small damping rate Γ, it is reason- (41) able to question if this corresponds to a loss of quantum The quantity J(ρSXf ) defines the upper limit of the information from S + P , since the plateau sits approxi- Holevo bound [28]—the accessible information—which mately at the classical limit with most information lost provides the maximum classical data that can be trans- from global correlations. Far from this case—particularly mitted through the quantum channel. Accordingly, the within the lossy regime—it is unclear if the redundancy conditional entropy S(ρ M Xf ) of the bipartite system stems from local classical knowledge on the pointer states S|{ j } of the qubit, since the plateau falls well below this bound. is written as Xf X S(ρS Mj ) = pjS ρ Xf , (42) |{ } S Mj j |

which expresses the lack of knowledge in determining ρS

when ρXf is known. A measurement on the subsystem Xf Xf is formulated in terms of the projectors Mj , where B. Accessible information and quantum discord the post-measurement state of the qubit is

1 h Xf Xf i ρ Xf = TrX M ρSX M , (43) S M f j f j We shed light on the above discussion by considering | j pj the following definition of the QMI [26, 27]: with an outcome j obtained with probability pj = h i Xf ¯ TrS,Xf Mj ρSXf . The second quantity δ(ρSXf ) de- ¯ I(ρSXf ) = J(ρSXf ) + δ(ρSXf ), (39) fines a general measure of quantum correlations between 10 the two subsystems, known as the quantum discord [29]. Γ+ = Ω0 Γ+ = 10Ω0 Note the bar is used to distinguish the discord from the 1 (a) (b) information deficit δ. 0.8 ¯ It is emphasised that the accessible information and δ(ρSP )/S(ρS ) 0.6 discord are optimised through a choice of positive J(ρ )/S(ρ ) operator-valued measure (POVM) M Xf . Our moti- 0.4 SP S { j } vation for minimizing the quantum discord stems from Correlations 0.2 wanting to examine the correlations of the state least 0 disturbed by measurement. Here, the measurement is 0 5 10 15 20 2 4 6 formulated by mapping the relevant system fragment to Ω0t Ω0t an effective two-qubit state, as was done with Eqs. (26) 1.2 (c) (d) Xf 1 and (34). The POVM Mj then makes up a set of { } 0.8 orthogonal projectors from the qubit states of Xf (see appendix B). 0.6 Initially we compute Eqs. (40-41) for the full system- 0.4 classical limit environment (f = 1) of case (i) and find that the QMI Correlations 0.2 0 is always shared equally between classical and quantum 0 5 10 15 20 25 4 8 12 16 20 correlations when I(ρSE) > 0. This intuitively follows Ω0t Ω0t since the information encoded by classical data is limited to S(ρS). The remaining information out of S + E then FIG. 8. Dynamics of the accessible information (blue solid has to make up the discord in equal amount assuming curve) and quantum discord (red dashed curve) between qubit the state is pure, based on the global entanglement of and pseudomodes (X = P ), taken from Eqs. (40) and (41) the system and bath. Alternatively, for fractional states with ∆ = 0 (solutions in appendix B). Panels in the lefthand (f < 1) we find a more interesting albeit complicated in- column are plotted for Γ+ = Ω0, while those in the right- terplay between classical and quantum correlations. The hand column are plotted for Γ+ = 10Ω0. (a), (b) Γ = 0.4Γ+: classical correlations decay on a fast time scale and quickly quantities of interest here are the (averaged) partial ac- 3 cessible information J(ρ ) and partial quantum dis- approach zero. (c), (d) Γ = 10− Γ+: classical correlations h SXf i decay much slower with a large separation of time scales. cord δ¯(ρ ) , which from Eq. (39), fulfil the relation h SXf i

¯ I(f) = J(ρSXf ) + δ(ρSXf ) . (44) of Markovian dynamics—we recall that the effect of noise h i h i h i is more substantial with a higher rate of increase in vac- In Fig. 7 we show plots of the average correlations for uum population, which here increases the damping rate case (i). The most striking feature is the sharp rise in of the classical information by a factor proportional to partial quantum discord around small fraction sizes. As Γ+/Ω0 (e.g., a roughly ten times larger gradient between quantum correlations generally decline in value for larger Γ+ = Ω0 and 10Ω0). In contrast, the quantum discord fractions, the accessible information grows linearly and begins to decay at a faster rate. At longer times it can be as such is characteristic of non-redundant classical infor- seen that the quantum correlations become better pro- mation—i.e., its partial information plot does not have tected against decoherence when the discord decreases a flat plateau shape. Note also that the distribution of more slowly. We examine the limiting case of this be- classical correlations between different arrangements of havior in Sec. IV C. Our current observation is that the fragments ( J(ρSXf ) vs. f) is essentially static over dynamical behavior of the classical correlations is qual- time and independenth i of system-environment coupling itatively the same for both weak and strong coupling. strength. The discord, which takes large values in the The only particular difference is the presence of memory majority of fractions, therefore indicates a clear distur- effects in the latter. bance to the overall state from performing local measure- In the regime Γ Γ+ a somewhat opposite effect oc- ments on the pseudomodes. This behavior reveals the curs with respect to≈ the full qubit-pseudomode informa- interaction does not produce a class of states exhibiting tion. In this instance, classical correlations disappear complete Darwinism. asymptotically in time so that eventually the quantum Now turning our attention to case (ii), we address discord makes up all of the QMI. As almost no classical the dynamical behavior of the correlations with respect information is present even within the full memory re- to the full fraction of pseudomodes, displayed in Fig. 8. gion of the environment, we find a case where quantum Let us start by considering the regime Γ Γ+. Re- information is in fact redundant. This is confirmed in markably, when the dynamics are non-Markovian, Eq. Fig. 8(b) where the average correlation content of a frag- (40) stays close to its maximum value over the course of ment is shown with varying f. Here, δ¯(ρ ) I(f) h SPf i ≈ h i the interaction, and hence the classical correlations are and J(ρSPf ) 0 regardless of the size of the fragment. robust to the noise influence of the Markovian environ- Moreover,h Fig.i ≈ 9(a) shows these same quantities plot- ment. This is also true but to a lesser extent in the case ted for a large separation of time scales. We see that 11

X = P C. Maximization of classical correlations 2 (a) 1.5 So far, in studying case (ii) we have found that for a Lorentzian (14) with a highly peaked internal structure, 1 Γ Γ+, the accessible information is non-redundant and signiﬁcantly delocalized across the environment. As 0.5 a corollary to the results of Sec. II B, we examine the

Ave. correlations circumstances under which the classical correlations are maximised against the quantum discord in “global” cor- 0 relations, e.g., for f = 1. (b) f Whether classical or quantum correlations are predom- 1.5 inant has been shown not to depend on the presence of memory eﬀects in the dynamics. This suggests it de- 1 pends only on the degree of separation of the timescales. In fact, numerical evidence shown in Fig. 10 reveals that 0.5 decreasing the ratio Γ/Γ+ further slows down the decay

Ave. correlations of classical correlations compared to the plots shown in the bottom row of Fig. 8. If we observe the behavior 0 0 0.2 0.4 0.6 0.8 1 J(ρSP ) at a fixed time, we see it grows larger by decreas- f ing the value of Γ. This behavior lies in contrast to the quantum correlations which tend to fall off more quickly, FIG. 9. Total partial information (violet solid curve), but can still make up a larger proportion of the total cor- partial accessible information (blue long-dashed curve), and relations given the QMI also dissipates more slowly. We partial discord (red dashed curve) between the qubit and postulate that in the idealised limit of Eq. (19), i.e., with pseudomodes plotted over S(ρS ) at time Ω0t = 50, where a large separation of time scales, 3 Γ = 10− Γ+. (a), (b) Same parameters as those in Fig. 7, i.e., (a) Γ+ = Ω0 and (b) Γ+ = 10Ω0. Open points show the sum of the averaged classical correlations and discord, as in Γt 0, Γ+t 1, Ω0/Γ+ = fixed, (45) Eq. (44). −→ the accessible information converges towards its maximum, thereby revealing that the full memory region of the environment acquires (almost all) classical data on the qubit state. Notice the limit Γ+t is avoided, the quantum and classical correlations mimic those in as here the qubit would approach the ground→ ∞ state with Fig. 7 for case (i), though without a sudden increase zero entropy, resulting in all correlations being lost. Let in the discord at large f since the maximum available us now assume Eq. (45) holds. As t increases fur- information via measurement is limited when the state ther, what we expect is for the discord to make up is mixed—even for the eigenstates of a global observable an increasingly smaller proportion of the QMI. Once which coincide with the eigenbasis of ρSP (i.e., partial J(ρ ) δ¯(ρ ) in the very long time limit, the state SP SP information plots are not anti-symmetric). ρSP then shows robustness under non-selective measure- Overall, we conclude that the emergence of a classical ments M P (f = 1). Writing this in terms of a local { j } plateau—in the partial information plots of either case operation on P , (ΛP 1S)ρSP , we should have (i) or (ii)—does not guarantee that classical information ⊗ is redundant. This reveals significant underlying differ- X P P ences between the partial information plots presented in (Λ 1 ) ρ = 1 M f ρ M f 1 P ⊗ S SP S ⊗ j SP j ⊗ S Figs. 3—5 and those found in Refs. [30, 31]. For ex- j ample, with an Ohmic environment interacting with a ρ . (46) quantum Brownian oscillator, the quantum correlations ≈ SP (entanglement) are found to be suppressed for all but very large fraction sizes [30], meaning that the partial Of course, the finite nature of the quantum correlations information plots alone reveal the presence of redundant means our results do not provide an example of complete classical correlations in the system. Here, we find that einselection [27], yet it can be appreciated that the state the same is not a sufficient condition for the redundancy attains its most classical-like form when the dynamics of classical information, against what we originally inter- fulfill Eq. (45). It is also interesting to note how the slow preted as “successful” Darwinism in Sec. IV A. A similar loss of classical correlations occurs in line with the slow point regarding the nonunique association of the classi- decay in the pseudomode population, which as we recall cal plateau to purely classically correlated states has been from Sec. II B occurs past the crossover in dynamics at made in Ref. [32]. times t O(1/Γ ). ∼ + 12

1 where A = √A†A. Under a divisible map (48) the rate of change| | in the trace distance over any time interval 0.8 is always negative. One can then show that the distinguishability of the pair of states is a monotonically J(ρSP )/S(ρS ) decreasing function in time—overall, corresponding to a 0.6 continual loss of information from S to E [35]. Variation from this behavior, i.e., when the trace distance tempo- δ¯(ρ )/S(ρ ) 0.4 SP S rally increases, indicates that the process is nondivisible Correlations through reverse ﬂow of information back to the open sys- 0.2 tem. In terms of the rate of change of the trace distance, 1,2 ˙ 1 2 σ(ρ (0), t) = D(ρS, ρS), (50) 0 S 5 4 3 2 1 0 10− 10− 10− 10− 10− 10 the quantum process is non-Markovian if and only if Γ/Γ+ σ(t) > 0 at some point during the evolution of the den- 1,2 1,2 sity matrices ρS (t) = Φ(t, 0)ρ (0). FIG. 10. Full accessible information [Eq. (40)] and quantum Concerning our model, the trace distance can always discord [Eq. (41)] for case (ii), plotted as a function of Γ at be expressed analytically as time Ω0t = 50 (∆ = 0). The blue solid and violet dashed curves are for Γ+ = Ω0, while the red long-dashed and black 1 2 p 2 2 2 D(ρS, ρS) = G(t) G(t) a + b , (51) dotted curves are for Γ+ = 10Ω0. | | | | | | 1 2 where a = e ρS(0) e e ρS(0) e and b = 1 h |2 | i − h | | i D. Non-Markovianity e ρS(0) g e ρS(0) g . By taking the time deriva- htive| of the| i above − h | [36], | i

Now we consider the relation between the non- 2 2 2 1,2 2 G(t) a + b ˙ Markovianity of the qubit dynamics and the redundancy σ(t, ρS (0)) = p| | | | G(t) , (52) G(t) 2a2 + b 2 | | measure (23). Our model provides a good foundation to | | | | investigate this connection as there is clear delineation we see that σ(t) is only positive if G˙ (t) > 0 at any of memory effects in terms of information backflow to | | the open quantum system. time t, which here sets the condition for nondivisibility We illustrate the concepts surrounding non- and thus non-Markovianity from the BLP measure [37]. Markovianity in this model by starting with the Hence information flows back to the qubit at times when following definition. Let us state that if the family of there are temporary revivals in its population. Since the dynamical maps Φ(t, 0) governing the evolution master equation (17) also has a time-local structure, we can use Eqs. (18) and (52) to prove that ρS(0) ρS(t) = Φ(t, 0)ρS(0), t 0 (47) 7→ ≥ σ(t, ρ1,2(0)) > 0 γ(t) < 0, t 0. (53) is divisible into two completely positive and trace pre- S ←→ ∀ ≥ serving maps (CPTP), that is, It is therefore reasonable to suspect that memory effects

Φ(t2, 0) = Φ(t2, t1)Φ(t1, 0), t2 t1 0, (48) will appear in the partial information plots at such times ∀ ≥ ≥ when the dynamics are non-Markovian—specifically, in then ρS undergoes a Markovian evolution. Here the no- instances when the decay rate (15) becomes negative. tion of divisibility is enough to distinguish between what How the redundancy behaves with respect to changes in is considered Markovian and non-Markovian behavior the trace distance (51) and γ(t) is precisely the connec- [33]. However, it should be stressed that CPTP maps as- tion we aim to make with our results. sociated with time-dependent Markov processes are cat- In Fig. 11, we show various values of the redundancy egorically different from those which form a dynamical computed in both the strong- and weak-coupling limits. semigroup [3]. In a similar way, the situation where Note for Γ+ = Ω0 the detuning is taken to be non-zero, a given amount of information leaks back to the open for the reason that the map Φ(t, 0) is noninvertible for system has been shown to be intimately related to the ∆ = 0, and so no strict definition of divisibility exists in non-Markovian properties of the quantum channel. The this case [35]. Here, we are not necessarily interested in Breuer-Laine-Piilo (BLP) measure of non-Markovianity the exact numerical value of the redundancy, but rather [34], as an example, is rooted in the particular interpre- its dynamical behavior with respect to the decay rate. tation of shared information as the distinguishability of Since, from Eq. (18), we have established that the qubit a pair of input states ρ1 , ρ2 to the channel. Their { S S} receives back population directly from the depletion of distinguishability is characterized by the trace distance the pseudomodes when the rate in the master equation 1 2 D(ρS, ρS), given by is negative, it is plausible to think memory effects will 1 also influence correlations between S and P . Therefore, D(ρ1 , ρ2 ) = Tr ρ1 (t) ρ2 (t) , (49) S S 2 S − S we have additionally computed the redundancy for case 13

Γ+ = Ω0; ∆ = 0.05Γ+ Alternatively, in the weak-coupling regime the plateau 15 40 only continuously grows in length while the qubit un- t1 t2 (a) dergoes a Markovian evolution [in both cases (i) and

0 20

Ω (ii)]. This ﬁrmly suggests that the nonmonotonicity of /

) 0

t the redundancy captures the non-Markovian dynamics (

10 γ 20 − of the qubit—which, in this context, is clearly based on )

t 40 the same effective behavior in the trace distance (51). ( − 0 2 4 6 8 10 δ Ω0t We also point out that the connection between quantum R Darwinism and non-Markovianity has also been studied 5 recently in Refs. [15, 16]. The authors find a similar effect, where information backflow to the open system translates into poor Darwinism, i.e., a worsening of the t t 1 2 plateau at these times. 0 0 5 10 15 20 25 Furthermore, the fact that the redundancy of cases (i) Ω0t and (ii) shows similar dynamical behavior suggests that the rollback of the plateau occurs specifically because of Γ+ = 10Ω0 15 information backflow from the pseudomodes (memory) to 0.5 the qubit. Indeed, consider the example of an initially ex- 0.4 (b) 0 cited qubit for the same parameters as Fig. 11(a). What Ω 0.3

/ 2

) we ﬁnd is that at times when G(t) starts to increase t 0.2 ( | |

10 γ there is an accompanied increase in QMI of the qubit 0.1

) and environment from its minimum zero value. This in- t 0 ( 0 1 2 3 4 5 δ dicates that correlations previously removed by dissipa- Ω0t R tion re-develop because of information flow back at times 5 when the redundancy decreases. Now, within the regime Γ Γ (shown in Fig. 11), we observe that the dy- + namics of QMI between the qubit and memory I(ρSP ) faithfully coincides with that of I(ρSE). Thus we can as- 0 0 5 10 15 20 25 30 35 sociate the same re-correlation effect with revivals in the Ω0t qubit population, which, from Eq. (18), simultaneously comes from losses in the pseudomodes. In this sense the energy/information received back by the qubit from the FIG. 11. Redundancy Rδ(t), Eq. (23), plotted as a function of time for cases (i) (red triangles) and (ii) (blue marked pseudomodes can provide the physical mechanism for the curve) with δ = 0.15. Insets: Decay rate γ(t) from Eq. (15) drop in the classical plateau. for each set of parameters. (a) Γ+ = Ω0 and ∆ = 0.05Γ+: the dashed lines indicate times at which the redundancy peaks. (b): Γ+ = 10Ω0 and ∆ = 0: in the weak-coupling limit, V. SUMMARY AND DISCUSSION redundancy monotonically increases in time. In summary, we have provided a detailed investiga- tion into emergent features of quantum Darwinism by (ii)—using a large separation of time scales—to compare applying the framework to a qubit interacting with many with the those of case (i). subenvironments of bosons. The basis of our paper de- In the strong-coupling regime, the key indication from rives from the idea that the original environment maps our results is that the redundancy has peaks and troughs to a bipartite structure containing a memory and non- exactly in line with the decay rate. Let us first consider memory part. From this we have examined how informa- case (ii) at time intervals during which γ(t) > 0. Here, tion is encoded into fractions of the environment in two the redundancy is seen to increase up until the point at separate cases: one where we constructed random frag- which the decay rate is discontinuous and subsequently ments out of subenvironments [case (i)], and the other becomes negative. It is noticed that the plateau grows in where we constructed random fragments of independent length as the open system monotonously loses informa- parts of the full memory [case (ii)]. Our main effort has tion into the environment. Then, as γ(t) begins to grow been to recognize whether the emergence of redundant in- from negative values, that is, when information flows formation occurs, and, if so, where into the environment back to the qubit, the redundancy plateau is suppressed such information is proliferated. By considering differ- considerably before increasing again at times when the ent dynamical regimes we identify instances in cases (i) decay rate becomes positive. The same type of behavior and (ii) where redundant information forms close to the can also be seen to occur from the perspective of case (i), classical bound, implying “successful” Darwinism. where memory effects are visible during times when the Despite the formation of the classical plateau, which plateau emerges. usually is considered as a hallmark of successful quantum 14

Darwinism, our results demonstrate a scenario where winism) in a similar way as Refs. [15, 16]. classical information is precisely non-redundant. Con- VI. ACKNOWLEDGEMENTS sequently, we found the quantum discord—taken from the partial information—to obtain relatively large val- This paper has been supported by the UK Engineer- ues in small fractions of pseudomodes and/or subenvi- ing and Physical Sciences Research Council (Grant No. ronments, realizing the highly non-classical nature of a EP/L505109/1). typical system-fragment state based on the fact that it is disturbed significantly under local (projective) measurements. In parallel we have analyzed the dynamics of the Appendix A: Exact solutions for the state classical and quantum correlations between the qubit and coefficients memory region. In both cases—either when considering the partial correlations from fractions of the environment For a sufficiently large number of subenvironments #E or full collection of pseudomodes—we have found qual- the weights wk may be replaced across the interval dξk itatively similar behavior in the results across both the by strong- and weak-coupling regimes. Substantial differences are only introduced through relatively varying the wk = W (ξk)dξk, (A1) decay rates of the pseudomode population. For example, in the lossy regime, that is, where correlations and with W (ξ) modeling the underlying pseudomode dis- population are significantly damped in time, the QMI of tribution in Eq. (6). Imposing the continuum limit the qubit-pseudomode system shows asymptotically de- #E , the memory kernel is written as caying classical correlations with prevalent discord. A → ∞ regime where the pseudomodes maximize their classical Z ∞ f(t) = Ω2 dξ W (ξ) exp[ i(ξ ω )t Γt/2]. (A2) correlations over the course of the dynamics has also been 0 − − 0 − identified. −∞ Finally, we have sought to cement the connection be- Note that W (ξ) is in principle arbitrary, apart from hav- R tween the emergence of redundancy in the quantum Dar- ing to fulfill the normalization condition ∞ dξW (ξ) = winism framework and non-Markovianity. Memory ef- 1. For simplicity, we take −∞ fects are characterized by the backflow of information (and population), which in turn reflects the nondivisibil- 1 ΓW /2 W (ξ) = 2 2 , (A3) ity of the dynamical map. We show that the redundancy π (ω0 ∆ ξ) + (ΓW /2) plateau in the partial information plots (see Fig. 11) − − is suppressed at times when information flows from the with the parameters being defined in the main text. The pseudomodes to the qubit. Remarkably, the redundancy function contains a single pole in the lower complex ξ- acts as a witness to non-Markovian behavior in directly plane such that Eq. (A2) can be evaluated using the the same way as the trace distance does in the BLP mea- residue theorem. The resulting equation for G(t) is solved sure. using Laplace transforms, where Looking ahead, our results support the possibility of Ωt (i∆ Γ /2) Ωt developing a novel quantifier of non-Markovianity from (i∆/2 Γ+/4)t + G(t) = e − cos − sin , the perspective of information redundancy in line with 2 − Ω 2 Ref. [16]. While the redundancy Rδ(t) does not strictly (A4) p 2 2 detect memory effects on the basis of divisibility but in- and Ω = 4Ω0 (i∆ Γ+/2) . For the sake of com- stead from information flow between the system and en- pleteness, we note− the− same can be derived by noting vironment, these two concepts have been shown to be that the structure is given by the convolution D(ω) = closely connected [38, 39] through measures of informa- (W L)(ω), where tion (e.g., the QMI) similar to those employed in the ∗ Z current paper. This could lead to comparisons between ∞ (W L)(ω) = dξ W (ξ)L(ω ξ), (A5) related measures beyond the current model. Alterna- ∗ − tively, the methods employed here could be applied to −∞ practical models of cavity QED systems to test for simi- and L(ω) = Γ/[ω2 +(Γ/2)2]. The result is simply another lar effects encountered in this paper. Lorentzian of width Γ+, as given in Eq. (14). Note added in proof. Recently, we became aware of The coefficients ck,λ(t) are found by substituting Eq. [40], which examines the effect of non-Markovianity on (A4) into the Schrödingerequation, while bk(t) are ob- the emergence of classicality (and hence quantum Dar- tained through Eq. (11). Their solutions are given by 15

( 4Ωk ce(0) ΓW Γt/2 bk(t) = 2 ∆k + ∆ + i e− − Γ ΓW 2 2 i(2∆k + ∆) + −2 + Ω " i∆ t (i∆/2 Γ /4)t ΓW Ωt e k e − + ∆ + ∆ + i cos (A6) − k 2 2 #) i∆ Γ /2 Γ Γ Ω Ωt − + ∆ + ∆/2 i − W + i sin , − Ω k − 4 2 2 ( 4g c (0) Γ c (t) = k,λ e δ + ∆ + i + k,λ −(i(2δ + ∆) Γ /2)2 + Ω2 λ 2 λ − + " iδ t (i∆/2 Γ /4)t Γ+ Ωt e λ e − + δ + ∆ + i cos (A7) − λ 2 2 #) i∆ Γ /2 Ω Ωt − + (δ + ∆/2 + iΓ /4) + i sin , − Ω λ + 2 2

where δλ = ωλ ω0. To compute the relevant von Neu- erroneous results. The POVM that fulﬁls this condition mann entropies− we trace over the degrees of freedom ex- has been shown by Datta to be formulated using rank-1 cluded from Xf using the two-qubit representation of the projectors [41]. density matrix ρSXf . For example, a single realization of In order to realize the measurement in terms of such

ρSXf is composed by eliminating (#E m) randomly se- projectors, we make use of the fact that both the suben- lected (i) subenvironments from the pure− state (7) or (ii) vironments and pseudomodes—or fractions thereof—can pseudomodes from the mixed state solution of Eq. (12). collectively be mapped to a single qubit. As stated in the The latter is given by main text, the ground state of the qubit is deﬁned from the vacuum of Xf : e0 = 0 , while the excited | iXf |{ }iXf ρSP = Πp(t) g g ( 0 0 )P + ( ψe(t) ψe(t) )SP , (A8) | i h | ⊗ | i h | | i h | state e1 is formed through Eqs. (26) and (34). Let us | iXf write the complete set of local orthogonal projectors in where 0 P denotes the collective vacuum of the pseudomodes.| Ini addition, the the un-normalized state vector terms of the qubit states: ψe(t) reads | i 1 X Xf 1 ψe(t) = cg g, 0 + ce(t) e, 0 + bk(t) g, 1k , (A9) M = Xf + r σ , (B1) | i | i | i | i 1 2 · k 1 Xf 1 M2 = Xf r σ , (B2) where g, 1k indicates a single excitation in the k pseu- 2 − · domode.| Thei pseudomode vacuum population is given by where r = (sin θ cos φ, sin θ sin φ, cos θ)T is the Bloch vec- Z t T X 2 tor, and σ = (σx, σy, σz) contains the Pauli operators Πp(t) = Γ ds bk(s) . (A10) | | constructed from the basis 0 , 1 , along with the 0 k e Xf e Xf 1 {| i | i } identity Xf . The accessible information and discord are then extremized with respect to the free choice of angles Appendix B: Accessible information and quantum θ [0, π) and φ [0, 2π). ∈ ∈ discord Remembering that the conditional state is ρ X = S M f | j Tr [(1 M Xf )ρ (M Xf 1 )]/p (j = 1, 2), for each To gain a clear understanding on the distinction be- Xf S ⊗ j SXf j ⊗ S j tween classical and quantum information, we use the measurement one obtains Holevo quantity in Eq. (40) to gauge the information accessible via measurements on Xf . This information is 1 limited by the type of measurement used. Because the ρ Xf = Aj(θ) g g + Cj(θ) e e S Mj p | i h | | i h | ¯ | j QMI is invariant to how J(ρSXf ) and δ(ρSXf ) are as- signed, the quantum discord, in turn, is required to be + Bj(θ, φ) e g + h.c. (B3) minimized over all measurement bases M X to avoid | i h | { j } 16 where where k X1 f denotes summation over objects not in the fragment.∈ − 1n 2 Aj(θ) = ΠX (t) + η (t) Finally, by diagonalizing Eq. (B3) its eigenvalues are 2 f Xf h io obtained and substituted into Eq. (42) to evaluate the j 1 2 conditional entropy. We get + ( 1) − cos θ ηXf (t) ΠXf (t) , − − (B4) 1 j 1 iφ B (θ, φ) = ( 1) − sin θ e− η (t)c (t), j 2 − Xf e 1 C (θ) = c (t) 2 1 + ( 1)j cos θ , X j e S(ρ Xf ) = pjλi,j ln λi,j, (B7) 2| | − S Mj − |{ } i,j=1,2 1 with probabilities pj = 2 1 + ( 1)j cos θ σS + 2Π (t) . The coeﬃcients η (t) − h z i Xf Xf and ΠXf (t) are given by where (P 2 k E ,λ ck,λ(t) , if X = E, ∈ f | | ηXf (t) = P 2 (B5) bk(t) , if X = P, k Pf | | ∈ q i 2 2 and Aj(θ) + Cj(θ) + ( 1) [Aj(θ) Cj(θ)] + 4 Bj(θ, φ) λ = − − | | . ( i,j 2p c 2 + P c (t) 2, if X = E, j g k E − ,λ k,λ Π (t) = | | ∈ 1 f | | (B8) Xf c 2 + P b (t) 2 + Π (t), if X = P, g k P − k p From the above expression it is easy to see that the con- | | ∈ 1 f | | (B6) ditional entropy is invariant with respect to φ.

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