Generic emergence of classical features in quantum Darwinism

Quantum Darwinism explains the emergence of classical reality from the underlying quantum reality by the fact that a quantum system is observed indirectly, by looking at parts of its environment, so that only specific information about the system that is redundantly proliferated to many parts of the environment becomes accessible and objective. However it is not clear under what conditions this mechanism holds true. Here we rigorously prove that the emergence of classicality is a general feature of any quantum dynamics: observers who acquire information about a quantum system indirectly have access at most to classical information about one and the same measurement of the quantum system; moreover, if such information is available to many observers, they necessarily agree. Remarkably, our analysis goes beyond the system-environment categorization. We also provide a full characterization of the so-called quantum discord in terms of local redistribution of correlations.


I. INTRODUCTION
Our best theory of the fundamental laws of physics, quantum mechanics, has counter-intuitive features that are not directly observed in our everyday classical reality (e.g. the superposition principle, complementarity, and non-locality). Furthermore, the postulates of quantum mechanics reserve a special treatment to the act of observation, which contrary to its classical counterpart is not a passive act. The following fundamental questions then naturally emerge: Through what process does the quantum information contained in a quantum system become classical to an observer? And how come different observers agree on what they see?
The issues of the so-called quantum-classical boundary and of the related measurement problem dominated large part of the discussions of the early days of quantum mechanics. Indeed the debate between Bohr and Einstein on the meaning and correctness of quantum mechanics often revolved around the level where quantum effects would disappear -ranging from the microscopic system observed, up to the observer himself. From a practical perspective, our current understanding of the matter is nicely summarized by the quote (attributed to Zeilinger [1]): "the border between classical and quantum phenomena is just a question of money and technological innovations". Indeed our ability to manipulate quantum systems preserving their quantum features has made enormous progresses in recent years. However, even if we are somewhat pushing the location of the quantum-classical border thanks to our increased experimental ability, a fully satisfactory analysis of the quantum-to-classical transition is still lacking. Such an analysis would both deepen our understanding of the world and conceivably lead to improved technological control over quantum features.
Substantial progress towards the understanding of the disappearance of quantum features was made through the study of decoherence [2], where information is lost to an environment. This typically leads to the selection of persistent pointer states [2], while superpositions of such pointers states are suppressed. Pointer states -and convex combinations thereof -then become natural candidates for classical states. However decoherence by itself does not explain how information about the pointer states reaches the observers, and how such information becomes objective, i.e. agreed upon by several observers. A possible solution to these questions comes from an interesting idea termed quantum Darwinism [3][4][5][6][7][8][9][10][11][12][13][14][15], which promotes the environment from a passive sink of coherence for a quantum system to the active carrier of information about the system. There, pointer observables correspond to information about a physical system that the environmentthe same environment responsible for decoherence -selects and proliferates, allowing potentially many observers to have access to it. The main predictions of quantum Darwinism can be summarised as follows: • (objectivity of observables) Observers that access a quantum system by probing part of the environment of the system can only learn about the measurement of a preferred observable (usually associated to a measurement on the pointer basis determined by the systemenvironment interaction [3]). The preferred observable should be independent of which part of the environment is being probed.
• (objectivity of outcomes) Different observers that access different parts of the environment have (close to) full access to the information about the preferred observable and will agree on the outcome obtained.
The two properties above imply that information about the quantum system becomes objective, being accessible simultaneously to many observers.
The ideas of quantum Darwinism are beautiful and physically appealing. However we are still far from understanding how generally they apply. Significant progress was achieved in a sequence of papers [3][4][5][6][7][8][9][10][11][12][13][14][15], providing both general, but not fully rigorous, arguments and analysing specific examples where the emergence of objectivity can be studied in detail. In this paper we address the question of determining the circumstances under which quantum Darwinism takes place. We do so from the dynamical point of view, but without any specific assumptions on the dynamics. Surprisingly, we find that central features of quantum Darwinism are completely general, being consequences only of the basic structure of quantum mechanics.

II. RESULTS
We consider a general quantum dynamics from a quantum system A to n systems B 1 , . . . , B n . We interpret the B subsystems as parts of the environment, but do not assume anything else about them. We represent the dynamics from A to B 1 , . . . , B n by a completely positive trace-preserving (cptp) map Λ : D(A) → D(B 1 ⊗ . . . ⊗ B n ) (with D(X) the set of density matrices over the Hilbert space X).
Given two quantum operations (cptp maps) Λ 1 and Λ 2 , the diamond norm of their difference Λ 1 − Λ 2 ✸ gives the optimal bias of distinguishing the two operations by any process allowed by quantum mechanics (i.e. choosing the best possible initial state, applying one of the quantum operations to it, and performing the best possible measurement to distinguish the two possibilities) [16]. Thus if Λ 1 − Λ 2 ✸ ≤ ε, the two maps represent the same physical dynamics, up to error ε 1 . Let tr \X be the partial trace of all subsystems except X.

A. Objectivity of Observables
Our main result is the following: with for states σ j,k ∈ D(B j ). Here d A is the dimension of the space A.
As we mentioned before, the diamond-norm distance on the left-hand side of Eq. (1) represents how different the two physical processes Λ j and E j are: the smaller the diamond norm, the more similar the processes, to the extent that they can become indistinguishable for all practical purposes. The right-hand side of Eq. (1) is a bound on such a distinguishability that for fixed δ-or even for δ decreasing with n but not too fast 2 -becomes smaller and smaller as n increases. So, for fixed d A , in the case where we consider an environment with a large number of parts n (e.g. 10 15 ) for all environment parts but δn of them the bound on the right-hand side of (1) is very close to zero, i.e. the effective dynamics is E j for all practical purposes.
The operation E j in Eq. (2) is termed a measure-and-prepare map, since it can be implemented by first measuring the system with the POVM {M k } k and then preparing a state σ j,k depending on the outcome obtained. It is clear that an observer that has access to E j (ρ) can at most learn about the measurement of the POVM {M k } k on ρ (but possibly not even that if the states {σ j,k } k are not well distinguishable).
A key aspect of the theorem is that the measurement {M k } k is independent of j. In words, the theorem says that the effective dynamics from A to B j , for almost all j ∈ {1, . . . , n}, is close to a measure-and-prepare channel E j , with the associated measurement {M k } k the same for all such j. From the perspective of single observers, the evolution Λ is well approximated by a measurement of A, followed by the distribution of the classical result, which is finally "degraded" by a local encoding that, for each B j , produces a quantum state σ j,k upon receiving the result k.
Therefore the first feature of quantum Darwinism (objectivity of observables) is completely general! We can interpret {M k } k as the pointer observable of the interaction Λ. Note also that the bound is independent of the dimensions of the B subsystems, being therefore very general 3 .

B. Proof Idea
The proof of Theorem 1 is given in Section III A. It is based on quantum information-theoretic arguments along the lines of recent work by Harrow and one of us [17,18]. We develop the 2 E.g., for δ = n − 1−η 3 , for any 0 < η < 1. 3 Note, however, the dependence on the dimension dA of the system A. Although the functional form of this dependence might be improved, it is clear that no bound independent of dA can exist. Indeed, suppose A = A1, . . . , An and consider the noiseless channel from A to B1, . . . , Bn. It is clear that a dimension-independent statement of the theorem would fail. methods of [17,18] further to show that not only the effective channels E j are close to a measureand-prepare channel for most j, but that the POVM defining the channels is the same for all j. This latter feature was not appreciated in [17,18], but is fundamental in the context of quantum Darwinism. The rough idea of the proof is to consider the state obtained by applying the general dynamics on half of a maximally entangled state of the system A and an ancillary system. This gives the state ρ AB 1 ...Bn on AB 1 . . . B n . Then we consider the effect of measuring (in an appropriate basis that must be optimized over and is not given explicitly) a few of the B j systems of the state ρ AB 1 ...Bn , for randomly chosen j ′ s. We argue that the statistics of such measurement and the form of the postselected state in system A specifies a POVM {M k } k for which Eq. (1) holds true. As we will show, this is a consequence of an important property of the quantum mutual information: the chain rule. Intuitively this process shows that by probing a small part of the environment (with the appropriate measurement) and by considering the effect on the system A, the pointer POVM {M k } k is fully determined.
The argument has connections with the phenomenon of entanglement monogamy, which intuitively says that ρ AB j must be close to a separable state for most j 4 , and thus by the Choi-Jamiolkowski isomorphism [16] the associated channel E j must be close to a measure-and-prepare map. But it goes beyond what we expect from entanglement monogamy by showing the existence of the common pointer POVM for most E j (which is equivalent to saying that ρ AB j is close

C. Objectivity of Outcomes
We note that Theorem 1 does not say anything about the second part of quantum Darwinism, namely objectivity of outcomes. It is clear that in full generality this latter feature does not hold true. Indeed, as observed already in Ref. [10], if Λ is a Haar random isometry from A to B 1 , . . . , B n , then for any i for which B i has less than half the total size of the environment, the effective dynamics from A to B i will be very close to a completely depolarizing one, mapping any state to the maximally mixed state. Therefore objectivity of outcomes must be a consequence of the special type of interactions we have in nature, instead of a consequence of the basic rules of quantum mechanics (in contrast, Theorem 1 shows that objectivity of observables is a consequence only of the structure of quantum mechanics).
Can we understand better the conditions under which objectivity of outcomes holds true? First let us present a strengthening of Theorem 1, where we consider subsets of the environment parts. Let [n] := {1, . . . , n}. with for states σ St,k ∈ D( l∈St B l ).
Theorem 2 says that the effective dynamics to B j 1 , . . . B jt is close to a measure-and-prepare channel, for most groups of parts of the environment (j 1 , . . . , j t ). Let us discuss the relevance of this generalization to the objectivity of outcomes question.
Let B j 1 , . . . , B jt be a block of sites such that the effective dynamics from A to B j 1 , . . . , B jt is well approximated by for the pointer POVM {M k } k and states {σ B j 1 ,...,B j t ,k } k . From Theorem 2 we know that this will be the case for most of the choices of B j 1 , . . . , B jt . As we mentioned before, for many Λ the information about the pointer observable is hidden from any small part of the environment and thus outcome objectivity fails. Suppose however that the t observers having access to B j 1 , . . . , B jt have close to full information about the pointer observable. We now argue that this assumption implies objectivity of outcomes.
To formalize it we consider the guessing probability of an ensemble {p i , ρ i } defined by where the maximization is taken over POVMs {N i } i . If the probability of guessing is close to one, then one can with high probability learn the label i by measuring the ρ i 's. We have Then there exists POVMs Eq. (7) is equivalent to saying that the information about the pointer-observable {M k } k is available to each B j i , i ∈ {1, . . . , t}. Assuming the validity of Eq. (7), the proposition shows that if observers on B j 1 , . . . , B jt measure independently the POVMs {N B j 1 ,k } k , . . . , {N B j t ,k } k , they will with high probability observe the same outcome. Therefore, while objectivity of outcomes generally fails, we see that whenever the dynamics is such that the information about the pointer observable is available to many observers probing different parts of the environment, then they will agree on the outcomes obtained.

D. Deriving Quantum Discord from Natural Assumptions
Let us now turn to a different consequence of Theorem 1. In the attempt to clarify and quantify how quantum correlations differ from correlations in a classical scenario, Ollivier and Zurek [19] (see also [20]) defined the discord of a bipartite quantum state ρ AB as 5 where I(A : −tr ρ X log ρ X is the von Neumann entropy, and the maximum is taken over quantum-classical (QC) channels Λ(X) = k tr(M k X)|k k|, with a POVM {M k } k . The discord quantifies the correlations -as measured by mutual information -between A and B in ρ AB that are inevitably lost if one of the parties (in the definition above, Bob) tries to encode his share of the correlations in a classical system. Alternatively, quantum discord quantifies the minimum amount of correlations lost under local decoherence, possibily after embedding, and in this sense can be linked to the notion of pointer states [19]. As such, quantum discord is often seen as the purely quantum part of correlations, with the part of correlations that can be transferred to a classical system -alternatively, surviving decoherence -deemed the classical part [19][20][21][22].
Recently there has been a burst of activity in the study of quantum discord (see [21]). Despite the recent efforts, the evidence for a clear-cut role of discord in an operational settings is still limited [21]. Hence it is important to identify situations where discord emerges naturally as the key relevant property of correlations. Here we identify one such setting in the study of the distribution of quantum information to many parties, intimately related to the no-local-broadcasting theorem [22,23]. Indeed a corollary of Theorem 1 is the following: , h 2 is the binary entropy function, and the maximum on the righthand side is over quantum-classical channels Λ(X) = l tr(N l X)|l l|, with {N l } l a POVM and {|l } l a set of orthogonal states.
As a consequence, for every ρ AB , with Ej X j = 1 n N i=1 X j , and the maximum on the left-hand side taken over any quantum operation Therefore we can see the discord of ρ AB as the asymptotic minimum average loss in correlations when one of the parties (Bob, in this case) locally redistributes his share of correlations: Other operational approaches to quantum discord, in particular from a quantum information perspective, have been proposed, but we feel Corollary 4 stands out in comparison to them. First, Corollary 4 does not introduce from the start local measurements, which not so surprisingly would lead to the appearance of discord (as per its definition given in Eq. (9)); in contrast, measurements appear as "effective measurements " due to the presence of other B's. Second, Corollary 4 links quantum discord to the the redistribution of quantum systems and quantum correlations in a general and natural way. Notice that this is different from [25], where operational interpretations of discord are given that are more involved, and from [26], where discord is given an interpretation in quantum communication scenarios that does not really go much beyond its definition. Corollary 4 also has full validity, applying both to the case where ρ AB is a pure state and when it is mixed. As we will see in Section II E, in particular this removes the limitations of a recent related work by Streltsov and Zurek [31].

E. Relation to previous work
It is instructive to compare our result to previous work on the subject. In the pioneering works on quantum Darwinism [3][4][5][6][7][8][9][10][11][12][13][14], the focus was on studying specific examples where the emergence of objectivity could be analysed in detail. We regard Theorem 1 as providing a rigorous justification to some of the claims of those works (namely observable objectivity and some aspects of outcome objectivity).
The proliferation of information can intuitively be connected to the idea of cloning of information. The no-cloning theorem [27] is one of the hallmarks of quantum mechanics, stating that only classical information can be perfectly and infinitely cloned. Based on this intuition, in a beautiful paper [28] Chiribella and D'Ariano obtained the closest result to Theorem 1 previously known (building on [29,30]). They proved a variant of Theorem 1 for dynamics Λ in which all the B subsystems are permutation-symmetric, i.e. the information is symmetrically distributed in the environment. In this case Chiribella and D'Ariano obtained a similar bound to Eq. (1), but with the dimension of the B systems in place of the dimension of the A system. Therefore whether the assumption of permutation-symmetry of the B systems (which is hard to justify) was needed, and whether the bound had to depend on the dimensions of the outputs (which limits its applicability), were left as open questions until now.
Corollary 4 has a similar flavour to a result due to Streltsov and Zurek [31] regarding the role of quantum discord in the redistribution of correlations 6 . However Streltsov and Zurek were only able to treat the case where the initial state shared by Alice and Bob is pure. In such a case is was shown that Eq. (12) holds even without the need to consider asymptotics, i.e. without the limit on the right-hand-side of Eq. (12).
We remark that one can take an alternative approach to the study of the validity of the objectivity conditions of quantum Darwinism, not referring at all to the dynamics -as we instead do in this paper -and rather focusing on the properties of the (final) system-environment state. Such an approach was recently considered in [15] by asking what properties the final state of system plus environment should have to satisfy the conditions of "objectivity" in terms of quantum measurement theory. It turns out that from a few assumptions, including Bohr's non-disturbance principle, full objectivity requires the so-called broadcast structure. The latter has been explicitly shown to be compatible with what a canonical physical model involving photon scattering predicts [11] and with the standard classical information transmission perspective in terms of accessible information [14].

III. PROOFS
In this section we prove Theorems 1 and 2, Proposition 3, and Corollary 4. The proofs will follow the information-theoretic approach used in [17,18] for deriving new quantum de Finetti Theorems.
We will make use of the following properties of the mutual information: • Positivity of conditional mutual information: This is equivalent to strong subadditivity and to monotonicity of mutual information under local operations [38].
• For a general state ρ AB it holds [38] with the more stringent bound for a separable state σ sep AB [39].

A. Proof of Theorem 1
We will make use of the two following simple lemmas.

Lemma 5. (Lemma 20 of [34]) For every
Proof. The second inequality in (20) is trivial, as the diamond norm between two cptp maps is defined through a maximization over input states, while J(Λ 0 ) − J(Λ 1 ) 1 corresponds to the bias in distinguishing the two operations Λ 0 and Λ 1 by using the maximally entangled state Φ AA ′ as input. The first inequality can be derived as follows. Any pure state |ψ AA ′ can be obtained by means of a local filtering of the maximally entangled state, i.e., for a suitable C ∈ B(C d A ), which, for a normalized |ψ AA ′ satisfies tr(C † C) = 1. From the latter condition, we have that C ∞ ≤ 1. Let |ψ AA ′ be a normalized pure state optimal for the sake of the diamond norm between Λ 0 and Λ 1 . We find where we used (twice) Hölder's inequality M N 1 ≤ min{ M ∞ N 1 , M 1 N ∞ } in the first inequality, and C ∞ ≤ 1 in the second inequality.
⊓ ⊔ We are in position to prove the main theorem, which we restate for the convenience of the reader. with for states σ j,k ∈ D(B j ). Here d A is the dimension of the space A. We will proceed in two steps. In the first we show that conditioned on measuring a few of the B ′ s of ρ AB 1 ,...,Bn , the conditional mutual information of A and B i (on average over i) is small. In the second we show that this implies that the reduced state ρ AB i is close to a separable state z p(z)ρ z,A ⊗ ρ B i ,z , with the ensemble {p(z), ρ z,A } independent of i. We will conclude showing that by the properties of the Choi-Jamiolkowski isomorphism, this implies that the effective channel from A to B i is close to a measure-and-prepare channel with a POVM independent of i.
Let µ be the uniform distribution over [n] and define µ ∧k as the distribution on [n] k obtained by sampling m times without replacement according to µ; i.e. The inequality comes from the fact that π is separable between A and B 1 B 2 . . . B n because of the action of the quantum-classical channels M 1 , . . . , M n . The second line follows from the chain rule of mutual information given by Eq. (16). Define J k := {j 1 , . . . , j k−1 }. We have where (i) follows since only I(A : B j k |B j 1 , . . . , B j k−1 ) π depends on M j k ; (ii) by convexity of the maximum function; (iii) again because all the other terms in the sum are independent of j k ; (iv) directly by inspection and linearity of expectation; and (v) by the definition of f (k) in Eq. (24). From Eqs. (24) and (25), we obtain and so there exists a q ≤ k such that where we relabelled j q → j. Thus there exists a (q −1)-tuple J := (j 1 , . . . , j q−1 ) and measurements M j 1 , . . . , M j q−1 such that Let ρ z AB j be the post-measurement state on AB j conditioned on obtaining z-a short-hand notation for the ordered collection of the local results-when measuring M j 1 , . . . , M j q−1 in the subsystems B j 1 , . . . , B j q−1 of ρ. Note that ρ z A is independent of B j (for j / ∈ J). By Pinsker's inequality (17), convexity of x → x 2 , and Eq. (18), By Eq. (28) and convexity of x → x 2 , By Lemma 5 and so Note that E z ρ z A ⊗ ρ z B j = z p(z)ρ z A ⊗ ρ z B j is the Choi-Jamiolkowski state of a measure-andprepare channel E j [40], since E z ρ z A = ρ A = 1 1/d A . It is explicitly given by Note that the POVM {d A p(z)ρ z A } is independent of j. Thanks to Lemma 6, we can now bound the distance of two maps by the distance of their Choi-Jamiolkowski states to find Then where we used that the diamond norm between two cptp maps is upper-bounded by 2.
Choosing k to minimize the latter bound we obtain 7 Finally applying Markov's inequality, ⊓ ⊔ 7 The expression a/ √ k + bk is minimal for k = ( a 2b ) 2/3 . We further use that for b = 2/n < 1 it holds b 1/3 ≥ b 5/6 .

B. Proof of Theorem 2
The proof of Theorem 2 follows along the same lines as Theorem 1:
Proof. Since the proof is very similar to the proof of Theorem 1, we will only point out the differences. Let ρ AB 1 ,...,Bn := id A ⊗ Λ(Φ) be the Choi-Jamiolkowski state of Λ and C = {C 1 , . . . , C n/t } be a partition of [n] into n/t sets of t elements each. Define π C := id A ⊗M 1 ⊗. . .⊗M n/t (ρ), for quantumclassical channels M 1 , . . . , M n/t defined as M i (X) := l tr(N i,l X)|l l|, for a POVM {N i,l } l , with M i acting on ∪ j∈C i B j .
As in the proof of Theorem 1, by the chain rule, where the expectation is taken uniformly over the choice of non-overlapping sets C j 1 , . . . , C j k ∈ [n] t . We have From Eqs. (41) and (42), we obtain and so there exists a q ≤ k such that where we relabelled j q → j. Thus there exists a (q − 1)-tuple of sets C := {C j 1 , . . . , C j q−1 } and measurements M j 1 , . . . , M j q−1 such that Here we can follow the proof of Theorem 1 without any modifications to obtain that Choosing k to minimize the right-hand side as done in the proof of Theorem 1 and applying Markov's inequality, we obtain the result. ⊓ ⊔

C. Proof of Proposition 3
We will make use the following well-known lemma: [35]) Let ρ be a density matrix and N an operator such that 0 ≤ N ≤ 1 1 and tr(N ρ) ≥ 1 − δ. Then Proposition 3 (restatement). Let E be the channel given by Eq. (5). Suppose that for every i = {1, . . . , t} and δ > 0, Then there exists POVMs Proof. For simplicity we will prove the claim for t = 2. The general case follows by a similar argument.

D. Proof of Corollary 4
Corollary 4 will follow from Theorem 1 and the following well-known continuity relation for mutual information: [37]) For ρ AB ,

1/3
, h 2 is the binary entropy function, and the maximum on the righthand side is over quantum-classical channels Λ(X) = l tr(N l X)|l l|, with {N l } l a POVM and {|l } l a set of orthogonal states.
As a consequence, for every ρ AB , with Ej X j = 1 n N i=1 X j , and the maximum on the left-hand side taken over any quantum operation Λ : Proof. By definition, for all cptp maps Λ and E acting on B, and for any state ρ AB , it holds Combining Theorem 1 and Lemma 8 (specifically, Eq. (57)), we have that for every δ > 0 there exist a measurement {M k } k and a set S ⊆ [n] with |S| ≥ (1 − δ)n such that for all j ∈ S and all states ρ A ′ A it holds with and The claim is then a simple consequence of substituting E j with an optimal quantum-classical channel.
We now turn to the proof of Eq. (59). That the left-hand side of Eq. (59) is larger than the righthand side is trivial. Indeed one can pick Λ = Λ B→B 1 B 2 ...Bn as the quantum-classical map that uses the POVM {N l } l that achieves the accessible information I(A : B c ) := max Λ∈QC I(A : B) id⊗Λ(ρ AB ) with measurement on B and stores the result in n classical registers, one for each B i : Λ(X) = l tr(N l X)|l l| ⊗n . To prove that the left-hand side of Eq. (59) is smaller than the right-hand side it is sufficient to use Eq. (60) for the choice δ = n − 1−η 3 , for any 0 < η < 1. Then one obtains,

IV. CONCLUSIONS
The problem of the quantum-to-classical transition -and in particular, the problem of the origin of classical objectivity -is fascinating. The framework of quantum Darwinism appears as an intriguing possible explanation. Quantum Darwinism makes two predictions on the information about a system that is spread to many observers via the environment that interacts with the system and decoheres it. In this picture, the observers are imagined to acquire information about the system by each having independent access to some part of the environment. The first prediction of quantum Darwinism is objectivity of observables, which states that the environment selects the same specific classical information (i.e. information about one specific POVM) to be made potentially available to all the observers. The second is objectivity of outcomes -the fact that the aforementioned observers will (almost) all have access to the outcome of the observation and agree on it.
So far, the validity and applicability of the quantum Darwinism approach to the problem of the quantum-to-classical transition were only partially understood. In this work we have rigorously proven that the first property -objectivity of observables -is completely general, being a consequence of quantum formalism only (in particular of properties related to the monogamy of entanglement, but going beyond it). On the other hand, the validity of objectivity of outcomes does seem to depend on the details of the evolution, and we are only able to provide partial results about such a feature. Having in mind in particular the latter issue, we believe that establishing formal and mathematical (and at the same time, physically relevant) criteria for objectivity of outcomes would be very interesting.
Our results focus on the dynamics, but as a corollary we have also derived a clear-cut operational interpretation to quantum discord, which was originally introduced to capture the quantumness of correlations in information-theoretic terms. We proved that quantum discord corresponds to the asymptotic average loss in mutual information, when one of the parties, e.g. Bob, attempts to distribute his share of the correlations with Alice to many parties. From the perspective of quantum Darwinism, this means that the many observers having each access to only a part of the environment will, on average, only be able to establish at most classical correlations with the system of interest -the system that "gets measured by the environment". In this sense, we have fully generalized the results of [14] and [31] , that were limited to pure states.