Generic emergence of classical features in quantum darwinism

ABSTRACT Quantum Darwinism posits that only specific information about a quantum system that is redundantly proliferated to many parts of its environment becomes accessible and objective,

leading to the emergence of classical reality. However, it is not clear under what conditions this mechanism holds true. Here we prove that the emergence of classical features along the

lines of quantum Darwinism is a general feature of any quantum dynamics: observers who acquire information indirectly through the environment have effective access at most to classical

information about one and the same measurement of the quantum system. Our analysis does not rely on a strict conceptual splitting between a system-of-interest and its environment, and allows

one to interpret any system as part of the environment of any other system. Finally, our approach leads to a full operational characterization of quantum discord in terms of local

redistribution of correlations. You have full access to this article via your institution. Download PDF SIMILAR CONTENT BEING VIEWED BY OTHERS A STRONG NO-GO THEOREM ON THE WIGNER’S FRIEND

PARADOX Article 17 August 2020 A DYNAMICAL QUANTUM CHESHIRE CAT EFFECT AND IMPLICATIONS FOR COUNTERFACTUAL COMMUNICATION Article Open access 06 August 2021 TEMPORAL SELF-SIMILARITY OF

QUANTUM DYNAMICAL MAPS AS A CONCEPT OF MEMORYLESSNESS Article Open access 14 September 2020 INTRODUCTION Our best theory of the fundamental laws of physics, quantum mechanics, has

counter-intuitive features (for example, the superposition principle, complementarity and non-locality) that are not directly observed in our everyday classical reality. 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 (QC) 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 ability to manipulate quantum systems preserving their quantum features has made enormous progress

in recent years—enough to purportedly lead A. Zeilinger to state that ‘the border between classical and quantum phenomena is just a question of money’1. However, even if we are somewhat

pushing the location of the QC 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 decoherence3,2, where information is lost to an environment. This typically leads to the selection of persistent pointer

states3, 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, that is, agreed upon by several observers. A possible

solution to these questions comes from an intriguing idea termed quantum Darwinism4,5,6,7,8,9,10,11,12,13,14,15,16,17,18, which promotes the environment from passive sink of coherence for a

quantum system to the active carrier of information about the system (see Fig. 1). In this view, pointer observables correspond to information about a physical system that the

environment—the same environment responsible for decoherence—selects and proliferates, allowing potentially many observers to have access to it. The ideas of quantum Darwinism are beautiful

and physically appealing. Significant progress was achieved in a sequence of papers4,5,6,7,8,9,10,11,12,13,14,15,16,17,18. However, we are still far from understanding how generally the

ideas of quantum Darwinism apply. In particular, for example, given any specific interaction Hamiltonian it is not clear whether and to what extent classicality sets in. A careful and

far-from-trivial analysis must in principle be separately performed for each specific model (see the papers cited above). In this paper, we explore whether classical aspects along the

quantum Darwinism ideas emerge that are totally independent of the specific dynamics. We prove that some features that are a prerequisite for the validity of quantum Darwinism as explanation

for the emergence of classicality are indeed generic. One main consequence of our results is a deep qualitative change in the study of the emergence of classicality: from proving it in

given models to showing that it is present in some specific sense in any model involving sufficiently many subsystems of discrete variables. Our results apply beyond the system–environment

categorization: in a global system composed of many initially uncorrelated subsystems, any subsystem is being objectively measured by the other ones (see Fig. 2). Most importantly, our

approach allows to exactly identify which aspects of emergent objectivity are independent from the specific evolution/interaction, and which do instead depend on the model. Indeed the

present analysis splits the concept of emergent objectivity into two elements: objectivity of observables and objectivity of outcomes (see Results for detailed definitions). Such two

properties ensure that the information about the quantum system does become objective, being accessible simultaneously to many observers and agreed upon. As we report in the Results, the

first aspect of objectivity—objectivity of observables—is effectively always present. More specifically, there always exists an explanation for what (most of) the observers see such that

they only have at most access to the same classical information about one observable of the system observed. On the other hand, the validity of the objectivity of outcomes depends on how

much knowledge about the shared observable is available to the elementary subsystems. Finally, we make use of our techniques to prove in full generality (that is, going well beyond the

pure-state case treated in ref. 19) that when information is distributed to many parties the minimal average loss in correlations is equal to the quantum discord20, a quantity that has

recently attracted much attention but was still missing a full clear-cut operational characterization. Our results are related to (and in a sense, an extension of) the no-cloning theorem21,

and intuitively based on the monogamy of entanglement, as discussed more specifically in Methods. RESULTS PHYSICAL MOTIVATION AND NOTATION We want to analyse how the quantum information

content of a physical system spreads to (many parts of) its environment. To model this, although our mathematical description in terms of quantum channels (see shortly below) allows for a

more general scenario, consider _n_+1 systems _S_1,…,_S__n_+1 (see Fig. 2a). These may constitute a closed system, or be part of a larger system. We focus our attention on one system _S__i_,

which we shall call _A_ (see, for example, Fig. 2b), and we think of the others systems, now denoted _B_1,…,_B__n_, as of fragments of its environment. All our results assume that _A_ is

finite-dimensional, with dimension _d__A_, but we do not need such an assumption for the systems _B_1,…,_B__n_. Suppose that _A_ is initially decorrelated from _B_1,…,_B__n_. Independently

of any detail of the closed (that is, unitary) or open dynamics of _S_1,…,_S__n_+1, this condition ensures that the effective transfer of quantum information from _A_ to _B_1,…,_B__n_ is

represented by a quantum channel (also called a quantum operation)—a completely positive trace-preserving (cptp) map—, with the set of density matrices over the Hilbert space _X_ (see Fig.

3)22. We remark that the role of _A_ can be taken by any _S__i_, as long as it satisfies the condition of being finite-dimensional and initially uncorrelated from the other systems (compare

Fig. 2b with Fig. 2c). Given two quantum operations Λ1 and Λ2, the diamond norm of their difference is defined as , with the trace norm ||_X_||1:=tr((_X_†_X_)1/2). The diamond norm gives the

optimal bias of distinguishing the two operations Λ1 and Λ2 by any process allowed by quantum mechanics (that is, choosing the best possible initial state of the system-of-interest and of

an ancilla system, applying one of the quantum operations to the first system, and performing the best possible measurement to distinguish the two possibilities)23. Thus if , the two maps

represent the same physical dynamics, up to error _ɛ_. Finally, let tr\_X_ be the partial trace of all subsystems except _X_. OBJECTIVITY OF OBSERVABLES AND OBJECTIVITY OF OUTCOMES We will

adopt the following two notions for emergent objectivity: * Objectivity of observables: different observers that access a quantum system by probing part of its environment can only learn

information extractable from a single observable of the system (often associated to a measurement on the pointer basis determined by the system–environment interaction4) that, although

possibly non-unique, is 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 above observable and will agree on the outcome obtained (_cf._ the agreement condition of ref. 16). Our main result about objectivity of

observables is as follows (see Fig. 4): _Theorem 1:_ let be a cptp map. Define as the effective dynamics from and fix a number 1>_δ_>0. Then there exists a measurement, described by a

positive-operator-valued measure (POVM) {_M__k_}_k_ (_M__k_≥0, ∑_k__M__k_=I (ref. 22), and a set _S_⊆{1,…,_n_} with |_S_|≥(1−_δ_)_n_ such that for all _j_∈_S_, with for states . Here _d__A_

is the dimension of the space _A_. As we mentioned before, the diamond-norm distance on the left-hand side of equation (1) represents how different the two physical processes Λ_j_ and are:

the smaller the diamond norm, the more similar the processes, to the extent that they can become indistinguishable. The right-hand side of equation (1) is a bound on such a

distinguishability that for fixed _δ_—or even for _δ_ decreasing with _n_ but not too fast, for example, for , for any 0<_η_<1—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_ (for example, 1015), for all environment parts but _δn_ of them the bound on the right-hand side

of equation (1) is very close to zero, that is, the effective dynamics is for all practical purposes. The operation in equation (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 obtained24. It is clear that an observer who has access to

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_. We can interpret {_M__k_}_k_ as the pointer observable of the interaction Λ (see also ref. 18 in this context). 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 , 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_ on receiving the result _k_. Note that the bound is independent of the dimensions of the _B_

subsystems, being therefore very general. Note, however, the dependence on the dimension _d__A_ of the system _A_. Although the functional form of this dependence might be improved, it is

clear that no bound independent of _d__A_ can exist. Indeed, suppose _A_=_A_1, …, _A__n_ and consider the noiseless channel from _A_ to _B_1,…,_B__n_, the it is clear that a

dimension-independent statement of the theorem would fail. 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. 11, 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_}. _Theorem 2:_ Let be a cptp

map. For any subset _S__t_⊆[_n_] of _t_ elements, define as the effective channel from to . Then for every 1>_δ_>0, there exists a measurement {_M__k_}_k_ (_M__k_≥0, ∑_k__M__k_=_I_)

such that for more than a (1−_δ_) fraction of the subsets _S__t_⊆[_n_], with for states . Theorem 2 says that the effective dynamics to B j 1 , . . . B j t 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 be a block of sites

such that the effective dynamics from _A_ to is well-approximated by for the pointer POVM {_M__k_}_k_ and states . From Theorem 2, we know that this will be the case for most of the choices

of . 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 do 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 states _ρ__i_. We have _Proposition 3:_ let be the channel given by equation (5). Suppose that for every _i_={1,…,_t_} and

1>_δ_>0, Then there exists POVMs such that Equation (7) is equivalent to saying that the information about the pointer observable {_M__k_}_k_ is available to each . Assuming the

validity of equation (7), the proposition shows that if observers on measure independently the POVMs , 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. 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 Zurek25 (see also26) defined the discord of a bipartite quantum state _ρ__AB_ as

where is the mutual information, is the von Neumann entropy, and the maximum is taken over QC channels ΓQC(_X_)=∑_k_tr(_M__k__X_)|_k_〉〈_k_|, with a POVM {_M__k_}_k_ and orthogonal |_k_〉s,

which go from the system _B_ to a classical register. These are a special case of the measure-and-prepare channels of equation (2), with the _σ_ states pure and orthogonal. Notice that

Ollivier and Zurek originally25 defined discord in terms of projective measurement rather than general POVMs. 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

states25. 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 part20,25,26,27. Recently there has been a burst of activity in the study of quantum discord (see ref. 20). Despite the recent efforts, the

evidence for a clear-cut role of discord in an operational setting is still limited20. 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

theorem27,28. Indeed a corollary of Theorem 1 is the following (see Fig. 5; notice that in the following _B_ is a generic system, and so are the systems _B_1,_B_2,…,_B__n_): _Corollary 4:_

Let be a cptp map. Define ∘Λ as the effective dynamics from . Then for every 1>_δ_>0 there exists a set _S_⊆[_n_] with |_S_|≥(1−_δ_)_n_ such that for all _j_∈_S_ and all states _ρ__AB_

it holds where , _h_2 is the binary entropy function, _h_2(_x_)=−_x_log_x_−(1−_x_)log(1−_x_), and the maximum on the right-hand side is over QC channels

ΓQC(_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 , 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 equation (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 what done in ref. 29, where operational interpretations of discord

are given that are somewhat more involved, and in ref. 30, 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. In particular this removes the limitations of a recent related work by

Streltsov and Zurek19. RELATION TO PREVIOUS WORK It is instructive to compare our result to previous work on the subject. In the pioneering works on quantum

Darwinism4,5,6,7,8,9,10,11,12,13,14, 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 the basis of the objectivity observables and some aspects of outcome objectivity). The proliferation of information can intuitively

be connected to the idea of cloning of information. The no-cloning theorem21 is one of the hallmarks of quantum mechanics, stating that only classical information can be perfectly and

infinitely cloned. Based on this intuition, in two beautiful papers first Chiribella and D’Ariano31 and later Chiribella32 obtained the closest results to Theorem 1 previously known

(building on refs 33, 34). In those works a variant of Theorem 1 is proven for a dynamics Λ in which all the _B_ subsystems are permutation-symmetric, that is, the information is

symmetrically distributed in the environment. In particular, bounds similar to equation (1) were provided, 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 to be justified) 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 by Streltsov and Zurek19 regarding the role of quantum discord

in the redistribution of correlations. However, Streltsov and Zurek19 were only able to treat the case where the initial state shared by Alice and Bob is pure. In such a case it was shown

that equation (12) holds even without the need to consider asymptotics, that is, without the limit on the right-hand side of equation (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 ref. 16 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 shown17 to be compatible with what a canonical physical model involving photon scattering predicts12 and with the

standard classical information transmission perspective in terms of accessible information15 (see also ref. 35 for a general perspective). DISCUSSION 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 for it.

Quantum Darwinism makes two predictions (which, one could say, constitute its two pillars) 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 (that is, information

about one specific measurement of the system) to be made potentially available to all the observers. The second prediction is objectivity of outcomes, that is, the fact that the

aforementioned observers will (almost) all have access to the outcome of the observation and agree on it. The validity and applicability of the quantum Darwinism approach to the problem of

the quantum-to-classical transition were so far only partially understood. The fundamental conclusion of quantum Darwinism theory4,5,6,7,8,9,10,11,12,13,14,15 has so far been that the

conjunction ‘objectivity of measurements and objectivity of outcomes’ occurs typically in nature because of the specific character of local Hamiltonian interactions. In this work, we have

rigorously proven that the first pillar of quantum Darwinism—objectivity of observables—has solid foundations as a consequence of quantum formalism alone (in particular of the monogamy of

entanglement36, but going beyond the latter). There is always an effective explanation for what the individual observers see that is based on the measurement of one and the same observable

(more precisely, one POVM) for (almost) all observers, with the information about the outcome of such observation communicated to each observer in such a way that said information is at most

degraded—that is, not fully accessible. Notice that the observers could try to infer information about other observables, but such inference cannot be more successful than trying to infer

the value of such observables with processing performed on the quantum signal states that encode the outcome of the measurement of the specified observable. This consideration is in

particular relevant for cases where there exist several possibile ‘alternative’ decompositions with the structure of equation (2), and particular some freedom in the identification of the

shared observable. Our main result ensures that there is at least one objective observable. This means, on one hand, that, as mentioned, any attempt by the observers at obtaining information

about other observables is effectively the same as trying to infer information on such observables from the available information on the one fixed observable. On the other hand, our

result—the existence of at least one common observable—is a prerequisite for the selection of a unique observable. The latter uniqueness depends on the properties of the POVM elements _M__k_

and of the signal states _σ_ in equation (2), for example, on their (mathematical) linear independence. Thus, while the effective existence of at least a shared observable—from which

information about all other observables must be obtained—is independent of the details of the dynamics, its uniqueness does seem to depend on such details. Also the validity of objectivity

of outcomes does seem to depend on the details of the interaction, and we are only able to provide partial results about such a feature. Our results seem to indicate that the two pillars of

quantum Darwinism are qualitatively different, with the pillar of objectivity of observables a more solid one, because it is deeply rooted in the formalism of quantum mechanics. Our findings

suggest that future research should focus on understanding the minimal assumptions needed to ensure the uniqueness of the shared observable and the objectivity of outcomes. Another striking

aspect of the generality of our results is that, as mentioned already in the introduction, they actually allow us to go beyond the system–environment categorization. The key point here is

that our analysis does not rely on any symmetry assumption about the interaction between the systems _S_1,…,_S__n_+1 introduced in the first part of the Results section, or about the systems

themselves; the conditions of independence and of finite-dimensionality mentioned there suffice to ensure that every system is objectively measured by the others. Up to our knowledge, this

is the first result of this generality. A key question is how the present approach can be further generalized to an infinite-dimensional system. This will likely require the consideration of

bounds of energy and energy fluctuations, leading to the consideration of an effective dimension for physical systems. Finally, we remark that 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, for example, Bob, attempts to distribute his share of the correlations with Alice to many

parties. From the perspective of quantum Darwinism, one can interpret this result as the fact 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 refs 15, 19, that were limited to pure states. METHODS PROOFS The detailed proofs of Theorems 1 and 2, Proposition 3 and Corollary 4 are presented in Supplementary Note 1, Supplementary

Note 2, Supplementary Note 3 and Supplementary Note 4, respectively. The proofs rely principally on properties of mutual information that are listed in Supplementary Methods, and make also

use of refs 41, 42. Here we only provide the proof ideas of Theorem 1, Proposition 3 and Corollary 4. PROOF IDEA OF THEOREM 1 The proof of Theorem 1 is based on quantum information-theoretic

arguments along the lines of recent work by F.G.S.L.B. and Harrow37,38 for deriving new quantum de Finetti theorems. We develop the methods of F.G.S.L.B. and Harrow37,38 further to show

that not only the effective channels are close to a measure-and-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 refs 37, 38, 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 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 , 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 equation (1) holds true. This is a consequence of an important property of the quantum mutual information: the chain

rule22. 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 determined. The argument has connections with the phenomenon of entanglement monogamy36, which intuitively says that must be close to a separable state for most _j_. A state

is separable if it can be written as a convex combination of product states: . Thus, by the Choi–Jamiołkowski isomorphism23, the associated channel must be close to a measure-and-prepare

map. Our results go beyond what we simply expect from entanglement monogamy, by showing the existence of the common pointer POVM for most (which is equivalent to saying that is close to for

an ensemble independent of _j_). PROOF IDEA OF PROPOSITION 3 The proof of Proposition 3 is based on the Gentle Measurement lemma39. The latter essentially states that, if a specific outcome

of a measurement on a quantum system in a state _ρ_ is highly probable, then, on obtaining that outcome, the state of the system is not changed too much. In the case of the hypotheses of

Proposition 3, by distinguishing the single states , an observer can guess with high probability correctly the outcome _k_. The Gentle Measurement lemma is used to prove that this does not

compromise too much the ability of another observer to also obtain the correct result, and hence agree with the first observer. This is generalized to many observers. PROOF IDEA OF COROLLARY

4 The Corollary is based on Theorem 1. The corollary does not use the full power of Theorem 1, as the only important feature in this case is that most of the effective channels to each

receiver are close to measure-and-prepare channels. Thus, for a large number of receivers, for most of the latter the amount of correlations with _A_ cannot be greater than the amount of

correlations that can be established through a measurement. The additional key property used is the continuity of quantum mutual information, based on the Alicki–Fannes inequality40; this is

combined with an opportune scaling of _δ_ with _n_, so that the averaging over the broadcast receivers makes irrelevant the contribution coming from the (relatively few, that is, sublinear

in _n_) receivers who potentially have access to a larger amount of correlations. ADDITIONAL INFORMATION HOW TO CITE THIS ARTICLE: Brandão, F. G. S. L. _et al._ Generic emergence of

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disordered globally than locally. _Phys. Rev. Lett._ 86, 5184–5187 (2001) . Article  ADS  CAS  PubMed  Google Scholar  Download references ACKNOWLEDGEMENTS F.G.S.L.B. thanks David Poulin for

introducing him to quantum Darwinism and for useful correspondence on the subject. M.P. thanks Giulio Chiribella, Robert Koening, Jarek Korbicz, Alexander Streltsov, Wojciech Zurek and

Michael Zwolak for useful and stimulating discussions. P.H. thanks Jarek Korbicz, Ryszard Horodecki and Jess Riedel for discussions on quantum Darwinism. F.G.S.L.B. was funded by an ESPRC

Early Career Fellowship. M.P. acknowledges support from NSERC, CIFAR, DARPA and Ontario Centres of Excellence. P.H. is supported by the National Science Centre project Maestro

DEC-2011/02/A/ST2/00305. Partial support of this work by the Polish Ministry of Science and Higher Education Grant no. IdP2011 000361 is also acknowledged. AUTHOR INFORMATION AUTHORS AND

AFFILIATIONS * Quantum Architectures and Computation Group, Microsoft Research, Redmond, 98052, WA, USA Fernando G. S. L. Brandão * Department of Computer Science, University College London,

London, WC1E 6BT, UK Fernando G. S. L. Brandão * Department of Physics & Astronomy and Institute for Quantum Computing, University of Waterloo, Waterloo, N2L 3G1, Ontario, Canada Marco

Piani * SUPA and Department of Physics, University of Strathclyde, Glasgow, G4 0NG, UK Marco Piani * National Quantum Information Center of Gdańsk, Sopot, 81-824, Poland Paweł Horodecki *

Faculty of Applied Physics and Mathematics, Technical University of Gdańsk, Gdańsk, 80-233, Poland Paweł Horodecki Authors * Fernando G. S. L. Brandão View author publications You can also

search for this author inPubMed Google Scholar * Marco Piani View author publications You can also search for this author inPubMed Google Scholar * Paweł Horodecki View author publications

You can also search for this author inPubMed Google Scholar CONTRIBUTIONS All authors contributed to all aspects of this work. CORRESPONDING AUTHOR Correspondence to Marco Piani. ETHICS

DECLARATIONS COMPETING INTERESTS The authors declare no competing financial interests. SUPPLEMENTARY INFORMATION SUPPLEMENTARY INFORMATION Supplementary Notes 1-4, Supplementary Methods and

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classical features in quantum Darwinism. _Nat Commun_ 6, 7908 (2015). https://doi.org/10.1038/ncomms8908 Download citation * Received: 10 October 2014 * Accepted: 24 June 2015 * Published:

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