Quantum Enabled Observers and Quantum Shannon Theory

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Quantum enabled observers and quantum Shannon theory

A. Feller1, B. Roussel1, I. Frérot4, O.Fawzi3, and P. Degiovanni2 (1) European Space Agency - Advanced Concepts Team, ESTEC, Keplerlaan 1, 2201 AZ Noordwijk, The Netherlands. (2) Univ Lyon, Ens de Lyon, Université Claude Bernard Lyon 1, CNRS, Laboratoire de Physique, F-69342 Lyon, France (3) Univ Lyon, Ens de Lyon, Université Claude Bernard Lyon 1, CNRS, INRIA Laboratoire d’Informatique du Parallélisme, F-69342 Lyon, France and (4) ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, Av. Carl Friedrich Gauss 3, 08860, Castelldefels, (Barcelona), Spain (Dated: December 6, 2020) Quantum theory is the fundamental framework which describes all of modern physics except gravity. Despite its tremendous successes, fundamental questions regarding its foundations and interpretation are still actively investigated. A key question is to understand the quantum-to- classical transition without relying on ad-hoc postulates. The decoherence program, a significant step towards answering this question, has recently been renewed by considering the information flow from a system into its environment. This led to a paradigm shift called "quantum Darwinism" in which the environment is now seen as a quantum communication channels between the system and observers. In this study, we propose to push this idea forward by considering the full (quantum) network of observers in the quantum Shannon theory framework. Our main result is the clarification of the information theoretical criteria for independent and shared objectivity as well as the identification of the corresponding shared quantum states between the system and the quantum-enabled observers. This leads to a (non-trivial) separation between the classical information broadcasted to the observers by the system and the (correlated) quantum noise they experience, clarifying the structure of the global correlations in the "observed system/many observers" quantum system.

PACS numbers: Keywords: quantum Shannon theory, quantum observers, decoherence, classicality

Contents 3. Bipartite state typology 11 B. Quantum discord 13 I. Introduction 2 1. Basic definition 13 2. Simple examples and bounds 13 II. Quantum enabled observers 2 3. Operational meaning via quantum state A. Definition and capabilities2 merging 14 1. Definition2 C. Quantum Darwinism 15 2. Measurements2 1. A short review 15 3. Communication capabilities2 2. Perfect coding and decoding 16 B. Extracting a classical image2 3. Perfect redundancy 17 1. Ideal classical observation3 4. Discord in quantum Darwinism 18 2. Noisy classical measurements3 IV. Many-observer structures 20 3. Many classical observers4 A. Statement of the problems 20 4. Classical image extraction by B. Objectivity of observables 20 quantum-enabled observers5 C. Hierarchy of objectivity 21 C. Questions to be addressed6 1. Independent objectivity 21 1. What is to be seen vs how it can be seen6 2. Shared objectivity 22 2. Reconstructing an objective classical image7 3. Observer capabilities and accessible V. Conclusion 23 informations8 A. Summary 23 4. Guessing probability and min entropy.8 B. Perspectives and open questions 24 5. Accessible information.9 A. Quantum measurements 24 III. Correlations and quantum to classical 1. General measurements and POVMs 24 transition 9 2. Ideal measurements 24 A. Classical vs quantum correlations9 1. The mutual information9 B. Operational meaning of the mutual 2. Information gain through a measurement 10 information 25 2

C. Classical and quantum bounds on entropies26 When the observer Oj performs such a generalized mea- 1. Sub-additivity and lower bounds 26 surement and obtains the result α ∈ Rj, the state |Ψtoti 2. Strong subadditivity 26 of the system and all the observer’s quantum degree of 3. Suﬃcient criterion for entanglement 26 freedom jumps to the relative state with respect to this 4. Behavior under temporal evolution 27 measurement result α:

I. INTRODUCTION Such a collection of at most one experimental result collected by each observer is called a measurement record. II. QUANTUM ENABLED OBSERVERS Note that the relative state depends on the full measurement record. The measurement records can be stored in the memories of the observers and then processed clas- A. Definition and capabilities sically or communicated among the various observers. We thus have the notion of a quantum trajectory for 1. Definition the full quantum state associated to the experimental records collected by the observers. Note that this no- A quantum-enabled observer is an agent capable of lo- tion only depends on the data that the observer is using cal quantum and classical operations as well as classical to retro-compute the quantum trajectory of the system or quantum communications. It is realized as a physi- with respect to the records he is considering. Consider- cal subsystem that is macroscopic enough to be partly ing all the data collected by all the observers gives access described as classical, thereby explaning why it has clas- to the most refined quantum trajectory, which is then sical memory and computing capabilities as well as classi- independent on the observer as soon as they have access cal communication capabilities. But it has also quantum to classical communications, and therefore to the same degrees of freedom that can be completly isolated from measurement records. the rest on demand. The observer can then perform any unitary quantum operation on these degrees of freedom. Moreover, these degrees of freedom can be extended by 3. Communication capabilities adding more qubits or using the quantum communications links with the other observers. The observers are capable of classical communications Among these degrees of freedom, the quantum mea- but also of quantum communications. This means that surement apparatus of each observer is directly or indi- they can establish ideal quantum communication chan- rectly coupled to the system and therefore, each observer nels between them and send parts of their quantum de- can be thought of as having access to a part of the sys- grees of freedom along these ideal quantum channels. Of tem’s environment, exactly as in quantum Darwinism. course, they have the same capabilities for classical com- Consequently, after the phase of pre-measurement, the munications channels. Observers can also at will intro- system as well as the observer’s quantum degree of free- duce new quantum degrees of freedom into the game and dom are in a global entangled state |Ψtoti. therefore, can also generate on demand maximally entangled qubit pairs (called EPR pairs) between themselves. Theoretical quantum and classical information quanti- 2. Measurements ties will be used to quantify the amount of quantum as well as classical communication resources that they can Measurement denotes the process by which any ob- use. We will use the notations of resource calculus to server extracts classical information from the quantum describe protocols and relations between them. degrees of freedom it has access to. Here, we will consider that the observer Oj has access to general measurements which are described by a set of B. Extracting a classical image Kraus operators Mj,α where α ∈ Rj denotes the possible results. These operators act only on Hj and satisfy: What does it mean for a quantum-enabled observer to extract a classical image through observation? To X † 1 Mj,αMj,α = Hj . (1) address this question, we will start by reviewing mea- α∈Rj surement in classical physics, first by discussing the ideal 3 measurement framework that underlies classical theo- panel of Fig.1. In this approach, the system S, charac- ries, starting from classical mechanics up to relativity1 terized classically by its microstate s, broadcasts infor- Then, we will discuss briefly noisy classical measure- mation on the value X(s) of a certain observable X. The ments. And finally, we will discuss measurements by system is thus placed on the encoder end of a noisy classi- quantum-enabled observers. cal communication channel C, which represents both the experimental apparatus and noise induced by the environment. On the receiver end of the channel, an ideal 1. Ideal classical observation measurement terminal together with a signal processor play the role of the decoder. The signal procesing task Classical observers having complete access to the state consists in an estimation algorithm, which provides an X of a classical system will record it in their own memory estimate for x = X(s) from the measured values m, de- as XO for observer O. If several observers have access noted Xest(m). to the same part of the space of states for the system, The noisy communication channel plays the role of the we can define a transition function FO,O0 that sends XO experiment with its imperfections and the decoder plays onto XO0 . These transition functions satisfy the following the role of the classical observer. Notice that here, the compatibility condition: encoder is fixed. The system S can be in various microstates that are characterized by a probability distri-

FO1,O3 = FO2,O3 ◦ FO1,O2 . (4) bution pS(s) but the encoding stage of this system is the same, regardless of the microstate. Note that the state of the system and the observations The noisy communication channel is characterized by maps which map it onto the observer’s own descriptions the noise probability matrix formed by the conditional are in a sense tautological: they express that the com- propabilities p(m|x) for the measurement results, know- patibility conditions define a geometric structure which ing the value of the observable X. This represents the is locally mapped by each observer in its own memory imperfection of the measurement apparatus. records. In a measurement oriented view, only the tran- In full generality, the observer wishes to reconstruct sition functions F between the measurement records Oi,Oj the value of the observable from the measurements. The of the different observers satisfying the compatibility con- probability for guessing the correct result is defined as dition (4) are considered. The transition function between two observers tells us which pairs of quantities are totally correlated. More specifically, in the case of classi- pguess(x|Xest, C) = Em [δ(Xest(m) − x)] (5) cal field theory, specifying these compatibility conditions correspond to mathematically define space-time as a (dif- in which m = M(x) is a random variable distributed ac- ferential) manifold and field space as a fiber bundle over cording to p(m|x). Note that this quantity depends on space time as well as field configurations as (local) sec- the results we wish to estimate but most importantly it tions of this fiber bundle. depends on the estimation algorithm Xest and of course, Of course, this is a (very compact) review of the kine- on the imperfect measurement, id est on the noisy com- matic framework of observation in classical physics for munication channel C. the specific case of ideal observers having a complete ac- Various estimation scores can be defined. The most cess to all the system’s degree of freedom with infinite natural one for physicists is the average on all the possible precision. In a more realistic framework, one could imag- values of x: ine considering observers that do not have a complete access to the system, one measuring partially or totally Q[Xest|C] = Ex [pguess(x|Xest, C)] (6) coarse-grained quantities with respect to the measurements performed by the others. One should also take in which the average over x is taken according to the into account the noise introduced by the measurement macro-state of S which induces a probability distribu- itself. tion p(x) for the values of X. It quantifies the quality of the estimation algorithm Xest for a given imperfect measurement C. In computer science as well as in all critical 2. Noisy classical measurements applications, one considers the worst case scenario score:

The problem of a noisy classical measurement can be viewed as an estimation problem as depicted on the left W[Xest|C] = min [pguess(x|Xest, C)] . (7) x

It is easy to prove that the quality of the estimator Q[Xest|C] is maximized by the maximum- 1 This framework was mainly introduced to discuss special and then later general relativity by A. Einstein but it can also be likelihood estimator Xest(m) = argmax[p(x|m)] in used to discuss any type of classical theory, even non-relativistic which the likelihood of x for a given m is p(x|m) = P mechanics. p(m|x)p(x)/( x p(x)p(m|x)), deﬁned using Bayes rule. 4

For such a choice, • We can allow independant measurements, which amounts ro replacing M by n-independant copies X ⊗N Q[Xest|C] = p(x)p(m|x)δx,Xest(m) (8a) of M, or collective measurements, replacing M m,x by MN acting on the N outputs of the physical X = p(m, X (m)) noisy channels. This clearly makes sense in quan- est tum theory as we shall see later. m X In the case of classical systems, this seems less nat- = p(m)p(Xest(m)|m) . (8b) ⊗N m ural so we will restrict ourselves to M : this en- sures that we have the same encoding stage giving This last quantity is maximized if, for all m, we choose X(s) for a given micro-state and the same measure- Xest.(m) = argmax[p(x|m)] which is the maximum like- ment apparatus used on all realizations. hood estimator denoted by ML. Of course considering this optimal estimation aglorithm gives the highest • The estimation algorithm Xest,N takes the N mea- guessing probability for the noisy channel, or equivalently surement results (m1, ··· , mN ) and gives an es- noisy measurement, considered. This guessing probabil- timate of x which is identical for the N realiza- ity then only depends on the channel and is denoted by tions. It can consist of N parallel use of the sin- pguess(C). gle shot estimation algorithm and a décision pro- The inverse of pguess(C) represents the typical number cess that computes the final estimation from these of independent trials needed to obtain a correct estimate N single shot estimation processes, for example of the quantity X(s) for a given ss. We can then define through a majority vote. Or it can work directly the “min-entropy” as on (m1, ··· , mN ) and compute an estimate by exploiting all these values. Hmin(S|M) = − log2(pguess(C)) (9) From these, a guessing probability and an estimation which represents the size of the register needed to store score can also be defined which depend on EstN and on the data associated with the typical number of indepen- C⊗N , the N-time use of the imperfect measurement ap- dent trials to get a correct estimate. But before showing paratus C. how this estimation-based entropy connects to information theoretical entropies, let us make things more ex- plicit by discussing the simple example of the noisy esti- 3. Many classical observers mation of a binary valued quantity. Let us consider the bit-flip channel C in which a classi- p The case of several observers can be viewed as a gen- cal bit is flipped with probability p. We assume that both eralization of the situation we have just considered. Here calues of x are equiprobable. The Bayesian likehood for also a single source (the system) is connected to several having x given a result y is then p (x|y) = p(y|x) which B noisy communication channels in a star topology, but is 1 − p whenever x = y and p whenever x 6= y. Conse- they are allowed to measure different observables: instead quently, for p < 1/2, the maximum likehood algorithm of having C⊗N , we have N different channels (C , ··· , C ) is X (0) = 0 and X (1) = 1. This enables us to es- 1 N ML ML as depicted on the right panel of Fig.1. And of course, timate the guessing probabilities: p (0) = p (1) = guess guess each of them has his own decoding stage, corresponding 1 − p which is the probability for obtaining y = 0 (resp. to a different observer, which we will now discuss. y = 1) for x = 0 (resp. x = 1). Consequently, aver- Note that all the channels are fed with the same s aging over x leads to p (ML|C ) = 1 − p and thus guess p which gives (X (s), ··· ,X (s)), the true values of the H (ML|C ) = − log (1 − p) for p < 1/2. In the case 1 N min p 2 observables for the micro-state s. In this case, the point p > 1/2, then the estimation algorithm gives X (0) = 1 ML for the observers is to try to recover the microstate s. and X (1) = 0. When x = 0 is emitted, the probability ML One could imagine that the obserables X are multi- for having the correct estimation thus corresponds to the j dimensional so that each of the estimation algorithms case y = 1 which occurs withj probabiloty p and thus aim at reconstructing an estimate for s. Therefore, we p (0) = p (1) = p, which leads to p (ML|C ) = guess guess guess p are left with the question of consensus: can they agree p or equlivalently H (ML|C ) = − log (p) for p > 1/2. min p 2 on the same value of s, or, in estimation theory terms, Note that, so far, we have defined a one-shot estimation can they add a collective estimation stage that will give problem: given one realization of the microstate s and them a consensual estimation whose quality can then be therefore, a single value of X, the guessing probability is assessed along the lines sketched above ? defined. This can naturally be generalized to the case of an N-shot estimation problem as follows: But one could also imagine that the various observers cannot reconstruct s individually because they don’t ac- • Instead of considering a single use of the noisy com- cess all the necessary obserbables, so they have to use a munication channel, one considers N uses of the collective estimation algorithm. communication channel, fed with the same value of In both cases, they have to exchange information and x. that’s why, in the present framework, we have to endow 5

O1 Noisy channel O2 System Sig. Proc. xest System X M O . ..

FIG. 1: A Shannon view at noisy classical measurements: (Left Panel) Recasting of measurement by a single observer in terms of a noisy communication channel. The system S is placed on the emitting side of a noisy communication channel into which it injects values of the observable X. The noise represent the experimental noise from the experimental system (not the system) to the imperfection of the measurement apparatus. The decoder combines an ideal noiseless measurement device M and a signal processor Sig. Proc. which together lead to an estimate xest = Est(m) for the value of X from the measurement results m. This end of the noisy channel is the classical observer O. Right panel: when several classical observers O1,..., ON are present, the system is connected to several noisy communication channels (in black). Comparing the result of the post-processed data obtained by the observers is done by using ideal communication channels connecting them (in grey).

not included in any observer will depend on the measurement results. If a set of n quantum-enabled ob- C1 O1 C1 O1 servers O1,...,n = (O1,..., On) perform generalized mea- SX SX surements, we are left with a relative state of the system and other untouched degrees of freedom – the environ- C2 O2 C2 O2 ment of (S, O1,...,n) – with respect to the results of these n generalized measurements. Of course, the measurement (b) (b) results are random and therefore, we do not recover an ideal classical measurement. Moreover, generically, the measurement results cannot be described using a classical noise model, which amounts to assume that the corre- FIG. 2: Two classical observers O1 and O2 are measuring the same observable X of the system S through noisy channels C lations between the various measurements obey Bell like 1 inequalities. and C2: (a) the observers are not allowed to communicate (b) O1 and O2 are allowed to use a perfect classical communica- The question is precisely to determine under which tion channel (in grey) and exploit correlations between their conditions there is a classical reality to be recovered and, imperfect measurements. in such a case, how it can be recovered by the observers. This clearly depends on the choice of generalized measurements by the observers, on their communication ca- them with classical perfect communication channels as pabilities and on the global entangled state generated by shown on Fig.1. The collective estimation algorithm can the quantum pre-measurement interaction. be delegated to one of the observers, so we don’t have to Instead of digging directly into that problem, let us show it on the ﬁgure. discuss what the quantum-enabled observers perspective can bring us. From the point of view of quantum-enabled observers, allowing only local operations and classical ex- 4. Classical image extraction by quantum-enabled observers change of the measurements results (LO) corresponds to what the observers discussed in Section IIB2 were do- As explained before, when performing measurements, ing: they perform measurements and then compare their quantum-enabled observers alter the total state in a mea- results. We shall call them LO-observers. surement dependent way. This measurement induced A more powerful set of capabilities is obtained by al- quantum back-action has no equivalent in the classical lowing the observers to condition the choice of measure- world in which the state of the system is, in principle, ments they are performing to the results of other ob- left invariant by the act of measuring. servers. This is the full power of local quantum opera- For a quantum-enabled observer, extracting a classi- tion and classical communication (LOCC) which enables cal image of the world consists in performing a general- them to perform a kind of collective adaptative measure- ized measurement on the quantum degrees of freedom ment (see Fig.3). it has access to. However, by doing so, the relative By contrast, sharing entangled quantum states on top state of the other observers and of the system will de- of being able to perform all LOCC operations enables to pend on its measurement result. If all the observers per- do more than with LOCC only. For example, the quan- form such measurements, the relative state of the system tum teleportation protocol expresses that sharing entan- and of the environmental degrees of freedom that are gled pairs enables to simulate quantum communication 6

constructing a quantum reality, a problem usually called quantum tomography except that, here, it is addressed using several observers instead of only one. Concerning the first question (Q1) in the perspective of reconstructing a classical reality, the issue of objectivity can be formulated as the existence of a classical quantity, A B S A B S independent of the observers. Whether or not and how the observers have the capability to access it, to reach FIG. 3: Circuit representation of operations performed by a consensus about it, heavily depends on the observers observers: boxes represent classical measurements, double and, therefore, is part of the second question (Q2) we lines represent classical communication and oblique solid lines have formulated. More precisely, questions about con- are perfect quantum channels. Left part: Alice and Bob who sensus between the observers often implicitly refer to a are two quantum-enabled observers entangled with a system comparison between reconstructed classical images of the S, are only using LOCC. Right part: by using quantum com- world by each observer. In this case, it implies a compar- munication, Alice and Bob are performing a state transfer ison between the observer’s experimental data. As such concentrating all their shared quantum information on Bob it makes sense when considering LO-observers. In the without affecting the system and possibly leaving some en- case of LOCC-observers fully exploiting the possibility tangled pairs free for later use between them. to design cooperative adaptative measurements, a consensus emerges automatically, almost by definition, from their collective work to reconstruct a classical image of between two observers. Therefore, a quantum-enabled the world. They can share it after their collaborative observer entitled with LOCC and shared entanglement reconstruction process. This difference between LO and with the other observers can use quantum communica- LOCC-observers shows that there is a hierarchy of notion protocols (QCPs) to transfer his share of the global tions of objectivity depending on the constraints we put quantum state as well as its correlations with the sys- on information exchange. tem and with the other observers away from him [1] (see In principle, questions (Q1) and (Q2) can also be ad- Fig.3). The quantum-enabled observer which is the re- dressed in the perspective of reconstructing a quantum ceiver in such a protocol can then performed general- reality, that is the quantum state of the system. How- ized measurements on the global state he now has in ever, one should be careful that the state of the system is his hands. This means that he potentially has access not absolute but relative to the observer [14, 15]. Several to all the correlations that were shared between him and subtleties must therefore be addressed when considering the sender of the quantum state transfer protocol before questions (Q1) and (Q2): the role of the backaction of it was executed. Consequently, by using quantum non- each measurement performed by each observer on the local resources, the observers can access quantum corre- entangled state of the system must be taken into account lations between themselves and even can characterize, in and it is not obvious at all that a cooperative recon- the large number of realization limits, the quantum state struction of the reduced state of the system before any they share by transferring correlations to one of them. measurement takes place is always possible. Secondly, This is why the framework of quantum Shannon the- the notion of objectivity of the reconstructed quantum ory applied to quantum-enabled observers appears as the state has to be specified. As of December 6, 2020, this proper framework to understand the emergence of a clas- is still left as a perspective and most of this report will sical picture for physical observers embedded within a address questions (Q1) and (Q2) in the perspective of quantum universe. reconstructing a classical reality. To address these questions, three point of views will be used throughout these notes, each of them with a dif- C. Questions to be addressed ferent focus:

1. What is to be seen vs how it can be seen • Focusing on the structure of the state of the system and the observers will enable us to understand which information they can access. This will be As explained before, the main questions addressed in closely related to the nature of correlations between these notes is the question of (Q1) understanding which the system and the observers and between the ob- “image of the world” can be accessed to the observers and servers themselves. Depending on their capacities, (Q2) how it can be retrieved. These questions are basi- they will be able to extract all, part or maybe none cally diﬀerent : the ﬁrst question involves the structure of the information they are looking for on the sys- of the global quantum state for the system and the ob- tem. servers whereas answering the second one is relative to the operational capacities of the observers. • We shall use information theoretical quantities to Both questions can be declined on the problem of re- distinguish the various classes of system/observer constructing a classical reality or on the problem of re- states and we shall also use information theoretical 7

measures of the abilities of the observers to recover |si of S. The orthonormal basis (|si)s then defines an information on the system given their capacities. observable for S which is the same for all the observers in F . • How the observers extract an image of the world, The same definition can be given for a generalized mea- either independently or collectively is an estima- surement M on S with values s : in this case, there exist tion problem the answer to which depends on the mutually orthonormal registrer states |si ∈ HE\F and observers capabilities. Of course, it would be in- normed but non orthonomal states |ψ i ∈ H such that teresting to quantify the minimal communication s S and computation resources needed to perform such X ρ = p(s) |sihs| ⊗ |ψ i hψ | ⊗ ρ(F |s) (12) a task. RSF s s s Of course, since information theoretical quantities are notoriously difficult to compute, it might be interesting to where R denotes the space of register states generated by address these questions within a more operational frame- the |si. Taking the trace over S, the quantum state of work, that is by considering only specific set of observ- RF is then ables accessible to the observers and focusing on corre- X ρ = p(s) |sihs| ⊗ ρ(F |s) (13) lations between these observables. Whenever incompat- RF ible observers are to be considered introduces an essen- s tial difference between LO and LOCC observers whereas and consequently, we are in the same situation as before considering only compatible observables means that con- except that we have traded S for the register’s degrees of sidering adaptative measurements is not different from freedom. States of the form given by Eqs. (11) and (13) data processing the results of all the measurements. For are called classical–quantum states. As we shall see in such families of compatible observers, LO and LOCC are SectionIII, not all bipartite (S, F ) states are of this form. indeed identical. A decomposition of the form (11) does not tell us if the various observers in F can identify unambiguously each value of s. For instance, it could be the case that 2. Reconstructing an objective classical image for all s, ρ(F |s) = ρ0(F ) with ρ0(F ) some state independent of s; in this case, the global state would be of the Let us first discuss the problem of reconstructing an form ρSF = ρS ⊗ ρ0(F ), so that the observers degrees objective classical image. First of all, in quantum the- of freedom are uncorrelated with the system – no infor- ory, classical configurations correspond to perfectly dis- mation about s is available in F . But also in the other tinguishable quantum states, that is to mutually orthog- extreme case where all ρ(F |s) are perfectly distinguish- onal quantum states. able for different values of s – so that the information Let us assume that such states of the system S are about s is, in principle, available in F –, it does not tell denoted by |si, indexed by s and that the whole envi- us how the observers can, in practice, recover this in- ronment of S is denoted by E and contains the observers formation. In particular, it does not tell us whether this quantum degrees of freedom which are denoted by F . information about s is available, in principle, to every ob- Then reconstructing a classical image of s means being server individually, or to all observers only if they share able to distinguish between s 6= s0 by performing mea- some information they extracted individually, or to all surements on F. In order for this to be possible, we need observers only if they perform a global collective mea- the state of F relative to the various values of s to be surement on all their degrees of freedom. To clarify this orthogonal. These conditional states are defined by hierarchy of situations, we first have to distinguish two different notions of objectivity: ρ(F |s) = trS (ΠsρSF ) (10) • Objectivity of observables: different observers prob- where Πs = |si hs| and ρSF denotes the joint state of ing independent parts of the environment have ac- S and F . Note that in this case, ρSF may still contain coherences between different values of s and s0. Assuming cess to only one observable of the system. that no such coherences exist is an extra hypothesis which • Objectivity of outcomes: different observers prob- amount to saying that there are extra quantum degrees ing independent parts of the environment have full of freedom, not in S not in F which select the orthogonal access to the above observable and agree on the basis |si for F . This einselection is precisely what defines outcome. the objectivity of the basis |si of S for F : there is no ambiguity in the fact that |si is the proper basis of Hs. The objectivity of observables states that the environ- This implies that: ment selects one specific observable of the system (one X specific measurement) which may be accessible by per- ρSF = p(s)Πs ⊗ ρ(F |s) . (11) forming a generalized measurement on F . In other words, s this is the pointer observable of S induced by its environ- in which s 7→ p(s) is a probability distribution and ρ(F |s) ment. The second aspect of objectivity states that, given denotes the conditional state of F relative to the state this pointer observable, all the observers agree on one 8 specific outcome (one specific measurement result). This [Pascal]: Reprendre cela en définissant les quantités is the real consensus on memory records among many ob- dans le cas quantique en homogénieisant bien les no- servers. Of course, this is a much stronger requirement. tations. J’ai commenté la discussion classique. As will be discussed more precisely later, it has be shown [8] that the emergence of an objective observable is inherent to the structure of quantum theory while the emergence of one specific result is dependent on the 4. Guessing probability and min entropy. global ρSF state and on the observer’s capabilities. The first important point is that a single generalized measurement on F can unambiguously distinguish between In order to quantify the ability for the observers to infer |si ⊥ |s0i for s 6= s0 if and only if ρ(F |s) ⊥ ρ(F |s0) for s, we introduce the probability Pguess that the observer, s 6= s0. This is the global orthogonality condition which receiving the quantum state ρ(F |s), correctly “guesses” states that quantum-enabled observers using QCPs pro- the actual value of s. tocols between them can, in principle, cooperatively dis- More generally, if the classical variable s is broadcasted tinguish between s and s0 6= s. As we shall see now, even through a quantum channel, the oberver receives the con- when the global orthogonality condition is satisfied, it ditional or relative quantum state ρ(F |s) with probability is a non-trivial problem to assess whether more limited ps as in Eq. (11). Here, the quantum state ρ(F |s) is in observers can distinguish the various states |si. general mixed, so that the classical case is recovered if all ρ(F |s) are diagonal in the same basis |ai, and can be viewed as classical probability distributions p(a|s) over 3. Observer capabilities and accessible informations these states thanks to: X X From a quantum communication point of view, we are ρSF = p(s) |si hs| ⊗ p(a|s) |ai ha| (14a) dealing with a quantum communication channel that en- s a X codes the classical information (s, p(s)) using quantum = p(a, s) |si hs| ⊗ |ai ha| . (14b) states ρ(F |s) received by the observers. In cryptographic a,s terms, we are encoding a secret. The ability to uncover this secret is a function of the decoder’s capability at However, note that this is not generally the case: we the reception’s end of the quantum communication chan- are left with Eq. (11) in which the relative states ρ(F |s) nel. We introduce four different classes of observers cor- cannot be diagonalized in a common basis. responding to different decoding capabilities: In the present setting, the guessing probability is de- • Quantum enabled observers (QEO-observers): each fined by averaging over s a quantum guessing probability observer can apply any local quantum operation for each s, corresponding to the best choise of POVM and is allowed to use any quantum non local com- {Ns}s by the observer: munication resource. ! • LOCC-observers: each observer can apply a lo- X Pguess[{ps, ρ(F |s)}] = max ps tr[Nsρ(F |s)] . cal quantum operation and classical communica- {Ns} s tion is allowed between the observers. Therefore, (15) global adaptative protocols based on measurements We then introduce the “min-entropy” as Hmin/S|F ) = and quantum operations conditioned to classical in- − log Pguess[{ps, ρ(F |s)}]. In general, as in the classical formation is allowed. Equivalently, the outcome domain, we have the inequality: ρ(F |s) is split over the Fi which act as quantum decoders only allowed to use classical communica- Hmin(S|F ) ≤ H(S|F ) (16) tion. • LO-observers: observers are allowed to apply local where the relative or conditional entropy is defined as operations but classical communication is only al- H(F |S) = H(ρSF ) − H(ρF ) with H the von Neumann lowed for comparison of the results. No adaptative entropy, and ρSF is the classical-quantum state defined collective measurement is allowed. In cryptograph- in Eq. (11). An important result, called the quantum ics terms, this corresponds to 1-way decoding by a equipartition property, is that the bound can be reached set of independant decoders allowed to use classical by considering many copies of ρSF : communication only. 1 ⊗n Each of these observer classes therefore corresponds to Hmin(S1 ...Sn|F1 ...Fn)ρ → H(S|F )ρSF (17) a set of allowed generalized measurements. n SF • 1 − Obs: finally, we shall consider the ability of a where we allow for an -deformation of the quantum state ⊗n single observer, not allowed to communicate with ρSF , and maximize the l.h.s over all such -deformed other observers, to recover s. states. 9

5. Accessible information. needed to address in a rigourous and quantitative way the questions that have been formulated in SectionIIC. [Omar]: Il faudrait etre consistent sur la As we shall see in SectionIIIA, classical and quantum notation pour accessible info. Je propose correlations must be distinguished. Classical correlations I (S, F ),I (S, F ; LO), ... correspond to correlations between classical random vari- acc acc ables. The corresponding correlation measure is the mu- From an information theoretical point of view, we can tual information of Shannon’s classical information the- define an accessible information for each measurement ory. But for two quantum system, as is well known from class X by Bell inequalities, correlations between measurement results can be higher than authorized by classical physics. AccX (ρSF ) = max(I[S, M]) (18) In the quantum case, von Neumann mutual information M∈X is the relevant information-theoretical measure of corre- in which I[S, M] represents the von Neumann mutual lations. information of the reduced density operator after the We then introduce a variant of the mutual informa- M measurement has been performed (see SectionIII). tion that corresponds to the information gain through a From an information theoretical perspective, this quan- quantum measurement. This notion, which generalizes tity is the classical capacity of the quantum channel the corresponding one from classical Shannon theory, is which has S on its emitting side and a receiver con- subtle due to the backaction associated with the mea- strained to use measurement in the X measurement side surement when it is performed on the system on which (see REF). The classical information transmitted along we want to learn something. These notions of mutual this quantum channel consists in the values s considered information are then used to discuss a typology of bipar- in Eq. (11). In particular, for the Quantum Enabled Ob- tite state which consists in identifying the various forms servers (QEO), AccQEO(ρFS) is nothing but the full ac- under which classical correlations can be present in non- cessible information on the F side of the quantum chan- entangled bipartite states. nel that encodes s into ρ(F |s). Finally, the quantum discord, initially introduced by For a single observer, we consider the average over the Ollivier and Zurek [28], is a quantitative measure of non- observers Fi of the accessible information using general- classical correlations between two systems. It will be dis- ized measurements on Fi. Then, the definitions imply cussed in SectionIIIB. that [Pascal]: Rappeler mieux le role de la discorde comme critère pour les états CQ, QC... Là on passe trop vite AccQEO(ρSF ) ≥ AccLOCC(ρSF ) dessus. ≥ AccLO(ρSF ) ≥ Acc1−Obs(ρSF ) (19) [Alex]: La section Darwinisme n’est pas presentee. These accessible informations correspond to the communication capacity of the full communication system, from the source s to the decoding by the Fi’s. A. Classical vs quantum correlations [Alex]: Prendre la moyenne n’est sans doute pas an- odin. Le rendre explicite dans la notation hi. We consider a bi-partite quantum system AB, described by a generic quantum state ρAB (i.e. non- [Irénée]: Plutôt, on peut dire que c’est vrai pour necessarily a pure state). We will discuss the nature chaque observateur. of correlations between subsystems A and B, viewed as The global orthogonality condition implies that sources of classical information (encoded in classical vari- AccQEO(ρSF ) = S[p(s)] –namely, there exists a global ables s, e.g. classical bits), or of quantum information measurement on F whose outcome identifies a unique (encoded in quantum states ρ(s), e.g. qubits). value of s–, but it does not imply that the information about s can be accessed through LOCC or LO protocols. 1. The mutual information

III. CORRELATIONS AND QUANTUM TO In classical information theory, correlations between CLASSICAL TRANSITION two sources are quantiﬁed by the (classical) mutual information In this section, we discuss correlations between two quantum systems and more precisely, the diﬀerence be- I[A, B] = S[A] + S[B] − S[A, B] . (20) tween such correlations and the ones that appear between P two classically correlated classical systems. We also dis- In this equation S[A] = − a p(sa) log p(sa) denotes the cuss quite extensively the information gain associated Shannon entropy of the marginal probability on A (and with a quantum measurement. This discussion will in- similarly for S[B]), whereas S[A, B] denotes the Shannon deed provide us with the essential notions and quantities entropy for the joint probability distribution p(sa, sb). 10

I[A, B]/S[A] in a statistical ensemble characterized by a probability 2 distribution a 7→ pa on the ensemble of all microstates. Bob then measures a quantity XB, thereby obtaining results x which are randomly distributed even for a given 1 a since the measurement is noisy. Exactly as explained 1/2 in [11, Sec. 2.4.3], the information gain is the diﬀerence 0 S[B]/S[A] between the initial entropy S[A] and the average entropy 1 on a that can be obtained for a ﬁxed x: X FIG. 4: Bounds for the Shannon and von Neumann mutual S[A|XB] = p(x)S[A|x] (22) informations in units of S[A] in terms of S[B]/S[A]. The x dark grey zone corresponds to the classical bounds whereas the light grey zone corresponds to values of I[A, B]/S[A] that in which S[A|x] is the entropy of the Bayesian probabil- are not allowed in a classically correlated system and even not ity distribution p(a|x) = p(a, x)/p(x). Consequently, the in a separable state as shown in AppendixC3. information gain in this case is

S[A] − S[A|XB] = S[A] + S[XB] − S[A, XB] = I[A, X] . [Irénée]: Du coup, est-ce que Slepian Wolf est perti- (23) nent dans notre contexte? Equivalently, this is the difference between the total en- Its operational meaning in classical communication tropy of the observed results S[XB] and the average en- theory comes from the result of Slepian Wolf (see Ap- tropy of the noise which is pendixB): I[A, B] corresponds the economy in classi- X cal information transmission rate that can be made by S[XB|A] = paS[XB|a] (24) using correlations in a distributed compression protocol a thereby justifying its use as a quantitative measure of where S[XB|a] denotes the Shanon entropy of the condi- classical correlations. tional probability distribution p(x|a) that gives the dis- In the quantum case, the same formal definition is used persion of the results for Bob (noise). Consequently, the but this time with von Neumann entropies: information gain is given by the mutual information between A (the system) and X (the results): Svn[ρ] = − tr(ρ log ρ) . (21) The corresponding von Neumann mutual information Igain[A; XB] = I[A, XB] . (25) will be denoted by I[A, B]. The basic properties sat- The case of noiseless measurements discussed in Ap- isfied by this quantity are recalled in AppendixC. The pendixA2 is the one where p(x|a) = δ and there- key point is that although both the Shannon and the x,X(a) fore where the noise entropy vanishes S[X |A] = 0. This von Neumann mutual information have the same lower B leads to an information gain S[X ] as obtained in the bound, equal to zero when the two systems are in a B AppendixA2. This is also I[A, X] as expected from product state, they differ by their upper bound which Eq. (25). is equal to min(S[A],S[B]) for the Shannon entropy and Let us now move on to quantum systems. In order to 2 min(S[A],S[B]) for the von Neumann entropy. mimic the context in which we are working, let us first Figure4 shows the region, in the I[A, B]/S[A] and consider the case where the measurements are performed S[B]/S[A] variables in which correlations cannot be de- by Bob who accesses quantum degrees of freedom that scribed classically because the von Neumann mutual in- are not the same as the ones accessible to Alice. Given formation is larger than the upper bound for the clas- ρ an initial density operator for Alice and Bob, and sical Shannon information. In this region, one of the AB a generalized measurement M performed by Bob, after two conditional entropies S[A|B] = S[A, B] − S[A] or B the measurement, the reduced density operator for Alice S[B|A] = S[A, B] − S[A] has to be negative. is the statistical mixture of the

† TrB(MxρABMx) 2. Information gain through a measurement ρ(A|x) = † (26) hMxMxiρAB

Before considering the various types of bipartite states, in which the Mx operators act on HB, satisfy let us revisit the problem of quantifying the information P † 1 † x MxMx = B with probabilities px = hMxMxiρAB . It gained in a measurement. This problem has to be con- therefore seems natural to define the final entropy as the sidered in the classical domain but also in the quantum average over the measurements results of the entropies of domain. these density operators: Let us first consider the case of an imperfect measurement in the classical domain. This is exactly the situa- X S[A|MB] = pxS[ρ(A|x)] . (27) tion considered in SectionIIB2. Alice prepares a system x 11

(init) Note that this definition assumes that Bob communicates in which ρA denotes the density operator of A before the results of its measurements to Alice so that she can the measurement is performed. Note that, because of deduce the reduced density operators ρ(A|x). the measurement back-action on the system itself, ρ(init) Since here the measurement is performed by Bob us- is not of the form given by Eq. (31) and therefore, this is ing operators that commute with any operator on Alice’s not the Holevo quantity. side, ρA does not change during the measurement pro- In a classical system, this quantity is always positive cess. Although this is quite obvious from no-signaling, but, in quantum systems, it can indeed be negative. This the formal proof is simple: after a generalized measure- does not occur for a projective measurement, for which ment performed by Alice, the total density operator be- the ρ(A|x) are all projectors since in this case S[ρ(A|x)] = comes (init) 0 for all x and the information gain is S[ρA ] reflecting (post) X † the fact that a specific pure quantum state is associated ρAB = (1A ⊗ Mx)ρAB(1A ⊗ Mx) (28) x to each measurement result. In order to obtain a negative result, we need the entropies S[ρ(A|x)] pondered by p Taking the trace over Bob, leads to x to add up to more than S[ρA] which is a rather strong (post) (post) (init) constraint. It is apparently proven in Ref. [29], that ρA = TrB ρAB = TrB (ρAB) = ρA . (29) I[ρA,MA|ρA] ≥ 0 for any generalized measurement. Consequently, the quantity However, there exist general quantum measurements associated with a POVM whose associated state evolu- I[A, MB|ρAB] = S[A] − S[A|MB] . (30) tion of a more general form than generalized quantum measurements (see AppendixA1) for which the r.h.s. of in which S[A] = S[ρA] with ρA = TrB(ρAB) represents the amount of information gained by Alice from the mea- (33) is negative. This has lead to the introduction of a different definition of the information gain in the quan- surement MB performed by Bob: it is the difference between the initial von Neumann entropy of A and the av- tum measurement, which reduces to the one discussed P here, but which is always positive [6]. erage post-measurement entropy x pxS[ρ(A|x)] of A, obtained by the process of conditionning to Bob’s results Let us wrap this discussion on the measurement per- on Alice’s side. The quantity defined by Eq. (30) is the formed on A itself by focusing on the the case of pro- mutual information between Alice and Bob with respect jective measurements. The information gain can then be to the measurement MB as defined in [28]. Note that it lower than the difference between the maximal potential depends on ρAB which precisely contains the correlations information gain, given by the Holevo quantity associated that enable to learn something on A from a measurement with the relative density operators ρ(S|α) associated to performed on B. each measurement results, ponderated by their proba- In the present case, because the measurement is per- bilities. The potential loss, which represents the price to formed on Bob’s side, we have pay due to the quantum measurement’s back-action on A X itself, is given by the entropy increase due to the decoher- ρA = pxρ(A|x) (31) ence induced by the measurement process as explained in x AppendixA2. Note that this is not the case when the in which the ρ(A|x) are not necessarily orthogonal. Con- measurement is performed on a different system B (see sequently, the final expression for the information gain Eq. (32)) because, in this case, the price is already paid: correlations present in ρAB already imply a decoherence " # of A. X X [Pascal]: Voir enfin le papier [6] pour une interpré- I[A, MB|ρAB] = S pxρ(A|x) − pxS[ρ(A|x)] . x x tation théorie de la communication... Peut être à dis- (32) cuter plus loin ? This is exactly the Holevo quantity χ[(ρ(A|x), px)] which bounds the amount of information that can be recovered if Alice encodes the values of x distributed along px using mixted states ρ(A|x) [19]. Note that, by concavity of the 3. Bipartite state typology von Naemann entropy, this quantity is always positive. Moreover, when the relative states ρ(A|x) are mutually To understand the various degree of correlations be- orthogonal, it reduces to the Shannon entropy of the re- tween two systems, it is first useful to see how classical sults as would be expected since we are dealing with a correlations can be described within the quantum frame- maximally efficient generalized measurement. work. A classically correlated bipartite state (also called The definition of the information gain can be extended a classical–classical or CC-state) is a density operator of to the case where the measurement is performed on A the form itself, therefore leading to [17]: (cc) X (init) X ρ = p |iihi| ⊗ |jihj| (34) I[A, MA|ρA] = S[ρA ] − pxS[ρ(A|x)] (33) AB ij x i,j 12 with (|ii)i and (|ji)j being two orthonormal basis and Bob’s) side whereas the third one represent the maximal pij a probability distribution for the (i, j). For such a information that can be obtained using measurements state, the quantum mutual information coincides with performed independently on both sides. Note that, in the Shannon mutual information associated with the this third case, the definition is the one given by Eq. (33) probability pij. Consequently, bipartite states with quan- applied to the whole system (A, B). These three mea- tum mutual information exceeding the classical upper- sures can be used to differentiate among the various bi- bound min(S[A],S[B]) are not classically correlated in partite states introduced above: the sense of Eq. (34). As shown in AppendixC3, they are indeed entangled. Theorem A.1. The state ρAB is a classical-classical Classical-classical states are a particular case of classi- state if and only of I[A, B] = Icc[ρAB]. cal conditionning via a measurement, also called classical- Proof. The proof uses two preliminary results that we quantum or CQ-states: admit: X ρAB = pi |iihi| ⊗ ρ(B|i) (35) Lemma A.1. Given a classical-quantum state ρAB = P B B i i pi |iihi|⊗σi , we have I[A, B] = Icq[ρAB] = χ(pi, σi ). B Moreover, Icc[ρ] = I[A, B] if and only if the states σi in which |ii an orthonormal basis and the pis define a commutes thus ρAB be classical-classical. probability distribution. This basis defines a measure- Lemma A.2. Given a state ρ and two local operations ment MA performed on Alice’s side. Equivalently, this is AB a generalized measurement performed on Bob by Alice. ΛA and ΛB, if I[(ΛA ⊗ ΛB)(ρAB)] = I[A, B], then there ∗ ∗ ∗ ∗ The first thing to notice is the asymmetry of this exists inverse maps ΛA and ΛB such that (ΛA⊗ΛB)(ΛA⊗ definition: a state could be classical-quantum but not ΛB)(ρAB) = ρAB. quantum-classical. The point is that, in the classical- First if the state is CC, the result is straightforward. quantum state, we do not require the conditional states cc Now suppose that I[A, B] = Icc(ρAB) = Icc(ρAB) with ρ(B|i) to be mutually orthogonal nor mutually com- cc MA and MB two optimal measurements and ρ = muting. To understand the difference with classically AB ρAB(MA ⊗MB). Then by the second lemma, we have two correlated states, let us diagonalize each ρ(B|i) = ∗ ∗ P inverse measurements MA and MB. Now we can consider pi(j) |j; iihj; i| with j 7→ pi(j) defining a probabil- qc cc ∗ j∈Ii the quantum-classical state ρAB = ρAB(MA ⊗ 1B). This ity distribution and (|j; ii)j being an orthonormal basis. QC QC is thus a CQ state such that ICQ(ρ ) = Icc[ρ ] = Then, we have AB AB Icc[ρAB] = I[A, B]. Thus by the second part of the first X X lemma we conclude that ρAB is a CQ state (commuta- ρAB = pipi(j) |iihi| ⊗ |j; iihj; i| (36) tion of the marginals) and again with the first lemma i j∈I i that ρAB is CC. but since the diagonalization bases Bi = (|j; ii)j may be Before moving on to discuss quantum discord, let us different for different i’s, the state may not be classical- mention that quantum correlations are not necessarily classical. In the next subsection, we will introduce the equivalent to quantum entanglement. General bipartite quantum discord, a quantity that precisely vanishes for entangled states are defined as the opposite of separable quantum-classical states and therefore encodes the devi- states and a bipartite state is called separable if and only ation to classical conditioning. And as one could expect, if it can be written as a statistical mixture of factorized we shall see that the quantum discord is also asymmetric. states Before doing so, the information gain in a quantum X A B measurement that has been introduced in Section IIIA2 ρAB = pi σi ⊗ σi (38) leads us to introduce two other measures of correlations i between Alice and Bob: where the pi’s define a probability distribution. Separa- Icq[ρAB] = max (I[B,MA|ρAB, ]) (37a) ble states form a convex set but there is no easy way to MA determine whether or not a bipartite state is separable Iqc[ρAB] = max (I[A, MB|ρAB]) (37b) or not: in full generality it is an NP-hard problem [18]. MB Entangled states are exploited in quantum communi- Icc[ρAB] = max (I[AB, MA ⊗ MB|ρAB]) . (37c) cation protocols such as quantum teleportation and su- MA,MB perdense coding. Pure entangled state also exhibit non- local characteristics such as violations of Bell inequalities [Alex]: Ces deux premières information ne sont que which express that the correlations they lead to cannot be des informations accessibles ? explained by any model based on local hidden variables All these mutual informations are non-increasing under [9]. But in the case of statistical mixtures, entangled local operations and are positive. The first (resp. second) states are not necessarily showing non-local correlations. one represent the maximum information that can be ob- The recent review [2] presents a clear discussion of the tained by performing a measurement on Alice’s (resp. different degrees of correlation between different systems 13 and an in depth review of the various measures of the Using I[A, B] = S[A] + S[B] − S[A, B] and Eq. (40), quantumness of correlations between two or more sub- we obtain equivalent definitions for the discord: systems. DB(ρAB) = S[ρB] − S[ρAB] + min (S[A, MB|ρAB]) MB B. Quantum discord (42a) = min (S[A, MB|ρAB]) − S[A|B] (42b) MB [Pascal]: Je vais aussi taper mes notes sur la dé- composition du diagramme des possibilités à trois en in which the B subscript recalls over which subsystem zones où les corrélations sont classiques et quantiques the measurements are considered. au sens donné par les bornes sur la mutuelle. Mais We can also define the discord relative to a spe- faut sans doute discuter plus précisément la discorde cific measurement MB without performing the maximum là dedans. J’ai pas une vision très claire. computation. This quantity, denoted by DMB (ρAB), is then I[A, B] − I[A, MB|ρAB] = S[A, MB|ρAB] − S[A|B] without the maximum in Eq. (40). It will be useful when 1. Basic definition discussing Quantum Darwinism in SectionIIIC. [Alex]: The asymmetric mutual information equals to In classical information theory, we have two equiva- mutual information when Bayes’s rule is valid, as for lent ways of writing the mutual information: I[A, B] = classical systems. S[A] + S[B] − S[A, B] and I[A, B] = S[A] − S[A|B]. Transposing the first definition to the quantum regime is without problem. However, we have to be careful about the second since, in the classical theory, it in- 2. Simple examples and bounds volves a conditional statement. In quantum theory, the von Neumann conditional entropy can be defined by The two simplest examples of discord are obtained S[A|B] = S[A, B] − S[A] but, contrarily to the classical by considering a qubit Bell state where we can check case, it does not always correspond to a classical condi- that DB[ΨAB] = 1 and a maximally mixed state of |00i tionning to the results of a measurement. and |11i for which DB[ρAB] = 0. More generally, for Consider a shared quantum state ρAB between Alice a pure bipartite global state |ΨABi, then S[A] = S[B] and Bob and assume that Bob performs a measurement and the mutual information I[A, B] reaches its maxi- MB obtaining the set of conditional states (ps, ρA|s) for mum possible value 2S[A]. But in this case, the Schmidt Alice. The conditional entropy of Alice knowing that decomposition of |ΨABi tells us that S[A] is the min- Bob made a measurement is defined by the straightfor- P imum value S[ρA|MB] over all the possible measure- ward adaptation of Eq. (27): S[ρA|MB] = s psS[ρA|s]. ments taken on B. Consequently, in this particular case We can then use this quantity to define the mutual infor- DB[|ψABi] = S[B] + S[A] = 2S[A] = I[A, B]. As ex- mation of ρAB relative to MB along the lines of Eq. (30): pected, all correlations between A and B are of quantum origin.

I[A, MB|ρAB] = S[A] − S[ρA|MB] . (39) However, as we shall see later, quantum discord goes beyond simple entanglement to characterize non classical To capture all the classical information in the quantum features of quantum states since it can be non zero for state ρAB, we have to maximize it on the set MB thus separable states. These considerations raise two immedi- arriving at: ate questions: what are the lower and upper bounds on

J[A, B] = max (S[ρA] − S[A, MB|ρAB]) . (40) the quantum discord and what does it mean when they MB are saturated? This asymmetric mutual information, which is always Because J[A, MB] ≥ 0, we first obtain that the dis- positive and smaller than S[ρA], represents the part of cord is bounded from above by I[A, B]: the quantum the correlations between A and B that can arise from a part of correlations cannot exceed the total correlation classical conditionning through measurement performed between A and B. Moreover, because of the upper bound on B. It is the maximum information that can be gained J[A, B] ≤ I[A, B], the quantum discord is positive, thereby performing such a measurement. The quantum dis- fore cord is then defined as the part of the total correlation that cannot be obtained in this way: 0 ≤ DB(ρAB) ≤ I[A, B] . (43) D (ρ ) = I[A, B] − J[A, B] . (41) B AB The upper bound is reached whenever all the correlations As such, it can be interpreted as the quantum part of are of quantum origin as in the simple example of a max- the correlations between A and B. However, note that a imally entangled Bell pair. The lower bound is reached priori, this quantity is not symmetric in A and B. in the case of a CC state defined in Section IIIA3. 14

E E a statistical mixture describe by ρAB which we purify by introducing the environment E and a tripartite pure state |ψABEi shared between Alice, Bob and E. The corresponding ressources that are needed to perform the |ψ i ABE state transfert are given by A B A [qq] B B 1 2 1 1 [q → q] B0 hψ i + I [A, E][q → q] ≥ I [A, B][qq] + hψ i . ABE 2 vn 2 vn B2E (45) FIG. 5: State transfer protocol: Alice and Bob share a cor- in which the state ψ⊗n carries the same information as related statistical mixture that can be purified using the envi- B2E ψ⊗n in the large n limit. ronment E into a pure state |ΨABE i. The state transfer pro- ABE tocol consists in transferring the quantum correlations shared The protocol leads to another ressource inequality between Alice and Bob (see left part), as well as between Alice which quantifies the amount of classical and quantum and the environment in the hands of Bob using only classical communication resources that need to be used to achieve and quantum communication between Alice and Bob and do- the desired result without extra entangled pairs: ing nothing on E. In the end (right part), |ΨABE i is shared between the environment E and some of the qubits B2 of Bob hψABEi + S[A|B][q → q] + I[A, B][c → c] ≥ hψB2Ei . whereas the protocol can leave us with extra entangled pairs (46) shared between Alice and Bob. This protocol, called the quantum Slepian-Wolf protocol expresses that, given the shared state |ψABEi, one has to 3. Operational meaning via quantum state merging transfer I[A, B] classical bits and S[A|B] quantum bits (when positive!) to Bob to achieve the desired result. If the qubit transfer is realized using quantum teleporta- In order to be really meaningful in the context of infor- tion, it means that the state merging protocol consumes mation theory, a quantity should be given an operational S[A|B] maximally entangled pairs. meaning, usually related to an information processing task. The purpose of this section is to discuss such a A conceptual subtlety arises in quantum theory since meaning for the discord. the von Neumann relative entropy S[A|B] can become This drawback about quantum discord has now re- negative. In this case, its opposite is called the coher- solved relating it to different quantum information tasks. ent information [23]. In this case, S[A|B][q → q] can be We describe here two operational interpretations [10, 24] put on the r.h.s. of the resource inequality (46) and the of the quantum discord using the quantum state merging operational meaning of the quantum relative entropy is protocol [20]. Most results here should be understood in theb that, the protocol leaves us with −S[A|B] possible the asymptotic regime of independently and identically future uses of a quantum channel. Since, by the quan- distributed systems: tum teleportation protocol, an ideal qubit transfer can be emulated by a shared EPR pair and two classical bit ⊗n D(ρAB) transfer, the state transfer protocol can be re-expressed D(ρAB) = lim = I[A, B] − J[A, B] . (44) n→+∞ n as the state maerging protocol resource inequality We first begin by reviewing the basics of the state hψABEi + I[A, E][c → c] ≥ −S[A|B][qq] + hψB Ei (47) merging protocol. Let’s consider, in a classical setting 2 first, Alice and Bob having access to incomplete infor- whenever S[A|B] < 0. In this case, the (positive) co- mation X and Y respectively (see Fig.5). How many herent information has an operational intepretation as bits of information does Alice have to send so that Bob the number of distilled maximally entangled pairs in the can reconstruct the whole information content of X? state merging protocol [20]. Naively, Alice could send H[X] bits of information In this perspective, the first operational definition of where H is the Shannon entropy. But this is not opti- the quantum discord is given by the following theo- mal. Indeed, it is possible to use the correlation between rem [24]: X and Y to lower the amount of bits Alice has to send. Slepian and Wolf have shown that she could in fact send Theorem B.1. The quantum discord DB(ρAB) is the H[X|Y ] = H[X,Y ] − H[Y ] bits of information. This is minimum increase in the cost of quantum communication intuitively satisfying since the relative entropy H[X|Y ] for state merging between A and B with a measurement quantifies the uncertainty left about X knowing Y . performed on the receiving end B. The quantum state merging protocol is the extension of this setup to the quantum case [20]. It states that, us- Intuitively, a measurement on B destroys quantum cor- ing a quantum communication channel, Alice can transfer relations between A and B and consequently increases all her share of correlations initially shared with Bob and the cost for A to merge the post-measurement state with the environment in Bob hands and that, in the process, B. The rigourous proof is given in AppendixD1. Alice and Bob may be left with extra maximally entan- This result sheds some light on the properties of the gled pairs shared together. Initially, Alice and Bob share discord. First, measurement on B may lead to a loss of 15 correlations2 and therefore increases the price for state vision, we allow for a quantum conditionning of A by B merging. This explains why the discord must be positive. and therefore the proper conditional entropy is S[A|B] Secondly, Bob can at most recover S[B] qubits which is whereas in the latter case, only conditioning by classi- therefore an upper bound on the discord. Finally, the cal measurement results is allowed and we have to use quantum discord is zero for a state of the form S[ρA|MB]. In the next section, we will derive a different operational characterization of the quantum discord that X ρAB = piρA|i ⊗ |biihbi| (48) does not rely at all from the start on a classical measure- i ment. [Pascal]: A voir si on met une discussion du mother where the states |bii diagonalize ρB. Measuring in this protocol ou pas... basis and forgetting the result generates the state ρAB. This means that the measurement causes no loss of information and all the correlations between A and B are preserved in the measurement process. C. Quantum Darwinism The second operational meaning of the quantum discord is obtained by extending the standard state merging [Pascal]: En cours de reprise. Je suis en train de re- protocol by considering the preparation of the input state lire le premier papier de Zurek dans Phys. Rev. A using LOCC and local ancilla which they forget once the pour bien voir si on n’a rien oublié. preparation is completed. We then have the theorem [10]: In the Quantum Darwinism approach to the emergence Theorem B.2. The quantum discord is the total entan- of classicality that has been initiated by Zurek [7, 26, 27], glement consumption in the extended state merging pro- the environment is considered as a set of complex systems tocol: collecting information on the state of the system. Each observer has then access to a specific fragment or sets of D(A|C) = S[A|B] + EF [A, B] . (49) fragments of this environment. The quantum Darwinism approach aims at establishing information theoretical cri- in which teria for the emergence of a consensual classical view of " # the state of the system by the observers. Thus, quan- X AB EF [A, B] = min piS[trA(ψi )] (50) tum Darwinism appears to be very close, in the spirit, (p ,|ψAB i) i i i to the discussion of classical measurement by many observers presented in Section IIB2. We shall therefore is the minimum amount of entanglement A and B have review it briefly here and discuss more specifically the to use to create the state ρAB by LOCC. In Eq. (50), quantumness of the correlations that are considered in the minimum is taken on all the representation of ρAB this approach. AB as a mixture of the pure states |ψi i. The quantity EF (A, B) is therefore called the entanglement of formation of the state ρ . Thus, S[A|B]+E [A, B] quantifies AB F 1. A short review the amount of entanglement the two agents have to con- sume to first prepare the state ρAB and then perform the state merging protocol. This-two stage protocol is called Quantum Darwinism considers situations in which a the extended state merging protocol. The rigourous proof quantum system S is probed by multiple observers having of this result is given in AppendixD2. access to independent sets of data through parts of the This formula for the discord illustrates directly its total environment E of the system as shown on Figure6 asymmetry. A difference between this characterization represents schematically the shift on how to model the and the previous one is that this one refers to one state environment. ρAB while the previous one referred to two different This framework is very simular to the one considered in states, ρAB and the one measured by Bob. Here, no Section IIB2 except that here, the system is also quan- measurement by Bob explicitly appears. tum as well as the environmental degrees of freedom each As a general remark, those two operational charac- observer can access. terization of the quantum discord, while fulfilling their To quantify the ability of one observer monotoring one purpose, do not really go beyond its mere definition specific fragment F of the environment, Zurek et al consider the von Neumann mutual information I[S, F ]. In DB(ρAB) = minMB (S[A|MB]) − S[A|B]. Indeed, they, in the end, compare Bob viewed as a purely quantum sys- the same way, discussing the consensus between two ob- tem with Bob viewed as performing measurements and servers respectively monitoring Fi and Fj is based on the thus being a source of classical information. In the first the mutual information I[Fi,Fj]. Let us begin by considering the ability of a given observer to access information on the state of the system. To quantify the classicality of a state , we focus on the 2 At best nothing is changed. region where the mutual information defined by ?? is 16

F1 I[S; fδ] E E E2 F2 2 E1 2S[S] E3 E1 E3 S S S E7 E7 E4 E4 F4

E E6 5 E5 E6 S[S] F3 (1 − δ)S[S]

FIG. 6: Different structures for the environment. While standard decoherence models consider the environment as a huge 0 f monolithic set of degrees of freedoms, the quantum Darwin- 1 0 fδ 1 2 ism approach focuses on the physics of fragments Fi of the environment and their relations to the systems. Figure extracted from [35]. FIG. 7: Two of the typical behaviors of the mutual information I[S, Ff ] as a function of the size f of the fragment. The blue curve is the expected behavior for a randomly chosen state in HS ⊗ HE while the red one with its plateau corresponds to a Darwinian situation. close the von Neuman entropy of the system. This quali- Figure7 shows the two limiting cases for the behavior tatively means that the observer has enough information of the mutual information. The most interesting case for to reconstruct classical information about the state of the the problem of the emergence of a consensus is when a system3. plateau around the Von Neumann entropy of the system forms for a very small sized fragment of the environment. Indeed, it means that, by probing a small set of degrees of freedom of the environment, an observer has access to all the classical information about the state of the system. Moreover, other observers can do the same by I[S, F ] ≥ (1 − δ)S[S] , (51) probing other degrees of freedom. Information accessible to observers is information redundantly stored in the environment. This is in this sense that the information is said to be objective. The redundancy Rδ is then defined as the inverse of the minimum size reach the inequality (51): where the small δ parameter quantifies a possible information deficit that happens in every realistic setup. An 1 R = . (53) average version of this quantity over the set of fragments δ f of the same size 0 < f ≤ 1 with respect to E is more δ convenient to have a general view of the behavior of the What kind of states maximize this redundancy? In- mutual information. Zurek et al thus consider tuitively, pointer states of the system will maximize this since by definition pointer states are states which are ro- bust to the interaction with the environment and can thus “live on” to proliferate their information into many environment channels.

I[S, Ff ] = hI[S, F ]i|F |/|E|=f (52)

2. Perfect coding and decoding

[Alex]: A discuter mais je ne suis pas convaincu que ca soit le bon endroit pour discuter ca. Je dirais plu- in which the average is taken over all fragments size f. tot qu’il faut le mettre à la fin de cette section, après toute la discussion sur la discord et la condition “for all practical purpose” de la fin qui n’est pas suff- isante et qui doit etre précisée par une condition sur 3 The distinction between quantum and classical correlations will be discussed later using the quantum discord. l’accessible. 17

[Pascal]: Je ne suis pas encore convaincu par ce que Moreover, the state ρFS is indeed classical/classical: j’ai écrit... Est-ce que quelqu’un pourrait vérifier que X ˜ j’ai pas écrit de bétise? ρSF = p(s)ρS(s) ⊗ ρ(F |fs) (56) s Let us assume here that the reduced density operator of the system corresponds to a statistical mixture in which the ρ(S|s) are mutually orthogonal as well as of mutually orthogonal states |si with von Neumann en- the ρ(F |f˜) which are coarsed grained of reduced density tropy S[S]. What does it mean if the capacity Iqc[ρSF ] = operators ρ(f, s), involving non-intersecting sets of values maxMF (I[S, MF |ρSF ]) is equal to S[S]? The answer is of f for different s, and therefore are mutually orthogonal given by the following theorem [27]: ˜ ˜ for fs 6= fs0 . Note that for the (S, F ) system, measuring S and finding s is then equivalent to measuring on F and Theorem C.1. Iqc[ρSF ] = S[S] if and only if there ex- ˜ finding fs: ists an observable MF on F such that S[S|MF ] = 0. Moreover, in this case, S[F |X ] = 0 where X denotes ˜ ˜ S S ρ(FS|s) = ρ(FS|fs) = ρ(S|s) ⊗ ρ(F |fs) . (57) any observable of S that commutes with ρS. We thus have a perfect classical channel connecting the Proof. We assume that I [ρ ] = S[S]. Consequently, ˜ qc SF classical values s and fs. Thus, given any observable XS there exists a generalized measurement M on F such F on S that admits |si as eigenvectors, S[F |XS] = 0. We that S[ρSF |MF ] = 0 since, by definition, I[S, MF |ρSF ] = thus have I[F,XS|ρSF ] = I[S, M˜ F |ρSF ] = S[S]. S[S] − S[S|MF ]. Conversely, if there exist a generalized measurement MF on F such that S[S|MF ] = 0, 0 0 [Alex]: Dans le théorème, il est demandé d’avoir then minM 0 (S[S|M ]) ≤ 0. But S[S|M ] ≥ 0 for any F F F I [ρ ] = S[S]]. Donc il y a un max sur les mesures generalized measurement on F as we have seen in ??. qc SF 0 sur F. Avec ce qui est montré ici dans la réciproque, Consequently we have minM 0 (S[S|M ]) = 0. Therefore F F on montre juste que I [ρ ] ≥ S[S] a priori. Con- 0 qc AB I[ρ ,M ] = S[S] − min 0 (S[S|M ]) = S[S]. SF F MF F clure en utilisant les inégalités sur la mutuelle clas- Let us now turn to the second part of the theorem. sique ? Assuming now that Iqc[ρSF ] = S[S], we will show that Mais vu que on prend un mélange statistique sur S, S[SF |X ] = 0 for any observable of S that commutes S est ce que le Iqc[ρSF ] ici n’est pas en fait un Icc[ρSF ] with ρS. Because measurements performed on S and F ? commute, we can always specify the action of the gen- Finalement ça montre que la donnée classique s est eralized measurement on F by its action on each ρ(S|s). reconstruite via des mesures sur F ssi S[S|MF ] = The post-measurement reduced density operator for F is 0 ie l’état de S est parfaitement connu en mesurant thus of the form: MF . Vu qu’on traite des données classiques ici, c’est rassurant et ça suit l’intuition de la formule. Mais (post MF ) X ρS,F (F |s) = p(f|s) ρ(F |f, s) ⊗ |fi hf| , (54) c’est bien de le montrer. f [Pascal]: 12/11/2020: Aujourd’hui j’ai un peu plus in which the f are the possible results of the measurement l’impression d’être convaincu de ce que j’ai écrit mais ce serait bien d’avoir une confirmation extérieure. MF . Here, |fi denote the auxiliary mutually orthogonal ancillary quantum states that keep the record of the mea- To summarize, in terms of communication theory, sured value f. Indeed, in the following, we shall include when Iqc[ρSF ] = S[S], the system sends some classical these degrees of freedom within F . Therefore, the total information s into its environment and the fragment F post-measurement density operator is: is such that there exist a way to decode it without any ambiguity by choosing the appropriate generalized mea- (post MF ) X surement on F ’s side. ρS,F = p(s) p(f|s) ρS(s) ⊗ ρ(F |f, s) , (55) (f,s) in which the ρ(F |f, s) and ρ(F |f 0, s0) are mutually or- 3. Perfect redundancy 0 thogonal as soon as f 6= f . The only way S[S|MF ] could be zero is by assuming that the set of possible values of Quantum darwinism involves another key notion which f is partitioned into disjoint sets Js, each of them asso- is redundancy. In the perfectly Darwinian situation, ciated with a value of s therfore defining a function that we expect that there are many fragrments of similar associates to f a given s so that indeed, conditionning to size that encode all the information about S, that is a value of MF leads to only one possible ρS(s) = |si hs|. Iqc[ρSFj |MFj ] = 0. We know from the Theorem C.1 that We can then coarse grain the measurement performed for each fragment Fj, there is an observable MF that en- ˜ j on F so that it associates a unique value fs to each s. ables a perfect reconstruction of the information about This coarse-graining defines a generalized measurement any observable that diagonalizes ρS. on F which we denote by M˜ F . After this coarse-graining, Equation (57) implies that measuring the observable ˜ P the correspondence between s and the fs is a bijection. s s |si hs| acting on S or measuring the coarse-grained 18

˜ observable MF on F has the same eﬀect on ρSF . Since [Alex]: L’état qui suit n’est pas classique-quantique. these operators are acting only on S or F , this means that De plus, vu qu’on parle de Dqrwinism/décohérence, they indeed have the same eﬀect on ρSE where E, which il vaut mieux dire qu’on est dans une situation où contains F , is the full environment of S. As pointed out l”’etat du système est complètement décohéré plutot in [27], measuring the observable diagonalizing ρS can be que de dire qu’on fait une mesure sur S. Comme écrit emulated by a measurement in F . dans la phrase d’avant. [Pascal]: Boucler cette discussion sauf ce qui est dans This can be done as follows: let us consider a mea- le papier [27] me semble inutilement compliqué. Il me surement MS performed on S so that, after the mea- semble qu’on voit directement que dans le cas où on surement, the joint (S, F ) state is classical/quantum (see a plusieurs fragments qui permettent de reconstruire Section IIIA3) of the form complètement, on reconstruit la même chose. Mais quelque chose doit m’échapper dans le la Section V de (post) X ρSF = psρS(s) ⊗ ρ(F |s) (60) [27]... s

[Pascal]: At ce stade, il faut probablement faire un where the ρS(s) are mutually orthogonal states of S and distingo entre la situation perfaitement Darwinienne ps denote the probability for obtaining the result s. Since et la situation Darwinienne mais pas parfaitement. S is totally decohered by its environment, we also assume C’est ce qui permet d’élargir la discussion à la nature that: des corrélations entre les fragments et le système... X et aussi à la discussion du début de la courbe ou de ρS = p(s)ρS(s) . (61) la situation non Darwinienne. C’est important pour s bien motiver que l’on va se mettre à regarder la discorde. Peut être qu’il faudrait intercaler une sous- Note that this does not imply that the ρ(F |s) are mutu- section entre une situation idéalement Darwinienne ally orthogonal but only that the ρ(E|s) are: information type GHZ et une où c’est imparfait. on s can be spread over all E. The non-orthogonality of the ρ(F |s) is precisely what prevents an observer access- ing only F to recover all the information about the values s. Because of Eq. (61), the mutual information between 4. Discord in quantum Darwinism S and F after this measurement process can be obtained as 4: " # Given a quantum channel from Alice to Bob, we know X X how to quantify the maximum amount of classical in- I[(S, F )post] = S psρ(F |s) − psS[ρ(F |s)] (62) formation that can be transmitted though it. Alice en- s s codes the classical messages a emitted with probabili- which is precisely the Holevo quantity for F prepared in ties pa within not necessarily orthogonal quantum states the mixture of the conditional states ρ(F |s) with proba- ρ(A|a), thereby leading to the statistical mixture bilities ps (see Eq. (59)). We will denote it by χ(MS,F ). The diﬀerence between the initial mutual information X ρ = p ρ(A|a) . (58) I[S, F ] and the post-measurement information χ(MS,F ) A a thus coincides with the quantity appearing in Eq. (41) a but without optimizing on the choice of the measurement on S. This is the quantum discord [28] relative to The amount of classical information that can be retrieved the measurement MS performed on S and it obeys [36]: by Bob using classical measurements is bounded from above by the Holevo quantity [19]: I[S, F ] = χ(MS,F ) + DMS (ρSF ) . (63)

" # The left hand side of (63) doesn’t depend on the measure- X X χ[(ρ(A|a), pa)] = S paρ(A|a) − paS[ρ(A|a)] . ment MS chosen by the observer while the decomposition a a does. The information characterized by χ(MS,F ) is a

(59) locally classical accessible information while DMS (ρSF ) In quantum Darwinism, the quantity of interest is the corresponds to the global quantum correlations. This total correlation, measured by mutual information, be- can be viewed as a kind of “conservation law” or alter- tween the system S and a fragment of the environment natively as the expression of the objectivity of the total F , assuming that S is totally decohered by its environment. It is then natural to decompose the mutual information into a “classical component” given by the Holevo quantity and a “quantum component”. 4 See discussion of Sec. 7.5.1 of our book. 19 correlation I[S, F ] between S and F with respect to the choice of the measurement MS performed on the system. We can have a reﬁned understanding of the general form of the mutual information as we vary the size of the fragment when we make the assumption that the global state ρSE is pure and that the measurement made on the system is projective (a rank one POVM). In this case, we have D(MS,E \ F ) = HS − χ(MS,F ). Indeed:

D(MS,E \ F ) = I(S, E \ F ) − χ(MS,E \ F ) X = S(S) − S(SE \ F ) + p(s)S(ρ(E \ F |s) s X = S(S) − S(F ) + p(s)S(ρ(F |s) s FIG. 8: Anti-symmetry between the quantum discord and the = S(S) − χ(M ,F ) , Holevo quantity around (1/2,S(S)/2) for a global pure state S and a projective measurement on S. where we used, for the second and last lines, the definitions of th mutual information and the Holevo quantity branching state or is said to have the spectrum broadcast and for the third that the global state is pure so that structure): S(SE \ F = S(F ) as well as the fact that for a rank one POVM, the conditional state is also pure. This equal- ! X √ O ity can also be written in the more symmetrical form as |ΨSEi = ps |si ⊗ |ψ(k|s)i , (66) follows: s k where s indicates the outcomes of a pointer measurment χ(MS,F ) − HS/2 = HS/2 − D(MS,E \ F ) . (64) MS and k denotes the various environmental channels which geometrically means that we have a symmetry within the environment E. Each fragment F corresponds to a certain set of values of k. Because of the factor- around HS/2. It shows that the larger the classical information is stored in the fragment F , the less quantum ized and pure form of ρ(E|s), the state of any frag- information is stored in the larger fragment E \ F . ment of the environment relative to the value s of the pointer observable is pure. Consequently, when con- For an arbitrary state ρSE, we have instead the inequality: sidering the Holevo quantity for the statistical mixture of the ρ(F |s) with probability ps, the Holevo quantity is equal to the entropy of the reduced density operator D(MS,E \ F ) ≤ HS − χ(MS,F ) . (65) P ρF = s psρ(F |s) because we subtract zero since all the Indeed, by considering a purification of the state from an ρ(F |s) are pure. By contrast, when considering another additional ancilla R, the previous computation gives the observable X of the system and decomposing the global entangled state |Ψ i relatively to another observable equality D(MS,E \ F ) = S(S) − χ(MS,FR). Now, from SE the data processing inequality, we know that the Holevo for the system, the state ρ(F |x) is no longer pure. Con- sequently χ[(ρ(F |x), p )] ≤ χ[(ρ(F |s), p )]. bound contracts when we trace over R giving χ(MS,F ) ≤ x s From the conservation law (63), we therefore expect χ(MS,FR) from which we obtain the inequality. the discord to be minimal when computed relatively to [Alex]: J’en suis lq. Faire une discussion de la sym- the pointer observable. More precisely, Eq. (63) tells us métrie autour de la moitié: dire que c’est valable que that the discord has the form en moyenne. Donner un contre exemple. Omar dis- ait que demander la symmétrie de l’état revenait au X DMX (ρSF ) = DMS (ρSF ) + pxS[ρ(F |x)] . (67) même grosso modo que la moyenne. Ecrire un truc la x dessus. The second term of the r.h.s. is always positive and If χ(M ,F ) ≥ (1 − δ)H then D(M ,E \ F ) ≤ δH . S S S S vanishes for the pointer basis MS showing that, for Thus again the more classical information we have access such a choice, the discord is minimum. In the Dar- in F, the less quantum information we have in E \ F .A winian case, the E \ F part of the environment is large world where classical information is present is a world enough to completely decohere (S, F ) thereby leading to where quantum information is only accessible to global a classical-quantum state of the form (60) for ρFS. On observers. the Darwinian plateau, I[S, F ] = S[S], thereby leading The first important result is that the Holevo quantity to I[S, F ] − S[ρF ] = S[S] − S[SF ]. is largest when computed using pointer states and the [Pascal]: La fin de l’argumentaire n’est pas très associated observable than when using another observ- claire... Je reprends le papier pour y voir plus clair. able. To show this, consider the state (sometimes called 20

For all practical purposes, in the darwinian case, the generic and inherent to the structure of quantum theory. discord vanishes and therefore, we have We will review this work in SectionIVB. [Alex]: Justement le “for all practical purposes” n’est The second aspect of objectivity concerns the observers pas suffisant. Il faut l’imposer pour avoir une situta- and states that, given this pointer observable, all the ob- tion objective. Et rajouter l’accessible. servers agree on one specific outcome (one specific measurement result). This is the real consensus on memory [Alex]: Donner le contre exemple Horo qui montre de records among many observers. Whether or not and how I=S insuffisant. Citer d’autres papiers. Conclusion: the observers can achieve this depends on the observer’s il existe des etats avec plateau où le plateau est ma- capabilities as explained in Section IIC3. joritairement composé de discord ie information pas First, we wan assume that a decomposition into ob- accessible classiquement. Dou la necessite d”aller plus servers is given and ask whether or not, and how the loin. observers can reconstruct the values of the observable. Elaborating on the discussion of SectionIIB4, we will introduce a hierarchy of objectivity notions. This amounts I[S, F ] = χ(MS,F ) = S[F ] . (68) to understanding the nature of correlations when reconstruction is possible. It will be discussed in SectionIVC. In the end, the information relative to a pointer state obtained from a small fragment F quickly saturates the Holevo bound, meaning that only classical information B. Objectivity of observables can be obtained from this fragment, even when we en- large it. Of course, things change at the end of the Dar- Let us consider a set of subsystems and chose one as winian plateau: a global measurement encompassing al- our system, calling it S. The rest of them forming its en- most the whole environment can provide more informa- vironment are denoted as F1,...,Fn. We suppose that S tion coming from the hidden quantum correlations char- is finite dimensional. We can model the dynamics of S as acterized by the discord. But in ther Darwinian case, one a completely positive trace preserving (CPTP) map de- has to go to the end of the plateau – id est to very large noted Λ. The fundamental result of Ref. [8] subsequently scales – to see quantum correlations (I[S, F ] exceeding improved in Ref. [30] is the following: S[S]). Theorem B.1. Let Λ: D(S) → D(F1 ⊗ · · · ⊗ Fn) be a quantum channel and Λj the reduced channel to the IV. MANY-OBSERVER STRUCTURES subsystem Fj. Given an integer q ≥ 1, there exists a POVM {Mx}x and a set Q ⊆ {1, . . . , n} of size q such that for all j ∈ {1, . . . , n} − Q: A. Statement of the problems s 2 ln d In this section, we consider what we call the many- 3 S kΛj − Ejk ≤ dS , (69) observers situation: instead of focusing on one specific q fragment of the environment, we consider all of them and try to understand the nature of correlations between the with Ej defined by fragments along a given decomposition in parts. X As mentioned in Section IIB4, the question of the Ej(X) = tr(MxX)σj,x , (70) emergence of objectivity can be addressed at two dif- x ferent levels: for states σj,x ∈ D(Fj) and dS the dimension of the space S.5 • Objectivity of observables: different observers probing independent parts of the environment have ac- The operation Ej is called a measure-and-prepare map cess to only one observable of the system. since it can be obtained by Fj first by measuring Mx and then preparing any state σj,x conditioned on the result • Objectivity of outcomes: different observers prob- x. The system F can at most recover the information ing independent parts of the environment have full j about the measurement {Mx}. access to the above observable and agree on the The remarkable point of the theorem is that the mea- outcome. surement Mx does not depend on the observers Fj. It can be thought as the pointer basis of the interaction Λ. The objectivity of observables states that the environment selects one specific observable of the state (one specific measurement) which is accessible to almost all observers. In other words, this is the pointer observable of 5 The norm k · k is the widely used “diamond” norm between S induced by its environment. It can be shown [8] that quantum channels. As its precise definition is not important for the emergence of an objective observable is indeed quite our discussion, we refer to [? ] 21

Thus almost all the observers have a dynamics very close T . However, this consensus strongly depends on the al- to a measure-and-prepare dynamics with the measure- lowed operations within the community. If we consider ment Mx independent of them. individual systems Fi for i ∈ T , they might not contain Another way of thinking about it is that the dynamics information about the outcome x and as a result the dif- is close to a situation where the system S is first mea- ferent observers Fi for i ∈ T do not necessarily agree on sured by Mx and only then the classical information is the outcome x. broadcast and locally degraded into the state σj,x for all the different observers Fj. Notice that the theorem does not say that the quantum states σj,x broadcasted to ob- C. Hierarchy of objectivity server i are perfectly distinguishable (tr(σj,xσj,x0 ) = 0 if x 6= x0). Therefore, in general, the observer j, who only In this section, we focus on the objectivity of outcomes. receives the statistical mixture Ej(X), has only a partial We assume that our observers F1 ...Fn can in principle information about the actual value of x. collectively reach a consensus on the value of X (which As a remark, keep in mind that this result states the we can think of as a measurement outcome). Mathe- existence of a least one pointer observable but does not matically, this can be written as Pguess(X|F1 ...Fn) = 1, say anything about its uniqueness. Depending on the dy- i.e., the optimal probability of correctly guessing X by namics, it is still allowed to have different possible pointer performing a measurement on F1 ...Fn is 1. We also re- observables. call the definition Hmin(X|F ) = − log Pguess(X|F ). Note An important point here is that even if all the observers that the procedure for determining X will in general Fj have access to the same measurement Mx, the theo- involve all the systems F1 ...Fn together. We are in- rem does not say anything about a consensus around the terested in determining under what conditions, the ob- actual outcome of the measurement. Indeed, we see from servers can locally determine this objective value X? Eq. (69) that for each j, the evolution is only close to The states we consider in this section have the follow- a mixture of the states σj,x. The possibility still exists ing form: that Fj can obtain the result xj while Fi with i 6= j can X (x) obtain xi 6= xj. The objectivity of outcomes is not yet ρ = p(x)|xihx| ⊗ ρ . (73) XF1...Fn F1...Fn quantified by this result. x Still we can go in this direction and obtain a slightly more refined theorem:

Theorem B.2. Let Λ: D(S) → D(F1 ⊗ · · · ⊗ Fn) be 1. Independent objectivity a quantum channel. Given integers q, t ≥ 1, there exists a POVM {Mx} and a subset Q ⊆ {1, . . . , n} of size q Here, we consider the setting where the observers such that for any subset T ⊆ {1, . . . , n} of size t that is are completely independent. We determine the condi- disjoint from Q, we have tions under which each observer can completely deter- s mine X. In mathematical terms, this means that for all 2(ln d )t i ∈ {1, . . . , n}, P (X|F ) = 1. kΛ − E k ≤ d3 S , (71) guess i T T S q Theorem C.1. Let ρXF1...Fn be a state as in (73). Then the following conditions are equivalent: with: X 1. Pguess(X|Fi) = 1 for all i ∈ {1, . . . , n} ET (X) = tr(MxX)σT,x , (72) x 2. Iacc(X,Fi) = S(X) for all i ∈ {1, . . . , n} for states σT,x ∈ D(⊗j∈T Fj) and dS the dimension of 3. For all i ∈ {1, . . . , n}, there exists an isometry Wi the space S. † ¯ (i.e., Wi Wi = I) that maps the space Fi to Xi⊗Ni, where X¯ is isomorphic to X, such that Note the diﬀerence of this result compared to the previ- i ous one. Here we can state that the evolution of a collec- n n O O † tion of systems of size t is close to a measure-and-prepare ( Wi)ρXF1...Fn ( Wi ) dynamics with measurement Mx still independent of the i=1 i=1 subsystems. However, now, we know that after a mea- n ! X O (x) = p(x)|xihx| ⊗ |xihx| ¯ ⊗ ρ . surement, the joint state of the ensemble of systems T is X Xi N1...Nn one of the σT,x with one outcome x: if the states σT,x x i=1 are distinguishable from each other for diﬀerent values of x, then considering the community T as a whole there Proof. We will prove that (3) ⇒ (2) ⇒ (1) ⇒ (3). is only one value of x that is consistent with the global For (3) ⇒ (2), the measurement on Fi can be taken to state. This means that in principle an internal consensus apply Wi followed by a measurement of the register X¯i can be reached between the members of the community in the basis {|xi}. 22

TABLE I: Hierarchy of objectivity for a classical data. The examples illustrate that the objectivity levels we deﬁne are all diﬀerent. The letter denotes the existence of a family of states (in general with growing dimension) such that the corresponding quantity can be made arbitrarily close to 0. Assumptions State Interpretation 0 I (X,F ...F ) = S(X) ∀x 6= x0, ρ(x) ⊥ ρ(x ) X collectively recovered from F ...F acc 1 n F1...Fn F1...Fn 1 n 0 (x) (x0) ∀i, Iacc(X,Fi) = S(X) ∀i, ∀x 6= x , ρ ⊥ ρ X individually recovered from Fi Fi Fi Iacc,LO(X,F1 ··· Fn) = S(X) X recovered by collecting local meas. outcomes Iacc,LOCC(X,F1 ··· Fn) = S(X) X recovered by collecting local adaptive meas. outcomes

Iacc(F1 ...Fn) ≥ Iacc,LOCC(X,F1 ··· Fn) ≥ Iacc,LO(X,F1 ··· Fn) ≥ Iacc(Fi) Global meas. Adaptive meas. Sharing meas. Indep. estimation Example states S[X] S[X] S[X] S[X] Fi have a copy of X S[X] S[X] S[X] 0 (secret sharing) S[X] S[X] (information locking) S[X] (data hiding)

For (2) ⇒ (1), we have 2. Shared objectivity

Iacc(X,Fi) It is simple to construct examples of states where inde- = S(X) − min S(X|Z) pendent objectivity does not hold, i.e., information about measurement {Mz }z with outcome Z X is stored in the correlations between the different fragments. Let X ∈ {0, 1} be uniformly distributed, and let ≤ S(X) − Hmin(X|Fi) . F1,...,Fn be classical systems with F1,...,Fn−1 be in- Ln−1 dependent and uniform bits, while Fn = X ⊕ i=1 Fi. So Iacc(X,Fi) = S(X) implies that Hmin(X|Fi) = 0 Then for n ≥ 2, we have I(X,Fi) = 0 for any i ∈ which is the same as Pguess(X|Fi) = 1. {1, . . . , n} whereas I(X,F1 ...Fn) = S(X) = 1. In fact, For (1) ⇒ (3), Pguess(X|Fi) = 1 implies that there ex- this example shows an even more extreme setting where ists a measurement {M i } on F such that tr(M i ρ(x)) = x x i x Fi information is stored in the correlations: any group of at i (x) 0 most n−1 fragments does not get any information about 1 for all x. This implies that M 0 ρ = 0 for x 6= x x Fi i (x) 0 X. This is a simple example of a well-studied topic in and thus M 0 ρ = 0 for x 6= x . Now we define the x F1...Fn information theory and cryptography called secret shar- P p i isometry Wi = x |xi ⊗ Mx. Thus, for any x, we have ing [4]. Considering quantum systems F1,...,Fn, it is natu- n n ral to consider limited classical communication abilities O (x) O ( W )ρ ( W †) and ask whether the consensus on the value X may be i F1...Fn i i=1 i=1 recovered by the fragments. The first setting that we con- n ! n ! sider is when each fragment i can perform a measurement O p (x) O p = |xi ⊗ M i ρ hx| ⊗ M i {M i} obtaining an outcome Z . When Z ...Z are suf- x F1...Fn x z i 1 n i=1 i=1 ficient to recover X, we say that we have local operations n ! n ! n ! (LO) objectivity. Observe that the secret sharing exam- O O p (x) O p = |xihx| ⊗ M i ρ M i . ple that we just described does satisfy LO objectivity, x F1...Fn x i=1 i=1 i=1 and in fact for classical states LO objectivity is equivalent to collective objectivity. But for general quantum This proves the desired result. states, this is not the case. We now state a result to characterize the states satisfying LO objectivity in the following theorem. The structure of this state has in fine a very natu- Theorem C.2 (Characterizations of LO-objectivity). ral structure and transparently shows its objective na- Let ρXF1...Fn be a state as in (73). Then the following ture: each observer has access to the complete informa- conditions are equivalent: tion about x and can do it independently of what the 1. For all i ∈ {1, . . . , n}, there exist mea- others are doing. Independent objectivity does not in i itself forbid the existence of a strongly correlated noise, surements {Mz} with outcome Zi such that potentially relative to x but requires that each observer Pguess(X|Z1 ...Zn) = 1 can filter x out of it. Note that nothing can a priori be 2. Iacc,LO(X,F1 ...Fn) = S(X) said on the hardness of this filtering process, it is just possible in principle. 3. For all i ∈ {1, . . . , n}, there exists an isometry Wi 23

† ¯ (i.e., Wi Wi = I) that maps the space Fi to Zi ⊗Ni F2 are qubit systems. such that 1 X b b n n ρXF F = |xihx|X ⊗ |bihb|F ⊗ H |xihx|F H , O O 1 2 4 1 2 † x∈{0,1} ( Wi)ρXF1...Fn ( Wi ) b∈{0,1} i=1 i=1 n ! (74) X O 0 (z,z0) = p(x)|xihx|X ⊗ |ziihz | ¯ ⊗ ρ , i Zi N1...Nn where H is the Hadamard transform. Note that from the 0 0 z1...zn,z ...z i=1 1 n point of view of F2, the state of system X is encoded in f(z1,...,zn)=x 0 0 f(z1,...,zn)=x a basis b chosen at random that F1 has access to. Be- cause we are encoding with two complementary bases, for some function f. without knowing the basis, the observer F2 cannot fully recover X. This is sometimes called conjugate coding [34] Proof. We will prove that (3) ⇒ (2) ⇒ (1) ⇒ (3). and can be quantified by uncertainty relations. Conju- For (3) ⇒ (2), the measurement on Fi can be taken gate coding is widely used in quantum information, it is to apply Wi followed by a measurement of the register at the basis of quantum key distribution protocols [5]. ¯ Zi in the basis {|zi}. Given the outcomes z1, . . . , zn, the Encoding with different incompatible basis is also called only compatible value of X is f(z1, . . . , zn) and hence information locking [12, 16]. I(X,Z1 ...Zn) = S(X). Observe that for the state in (74), if F1 and F2 are i For (2) ⇒ (1), there exists measurements {Mz}z with allowed to communicate before performing the measure- outcomes Zi such that ment, then X can be recovered: F1 could send the value of b to F2 who would then perform a measurement in the Iacc,LO(X,F1 ...Fn) basis b to extract X. This motivates our second shared = I(X,Z1 ...Zn) objectivity setting: the observers are allowed to communicate classically, and then perform measurements that = S(X) − S(X|Z1 ...Zn) depend on this communication, obtaining outcomes Zi ≤ S(X) − Hmin(X|Z1 ...Zn) . and collectively Z1 ...Zn are sufficient to recover X. We say in this case that we have local operations and classi- So I (X,F ...F ) = S(X) implies that acc,LO 1 n cal communication (LOCC) objectivity. As the example H (X|Z ...Z ) = 0 which implies that the out- min 1 n in (74) shows, LO objectivity can be a strictly stronger comes of measurements {M i} can be used to guess X z z requirement than LOCC objectivity. Being adaptive, perfectly. i LOCC operations are much more difficult to character- For (1) ⇒ (3), given the POVMs {Mz}z, we define the P p i ize, so we do not have for the moment a result about isometry Wi = |zii ¯ ⊗ M . As Z1,...,Zn can zi Zi zi the structure of LOCC objective states, similar to Theo- be used to guess X perfectly, let us call f the guess- rem C.2. ing function that maps z1, . . . , zn to the correspond- Note that LOCC objective states do not correspond to ing guess x. With this notation, we have for any x, all states with collective objectivity. In fact, considering P n i (x) z ...z :f(z ...z )=x tr (⊗i=1Mz )ρF ...F = 1. This im- 1 n 1 n i 1 n 1 n i (x) X (x) plies that (⊗ M )ρ = 0 whenever f(z1 . . . zn) 6= ρXF1F2 = |xihx|X ⊗ ρF F , (75) i=1 zi F1...Fn 2 1 2 x. As a result, x∈{0,1}

n n where ρ(0) = Πsym and ρ(1) = Πantisym , where O (x) O † F1F2 tr(Πsym) F1F2 tr(Πantisym) ( Wi)ρF ...F ( Wi ) 1 n Πsym is the projector onto the symmetric subspace of i=1 i=1 d d n ! n ! C ⊗ C (i.e., the span of the vectors of the form |ai ⊗ q q O X i (x) O X i |bi + |bi ⊗ |ai) and Πantisym is the projector onto the = |zii ⊗ Mz ρF ...F hzi| ⊗ Mz i 1 n i antisymmetric subspace of Cd ⊗ Cd (i.e., the span of the i=1 zi i=1 zi n vectors of the form |ai ⊗ |bi − |bi ⊗ |ai for a 6= b). This is X O 0 p (x) p the typical example of a data hiding state that are well- = |ziihz | ⊗ Mzρ Mz0 , i F1...Fn studied in quantum information theory, see [13, 21] and z ...z ,z0 ...z0 i=1 1 n 1 n the references therein for more details. f(z1,...,zn)=x 0 0 f(z1,...,zn)=x where we introduced the notation z = z1 . . . zn and Mz = V. CONCLUSION 0 √ √ Nn i (z,z ) (x) M . By deﬁning ρ = M ρ M 0 we i=1 zi N1...Nn z F1...Fn z get the desired result. A. Summary

Not all states that satisfy collective objectivity satisfy [Pascal]: Faire le résumé des principaux points. LO-objectivity. In fact, consider the state where F1 and 24

B. Perspectives and open questions full generality, it is described by a set of orthorgonal projectors Πα summing to 1 in HE. This leads to a POVM through [Pascal]: Mettre les directions futures † Eα = hE0| U ΠαU(|ψi ⊗ |E0i) . (A4)

If the projector Πα has rank strictly greater than one, we Appendix A: Quantum measurements can naturally write down a decomposition for the post- measurement relative state of the form Eq. (A2) by using 1. General measurements and POVMs |mα,ki an orthonormal basis of the projection space of Πα and defining In quantum theory, the most general form of measurement is described by positive valued operator measure- Mk,α |ψi = hmα,k| U(|ψi ⊗ |E0i) (A5) ments (POVMs). These involve a sequence of positive A straigtforward calculation shows that semi-definite operators Eα (hψ| |Eα| |ψi ≥ O for any state vector |ψi) summing to 1. The probability for obtaining † † Mα,kMα,k |ψi = hE0| U Πα,kU(|ψi ⊗ |E0i) (A6) the result α when the system is prepared in the statistical ensemble described by the density operator ρ is P † thereby implying that k Mα,kMα,k = Eα. Defining the post-measurmement state by Eq. (A2) amounts to p(α|ρ) = Tr(ρEα) . (A1) using a much finer generalized measurement defined by all the M operators and then coarse-graining the (α, k) Because of the conditions imposed on the E operators, α,k α through the forgetting of k. these probabilities are positive and sum to unity. The In the case where the initial state is pure, this lead key point is that the POVM only gives the probability to a relative state which is a statistical mixture whereas of obtaining a certain result but does not give the state √ considering M = U E can be viewed as the most of the system conditionned to this result: it does not α α α efficient measurement: no information is forgotten in the describe the quantum backaction. We need to specify it. result and therefore, the post-measurement relative state In the most general case, an initial (pre-measurement) is still pure. The maximally efficient general quantum state described a density operator ρ is sent onto a relative measurements correspond to the ones introduced in Sec- state described by a conditional density operator tion IIA2. [Pascal]: Compléter éventuellement... 1 X † ρ(S|α) = Mα,kρMα,k (A2) pα k in which the Mα,k operators obey 2. Ideal measurements

X † Ideal measurmeents are characterized by their repro- Mα,kMα,k = Eα . (A3) k ductibility: performing twice the same measurement without letting the system evolve between them will lead and pα = p(α|ρ) = Tr(ρEα) denotes the probability for to the same result. In classical physics, because there is obtaining the result α. no a essentiel back-action on the system, ideal measure- These fully general measurement can be viewed as ments are noiseless measurement: the result is a func- obtained from the so-called generalized measurement, tion of the microscopic state of the system. In quan- which are described by Kraus operators Mα such that tum physics, reproductible measurements are the projec- P † α MαMα = 1, by a process of coarse graining. Of tive ones, characterized by a set of orthogonal projectors course, given a POVM,√ one can always build Kraus op- which sum up 1. erators by taking Uα Eα in which Uα is a unitary oper- Here, we will recall the known results on the informa- ator that corresponds to a feedback action on the system tion gain through an ideal measurement process. As we determined by the measurement result. It is straightfor- shall see, the information gain is always positive but the † ward to check that MαMα = Eα. But as explained in balance equation for information satisfied in the classical the previous paragraph, this is not the only possibility. domain is not valid in the quantum realm because of the To understand it, let us come back to the construction essential backaction of the measurement on the system of Kraus operators in a purified picture: they are associ- [3]. ated with a projective measurmeent in the environment. Let us first consider the case of an ideal measurement More precisely, given |ψi ∈ HS, we first consider that the in the classical domain. A system is prepared in a statis- environment is prepared in a state |E0i and make the two tical ensemble characterized by a probability distribution systems through a unitary evolution operator acting on a 7→ pa of microstates. We then measure a quantity X, HS ⊗ HE thereby obtaining U(|ψi ⊗ |E0i). We then per- thereby obtaining the values x which are of course ran- form a projective measurement on the environment. In domly distributed. The pre-measurement entropy is the 25

P Shannon entropy of this ensemble S[A] = − a pa ln(pa). First of all, entropy conservation After measurement, the average entropy is h (unread)i (post) (results) X S ρ − S = S (A15) S[A|X] = pxS[A|x] (A7) x which is almost identical to Eq. (A9) except that the initial entropy has been replaced by the post-measurement where S[A|x] = S[pA(·|x)] is the Shannon entropy for the conditional probability distribution p(a|x) for As mi- entropy without reading the results. Consequently, the crostates conditionned to the results of the measurement. entropy of the results is not anymore the information Consequently, the information gain is gained by through the measurement process. The second result is that performing a projective ∆I = S[A] − S[A|X] (A8) meaurement will necessarily kills some coherences and therefore increases the entropy which is nothing but the mutual information I[A, X]. h i Since an ideal (noiseless) measurement performs a coarse S [ρ] ≥ S ρ(unread) (A16) graining of microstates according to the values of the measured quantity, we have whihc shows that the information gain S [ρ] − S(post) is X S[A] − S[A|X] = − p(x) ln(p(x)) (A9) smaller than the entropy of the results. Compared to x the classical case, this comes from the erasure of all the information associated with the observables incomatibles which expresses that the information gain is exactly equal with the one that is measured [3]. to the entropy of the results. This can be viewed as a The third result is that this price is never high enough conservation law for information: the results carry an to make the information gain negative. Finally, we obtain entropy which corresponds to the information gained on the general bounds the system. Let us now discuss quantum ideal measurements, 0 ≤ S[ρ] − S(post) ≤ S(results) (A17) which are projective measurements deﬁned by a set of orthogonal projection operators Πα summing to 1. Start- Note that, for projective measurements, the information ing from an initial density operator ρ, the projective mea- gain through the measurement can be expressed as surement generates the “unread” post-measurement density operator ∆I = χ [(p(α|ρ), ρ(S|α))α] − (∆S)dec (A18)

(unread) X in which the ﬁrst term is the Holevo quantity for the ρ = Παρ Πα (A10) α (p(α|ρ), ρ(S|α))α and (∆S)dec is the entropy increased induced by the decoherence associated with the projec- which corresponds to summing, over all the possible re- tive measurement process. This is the price to pay for not sults, the relative density operator conditionned to a spe- reaching the maximal possible information gain, which is ciﬁc measurement result here the Shannon entropy of the results. Π ρ Π ρ(S|α) = α α (A11) p(α|ρ) Appendix B: Operational meaning of the mutual weighted by the probability p(α|ρ) of obtaining α: information

p(α|ρ) = Tr(Παρ) (A12) The operational meaning of the Shannon mutual information appears in the classical Slepian-Wolf proto- The post-measurement entropy is then the average of the col [33]. In this protocol, we are considering two cor- entropies of these relative density operator: related sources of information and we want to ﬁnd the mainimal information rate that they need to use to (post) X S = p(α|ρ) S[ρ(S|α)] (A13) transmit their joint content. Ignoring their correlations α would require RA ≥ S[A] and RB ≥ S[B] bits of infor- which plays the analogue of Eq. (A7). Of course, the mation whereas one can achieve full transmission with Shannon entropy of the measurement results is RA ≥ S[A|B], RB ≥ S[B|A] and RA + RB ≥ S[A, B] by taking correlations into account. The gain is therefore X S(results) = − p(α|ρ) ln (p(α|ρ)) . (A14) S[A] + S[B] − S[A, B] = I[A, B]. In the quantum context, the analogous interpretation α is provided with the quantum Slepian-Wolf or state merg- The entropy conservation law (A9) is then replaced by ing protocol[20]: we want to compress a bipartite quan- three distinct results which are proved and discussed in tum source (A, B). Although the general solution to Ref. [3]. this problem is not known, it is known when we allow 26 free classical communication between the parties. The also by the von Neumann entropy, for which it is a far end result is that the full state can be compressed using less trivial result [22] although simpler proofs have been Svn[A, B] qubits, which is the Schumacher compression given recently [25, 31]. It states that rate [32] even in this situation where the source is split- ted in two. This can be done by ﬁrst transfering the S[A, B, C] + S[B] ≤ S[A, B] + S[B,C] (C6) information shared between A and B to Bob who already holds the B source. This requires the consumption which is equivalent to a statement on conditional entropies of Svn[A|B] qubits (when the conditional entropy is positive) and I [A, B] classical bits. Then, once Bob has the vn S[A|B,C] ≤ S[A|B] . (C7) equivalent of the full composite source in its hands, it can use S[A, B] to transfer the full quantum information to It thus expresses that forgetting a conditionning can only the third partner. increase the entropy. This bound translates into a bound on mutual information

Appendix C: Classical and quantum bounds on I[A;(B,C)] ≥ I[A; B] (C8) entropies stating that correlations between a smaller set of sub- 1. Sub-additivity and lower bounds systems can only be smaller than correlations between bigger sets. The conditional information S[A|B] defined as The strong-subadditivity can finally be recasted as a S[A, B] − S[B] satisfies the following inequalities. First statement on conditional mutual information. We first of all, both the Shanon and the von Neumann entropies need to define the conditional mutual information by gen- are sub-addidive: eralizing I[A; B] = S[A] − S[A|B] = S[B] − S[B|A]: S[A, B] ≤ S[A] + S[B] . (C1) I[A; B|C] = S[A|C] − S[A|B,C] (C9a) with equality if and only off the two sources A and B are = S[A, C] + S[B,C] − S[A, B, C] − S[C] uncorrelated. (C9b) The Shannon and von Neumann entropies of a com- = S[A|C] − S[A|B,C] = I[B,A|C] (C9c) posed system differ by the lower bounds they obey. The Shannon entropy of a composed system is always larger Then the strong-subadditivity expresses nothing but than the Shannon entropy of each of the subsystems the posititivity of the conditional mutual information: I[A, B|C] ≥ 0. S[A, B] ≥ max(S[A],S[B]) . (C2) The strong-subaddivity for the von Neumann entropy whereas the von Neumann entropy obeys the Araki-Lieb can be shown to be equivalent to other important prop- inequality erties. First it is equivalent to the decrease of relative entropy or Kullback-Leibler divergence under a completely Svn[A, B] ≥ |Svn[A] − Svn[B]| . (C3) positive trace preserving (CPTP) operation. This property, called Monotonicity of quantum relative entropy which enables the full system’s entropy to be lower than states that for any CPTP N and any pair of density the one of each component. operators ρ and ρ , we have This also leads to the following bounds on the mutual 1 2 information I[A; B] = S[A] + S[B] − S[A, B]. Both the D[N (ρ1)kN (ρ2)] ≤ D[ρ1kρ2] (C10) classical (Shannon) and quantum (von Neumann) mutual informations are positive as a consequence of Eq. (C1) This is also equivalent to the monotonicity of quantum but their upper bounds differ. For the classical case, the relative entropy under partial trace. Considering two mutual information cannot exceed the Shanon entropy of density operators ρ1 and ρ2 for a composed system shared any subcomponent by Alice and Bob, we have: I[A; B] ≤ min(S[A],S[B]) (C4) D[TrB(ρ1))kTrB(ρ2)] ≤ D[ρ1kρ2] (C11) whereas, it can reach twice that limit in the quantum Finally, the strong-subadditivity is equivalent to the joint case convexity of the quantum relative entropy. Ivn[A; B] ≤ 2 min(Svn[A],Svn[B]) . (C5)

3. Sufficient criterion for entanglement 2. Strong subadditivity By definition, bipartite entangled states are states Strong sub-additivity (SSA) is a property for tri- which are not factorizable, the latter being states whose partite systems. It is obeyed by the Shanon entropy but density operator can be written as (??). Let us show 27 that for factorized states, we have S[A|B] ≥ 0 and This version of strong additivity states that by disregard- S[B|A] ≥ 0. This shows that whenever S[A|B] or S[B|A] ing the system C, Alice and Bob have to share more in- is negative, the state is entangled. formation to apply the state merging protocol. The goal This statement follows from the concavity of ρAB 7→ of the argument is to relate this increase of the cost of S[ρAB] − S[TrA(ρAB)] since, starting from the separable the protocol to the discord. state Extend the Hilbert space so that measurement, that X we will denote M , can be modeled by a coupling to C. ρ = p ρ (i) ⊗ ρ (i) (C12) i AB i A B Suppose C to be initially in a pure state |0i. In the i purified setting, Mi is represented by a unitary opera- the concavity properties implies that tor U acting on C and B. For the moment, we initially X have have S(A, B) = S(A, BC). After the evolution, we S[A|B] ≥ p S[ρ (i) ⊗ ρ (i)|ρ (i)] i A B B have I(A, BC) = I(A0,B0C0). Discarding C0, we have i I(A0,B) ≤ I(A0,B0C0). Apply the state merging pro- X 0 0 0 ≥ piS[ρA(i)] ≥ 0 . (C13a) tocol. S(A|B) = S(A|BC) = S(A |B C ). Thus acting i on B with a unitary operator though an initially fac- Concavity of the quantum conditional entropy directly torized ancilla C doesn’t change the cost of the proto- 0 0 0 follows from strong subadditivity through thefollowing col. After forgetting C , we have I(A ,B ) ≤ I(A, B) or 0 S(A|B) ≥ S(A0|B0). argument. Let us introduce 0 ≤ λ ≤ 1 and ρAB and ρAB two density operators. The statistical mixture ρAB(λ) = So the cost on state merging induced by the measure- 0 0 0 ment Mi is given by D(ρAB|Mi) = I(A, B) − I(A ,B ). λρAB +(1−λ)ρAB is obtained by tracing over an arbitrary qubit, which we call C, the density operator This quantity, which still depends on the measurement performed by B, becomes the quantum discord when 0 ρABC (λ) = λρAB ⊗ Π0 + (1 − λ)ρAB ⊗ Π1 (C14) our operation is a measurement maximizing I(A0,B0). 0 0 where Π = |0i h0| and Π = |1i h1|. Applying the strong To see this, let’s compute explicitly I(A ,B ). We have 0 1 0 P ρ = pjρ ⊗ πj where πj is the projector on the subadditivity property (C6) to ρABC (λ) directly leads to AB j A|j the concavity of ρ 7→ S[A|B]. subspace of the result j. The unconditioned reduced AB 0 P Note that the positivity of both conditional entropies is states for A and B are respectively ρA = j pjρA|j = ρA 0 P what leads to I[A, B] ≤ min(S[A],S[B]) since I[A, B] = and ρB = j pjπj. Then, we have: S[A] − S[A|B] = S[B] − S[B|A]. Whenever I[A, B] exceeds this classical bounds means that S[A|B] or I(A0,B0) = S(A0) + S(B0) − S(A0,B0) S[B|A] ≥ 0 and therefore implies that the state is en- X = S(A0) + H(p) − (H(p) + p S(ρ )) tangled. j A|j However, note that this is not necessarily a necessary j 0 X condition for entanglement. = S(A ) − pjS(ρA|j) , (D2) j

4. Behavior under temporal evolution where H(p) is the Shannon entropy of the probability distribution pj and the second line is obtained using the mix- The quantum mutual information being nothing but ing property of the von Neumann entropy. After max- the relative quantum entropy or Kullback-Leibler diver- imization over the measurement MS, we obtain exactly 0 gence between the state of the full system ρAB and its J[ρAB], the asymmetric mutual information ??, ending marginals ρA and ρB, property (C10) implies that it de- the proof of the theorem by recalling the deﬁnition ?? of creases under temporal evolution when completely posi- the discord. tive trace preserving maps are separately applied ot each part. It also decreases (see Eq. (C8)) when one discards part of one of the subsystem. These results can be viewed 2. Proof of theorem B.2 as the leak of correlations between two systems into all the degrees of freedom they are connected to. The monogamy relation [? ] for a tripartite pure state |ψABC i reads: Appendix D: Alternative expressions of quantum discord S[B] = EF [A, B] + I[B,Cc] , (D3)

1. Proof of theorem B.1 where we used the notation Cc to mean that a measurement has been performed on the subsystem C. We then have: The argument starts with the inequality:

S[A|B,C] ≤ S[A|B] . (D1) EF [A, B] = S[B|Cc] = S[A|Cc] . (D4) 28

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