Theoretical and experimental aspects of quantum discord and related measures

Lucas C. Céleri,1, ∗ Jonas Maziero,1, † and Roberto M. Serra1, ‡ 1Centro de Ciências Naturais e Humanas, Universidade Federal do ABC, R. Santa Adélia 166, 09210-170 Santo André, São Paulo, Brazil

Correlations are a very important tool in the study of multipartite systems, both for classical and quantum ones. The discussion about the quantum nature of correlations permeates Physics since Einstein, Podolski and Rosen published their famous article criticizing quantum mechanics. Here we provide a short review about the quantum nature of correlations, discussing both its theoretical and experimental aspects. We focus on quantum discord and related measures. After discussing their fundamental aspects (theoretically and experimentally), we proceed by analysing the dynamical behaviour of correlations under decoherence as well as some applications in different scenarios, such as quantum computation and relativity, passing through critical and biological systems.

I. INTRODUCTION principle underlying quantum mechanics, known as non- classical (or quantum) correlations. As we shall see, this Correlations are ubiquitous in nature and have been is a very intricate issue for which we do not have a closed playing an important role in human life for a long time. theory yet. First we observe that the simplicity and gen- For instance, in economy correlations between price and erality of Eq. (1) allows its direct extension to the quan- demand are extremely important for a businessman (or tum domain as a measure of the total correlations in a even for a government) to decide his investment policy. bipartite system, In the field of biology, the genetic correlations are fun- I (A : B) ≡ S (A) + S (B) − S (A, B) , (2) damental to follow individual traits. The relationship between income distribution and crime rate is just one where S (X) ≡ −TrρX logρX is the von Neumann entropy example coming from the Social Sciences. Broadly speak- [2]. The reduced density matrix of partition X, ρX , is ing, correlation is a quantity that describes the degree obtained from the entire density operator by means of of relationship between two variables. In the classical the partial trace operation ρX ≡ TrY (ρXY ). However, domain, such a quantity can be measured within the differently from the classical scenario, here we face the framework of information theory, developed by Shannon problem of distinguishing between the various types of in 1948 [1]. correlations. As we shall see below, such problem does Let us consider first the classical scenario with two dis- not have a clear-cut solution hitherto. tinct random variables, A and B, with well defined prob- Since the early years of quantum mechanics, non-local ability distributions {pA} and {pB}, respectively. We correlations have been at the core of a long standing de- can think of A and B in terms of probability sets for the bate about the foundations of the theory. We may say outcome of measurements performed on the joint system that the discussion started in 1935, when Einstein, Podol- or on the two different subsystems, usually named Al- sky and Rosen (EPR) published their famous article en- ice and Bob. To say that A and B are correlated in titled “Can quantum-mechanical description of physical some form means that the joint probability distribution reality be considered complete?” [3]. They considered two {pAB} cannot be written in the product form {pA × pB}. spatially separated particles, A and B, having both per- To quantify the amount of correlations shared by the two fectly correlated positions and momenta. By defining a classical random variables, Shannon introduced the mu- complete theory as the one for which there is one element tual information [1]: of physical reality corresponding to each element of the theory and assuming local realism, i.e., shortly I (A : B) ≡ H (A) + H (B) − H (A, B) , (1) arXiv:1107.3428v1 [quant-ph] 18 Jul 2011 locality P • - There is no action at a distance; with H (X) ≡ − x∈X px log px being the well-known Shannon entropy. Here and throughout this article we • realism - “If, without disturbing a system, we can use the short-hand notation px ≡ pX=x, in which the predict with certainty the value of a physical quan- random variable X assumes the value x from the set X , tity, then there exists an element of physical reality and we use log to represent the binary logarithm. So, the corresponding to this physical quantity.”; correlations are quantified in bits. they argued that quantum mechanics would not be com- In this article we are interested in a very special kind plete (see [4] for a recent review about the EPR paradox). of correlation, the ones arising from the superposition In a sense, EPR implied that quantum mechanics should be extended, possibly through the introduction of hidden variables, making the new theory compatible with their ∗ [email protected] premise of local realism stated above. In other words, † [email protected] EPR demonstrated an inconsistency between local real- ‡ [email protected] ism and the completeness of quantum mechanics. 2

Schrödinger noted that at the core of the paradox lies must be abandoned. A recent review about the theoreti- the very structure of the Hilbert space [5]. He pointed cal and the experimental aspects of this theorem can be out that the state vector of the entire system (including found in Ref. [4]. We observe here that the extension both particles A and B) is entangled. This means that of the quantum theory proposed by Bell is classical, in we cannot ascribe an individual state vector for each par- the sense that the hidden variables are local in nature. ticle separately, only the entire system can have such a However, more recently, Colbeck and Renner have shown, mathematical representation. This fact leads to the def- under the assumption that the measurement setting can inition of entanglement. A pure state is separable if and be freely chosen by the observer, that the quantum the- only if it can be written in the product form: ory cannot be extended at all [8]. With concern to correlations, the scenario regarding |ψABi = |ψAi ⊗ |ψBi, (3) what was discussed until here is the following: The non- classical nature of correlations was attributed to states |ψ i |ψ i where A ( B ) is the state vector describing the sys- that violate some Bell inequality. This kind of correla- A B tem ( ) alone. Otherwise it is entangled. The argu- tion, also named non-local, cannot be explained by an ment of Schrödinger against EPR was that this entan- LHV model. However, this situation changed in 1989, A glement is degraded during the process by which and when Werner published an article showing that there are B become spatially separated [5], leading to the impos- mixed entangled states that do not violate any Bell in- Gedankenexperiment sibility of physically realizing the equality and therefore could be described by means of proposed by EPR. Although interesting, this argument an LHV model [9]. This was a surprising fact because it does not settle the paradox. indicates that there are non-separable (entangled) mixed Since then, entanglement has been at the core of the states that could be described in “classical terms” by an discussion about the non-local aspects of quantum the- LHV model. So, another kind of correlation arises, the ory, both in respect to its fundamental aspects as well one contained in an entangled state, but that do not vio- as for (more recently) technological applications. The ef- late any Bell inequality. This kind of correlation is indeed forts that lead us to the theory of entanglement have been non-classical since we disregard LHV models. This pic- mainly focused on two aspects of the problem: Given a ture remained unchanged until 2001, when it was iden- certain state, is it entangled? If yes, how much entangle- tified a new kind of quantum correlation, which may be ment is there? Besides many advances (both in the theo- present even in separable (non-entangled) states. retical and in the experimental fields) have been achieved, a complete theory describing entanglement is still lack- ing. We refer the reader to the reviews in Ref. [6] for II. QUANTUM DISCORD a more complete discussion about entanglement. Before we proceed, let us define the entanglement for more gen- eral states, the mixed states. For systems described by a This new scenario emerged from an nonequivalence be- density operator ρAB, the separability condition given in tween the classical and the quantum versions of informa- Eq. (3) becomes tion theory. We may explore this nonequivalence in sev- eral ways in order to reveal some non-classicality. In what X i i follows we will present a way to do it that we consider ρAB = piρA ⊗ ρB, (4) i very illustrative. From Bayes’ rule we can write the clas- sical conditional probability for obtaining the value a for i with {pi} being a probability distribution and ρX a den- the random variable A when the value of the random vari- sity operator for the subsystem X. In the case of con- able B is known to be b, pa|b, in the form pa|b ≡ pa,b/pb, tinuous variables, we must replace the summation by an which leads to an equivalent expression for the classical integral over the entire probability space. The state (4) mutual information (1) is the most general bipartite state that Alice and Bob can create using local quantum operations and classical J(A : B) ≡ H(A) − H(A|B), (5) communication (LOCC). Therefore, an entangled state where the classical conditional entropy reads H(A|B) ≡ is that one that cannot be prepared by LOCC. P Let us now go back to EPR. Despite all the progress − a,b pa,b log pa|b = H(A, B) − H(B). achieved in the path to understanding the EPR paradox, While Eq. (1) has a direct extension to the quantum it was only in 1964 that it was put on a mathematical territory, as shown in Eq. (2), here we are faced with ground, making possible its experimental investigation. a fundamental fact: A measurement generally disturbs Considering a model described by local hidden variables a quantum system. The problem arises when we try to (LHV), Bell proved a theorem, today consisting in the extend the classical concept of the conditional entropy so-called Bell inequalities, confronting the predictions of H(A|B) to the quantum domain. This quantity mea- quantum mechanics with those coming from the local re- sures the uncertainty about the random variable A after alistic assumption [7]. If a given state violates a Bell we measure B. Therefore, its extension to quantum me- inequality then, it is not possible to describe such a state chanics presents some ambiguity due to the fact that, by means of an LHV model, implying that local realism depending on the observable we choose to measure B, 3 the value of the conditional entropy would be different is convex), the optimization is performed on the ex- and may even be negative. treme points of the set of POVMs, which are rank 1 [12]. In order to obtain a quantum version of Eq. (5), let us For more general systems, this optimization can be per- (B) consider a set of measurements Πj on subsystem B of formed over the complete POVM set, which is a very the composite state ρAB. The reduced state of subsystem difficult problem, even numerically. A, after the measurement, is given by From the above definitions, it follows immediately that D(A : B) + C(A : B) = I(A : B). For pure states, j 1 n (B)  (B)o we have a special situation where the quantum discord ρA = TrB 1A ⊗ Πj ρAB 1A ⊗ Πj , (6) qj is equal to the entropy of entanglement and also equal to the Henderson-Vedral classical correlation. In other n (B) o where qj = TrAB 1A ⊗ Πj ρAB is the probability words, D(A : B) = C(A : B) = I(A : B)/ 2 [11, 13]. In for the measurement of the j-th state in subsystem B this case, the total amount of non-classical correlation and 1A is the identity operator for subsystem A. For a is captured by an entanglement measure. On the other n (B)o hand, for mixed states, the entanglement is only a part complete set of measurements Π , we can define the j of the non-classical aspects of correlations [10, 14, 15]. conditional entropy of subsystem A, for a known subsys- We may visualize the different “flavors” of non-classical tem B, as correlations considering the Bell-diagonal class of states, X  j  as depicted in Fig. 1. S (A|B) ≡ qjS ρA . (7) j

Thus, we can also define the following quantum extension for Eq. (5)

J (A : B) = S(A) − S (A|B) . (8)

While in the classical case we have the equivalence be- tween both definitions of the mutual information in Eqs. (1) and (5), i.e., I − J = 0, for a quantum state, Eqs. (2) and (8) are not equivalent in general any more. The difference,

D(A : B) ≡ I(A : B) − max J (A : B), (9) n (B)o Πj was called quantum discord (QD) by Ollivier and Zurek [10]. One can say that Eq. (9) reveals the quantumness of the correlations between partitions A and B, since it shows the departure between the quantum and the classical versions of information theory. An important observation is that QD captures the non-classical aspects of correlations contained in certain states, which includes entanglement. However, while an entanglement measure [6] vanishes for a separable state, QD may be non-zero for Figure 1. Geometric representation of the different types states of the form (4). For the simplest case of bipartite of correlations in parameter space of Bell-diagonal states 3 pure states, QD is equal to entanglement. ρ = 1 + P3 c σA ⊗ σB
/4 σA bd AB i=1 i i i , where i i=1 are the The information theoretic approach to study correla- usual Pauli matrices. Valid states are in the (yellow) tetra- tions also leaded us to a new definition of classical correla- hedron. Separable states are those in the (green) octahedron, tions. A quantum composite state may have the support while classical states comprise the (black) axis. States that of classical correlations, C(A : B), which, for bipartite violates the CHSH inequality are found in the blue regions quantum states, can be quantified via the measure pro- [16]. posed by Henderson and Vedral [11]: C(A : B) ≡ max [S(A) − S(A|B)] , n o (10) Π(B) j The discovery that bipartite mixed separable (non- entangled) states can have non-classical correlations [10, where the maximum is taken over the complete set of pos- n (B)o 17] has opened a new perspective in the study and com- itive operator valued measurements (POVM) Πj on prehension of quantum aspects of its nature. A recent subsystem B. For two-qubit systems, since the condi- result, that almost all quantum states have a non van- tional entropy is concave over the set of POVMs (which ishing quantum discord [18], shows up the relevance of 4 studying these correlations. Such quantum correlations, A. Thermodynamic interpretation of quantum that may be present in separable states, are conjectured discord to play a role in the quantum advantage of tasks in mixed state quantum information science [14, 15, 19]. In an attempt to understand the limitations of thermo- There are some important issues about quantification dynamics, Maxwell introduced a character (today named of non-classical correlations beyond entanglement. One Maxwell’s demon) that permeates physics since then. of the most relevant to be noted at this point is the Furthermore, Maxwell’s demon plays a role also in in- fact that almost all non-classicality measures, as QD, formation theory. In fact, thermodynamics and informa- are based on extremization procedures that constitute tion theory present several links. Developments in this a difficult problem, even numerically. Actually, analyti- direction have being achieved since the Brillouin treat- cal solutions for the QD were obtained only recently for a ment of the informational entropy (due to Shannon) and certain class of highly symmetrical states [20–22] (see also the thermodynamic entropy (due to Boltzmann) on the [19, 23–25]). Hence, an alternative, operational (without same footing [30]. An important relation between these any extremization procedure) quantifier is rather desir- two distinct sciences was explored by Landauer to finally able. “exorcise” Maxwell’s demon through his information era- Our aim in this article is to discuss some theoretical sure theorem [31]. For a recent review about the physics and experimental aspects of non-classical correlations, of Maxwell’s demons from the perspective of information and its classical counterpart. The remaining of the article theory see Ref. [32]. is organized as follows. In Sec. III we discuss some physi- Maxwell’s demon is usually modelled as a classical in- cal interpretations of the quantum discord (the most pop- formation processing device, whose role is to implement ular measure of quantum correlations). As we shall see, a suitable action based on previous information acquired the quantification of quantum correlations is still an open by means of a classical measurement. The final goal problem. Sec. IV is devoted to present other proposed of the demon is to extract work from the environment. measures for such correlations, the relationship between Zurek then asked if a quantum version of the demon them, their main features, and some analytical results. would be more efficient in accomplishing this task than Besides the quantification problem, there is another im- the classical one. This quantum demon was defined as portant issue, the one regarding the experimental detec- a being that is able to measure non-local states and im- tion of such correlations and Sec. V goes in this direction, plement quantum conditional operations. Zurek verified by addressing the problem of witnessing non-classical cor- that not only the quantum demon really could extract relations. The action of the decoherence on these corre- more work than the classical one, but also that the dif- lations is discussed in detail in Sec. VI, showing very ference in the work extraction is given by the QD [27]. interesting dynamical aspects. In Sec. VIII we review The machine considered by Zurek consists of a system the discussion about the role played by quantum correla- S correlated with a measurement apparatus M, and an tions in several scenarios, including quantum information environment E at temperature T . The demon reads the processing and critical systems, passing through biology outcome of M and then uses the acquired information to and relativistic effects on these correlations. Beyond the- extract work from E by letting S to expand throughout oretical developments, the experimental investigation of the available phase (or Hilbert) space. The difference these correlations and of its dynamics is discussed in Sec. in the work extraction by the classical and the quantum VII. A brief summary and our final discussions are pre- demons is given by [27] sented in Sec. IX. ∆W = kBT D(ρSM). (11) In this equation, the quantum discord is given by   III. PHYSICAL INTERPRETATIONS OF D(ρSM) = min S(˜ρM) + S(˜ρS|M) − S(ρSM) (12) QUANTUM CORRELATIONS {|Mki}

where {|Mki} is the measurement basis in the appara- Quantum Information Science (QIS) introduced a new tus state space and kB is the Boltzmann constant. The way to think about quantum mechanics (see [26] for a post-measurement entropies of the apparatus and sys- more profound discussion in this direction). As men- tem are given, respectively, by S(˜ρM) = H({pk}) and tioned in the last section, this new perspective motivates S(˜ρS|M), with states and (conditional) entropies defined the introduction of a measure to quantify quantum cor- analogously to Eq. (6) and (7). This result tells us that relations, the quantum discord. QD has been largely the efficiency of the demon is determined by the informa- studied in a great variety of situations and a physical in- tion about S that is accessible to the demon [27] and that terpretation of such a quantity is quite desirable for the it can acquire more information by means of a quantum very foundations of the quantum theory. Until now, we measurement when compared with a classical one. This have two proposed interpretations for QD, one in thermo- is a direct relation between a thermodynamic quantity dynamics [27] and the other [28, 29] from the information (work) and an information theoretic one (quantum dis- theoretic perspective through the state merging protocol. cord). In other words, the quantum discord is a measure 5 of the advantage obtained by means of the quantum dy- (see [37] and references therein for more details about the namics. In [33] it was shown that two different versions properties of I ). of quantum discord determine the difference between the Local information is a resource and the sound measure efficiencies of a Szilard’s engine under different sets of re- of information I can be used, with actions restricted to strictions. It is important to observe here that despite closed LOCC operations (CLOCC: unitary transforma- the quantum nature of the demon, his memory is still tions and sending systems down a complete dephasing classical. channel), to study how many product pure states (e.g., A closely related discussion was presented in Ref. [34], |0⊗ni⊗|0⊗mi) Alice and Bob can distil from a joint state where the authors considered the Landauer’s erasure ρAB, in the distant laboratory paradigm. The maximal principle, which states that in order to erase the infor- amount of local information that they can extract in this mation stored in a given system, we must perform an way is called localizable information and is defined as amount of work that is proportional to the entropy of that Il(ρAB) ≡ max [I (˜ρA) + I (˜ρB)] system, in the presence of a quantum memory. Then, Λ∈CLOCC they showed that the difference between the work cost = log dAB − min [S(˜ρA) + S(˜ρB)],(15) of the erasure procedure using a quantum and a classical Λ∈CLOCC memory is given by kBT ln2D(S : M), with the quan- with dAB = dim HAB and the extremization is taken tum discord being defined as the difference between the over the CLOCC operations. Besides ρ˜A and ρ˜B are the uncertainty about the system for an observer possess- reduced states of ρ˜AB = Λ(ρAB). The total information ing a quantum MQ and a classical MC memory, i.e., in ρAB, D(S : M) = S(S : MQ) − S(S : MC ) [34] . The role of information in thermodynamics has been I (ρAB) = log dAB − S(ρAB), (16) largely discussed in recent literature, with the main focus on the information formulation of thermodynamics (see can be used to obtain the information that cannot be [32] and, for a quantum formulation of thermodynamics, localized via CLOCC: see [35]). In this direction, a single-mode photo-Carnot ∆(ρAB) ≡ I (ρAB) − Il(ρAB) engine was considered [36]. The efficiency of the engine = min [S(˜ρA) + S(˜ρB)] − S(ρAB). (17) was related to the quantum discord and it was demon- Λ∈CLOCC strated that it can be larger than that of a purely classi- cal engine, showing that quantum correlations could be This quantity was named quantum information deficit an important ingredient in thermodynamics [36]. This (or quantum deficit or work deficit) and can be regarded result can be viewed as another thermodynamical inter- as a measure of the quantumness of correlations [37]. pretation of QD. The information that Alice and Bob can extract acting locally, without using a classical channel to communicate, reads B. Thermodynamics and the quantum information (ρ ) = (ρ ) + (ρ ) deficit Ilo AB I A I B = log dAB − S(ρA) − S(ρB). (18) A review on the paradigm concerning the connections Thus the classical information deficit: between quantum information theory and thermodynam- ics was presented in [37]. Information, in the form of pure ∆c(ρAB) ≡ Il(ρAB) − Ilo(ρAB) quantum states, can be used to extract work from a heat = S(ρA) + S(ρB) reservoir. In general, by using a system in the state ρ − min [S(˜ρA) + S(˜ρB)], (19) and in contact with a heat bath at temperature T , one Λ∈CLOCC can draw an amount can be seen as a measure of classical correlation, since it tells us how much more information can be obtained W = kT I (ρ) (13) by exploiting additional correlations in ρAB through a of work, where classical channel. For more details of this and related quantities we refer the reader to Ref. [37]. I (ρ) = log d − S(ρ) (14) is a measure of information in ρ and d is the dimension C. Information theoretic interpretations of QD of the system’s state space (d = dim H). The quantity I (ρ) determines the optimal rate of transitions between Let us now turn our attention to interpretations of QD states under noise operations (NO: unitary transforma- from the point of view of asymptotic information process- tions, partial trace, and addition of ancillary systems in ing tasks, the so-called information theoretic interpreta- maximal mixed states). That is, given n copies of ρ one tions. This kind of physical ground is relevant for the can distil nI (ρ) pure qubits. Conversely one can take consideration of QD as a measure of quantum correla- nI (ρ) pure states and produced n copies of ρ, using NO tions or as a presumed resource for quantum information 6 tasks. In two articles recently published [28, 29], it was of QD. This property tells us that having more prior in- given, independently, such an interpretation, by consid- formation makes the state merging protocol cheaper. On ering the state merging protocol [38]. Given an unknown the other hand, the loss of information will increase the quantum state distributed over two systems, the quan- communication cost of the protocol. What the authors tum communication needed to transfer the full state to in Ref. [29] have shown is that the minimum increase one of the systems is called partial quantum information. of communication cost due to all possible measurements The quantum state merging protocol was introduced to that can be performed on B is given by the QD between optimally transfer quantum partial information [38]. The A and B. protocol is based on prior information, shared between It is worthwhile to note that the state merging pro- both partners, as measured by the conditional entropy. tocol is inherently asymmetric (due to the definition of It is important to note that, in this section, when we say the conditional entropy), so the asymmetry of QD has a conditional entropy we are referring to a direct extension direct interpretation as well. of the classical conditional entropy, which means that we The fact that QD can have both thermodynamic and are considering H (A|B) → S (A|B) = S (A, B) − S (B). information-theoretic interpretations, as discussed above, Note that if there are no correlations between both part- touches in the important relationship between these two ners, the conditional entropy is just equal to the entropy (assumed) independent areas, a relation noted years be- of one of the partners. Therefore, the state merging fore by Brillouin [30]. This interesting research field has protocol takes advantage of the previous correlations be- attracted much attention in the recent literature and cer- tween both partners to reduce the communication cost. tainly deserves further investigations. For the classical case, the conditional entropy is always positive, implying a positive communication cost. How- ever, in the quantum domain, this quantity may become IV. MEASURES OF CORRELATIONS negative, and in that fact lies the core of the protocol as well as the information-theoretic interpretation of QD. It As stated in the Introduction, QD does not seems to be is important to note here that the protocol as well as the the definitive word for the quantification of non-classical interpretations of QD based on it must be taken in the correlations in a general way. This section explores some asymptotic limit of an infinity number of copies. In this of the proposed measures of such correlations, others case, we can define the regularized discord as than entanglement, as well as for their classical coun- ⊗n terparts. As we shall see, we do not have a closed theory ¯ D(ρAB) D(ρAB) = lim . (20) for the characterization and quantification of such cor- n→∞ n relations. Let us start with what we think that broadly In the remaining of this section we will be referring, un- defines quantum correlations, i.e., the impossibility of lo- less stated otherwise, to this generalized definition of QD. cally broadcasting it. Cavalcanti et al. [28] considered the state merging pro- No-Local-Broadcasting Theorem — M Piani, P. tocol as follows: A pure state ΨABC is shared between Horodecki and R. Horodecki have introduced a theo- the partners A, B, and C. The aim of the protocol is rem providing an operational classification of multipar- that A transfers her part to B by using classical commu- tite classical correlations and, hence, for its quantum nication and shared entanglement between them while counterpart [39]. Their theorem is based on the impos- maintaining the coherence with the reference state C. sibility of locally broadcasting, i.e., on the impossibility If the partial information is positive (as it always is in of locally distributing correlations in order to have many the classical case) A needs to send this number of qubits copies of the original state. Purely classical correlations to B. On the other hand, if the partial information is are those permitting such distribution. negative (which is permitted in the quantum domain), For the bipartite case they proved that the correlations A and B instead acquire the corresponding potential for (measured by the mutual information) that can be totally future quantum communication [38]. In Ref. [28] it was transferred from the quantum to the classical world are shown that the entanglement consumption in the proto- those present in a state of the form col is equal to the QD between A and C, with measure- X ments on C. An immediate consequence of these results ρAB = pij|iihi| ⊗ |jihj|. (21) is an interpretation of the asymmetric character of QD. i,j It is a measure of the difference in the resources needed for A and C to send their parts of the initial state to These are the so-called classical-classical states (CC). In B, through the state merging protocol. As a by-product, this equation, {|ii} and {|ji} are orthonormal basis for they have also noted that the quantum discord may be the subsystems A and B, respectively, and {pij} is a joint regarded as an indicator of the direction in which more probability distribution. As a consequence, if a given classical information can be sent through dense coding. state cannot be cast in the form (21) it must contain In a related article Madhok and Datta [29], using non-classical correlations. They then extended this defi- the strong sub-additivity property of quantum entropy, nition to the multipartite case. We note here a close re- S (A|B,C) ≤ S (A|B), provided another interpretation semblance between this definition and the one regarding 7 the separability problem. Although this result does not idea, based on a distance in a probability space, is as really quantify the amount of quantum correlations in a follows: A quantifier for a property is given by the dis- given state, it clearly indicates its presence or not. The tance, as measured by the relative entropy, from a given authors then go further by proposing a measure for such state (with the property) to the closest state without the correlations, in a clear analogy with QD. The proposed desired property [40]. The quantum relative entropy be- quantifier for non-classicality, for the bipartite case, is tween the states ρ and σ is given by given by S (ρkσ) ≡ Tr [ρ (logρ − log σ)] , (24) Q (ρAB) ≡ I (ρAB) − ICC (ρAB) (22) and it is a measure of the distance, in state space, be- with I (ρ) being the mutual information and tween these two states. It is important to observe that,   although the relative entropy is taken as a distance, it X A B
A B
† ICC (ρ) = max I  Πi ⊗ Πj ρAB Πi ⊗ Πj  does not satisfy all the properties of a real distance mea- {ΠA,ΠB } i j i,j sure; for instance, it is not symmetric. (23) Given a certain state ρ and the set Σ of all separable X the CC mutual information of state ρ. {Πi } is a POVM states, the entanglement is quantified by set for partition X. The extension of this quantity for ER (ρ) = min S (ρkσ) , (25) the multipartite case depends on the definition of mu- σ∈Σ tual information, which can be obtained by means of a distance in the probability space [40]. while the quantum discord (also named relative entropy Due to the symmetry of Eq. (21) (it is invariant un- of discord) assumes the form der permutation of A and B) we cannot directly translate QR (ρ) = min S (ρkχ) , (26) the interpretations of quantum discord of the last section, χ∈Ξ which are based on the asymmetric state merging proto- Ξ col, to the quantifier proposed in Eq. (22). However, this with being the set of CC states [defined by Eq. (21)]. quantity already has an information theoretic interpreta- Another quantifier introduced in Ref. [40] was the so- quantum dissonance ρ tion in terms of the impossibility of locally distributing called , which, for the state , is mea- correlations. This observation leaves open the question sured by the minimal relative entropy between the closest σ about the existence of a thermodynamic interpretation separable (denoted as ) and the classical states for this quantity. DR (ρ) = min S (σkκ) . (27) Equation (23) can be regarded as a symmetric version κ∈Ξ of QD. The symmetric aspects of correlations quantifiers This is a measure of quantum correlations in separable was discussed in Refs. [41, 42]. Employing orthonormal states, which is analogous to quantum discord, but ex- projective measures instead of the more general POVMs, cludes entanglement. Note that all these measures are an analytical expression for this quantity was also pro- valid for whatever the dimension of the system, as well vided in Refs. [41, 42], in the case of Bell-diagonal states. as for the multipartite case. Thermodynamic Approach — Based on the fact that Another interesting approach was given in Ref. [45]. information can be used to extract work from a heat bath, Having provided a condition for a state to have null QD, the authors in Ref. [43] (see also Sec. III A and Ref. [37]) the authors proposed the following expression to quantify have proposed a thermodynamic quantifier for quantum it correlations. Considering a scenario in which Alice and 2 2 Bob share a correlated state they defined the work deficit Qd (ρ) = min kρ − ζk = Tr (ρ − ζ) , (28) ∆ as the difference in the amount of work that can be ex- ζ∈Ω tracted by Alice and Bob through CLOCC and by one of with the minimum taken over the set Ω of zero discord the partners holding the whole state, i.e., the work that states. And k · k2 is the square Hilbert-Schmidt norm. A cannot be locally extracted. In other words, ∆ is a mea- related dual quantity, named measurement-induced non- sure of the difference between work that can be extracted locality, was proposed and studied in Ref. [46]. in the case where the information is distributed and in Measurement-Induced Disturbance — All the above the case where it is localized. For the case of pure states, mentioned quantifiers are based on a very difficult ex- ∆ is exactly equal to the amount of distillable entangle- tremization procedure involving the set of POVMs. ment, while for the general case, it is equal to the amount Walking through a different path, Luo has proposed of quantum correlations (including entanglement) shared another correlation quantifier based on the disturbance by the two partners [43]. ∆ is zero only for CC states. that the measurement processes causes in a system [47]. This beautiful result reveals, once more, the connection This quantity, called measurement-induced disturbance between the thermodynamic work and information, link- (MID), is defined as the difference between the quantum ing them in a measure of correlations. mutual information of the state, ρAB, and that one of Geometric Measure of QD — In an interesting article the completely dephased state, χAB Modi and co-authors [40] (see also [44]) provided a geo- metrical view for measuring quantum correlations. The MID (ρAB) = I (ρAB) − I (χAB) . (29) 8

As dephasing takes place in the marginal basis, the an extremization process that becomes intractable in the marginal states are left unchanged. The suitability of case of continuous variables. However, two independent MID and its relation to QD are discussed in Ref. [48]. works have extended the QD concept to the continuous It is worthwhile to mention that all quantum correlation variable scenario [50, 51] (for a review about quantum quantifiers presented above vanish for the CC states in information in continuous variables see [52]). Eq. (21). In Ref. [50], focusing in the bipartite case described by Now let us consider a multipartite system in the state two-mode Gaussian states, and within the domain of gen- ρ. We can quantify its (total) correlations using the dis- eralized Gaussian measurement [53], the authors defined tance between ρ and its marginals states in the product what they called Gaussian discord. They also proved form: that this quantity is invariant under local unitary opera- n tions and is zero only for product states. In Ref. [51] the X authors have computed the QD, as well as its classical I(ρ) ≡ S(ρkρ1 ⊗ ρ2 ⊗ · · · ⊗ ρn) = S(ρs) − S(ρ), (30) s=1 counterpart, for all two-mode Gaussian states. Another interesting article discussing the continu- where ρs is the reduced state of subsystem s, obtained ous variables problem studied, instead of QD, the via the partial trace operation. The state obtained from measurement-induced disturbance, shown in Eq. (29), ρ through local von Neumann measurements reads where the two-mode Gaussian state was analysed in some details[54]. dim H dim H dim H X 1 X 2 X n The relation among the quantifiers for entanglement, ρ˜ = ··· (Π ⊗ Π ⊗ · · · ⊗ Π ) i1 i2 in total correlation, classical correlation, and quantum dis- i1=1 i2=1 in=1 cord will not be treated here. The reader can find more × ρ (Π ⊗ Π ⊗ · · · ⊗ Π ) . i1 i2 in (31) details about this issues in Refs. [37, 55–64]. As stated in the introduction we do not have yet a Analogously, the correlation in ρ˜ (which can be inter- closed theory for the description of the so called non- preted as being the classical correlation in ρ because ρ˜ classical correlations. The developments in this direction can be locally broadcast) is given by lead to the appearance of several measures to quantify I(˜ρ) ≡ max S(˜ρkρ˜1 ⊗ ρ˜2 ⊗ · · · ⊗ ρ˜n) (and also characterize) these correlations, but none of {Π }dim Hs is is=1 them seems to be definitive. Beyond the quantities dis- n cussed in this section, we can find many other proposals X = max S(˜ρs) − S(˜ρ), (32) in literature (see, for example, the papers in Ref. [65]). {Π }dim Hs is is=1 s=1 Trying to fill this gap, C. H. Bennett and co-workers stated three postulates, based on reasonable physical as- It is natural to consider the difference [49] sumptions, which each measure or indicator of genuine Q(ρ) ≡ I(ρ) − I(˜ρ) multipartite correlation quantifiers should satisfy [66]. n The postulates, that apply to correlations in general (in- X = [S(˜ρ) − S(ρ)] − [S(˜ρs) − S(ρs)] (33) cluding multipartite entanglement or classical correla- i=1 tions) are stated as follows: First Postulate n as a measure of the quantum correlation present in ρ, • — “If an -partite state does not n since it quantifies the disturbance in the system due to have genuine -partite correlations and one adds a (n + 1) local measurements. As the von Neumann entropy is party in a product state, then the resulting - n non-decreasing under non-selective projective measure- partite state does not have genuine -partite corre- ments, we identify the two non-negative terms lations.” • Second Postulate — “If an n-partite state does not Q (ρ) ≡ S(˜ρ) − S(ρ) nl (34) have genuine n-partite correlations, then local oper- ations and unanimous post-selection (which math- and ematically correspond to the operation Λ1 ⊗ Λ2 ⊗ n · · · ⊗ Λ n X n, where is the number of parties and each Ql(ρ) ≡ [S(˜ρs) − S(ρs)] (35) Λi is a trace non-increasing operation acting on the s=1 i-th party’s subsystem) cannot generate genuine n- partite correlations.” as the non-local and local disturbances, respectively. Since our interest here is to measure quantum cor- • Third Postulate — “If an n-partite state does not relations, we propose that the local measurements have genuine n-partite correlations, then if one {Π }dim Hs is is=1 should be those that maximize (32). party splits his subsystem into two parts, keeping Quantum Discord for Continuous Variables — All the one part for himself and using the other to cre- quantifiers discussed so far were studied considering only ate a new (n + 1)-st subsystem, then the resulting the case of discrete variables (with finite dimension), (n + 1)-partite state does not have genuine (n + 1)- mainly due to the fact that all of them are based on partite correlations.” 9

The authors then proceed by proposing a definition of are the eigenvalues of ρbd. Also for Bell-diagonal states, it genuine n- and k-partite correlations that satisfy all three was shown that the formula for the symmetric quantum postulates [66]: discord [Eq. (22) with local von Neumann measurements instead of POVM] coincides with that in Eq. (38) for Definition 1 n • — A state of particles has genuine quantum discord [41, 42]. n -partite correlations if it is non-product in every A general two-qubit state can be parametrized as (see bipartite cut. e.g. [20] and references therein)

• Definition 2 — A state of n particles has genuine k- 1AB 1B 1A ρ = + ~x · ~σA ⊗ + ⊗ ~y · ~σB partite correlations if there exists a k-particle sub- AB 4 4 4 set whose reduced state has genuine k-partite cor- 3 1 X A B relations (according to Definition 1 ). + tijσ ⊗ σ , (40) 4 i j i,j=1 Here we note that one of the consequences of the pos- tulates stated above is that the existence of genuine n- where ~x and ~y are the Block vectors for the subsystem A A A A 3 partite quantum correlations without genuine n-partite A and B, respectively, ~σ = (σ1 , σ2 , σ3 ), and {tij}i,j=1 classical correlations is not justified [66]. The authors are the real elements of the correlation matrix T . By def- then proposed a measure of genuine multipartite correla- inition classical and quantum correlations are invariant tions based on work that can be drawn from local envi- under local unitary operations. Thus it is useful to put ronments by means of a multipartite state. state (40) into a simpler for: As we can see, there is a long way yet to be traversed 1 1 1 ρ˜ = AB + ~a · ~σA ⊗ B + A ⊗~b · ~σB in search of a complete theory to describe, both qualita- AB 4 4 4 tively and quantitatively, these correlations. 3 1 X A B + ciσ ⊗ σ . (41) 4 i i i=1 A. Analytical results for quantum correlations where ρ˜AB is obtained from ρAB through local uni- tary operations and, therefore, it follows that D(˜ρAB) = 1. Quantum discord D(ρAB). In Ref. [19], Girolami and Adesso regarded the normal form (41) and obtained a nice algorithm to com- As mentioned before, most quantum correlation quan- pute the quantum discord for any state of a two-qubit tifiers are based on extremization procedures that consti- system. They showed that the measurement direction tute a very difficult problem, even numerically. Here we (θ, ϕ) attaining the maximum in Eq. (9) can be obtained review some advances concerning analytical expressions by solving the following set of transcendental equations: for quantum correlations. Let us first consider the case of α/β α+β+γ + −
 +  2α two-qubit states with maximal mixed marginals (known λ1 /λ1 λ λ− = ; λ− = λ− 0 , (42) as Bell-diagonal states): 0 + −
α/β 1 0 λ− 1 + λ1 /λ1 0 3 ! 1 X A B where ρbd = 1AB + ciσi ⊗ σi , (36)   4 ± 1 | ~m−| i=1 λ = 1 ± 0 2 1−~b·X~

3 ± 1  | ~m+|  where 1AB is the identity matrix, {σi}i=1 are the Pauli λ = 1 ± (43) 3 1 2 1+~b·X~ spin matrices, and {ci}i=1 are real parameters con- j strained such that ρbd is a valid density matrix. For are the eigenvalues of ρA (j = 0, 1) and this class of states Luo solved the optimization problem " ∂(~b·X~ ) ∂(~b·X~ )# and obtained the Henderson-Vedral classical correlation α = det ∂θ ∂ϕ as [20]: ∂| ~m+| ∂| ~m+| ∂θ ∂ϕ 1 " ∂(~b·X~ ) ∂(~b·X~ )# 1 X  i   i  C(ρbd) = 1 + (−1) c log 1 + (−1) c , (37) β = det ∂θ ∂ϕ 2 ∂| ~m−| ∂| ~m−| i=0 ∂θ ∂ϕ

" ∂| ~m+| ∂| ~m+| # with c = max(|c1|, |c2|, |c3|). The quantum discord is γ = det ∂θ ∂ϕ , (44) thus given by ∂| ~m−| ∂| ~m−| ∂θ ∂ϕ 1 t X with ~m± = (a1 ± c1X1, a2 ± c2X2, a3 ± c3X3) and D(ρ ) = λ log 4λ − C(ρ ), bd ij ij bd (38) X~ = (2 sin θ cos θ cos ϕ, 2 sin θ cos θ sin ϕ, 2 cos2 θ − 1)t, i,j=0 where the superscript t denotes the transpose of vectors where or matrices. Further works concerning analytical devel- opments and numerical studies can be found in Refs. [23–  i i+j j  λij = 1 + (−1) c1 − (−1) c2 + (−1) c3 /4 (39) 25, 48, 67–69]. 10

2. Relative entropy of discord and quantum dissonance that is a classically correlated state. Therefore, for this class of states, the relative entropy of entanglement As discussed above, in Ref. [40] Modi et al. introduced and discord are equal: ER(|C4i) = QR(|C4i) = 2 and a unified framework to quantify the quantum correlations DR(|C4i) = 0. in any state ρ by means of the relative entropy of ρ and Relative entropy-based measures of quantum correla- its closest classical state χ (by classical states we mean tions were computed for superpositions of N-qubit GHZ those that can be locally broadcast), which is given by and W states in [70].

QR(ρ) = S(χ) − S(ρ). (45) 3. Geometric quantum discord We observe that QR encompasses all quantum correla- tions, including those contained in non-separable states. In Ref. [45] Dakić et al. introduced the geometric By its turn, the quantum dissonance (DR) measures the quantum correlations of separable states. If σ is the clos- quantum discord [Eq. (28)] and obtained an analytical formula for it considering general two-qubit states [see est separable state to ρ, DR is given by the relative en- tropy between σ and its closest classical state κ: Eq. (40)]:

D (ρ) = S(κ) − S(σ). 1 2 2
R (46) Qd(ρAB) = k~xk + kT k − λm , (50) 4 The quantifiers QR and DR were computed for some t classes of states as follows [40]: where λm is the largest eigenvalue of the matrix ~x~x + t Bell-diagonal states. First let us write the Bell- TT . A lower bound for the geometric quantum discord P4 diagonal states as ρ = i=1 λi|ΨiihΨi|, where λi [see Eq. of arbitrary bipartite states was provided in Ref. [71] (39)] are ordered in nonincreasing size and |Ψii are the (see also [72]). A numerical investigation of the relation four Bell’s states. The closest separable state to ρ is σ = between the original quantum discord and its geometric P4 i=1 pi|ΨiihΨi|, with p1 = 1/2 and pi = λi/[2(1 − λ1)] version was carried out in [73]. for i = 2, 3, 4. The closest classical state for both ρ and σ has the form [40]: q 4. Gaussian quantum discord ξ = (|Ψ ihΨ | + |Ψ ihΨ |) 2 1 1 2 2 1 − q As discussed before, the Gaussian quantum discord + (|Ψ3ihΨ3| + |Ψ4ihΨ4|) , (47) 2 (GQD) was defined in Ref. [51] (see also [50]), using generalized Gaussian positive operator valued measure- q = λ + λ ξ = χ q = p + p ξ = κ with 1 2 for and 1 2 for . ments, and was computed for general two-mode Gaussian Three-qubit W state. This multipartite pure state states ρAB. This class of states is specified by the co- is an entangled state and has the form: |W i = ˆ ˆ √ variance matrix (CM) Mc = {Tr(ρAB{Ri, Rj}+)}, where (|100i + |010i + |001i) / 3. Its closest separable state is ˆ R = (ˆxA, pˆA, xˆB, pˆB) is the vector of phase-space opera- 1 tors. All two-mode CMs can be transformed in a stan- σ = √ (8|000ih000| + |111ih111| 27 dard form ¯ ¯
  + 12|W ihW | + 6|W ihW | , (48) M1 M3 Mc = , (51) √ M3 M2 where |W¯ i = (|011i + |101i + |110i) / 3. The classi- cally correlated states χ and κ are obtained by dephas- through local unitary operations, with M1 = diag(a, a), ing |Ψi in the z-basis and σ in the x-basis, respec- M2 = diag(b, b) and M3 = diag(c, d). The CM Mc cor- tively. Quantitatively we have [40] QR(|W i) ' 1.58 and responds to a physical state if and only if A, B ≥ 1 and DR(|W i) ' 0.94. This is an interesting result once it ν± ≥ 1, where the symplectic invariants are defined as shows that, while for bipartite pure states discord and en- A = det M√1, B = det M2, C = det M3, D = det Mc and tanglement coincide, this is not necessarily true for mul- 2
ν = δ ± δ2 − 4D /2 with δ = A + B + 2C. Finally, tipartite pure states. There are separable multiparticle ± the Gaussian quantum discord for two-mode Gaussian pure states that present non-classical correlations. states is given by [51] Cluster state. For four qubits the cluster state, which is a resource used in measurement based quantum com- √ p D(ρAB) = f( B) − f(ν+) − f(ν−) + f( Emin), (52) putation, reads√ |C4i = (|0 + 0+i + |1 + 1+√ i + |0 − 1−i + |1 − 0−i)/ 4, where |±i = (|0i ± |1i)/ 2. The closest with separable state to |C4i is [40] 1 σ = (|0 + 0+ih0 + 0 + | + |1 + 1+ih1 + 1 + | X x + (−1)i x + (−1)i f(x) = (−1)i log (53) 2 2 + |0 − 1−ih0 − 1 − | + |1 − 0−ih1 − 0 − |) /4, (49) i=0 11 and an unknown quantum state. Such a protocol demands 2C2 + (B − 1)(D − A) four copies of the unknown state that may introduce ad- Emin = 2 ditional experimental tools. To define this witness, let (B − 1) us first define the so-called classical-quantum states. In p 2|C| C2 + (B − 1)(D − A) analogy with Eq. (21) that defines a classical-classical + (54) (B − 1)2 state, a classical-quantum (CQ) one is defined through the relation (D − AB)2 ≤ C2(B + 1)(D + A) if and X B ρAB = pi|iihi| ⊗ ρi , (57) 2 AB − C + D i E = min 2B p where {|ii} is an orthonormal basis for the subsystem A C4 + (D − AB)2 − 2C2(D + AB) ρB B − (55) and i are density matrices for partition . This defi- 2B nition is motivated by the fact that, when we compute otherwise. the QD of this state performing measurements on par- tition B [D(A : B)] a non-zero value is obtained while, for measurements performed on partition A, we always V. WITNESSING QUANTUM CORRELATIONS obtain D(B : A) = 0. An analogous definition follows for the quantum-classical states, i.e., states such that D(A : B) = 0. The no-unilocal-broadcasting theorem for Despite all the theoretical advances already achieved quantum correlations proved in Ref. [84] characterizes in this field, its experimental aspects are still in its the classical-quantum (quantum-classical) class of states early stages. As one can note, all the above mentioned as the only one whose correlations can be locally broad- measures of quantum correlations require the expensive casted by part A (B). knowledge of the entire density matrix of the system for The witness proposed in Ref. [75] constitutes a nec- their computation, a knowledge usually obtained in the essary and sufficient condition for the existence of the laboratory by means of quantum state tomography. It is non-classical correlations in the bipartite, but arbitrarily also necessary (in general) to perform demanding numer- (finite) dimensional, case (with the QD being computed ical optimization procedures to compute the amount of by measurements on partition A). In other words, the non-classical correlations in a given system (as discussed witness is zero for states of the form given in Eq. (57). in the last section). A possible way to partially solve this As a by-product, from the expectation value of the ob- difficulty is by using a correlation witness W , a quan- server that defines the witness one can provide a lower tity directly accessed in an experiment that can indicate bound for the QD. The witness is defined as the presence of correlations. Unlike the entanglement- separability paradigm, few witness for detecting such cor- 1  †  B B B B
W = XA + X V V − V V , (58) relations was proposed in the literature [45, 74–80], and 2 A 13 24 12 34 only three experiments were performed [81–83] until now. B P In this section we discuss the theoretical aspects of corre- where Vi,j = kl |klihlk|ij acts on the i-th and j- th copies of partition B of unknown state and XA = lation witnesses and leave the experimental tests for Sec. P VII. klmn |klmni hlmnk| is the cyclic permutation operator acting on partition A. The result presented in Ref. [75] In Ref. [74] it was shown that it is possible to de- ⊗4
tect non-classical correlations in a single run of a bulk- is that D(B : A) = 0 if and only if Tr W ρ = 0. ensemble experiment. The authors defined a witness map Considering the two-qubit case, it was reported in Ref. (W : S → <, with S being the state space of the system) [76] an experimentally friendly classicality witness for a possessing the following properties: (i) Wρ ≥ 0 for every broad class of such states. The witness is defined by state of the form given in Eq. (21) and (ii) there exists means of the following operators non-classically correlated states such that Wρ < 0. The A B Oi = σ ⊗ σ , (59) form of the proposed witness map is given by i i

A B Wρ = c − Tr (ρA1) Tr (ρA2) Tr (ρA3) ··· , (56) O4 = ~z · ~σ ⊗ 1B + 1A ⊗ ~w · ~σ , (60) where the Ai are Hermitian operators and c is a con- with σi being the Pauli matrix in direction i and ~z and stant that must be properly determined. The important ~w are arbitrary real vectors such that |~z| = |~w| = 1. The difference between this equation and the ones in the en- classicality witness is proposed as tanglement theory is that the present witness (as well as 3 4 all possible ones) is a nonlinear function of the measured X X W = |hOiihOji|. (61) operators. This is due to the fact that the non-classical i=1 j=1+i correlated states do not form a convex set [18, 74]. Another proposal was reported in Ref. [75], where The only way to obtain W = 0 is for a state of the the authors have provided a single observable witness of form given in Eq. (21), i.e., a classically correlated state. 12

This condition is only a sufficient one in general, being quantum discord only vanishes in the asymptotic limit. It necessary and sufficient in the case of the Bell-diagonal was analysed the case of two non-interacting qubits under states. This result was generalized for arbitrary (finite) the action of three, independent, decoherence channels, dimensions in Ref. [77]. i.e., the dephasing, the depolarizing and the generalized Studying the dynamics of the system-environment in- amplitude damping channels. Initial conditions that lead teraction, in Ref. [78] the authors have proposed a wit- to the entanglement sudden death were chosen and the ness for the correlations initially shared by system and dynamics of QD was investigated [88]. This result can environment, that can be determined through measure- be directly extended to more general cases, remembering ments only on the system. It is interesting to note that states of zero discord form a set of null measure and that, for the determination of the initial correlations, the that is nowhere dense in the state space [18]. knowledge of the structure of the environment or of the Another interesting result concerning decoherence was system-environment interaction is not required. obtained by Maziero and co-workers [22]. Studying the 2 2 dA dB Using the basis sets {Ai}i=1 and {Bj}j=1 (with dA = effect of the Pauli maps (phase, bit-flip and bit-phase- dimHA and dB = dimHB) in the Hilbert-Schmidt spaces flip channels) on two non-interacting qubits, the authors of Hermitian operators, one can write any bipartite state have predicted the sudden change (SC) phenomenon, as as well as the immunity against decoherence of correlations, both experimentally confirmed in Refs. [90, 91]. The SC X ρ = cijAi ⊗ Bj. (62) phenomenon consists in an abrupt change in the decay i,j rate of the correlations, highly dependent on the geome- try of the initial state. Initially it was thought that such In Ref. [45], in addition to the proposal of a geometric phenomenon will appear only under the dynamics of the measure for QD, the authors introduced a condition for Pauli maps, however it was found to occur in the presence the existence of non-zero quantum discord for any dimen- of a thermal bath [91, 92] and also in a squeezed com- sional bipartite states. They showed that if the rank of mon reservoir [93]. This peculiar behaviour was found the correlation matrix {cij} is greater than dA(dB), then again in the case of non-inertial qubits [94], being caused the quantum discord obtained by measuring the subsys- by the Unruh effect [95]. These results indicate that the tem A(B) is nonzero. This condition can be seem as a SC could be a dynamic characteristic of the correlations witness for QD and was experimentally verified in Ref. beyond entanglement, extending the ideas presented in [82]. Besides the witnesses discussed above other criteria Ref. [22]. However, a physical interpretation of such to infer the vanishing of quantum discord can be found behaviour is still lacking. in Refs. [85, 86]. On the other hand, considering the immunity against decoherence, it was found that, under certain conditions, the classical correlations can remain unaffected by deco- VI. DYNAMICS OF CORRELATIONS UNDER herence, while its quantum counterpart goes to zero [22]. DECOHERENCE This fact directly leads to an operational measure (with- out any extremization procedure) of correlations. The Besides the characterization and quantification of clas- quantum correlations of the initial state can be computed sical and quantum correlations, another interesting ques- through the difference between the initial mutual infor- tion is the behaviour of these correlations under the ac- mation and the mutual information of the completely de- tion of decoherence. This phenomenon, mainly caused cohered state [22]. Contrary to what happens with SC, by the injection of noise into the system, arising from its this phenomenon does not take place in a thermal envi- inevitable interaction with the surrounding environment, ronment, as experimentally demonstrated in Ref. [91]. is responsible for the loss of quantum coherence initially In a related work, it was found that, not only the clas- present in the system (see [87] for a modern treatment sical correlations, but also the quantum ones, can re- on this subject). This is a very important topic not only main, under certain circumstances and for a fixed pe- from a fundamental point of view, but also for practical riod of time, unaffected by decoherence (considering only purposes envisaging the development of quantum infor- Pauli maps) [96]. It was discovered that a class of two- mation protocols that make use of this kind of correla- qubit states for which the quantum correlations remain tions. In this section we will discuss the main theoretical unaffected until the transition time, after which they developments on this subject, letting the experimental began to decay while the classical correlations become investigation for Sec. VII. immune to decoherence [96]. A geometrical picture for the non-analytic behaviour of quantum correlations un- der decoherence, discussed above, was provided by Lang A. Markovian dynamics of correlations and Caves in Ref. [97] (see also [98]). In a different context, in Ref. [99] the authors have Considering the Markovian dynamics of two-qubit sys- investigated the role of the correlations between system tems, in Ref. [88] the authors showed that, in some cases and environment in their decoherent dynamics. It was where the entanglement sudden death [89] occurs, the regarded a two-qubit system under the action of two in- 13 dependent reservoirs, considering beyond the Pauli maps, the oscillators, the entanglement remains absent. In Ref. also the amplitude damping channel. The two main re- [105] the authors have analysed the case of two qubits sults reported in this reference are: (i) The decoherence interacting with a non-Markovian depolarizing channel. phenomenon may occur without bipartite entanglement The appearance of multiple sudden change was observed. between system and environment and (ii) the initial non- Similar results were experimentally observed in Ref. [106] classical correlations (presented in the two-qubit system) (see also Sec. VII). completely disappear, in some circumstances, being not transferred to the environments [99]. Concerning the continuous variables scenario, the dy- C. Quantum discord and completely positive maps namics of the Gaussian QD between two non-interacting modes under the action of a thermal reservoir was anal- Besides the results discussed above, there is an inter- ysed [100]. The dynamics of QD and the entanglement esting issue linking quantum discord and the evolution of was compared and, while entanglement suffers sudden open quantum systems. Rodrguez-Rosario et al. proved death (in certain cases), the QD decays asymptotically that if the system-environment initial state (ρSE) is a to zero, for the situations analysed in this reference. classical-quantum state, i.e., if the discord obtained by Common environments acting on bipartite system was measuring the system is null, then the system’s dynam- considered in Refs. [101, 102]. In Ref. [101], the au- ics is described by a completely positive map [107]. They thors considered two initially excited qubits, interacting also presented examples of separable initial states ρSE, with the modes of the electromagnetic field in the vac- with nonzero discord, for which the dynamical map de- uum state. This constitutes a common reservoir for both scribing the system’s time evolution is not positive [107]. qubits. It was demonstrated that the quantum and classi- This is a striking result concerning quantum discord, be- cal correlations are generated before entanglement. They cause it suggest something fundamental about correla- also studied the QD and the MID dynamics [101]. In Ref. tions, i.e., they can influence the dynamics of the system [102], the authors have considered an Ohmic thermal en- under the action of the environment in a profound way. vironment acting on two non-interacting qubits. It was In Ref. [108] two questions were faced: (i) Is the dy- demonstrated the existence of a stable amplification of namics of the open quantum system equivalent to a map QD for identical qubits, while for distinct qubits (with between the initial and the final sates? (ii) Is this map large detuning), the protection of QD from the environ- completely positive? The answer to the first question is ment action is possible in some cases. However, such affirmative, with the map being linear and Hermitian for phenomena can occur only before the sudden change time all initial conditions with the system-environment states [22]. Once more, we observe that the SC behaviour seems belonging to the so called special linear class of states: to be an important issue in the decoherence dynamics of X the quantum and classical correlations. ρSE = cij|iihj| ⊗ φij, (63) All the works discussed so far are dealing with Marko- ij vian environments, i.e., environments that do not present dim HS memory effects. Next we review some of the develop- where {φij}i,j=1 : HE 7→ HE with Trφij = 1 or φij = 0 ments that treat the dynamics of correlations under the dim HS ∀i, j, and {|ii}i=1 is an orthonormal basis for HS. The action of non-Markovian environments. answer for the second one is negative for initial special linear states presenting non vanishing quantum discord. Also with concern to fundamental aspects of the B. Non-Markovian dynamics of correlations system-environment dynamics, quantifiers for the classi- cal and quantum decorrelating capabilities of a quantum In Ref. [103] the authors have considered the case of operation were introduced and analysed in Ref. [109]. two non-interacting qubits under the action of two in- And it was shown in Ref. [110] that, under the presence dependent zero temperature non-Markovian thermal en- of any interaction with the environment, the quantum vironments. Comparing the dynamics of entanglement entropy rate of a system is zero if and only if the system- with that one for QD, it was discovered that while the environment state has zero quantum discord, with pro- first presents sudden death, the second only disappears at jective measurements of higher rank in the system. some times [103]. The case of two qubits interacting with a common reservoir was considered in Ref. [69], where SC was again observed. The results presented in this last VII. EXPERIMENTAL INVESTIGATIONS work seems to indicate that, for a general initial condi- tions, the SC will occur. Two non-interacting harmonic This section is devoted to the experimental aspects of oscillators under the action of independent and common quantum and classical correlations. Only a few exper- non-Markovian bosonic reservoirs were considered in the iments on this issue have been reported up-to-date in context of continuous variables in Ref. [104]. An inter- literature. esting result reported in this article is that, in the case The first experiment in this context was performed of a common reservoir, while QD can be created between to test the conjecture that the quantum correlations, as 14 measured by QD, are the figure of merit for the speed- relations. A modified version of the witness proposed in up in mixed-state quantum computation without or with Ref. [76] was presented and also implemented in an opti- little entanglement [14]. Using an all-optical architec- cal setup through the measurement of a single observable ture, the generation of correlations in the trace estima- in [83]. tion DQC1 (Deterministic quantum computation with The condition for non-zero quantum discord intro- one qubit) protocol was observed and no entanglement duced in Ref. [45] (based on the rank of the correlation has been found, despite the fact that an exponential matrix) was employed in the experiment reported in Ref. speed-up (in comparison with the best known classical [82] in the NMR setup, in order to demonstrate the non- protocol) is obtained with this protocol. However, a large classicality in the DQC1 algorithm using four qubits. amount of QD was generated, indicating that, although fully separable, mixed states can contain non-classical correlations that may be a valuable resource for quan- VIII. APPLICATIONS AND RELATED tum information technology [14]. EFFECTS From a more fundamental point of view, in Ref. [90] the authors have confirmed experimentally the sudden In this section, we will discuss some applications of change behaviour of correlations, as well as their immu- these general quantum and classical correlations in dif- nity against some sources of decoherence, both phenom- ferent contexts, like quantum computation and biological ena theoretically predicted in Ref. [22]. Also using an systems. As we shall see, there is an increasing interest in all-optical setup the authors followed the dynamics of such studies due to both fundamental and technological correlations under the action of a simulated dephasing reasons. environment in a controllable way. The experimental ob- servation showed an excellent agreement with the the- ory [90]. In this experiment, the environment is a non- A. The source of quantum advantage dissipative one, i.e., there is no energy exchange between system and environment. Only phase relations are lost. It is well accepted today that quantum mechanics can To fit this gap, in Ref. [91] the same phenomena was provide us a gain (in the computational time, or in the pa- demonstrated, but in this case, with the presence of a rameter estimation, for instance) over equivalent classical real dephasing channel and also of the amplitude damp- protocols. However, the source of such a gain, initially ing channel, responsible for the energy exchange between thought to be the entanglement, is not so clear [112, 113]. system and environment. Such phenomena was shown to In this subsection we shall present some recent develop- occur in a liquid state room temperature nuclear mag- ments in this direction. netic resonance (NMR) experiment [91]. This result in- In an article published in Physical Review A, Datta dicates that the SC may occur in general scenarios being and Gharibian [114] have analysed the role played by therefore a general property of quantum correlations. quantum correlations in the so-called mixed-state quan- The decoherence dynamics of correlations under the tum computation, focusing in particular on the DQC1 action of two independent amplitude damping channels model, proposed by Knill and Laflamme [115]. Con- (one for each qubit) and a global phase damping one was sidering separable states, the authors have found indi- studied in Ref. [111]. The experiment was performed in cations that the quantum correlations, as quantified by a solid state NMR system at room temperature. A the- MID, could be the source of the speed-up in the DQC1 ory to obtain the correlations from the NMR deviation model [114]. The QD was proposed as being the source matrix was also developed in this article, and an excel- of the quantum advantage in the DQC1 model in Ref. lent agreement with the experimental data was obtained [116] (see also [117]). Studying how correlations behave [111], uncovering thus the quantum nature of correlations in the Deutsch-Jozsa algorithm [118], the authors of Ref. in room temperature NMR systems. Another reported [119] have concluded that entanglement is not the unique experiment concerns the non-Markovian character of the signature of efficient quantum computation, and it is not decoherent dynamics of correlations [106]. By means of even necessary for protocols which present gain over their a controlled dephasing non-Markovian environment, the classical counterparts [119]. The prisoners dilemma and multiple sudden change behavior was observed. other quantum games were analysed in Ref. [120] with More recently, it was reported the first experimental similar conclusions. By analysing the mixed-state phase observation of a witness for quantum correlations [81], estimation, in Ref. [121] it was found that, although clas- theoretically proposed in [76]. Also employing the liquid sical correlations√ do not play an important role in this NMR architecture, the authors of Ref. [81] have mea- process, the N gain is still present when entanglement sured directly the witness in Eq. (61) with high precision. vanishes, but not the QD. It was shown in [122] that This result implies that it is possible to determine the quantum discord is responsible for the advantage in the quantum nature of correlations by means of only a few lo- quantum locking protocol, where entanglement is known cal measurements, without the necessity of the expensive to play no role. The connection between the mother pro- tomographic process and also avoiding the extremization tocol of quantum information theory and quantum dis- procedures involved in the quantification of quantum cor- cord was discussed in [123]. 15

Walking in the opposite direction, in Ref. [124], it was into the hole (the inertial qubit) while the other staying shown that the ability to create entanglement is neces- static (the accelerated one) near the event horizon. sary for the execution of bipartite quantum gates. In In another interesting article, the case of correlations other words, to execute such a gate, starting with the between modes of Dirac fields in non-inertial frames was LOCC should be supplemented by shared entanglement. considered [103]. It was found that the classical correla- QD is then related to the failure of the LOCC imple- tions decrease as the acceleration increases, which is in mentation of the gate [124]. A related work, linking QD sharp contrast with the scalar field case where the clas- and entanglement as a resource for quantum computation sical correlations do not depend on acceleration [130]. was presented in Ref. [125] (see also [45, 112]). In 1997, Moreover, the original correlations shared by Alice and Grover proposed an algorithm using quantum mechanics Rob (the accelerated Bob) from an inertial perspective to obtain a quadratic temporal speed-up in search appli- are redistributed between all the bipartite modes from a cations over unsorted data [126]. In a recent article the non-inertial perspective [131]. role of classical and quantum correlations was analysed Therefore, the dynamic behaviour of classical and in such scenario [127]. It was argued that entanglement quantum correlations can be very different depending on works as the indicator of the increasing rate of the success the system under consideration. This is an issue that cer- probability of the algorithm [127]. tainly deserves more study to permit us to understand its As we can see, despite all the progress, the debate dynamic fundamental aspects. Besides the fundamental about the source of quantum effectiveness is still an open questions, relativistic effects on correlations became im- question. portant for the attempt of constructing global communi- cations systems based on quantum mechanics.

B. Relativistic effects C. Critical systems Quantum theory and Relativity are the two main pil- lars on which rests the modern physics. Therefore, it is The study of critical systems has attracted the inter- natural to ask how the behaviour of quantum correla- est of a wide variety of scientists, both from the classical tions is changed when Relativity comes into play. This and quantum viewpoints. In general, the main element subsection only deals with relativistic effects on quan- for such studies is the concept of correlations. Indeed, tum discord. A discussion on the role of Relativity in changes in these correlations strongly affect the observ- information theory can be found in Ref. [128]. able properties of a many-body system and are responsi- Contrary to entanglement, there are only a few ar- ble, for instance, to quantum phase transitions (QPTs), ticles discussing the relativistic aspects of non-classical which is a critical change in the ground state of a quan- correlations so far. The first one is due to Datta [129]. tum system due to level crossings in its energy spectrum The system under consideration is a two, initially entan- (for an introduction on quantum phase transitions see gled, modes of the scalar field. The correlations shared [132]). by them were analysed in the case when the observer We will discuss here the developments obtained in the detecting one of the modes undergoes a constant acceler- context of quantum and classical correlations concepts ation, while the other stayed inertial. It was found that applied to quantum critical systems. We shall discuss while entanglement dies due to the Unruh effect, a con- only some of these developments and refer the reader to siderable amount of quantum discord still survives, even other references. in the limit of infinite acceleration [129], which is a very Quantum discord was applied to the study of QPTs in interesting result since, in this limit, the accelerated ob- the transverse field Ising model server should experience an infinite temperature thermal X x x z
HI = − λσ σ + σ (64) bath [95]. i i+1 i i The dynamics of classical and quantum correlations in a two-qubit system when one of them is uniformly accel- and in the antiferromagnetic XXZ spin 1/2 model erated during a finite amount of proper time was anal- X x x y y z z
ysed by Céleri and collaborators in Ref. [94]. In this case, HXXZ = σi σj + σi σj + ∆σi σj , (65) while the quantum correlations, as measured by symmet- hiji ric quantum discord [39, 42], is completely destroyed in the limit of infinite acceleration, the classical one remains with the sum running over the nearest-neighbours hiji nonzero [94]. It is interesting to note here that such cor- [133]. For both models, in the thermodynamic limit, the relations exhibit the SC behavior [22] as a function of amount of QD increases close to the critical points, indi- acceleration (the temperature of the associated Unruh cating that QD could be a good indicator of the existence thermal bath experienced by the qubit is directly pro- of such points in the system, as a qualitative detector. On portional to its proper acceleration [95]). These results the other hand, the derivatives of the QD (with respect were then extended to the case where the qubits lie in the to the parameter that drives the QPT) provide precise in- vicinity of a black hole, with one of them freely falling formation on the location and on the order of the QPTs 16

[133]. Such analysis was generalized by means of ana- Ref. [141] and the MID in Ref. [142]. The QD and entan- lytical expressions for the correlations in each model and glement were computed and compared for some systems also by discussing their finite-size behaviour in Ref. [21]. whose ground states can be expressed as matrix product Signatures of the critical behaviour of the system for the states in [143]. cases of first-order, second-order, and infinite order QPTs All the above results show that QD could play an im- was found in both classical correlations and QD [21]. portant role in critical systems from both point of views The pairwise correlation in the two-spin one- theoretical and experimental. So, it is a very interesting dimensional XYZ Heisenberg model, quantity to be explored in the condensed matter scenario.

z z x x y y HXYZ = B (σ1 + σ2 ) + Jxσ1 σ2 + Jyσ1 σ2 z z D. Biology + Jzσ1 σ2 , (66) at finite temperature was considered in Ref. [134]. The The quantum description of biological molecules and, authors have studied the dependence of QD on the tem- in particular, the role that quantum mechanics could play perature and also on the external magnetic field B acting in biological processes, have attracted much attention on both qubits. A very distinct behaviour from entangle- in the recent literature (see, for example, [144]). Al- ment was found and it was conjectured that QD is able though very controversial, some developments were ob- to detect a QPT even at finite temperature [134]. tained, mainly linking entanglement with the efficiency In Ref. [135] it was reported a study of the QD be- of a given process [145]. The aim of this subsection is to haviour in the thermodynamic limit of the anisotropic present the study of classical and quantum correlations XY spin-1/2 chains in a transverse magnetic field in such a systems, recently reported in Ref. [146]. The goal of this program is to identify whether quan- X λ H = − (1 + γ) σxσx + (1 − γ) σyσy  tum mechanical effects are present in biological systems, XY 2 i i+1 i i+1 i and in Ref. [146], the authors have analysed the role that X z QD, and its classical counterpart, play in the transfer of − σ (67) i an excitation from a chlorosome antenna to a reaction i center, by the so-called Fenna-Matthews-Olson protein considering both cases, for zero and finite temperatures. complex. The numerical simulations of Ref. [146] indi- After obtaining analytical expressions for pairwise corre- cate that a significant fraction of the total correlation in lations at any distant sites, it was shown that, at zero such a process is due to QD in the first picosecond (the temperature, QD can be able to detect a QPT even for relevant time scale for this dynamics). It is important to situations where entanglement fails in this task. For fi- note that this result was obtained both at cryogenic and nite temperatures, the authors showed that QD can in- physiological temperatures [146]. crease with temperature in the presence of a magnetic Despite very speculative, these works raise many ques- field [135]. tions about the role played by quantum mechanics in bi- A remarkable property of QD was provided in Ref. ological systems. Concerning quantum correlations we [136] where, analysing the XXZ model, it was shown could ask how the efficiency of a given process scales that, indeed, QD can indicate the critical points associ- with quantum discord (or other correlation measure)? ated with QPT. This result is in sharp contrast to entan- Another interesting question is if it is possible to model glement as well as to others thermodynamic quantities. some molecules, like the DNA, by means of a spin chain Another interesting result was recently reported in Ref. and, as a consequence, we could apply the results of the [137]. Pairwise QD was analysed as a function of dis- preceding subsection to these systems. tance between spins in the transverse field XY chain and Besides the developments presented here, there are in the XXZ chain in the presence of domain walls. It was many articles devoted to the study of classical and quan- found a long-range decay of quantum discord, that indi- tum correlations in several others scenarios. For instance, cate the critical points in these systems. This behaviour in Ref. [147] the dynamics of QD between two atoms is rather different from entanglement, which is typically interacting with a cavity field was analysed and the spin- short-ranged. Moreover, a clear change in the decay rate boson model in Ref. [148]. Applications in the solid state of correlations when the system crosses the critical point scenario can be found for the case of quantum dots in Ref. was predicted [137]. [149]. Beyond these developments, there are many other arti- cles in recent literature considering the quantum aspects of correlations in critical systems. For instance, the case IX. CONCLUDING REMARKS of the XY chain was considered in Ref. [138], the XX model in a nonuniform external magnetic field was re- Since the publication of the EPR criticism to quantum ported in Ref. [139] and the phenomenon of symmetry mechanics, the concept of classical and non-classical cor- breaking in a many body system in Ref. [140]. A com- relations has been greatly modified, as we have seen in parison of the QD with the work-deficit was provided in the introduction of this article. We have evolved from 17 correlations violating a Bell’s inequality to quantum cor- The experimental investigation of the behaviour of relations measured, for example, by the quantum discord, these correlations has been started only very recently. passing through entanglement. In this short review, we We have only a few experimental investigations at our focused on the non-classical aspects of correlations pre- disposal, and much more need to be done in order to sented in separable states, as the ones revealed by the lead us to a better test and comprehension about the quantum discord. We saw that there are several quanti- quantum nature of correlations in composed systems. fiers for such correlations and none of them seems to be The application of these correlations to another areas definitive. Despite the development achieved recently in like condensed matter physics and biological systems is this issue, the question concerning what is quantum in a in its early stages and a long way is still to be traversed correlated system, is still a bit open. in order to appreciate the true role of these correlations Beyond its fundamental aspects, another important in these different fields of knowledge. We hope that this and controversial subject was raised together with the review will call attention to the importance of discord- birth of the concept of quantum discord-like correlations, like correlations and will stimulate further research on the one concerning its application for quantum informa- the subject. tion processing. The main question in this direction is what is responsible for the advantages (over purely clas- sical systems) provided by quantum theory. Besides some controversial questions there are advances in this direc- ACKNOWLEDGMENTS tion. For instance, we certainly know that such kind of non-classicality plays an important role in communica- tion, as revealed by the non-local broadcasting [39], and The authors acknowledge financial support from in quantum metrology [121], among other scenarios. The UFABC, CAPES, and FAPESP. This work was per- investigation of the precise role of separable mixed states formed as part of the Brazilian National Institute for with non-classical correlations opens a very exiting av- Science and Technology of Quantum Information (INCT- enue of research. IQ).

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