Dynamics of Quantum Correlations: Entanglement and Beyond

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1 Dynamics of Quantum Correlations: Entanglement and Beyond Anna Muszkiewicz January 6, 2011 Just what kind of power is hidden in the Quantum Realm? This question has been addressed on numerous occasions by researchers in Quantum Physics, Quantum Information, Chemistry, and even Quantum Biology. In many cases finding a satisfactory answer has involved investigating the dynamics of quantum correlations in the systems of interest. Phenomena having no classical counterpart, such as entanglement sudden death or constant quantum discord in open systems have been predicted by theoreticians, and later on confirmed in the laboratory. In this survey I shall review these notions and outline the latest developments in this interdisciplinary field by investigating the time evolution of correlations such as entanglement and discord in bipartite continuous variable systems in a noisy background. Quantum correlations, or relationships between the outcomes of two or more trials which cannot be reproduced by any system obeying classical mechanics, have attracted considerable research interest in recent years. One of the best examples is perhaps quantum entanglement and quantum discord, whose time evolution in systems interacting with an external environment has been widely investigated. The motivation behind these research efforts is rather complex and is best understood in the context of a given field. On one hand, it is thought that through the discoveries in this area we will be able to push the boundaries of our knowledge of the world a bit further and hence understand the elusive quantum to classical transition [1, 2, 3]. On the other hand, it is also widely known that entanglement is crucial to the latest developments and novel applications of quantum information processing and quantum cryptography [4, 5]; it is also thought to be one of the necessary factors required for quantum computers to achieve a computational speedup over their classical analogues [6]. In the field of quantum information, quantum discord (which is viewed by many as a more general type of quantum correlations than entanglement [7]) has been shown to offer interesting research and application opportunities [8, 9]. Interestingly, these issues have also influenced a seemingly very distant field of chemical and biological sciences, thereby resulting in emergence of a new area of studies. This new discipline is termed Quantum Biology, and it primarily concerns itself with the effect of quantum correlations on certain biological processes. The most prominent example here is probably the research on the role played by entanglement and quantum discord in the process of photosynthesis [10, 11, 12]. This role has been studied extensively in recent years with hopes of elucidating the mechanism behind coherent energy transfer in biomolecules [13, 14], 1 which could in turn lead to e.g. improved techniques of harvesting solar energy. Some researchers have also looked at the possibilities of quantum coherence and dynamic entanglement in various molecular systems, however, this topic of research is currently in its infancy [15]. Quantum correlations are of paramount importance in many areas of science. Using them as a basis for new technologies requires extensive studies. Of particular interest are the effects of external factors on the system in question, since no quantum system can be completely isolated from its surroundings [2, 3]. Therefore it is important to know what the behaviour of the system s properties is when coupled to an environment. For this very reason there is currently a vast body of literature on the dynamics of quantum correlations in quantum open systems, which I have summarised below. This review is organised as follows: I first make a general outline of the models of environment and the systems under consideration typically used in these types of research. I then give the definition of entanglement and briefly describe the peculiarities of its dynamics in bipartite discrete variable systems, followed by a more detailed description of its behaviour in bipartite continuous variable systems. I then go on to discuss similar issues regarding quan- 1 In fact, quite a few different theoretical models for this process have been proposed lately, and currently experimentalists efforts are concentrated on confirming or disproving these. 1

2 tum discord. Afterwards I give a short account of the developments in the field of quantification and classification of quantum correlations and briefly discuss other measures of quantumness such as quantum dissonance and measurement induced disturbance. I conclude with an outline of possible developments and directions of future studies in the field of quantum correlations in continuous variable systems. Models of interest As remarked already, no quantum system can be fully isolated from the external environment. It is a simple matter to convince oneself of that. If we try to e.g. prepare two atoms in some non-trivial quantum state, typically we will need to shine specific laser pulses onto them. In this case the atoms, which constitute our system of interest, will be bombarded by photons. The latter act as an external heat bath. If, after we prepare the atoms in the desired state, we manage to put them into seclusion by e.g. placing them in an isolated cavity, there will still be factors such as spontaneous fluctuations of the vacuum, or (in any realistic scenario) thermal photons that will have the capacity of interacting with, and therefore changing the state of our system. Finally, should we at any point want to use our atoms for any task, we will typically need to perform a measurement on them. A measurement involves interaction with a much larger instrument, which we can again treat as a large heat bath. In most cases we do not have significant control over the surroundings of our system, and the latter needs to be modelled as an open system. In what follows, I outline the typical systems and environments that we will be concerned with. I also briefly discuss the ways in which the two can interact. Systems under investigation Since we are interested in implementing the developments of the quantum information theory (QIT), we shall be concerned with the systems that can be used for that purpose. Mathematically, the theory has been developed for two different types of systems that can be realised in the physical world, namely the discrete variable (DV) and the continuous variable (CV) systems. The DV systems are those with finite dimensional Hilbert space. By contrast, the Hilbert space for CV systems is infinite dimensional. The simplest example of a DV system is a qubit. This is a two level mathematical object which lives in a two dimensional Hilbert space. It is the fundamental unit of quantum information, and it is a quantum equivalent of a classical bit an analogous concept in classical information theory. The difference between the two is that while the bit is allowed to take one of only two possible values: 0 or 1, the states that a qubit can be in form a continuum this is due to the superposition principle in quantum mechanics. The general state of a qubit can therefore be written as ψ = α 0 + β 1 (where Dirac notation has been used), with the only restriction on the coefficients α and β resulting from the normalisation condition. 2 Qubits can be realised in any physical two level system: their state can be encoded in e.g. the spin of a spin 1/2 particle, the polarization of a photon, or the energy of a two level atom. As for CV systems, the simplest example of these is perhaps a Gaussian state. Mathematically, it is defined by a Gaussian probability distribution in phase space. Important examples of Gaussian states include coherent states, and single and two mode squeezed states. Coherent states are the quantum analogue of classical coherent electromagnetic waves, i.e. they are the states of minimum uncertainty in amplitude and phase. 3 They can be easily represented graphically with the aid of the phasor diagrams showing the two field quadratures. These are dimensionless quantities that can be related to the amplitude of the classical electric field, which in turn can be related to the position and momentum for an electromagnetic field mode. The uncertainties in the amplitude and phase can be translated into the uncertainties in the field quadratures. For coherent states, the uncertainties in the quadratures are the same. For a squeezed state, the uncertainty in one quadrature is decreased, while the uncertainty in the other one increases (so as to satisfy the uncertainty principle). The state therefore seems squeezed in one of the quadratures (Fig. 1). As mentioned above, the quadratures can be related to the position and momentum for electromagnetic field mode. In fact, electromagnetic field can be mathematically regarded as a set of harmonic oscillators, whose positions and momenta are directly related to the electric and magnetic components of the radiation modes. A single mode squeezed state then corresponds to one such oscillator with position x and momentum p squeezed along some particular axis. A two mode squeezed state consists of two oscillators, and the squeezing now corresponds to varying the uncertainty in their joint variables i.e. x 1 +x 2 and 2 More precisely, the two special states 0, 1 are called the spanning set or a basis. A state of a qubit can be any superposition of these two basis states. 3 In fact, in classical electromagnetism a coherent wave has a perfectly defined amplitude and phase. In the quantum regime, the uncertainty principle comes into play, and the two can no longer be specified with an arbitrary accuracy. 2

3 Figure 1: Phasor diagram for a coherent state (dotted circle) and a squeezed state (grey ellipse). Since X 1 = X 2 = 0, this is in fact a representation of a vacuum state. The uncertainties in the two field quadratures X 1 and X 2 are shown as well. Adapted from [16]. p 1 + p 2. Squeezing is usually quantified in terms of the squeezing parameter, r. Historically, first experiments implementing QIT protocols were conducted on DV systems. However, it was soon realised that it is easier to manipulate the physical equivalent of CV systems in the laboratory, and hence in practical applications. CV systems therefore drew a lot of interest in latest years, which is the primary reason for the review being focused on these [17]. Our system of interest may in principle consist of two or more elements (or parties): we could be dealing with e.g. many atoms in the DV case, or multi mode Gaussian states in the CV scenario. However, in the review I shall concentrate on bipartite systems, i.e. systems made up of two distinct parties. Possible environments By external environment, surroundings, or simply a heat bath we will mean everything that is not a part of our system, and yet can interact with it. In the simplest case of a system isolated from its environment and kept at zero temperature, vacuum fluctuations should be considered. In more realistic model, thermal excitations (or simply external multi mode electromagnetic field) need to be taken into account. In general, the structure of the environment will depend upon the physics problem in question. System environment interactions There are various tools to model the interaction of the system with the environment. Mathematically, the simplest one to consider is the case of a weak coupling between the two this corresponds to the Born approximation. The latter is frequently used in conjunction with the Markov approximation, which on its own implies that the environment does not retail any information on the states of the system preceding the current one. In other words, there are no memory effects in the system s dynamics. Markov approximation is valid in quite a few cases in quantum optics, and is used when the characteristic time scales of the system s dynamical evolution are much longer than those characteristic of the environment. When the two become comparable, Markovian treatment is no longer appropriate. In such cases the common practice is to use a non-markovian model in order to correctly picture the physical system in question. A non-markovian model is one in which the short time system environment interactions are taken into account (here the memory effects play an important role in the time evolution of the system). A yet more complete description of the physical world is achieved when we abandon the Born approximation and consider the strong coupling effects. Usually, the latter is mathematically formidable, and the resultant equations need to be solved numerically. Quantum correlations As described already, quantum correlations are the relations seen between outcomes of two or more trials that cannot be reproduced in any system described by classical mechanics. They are of uttermost importance to us, since many protocols in QIT were designed to make use of them. The most important types of these correlations are discussed in detail in this review. An important point to note, however, is that it is these correlations that are usually seen to be very fragile when the systems of interest interact with the external environment. During these interactions, the correlations between the two parties are lost, and the parties become correlated with their surroundings. This process is termed decoherence. It is hugely undesirable, since we have no control over the external environment and cannot make use of the correlations that the two parties of our system have established with their surroundings. What is entanglement? Entanglement has been in the centre of attention of quantum physicists since its accidental discovery in 1935 [18]. However, it was not until the 1990 s that it started being viewed as a valuable resource in quantum information and computation. Entanglement is perhaps best explained using a simple example. Imagine that a system of two qubits 3

4 A and B is described by the state vector of the form: Φ + = 1 2 ( 0 A 0 B + 1 A 1 B ) (1) with the qubit A belonging to Alice, and the qubit B to Bob. Phrased in the language of physics, the above equation states that if we make a measurement in the basis { 0, 1 } on a qubit held by Alice and obtain the result 0 A, then we are certain that Bob s qubit is in the state 0 B. Conversely, if the outcome of the first measurement is 1 A, it is certain that the second qubit is in the state 1 B. 4 When Alice makes individual measurements on her system without comparing any of her measurement outcomes with those obtained by Bob, she cannot spot any correlations. To her, the outcomes of her measurements are perfectly random: she obtains 0 and 1 with equal probability of 1 2 (provided she always begins her experiments with the state given in equation (1)). However, if the outcomes of Alice s measurements are compared to those of Bob s, a perfect correlation can be observed. Such a relation is non-classical in the sense that it cannot be reproduced in any system obeying classical mechanics. When the outcomes of the measurements on the subsystems reveal such a correlation, the two subsystems are said to be entangled. In order to continue our discussion, we need to define entanglement more rigorously. Such a definition in terms of a separability condition was formulated by Werner [19], who stated that a bipartite system comprised of two parties A and B is separable (or disentangled) when the density matrix ρ S of the whole system can be written as a sum of the tensor products of the density matrices ρ A and ρ B describing a given subsystem: ρ S = i p i ρ A i ρ B i, (2) weighted by their corresponding probability distributions p i 0. If this condition does not hold, the whole system is said to be in an inseparable, or entangled state. In order to visualise this mathematical relation, let us consider again the state of the two qubits given by (1). The density matrix for this system can be written as: ρ + = 1 2 (ρ sep + ρ int ) (3) where ρ sep = ( 0 0 )( 0 0 ) + ( 1 1 )( 1 1 ) and ρ int = ( 1 0 )( 1 0 ) + ( 0 1 )( 0 1 ), and the first bracket in each expression refers to the qubit held by Alice, while the second to that held by Bob. While ρ sep can be interpreted as a sum of density matrices of systems A and B tensored with each other, a similar conclusion cannot be drawn for ρ int. Due to the latter s presence in the overall expression for ρ +, the state of the system in (1) is inseparable. 5 Entanglement measures Although very useful from a pedagogical standpoint, the separability criterion in (2) is very hard to use in practice. For the purpose of research, different (and easier to compute) entanglement measures and separability criteria have been defined. Examples include maximal Schmidt rank [4], Peres-Horodecki (PPT) criterion [20], Duan criterion [21], Wootters concurrence [22], logarithmic negativity [23], and various schemes based on so-called entanglement witnesses [24]. A complete overview of entanglement measures is provided in two excellent review papers [5, 25]. Entanglement dynamics in discrete variable systems The theoretical background for discrete variable systems was developed long before the one describing continuous variable systems. Unsurprisingly, many discoveries in entanglement dynamics were initially made in DV systems, and subsequently extended to their CV analogues. For this reason I shall briefly discuss important landmarks in entanglement dynamics observed in DV systems for the first time, and afterwards move to the CV case, which shall be described in more detail. Entanglement sudden death The term entanglement sudden death (ESD, or less commonly, early-stage disentanglement [26], referring to disappearance of entanglement in finite time in bipartite systems coupled to external environments) was coined by Yu and Eberly, who were the first ones to discover and study this phenomenon from a theoretical viewpoint [27, 28, 29]. The discovery of ESD was quite surprising, as for the model considered in the studies (Markovian environments) the decay of correlations was expected to be asymptotic this law held for all of previous studies concerning any types of classical correlations [26]. In [29] an analysis of 4 This is true irrespectively of the distance between the two parties, which is why A. Einstein used to refer to entanglement as a spooky action at a distance. 5 In fact, ρ int in (3) encodes the quantum correlations between the two qubits. It is these correlations which are very fragile and undergo rapid decoherence when exposed to external environment. 4

5 the effects of decoherence on a quantum state of interest was performed. Starting with a simple model of two initially entangled two level atoms placed in separate cavities, the authors showed that a spontaneous emission of a photon may lead to complete disentanglement of the atoms in finite time (Fig. 2). The study above was conducted under the assumption that the walls of the cavities were made out of a perfectly absorbing material in the limit of T = 0K. The authors then decided to address a similar question, namely, what would change if the cavities were lossless? A completely opposite phenomenon was observed then. Entanglement sudden birth It was thus decided to examine entanglement dynamics in a double Jaynes-Cummings model, which consists of two two level atoms placed in separate lossless near resonant single mode cavities isolated from each other. As before, the zero temperature limit was considered. The study revealed that entanglement between the atoms oscillates periodically from a maximum to a minimum value. The latter corresponds to the system being completely disentangled, while the former to the atoms being fully entangled [30]. More importantly, the complete disentanglement of the atoms corresponded to the maximum entanglement of the cavity modes [31] (Fig. 3). The effect was termed entanglement sudden birth, or ESB, since entanglement was created between the initially uncorrelated modes of the two cavities [31]. This, however, should not come as a surprise: the information in the whole system was effectively allowed to flow from the atoms to the cavity modes and back, resulting in periodic rebirth of atom atom entanglement, as well as cavity cavity entanglement. A number of studies on ESD and ESB were conducted since the original discovery and subsequent insight by Yu and Eberly [32, 33]. Various models for the environment and initial states of the qubits were investigated, and a plethora of results emerged (examples include [34, 35, 36]). One of the most interesting conclusions was presented in [36], where it was shown that in any realistic model (in particular for T > 0K) entanglement always disappears in finite time when the subsystems are coupled to different cavities or heat baths. However, the study only focused on a special class of symmetric states, namely the X states, 6 and it is still unclear whether ESD always takes place for any general state of a quantum system. Figure 2: Entanglement measured with concurrence C as a function of time t and a dimensionless parameter a, which quantifies the probability for both atoms being in an excited state (for a = 0 the two atoms are in the ground state). ESD takes place for 1 a < 1, with exponential 3 decay observed otherwise. Adapted from [29]. Figure 3: Evolution of (a) atom atom, and (b) cavity cavity entanglement measured with concurrence C as a function of the rescaled time t t. The initial state of the atoms is eg cos( π ) + ge sin( π ), with e and g 4 4 corresponding to the atom being in excited and ground state. Instantaneous disappearance of atom atom entanglement corresponds to maximum cavity cavity entanglement, and vice versa. Adapted from [31]. 6 The X states are described by a relatively simple symmetric density matrix. See [36] for more details. 5

6 One of the most pertinent questions in the field is to determine the reasons behind the phenomena such as ESD and ESB. It has been partly addressed in [37], where it was pointed out that ESD is a result of incorporating dissipative terms in the Hamiltonian of the system, whereas ESB is an effect of a backaction terms in the Hamiltonian. Moreover, it is widely recognised that with increasing number of external electromagnetic field modes which is the usual environment chosen for the more realistic studies irreversible phenomena (such as ESD) occur due to the sheer amount of the external field modes [38]. When entanglement between the two parties of interest is transferred to a small amount of the field modes (as shown in [31]), it can then be transferred back to the system, however, when the number of external modes is very large (effectively infinite), initial bipartite entanglement is distributed between the modes and is never transferred back to the system when we exclude the possibility of memory effects. When various non-markovian models are used, ESD is usually followed by a few events of entanglement revival in the system of interest. These tend to occur after periods of complete disentanglement (these observations resulted in defining the self-explanatory concepts of separability and re-entanglement time, which were naturally found to vary depending on the model used). Despite these insights, a complete explanation of all of the intricacies of ESD and ESB continues to elude us. It can be said with a high level of certainty that for bipartite two level quantum systems prepared in an entangled state and placed in separate reservoirs isolated from each other, any realistic interaction with noisy environment leads to decoherence and subsequent disentanglement. As a result, many efforts have been employed to isolate the system of interest from the detrimental effects of the surroundings. However, some researchers have adopted a different approach and tried to investigate if the noise can be used to generate entanglement. A variety of interesting results emerged. Reservoir induced entanglement The term reservoir induced entanglement (or simply RIE) was first mentioned by Beige et al., who showed that it is possible for entanglement to be created between two atoms placed in a leaky cavity immersed into an environment containing a very large number of modes of electromagnetic radiation [39]. A number of different studies were then conducted. It was shown that entanglement can be created between two two level systems in separate single mode cavities driven by white noise [40, 41] and between two remote qubits in two cavities driven by an external electromagnetic field [42]. The former were extended to investigate the role of the common heat bath [43], and explicit dipole dipole interactions between the atoms [44, 45]. Other studies included investigation of non-markovian effects and various bath parameters [46]. In all cases entanglement was created between initially unentangled qubits for certain classes of initial conditions. These works demonstrate that it is possible for entanglement between two discrete systems to arise due to their indirect interactions mediated via the heat bath. This should not come as a complete surprise: the system s variables are coupled to the same bath variables, and some correlations should be expected to arise between the subsystems. Having outlined the landmarks in the time evolution of entanglement in DV scenario, I shall now describe its dynamics in CV systems. Entanglement dynamics in continuous variable systems Due to their significance in quantum optics and quantum information theory, there have been numerous studies conducted on entanglement dynamics in continuous variable systems, and in particular on Gaussian states. Advances in the field until the year 2007, including a detailed summary of mathematical description, entanglement quantification and properties for CV systems have been thoroughly analysed in two brilliant review papers [47, 48]. Moreover, many studies that had been conducted earlier were extended to more general ones in the last three years. The results of these are described in [49, 50, 51, 52, 53, 54], and we shall discuss them below. Suffice it to say that earlier works made frequent use of the Born Markov approximation (i.e. only the weak coupling limit was considered, and the short time correlations between the system and environment were ignored), and their conclusions need not apply in more general cases. One of the important studies in the field is reported in [49], where the methods previously used to study non-equilibrium systems were extended to cover the cases of strong coupling and memory effects on the system s dynamics. The authors numerically investigated the dynamics of initially entangled two mode squeezed states in a common bosonic reservoir. It was shown that the non-markovian effects prolong the initial separability time of the states, but have no considerable effect on the subsequent re-entangling time (this is intuitively expected, as the re-entangling 6

7 time is typically longer than the characteristic time scales of the system s dynamics). Entanglement dynamics was also investigated as a function of the coupling strength between the system and the environment. It was shown that the increase in the coupling corresponds to faster initial decoherence, and to faster subsequent entanglement generation, i.e. both the disentanglement and the re-entanglement time were found to decrease. The study above concentrated on the short time effects in entanglement dynamics. It was extended to include the asymptotic effects in [50, 51], where the weak coupling assumption was made. There, a non-markovian study of the time evolution of two resonant oscillators in a common bosonic reservoir was performed. The authors considered two models of the system bath interaction: a bilinear coupling in position and a symmetric coupling in position and momentum. The system s dynamics was investigated by means of an exact master equation for quantum Brownian motion derived in [49]. Three general types of the bath spectral density were considered, namely Ohmic, sub-ohmic and super-ohmic (corresponding to dissipative, strongly dissipative, and weakly dissipative environments, respectively). These are of the general form: ( ω ) (n 1) J(ω) γ 0 ω θ(λ ω) (4) Λ where Λ is the cut-off frequency, γ 0 is the coupling constant between the system in question and the environment, and ω is the angular frequency of oscillations (other proportionality terms have been omitted for clarity). The precise form of the spectral density depends on the value of the parameter n, which is n = 1, n < 1, and n > 1 for the three densities in question. In particular, in the studies [50, 51] n was 1 taken to be 1, 2, and 3. The authors investigated the long time entanglement for two mode squeezed and separable squeezed states. In the long time limit, three qualitative scenarios for entanglement dynamics emerge: (a) entanglement sudden death may take place (abbreviated as SD), (b) entanglement may undergo an infinite series of sudden deaths and revivals (abbreviated as SDR), or (c) it may have a finite nonzero value (denoted as NSD, for no sudden death). Mathematically, it can be shown that for certain parameters such as the asymptotic entropy of the final state, the environment acts as a supplier of entanglement, i.e. entanglement in the asymptotic state of the system may be larger than that in the initial one. In other cases the reservoir acts in a detrimental way on the correlations: initial entanglement is degraded, and may disappear in the long time limit. Figure 4: Phase diagram depicting asymptotic entanglement in (a) Ohmic and (b) sub-ohmic cases. Temperature T and the squeezing parameter r vary, while other variables characterising the environment and the system are kept constant. SD, NSD, and SDR acronyms stand for sudden death, no sudden death, and sudden death and revival. Adapted from [51]. The conclusions of the study may be summarised compactly on a phase diagram. The diagrams for both Ohmic and sub-ohmic spectral densities for the case where the oscillators are bilinearly coupled in position to the environment are shown in Fig. 4. A few interesting conclusions may be drawn from these diagrams. The horizontal axis corresponding to T = 0K contains finite asymptotic entangled states both for large and small values of r, with the SD phase being completely absent in this region. Therefore, finite asymptotic entanglement in NSD or infinite oscillations of entanglement in SDR constitute the two possible time evolutions of the correlations. The vertical line, corresponding to r = 0 and thus representing coherent states, covers the SD and NSD regions, thereby implying that there is a certain critical temperature T 0 below which the states never become disentangled, and above which entanglement always undergoes sudden death. The conditions necessary for these two scenarios are easily derived analytically. As we move away from the T = 0K and r = 0 lines into the low temperature low squeezing limit, we encounter the NSD phase. It is a purely non-markovian effect it becomes smaller with the increasing damping rate. For the states in this region of the phase diagram the final entanglement may be larger than that of the initial state. It is also clear that as long as the reservoir is cooled below a certain temperature and the initial squeezing is large enough, 7

8 the system will never enter the SD phase. In the high temperature limit, however, we see that the initial squeezing must be increased accordingly to prevent ESD from happening (which is what we might intuitively expect). In this regime, the coherent states never get entangled. Also, at any finite temperature, the SD and NSD phases are always separated by a narrow strand of the SDR phase this is yet another non-markovian effect. The phase diagrams for sub- Ohmic and Ohmic spectral densities are qualitatively similar, with the differences being that in the sub- Ohmic case the low temperture low squeezing NSD and SDR phases have a larger area. It is a curious observation, and the precise conditions for it can be derived mathematically. This, however, is beyond the scope of the review. No phase diagram for super-ohmic spectral density is provided for a simple reason: since the environment is now only weakly dissipative, 7 the system takes a very long time to reach equilibrium and oscillatory behaviour of entanglement is observed in the asymptotic limit, essentially irrespective of the initial temperature of the environment and the initial squeezing r. The case of the symmetric coupling in position and momentum is easier to analyse: the SDR phase is absent at all times. 8 Therefore only SD and NSD phases are present, and (unsurprisingly) as the temperature increases, the initial squeezing must be increased as well for finite entanglement to be present in the asymptotic limit (Fig. 5). This is the case for both Ohmic and sub-ohmic spectral densities. Figure 5: Phase diagram depicting asymptotic entanglement for the case of symmetric coupling in position and momentum in the Ohmic environment. Temperature T and the squeezing parameter r vary, while other variables characterising the environment and the system are kept constant. Adapted from [51]. However, for the super-ohmic case, various case studies show that (just like for the case of the position coupling) entanglement oscillations persist for very long times. This is again a consequence of the weak dissipation induced by the environment. The time required for equilibration of the states immersed in the Ohmic and sub-ohmic environments is much shorter than that for the super-ohmic case. The latter observation is in contrast to the conclusions reported in [55], where it is shown that the state of the system approaches equilibrium fastest in the super-ohmic environment. These results seem to be erroneous in the light of the more complete later study [51]. Finally, the authors show that for supra-ohmic environments (n > 1), the system never comes to equilibrium in the Markovian (or infinite cut-off) limit. These results are consistent with an earlier study on the influence of Markovianity on two mode squeezed states presented in [56]. The case of interacting oscillators is also briefly discussed, and it is shown that the oscillator oscillator coupling does not influence the phase diagrams to a large extent. This is yet another surprising observation, and one would expect for the entanglement properties of the system to be changed upon the introduction of such a coupling. The results of the study may be extended in several ways: (a) non-identical oscillators can be considered, and (b) oscillators in separate (uncorrelated) bosonic reservoirs may be studied. The latter case was partly addressed in [52], while the former was analysed in [53]. In [52], the authors point out that for the independent reservoir scenario in the high temperature regime entanglement always displays the sudden death phenomenon independently of the initial conditions and the characteristics of the environment. ESD occurs fastest for the sub-ohmic spectral density of the surroundings (dissipation is largest in this case). At intermediate values of the temperature, ESD occurs for small values of initial entanglement. For the larger values of the initial squeezing, entanglement oscillations may be observed (a series of disentangling and re-entangling events is seen). For the case of T = 0K, the disentanglement time approaches infinity when the Markovian approximation is employed (this is in agreement with the results reported in [56]). In the non-markovian case, the situation is slightly more complex: depending on the cutoff frequency, entanglement either undergoes sudden death, or it may approach the asymptotic value in 7 In fact, in the limit of infinite cut-off Λ, dissipation vanishes. 8 This is a mathematical consequence of the way the problem is set up. Physically, it is a result of the symmetry incorporated into the system. 8

9 the corresponding Markovian scenario (naturally, in the non-makrovian environment, entanglement oscillations are seen in the initial stage of the dynamical evolution). The case of entanglement dynamics in nonidentical oscillators was investigated in [53] in detail. The authors looked at both separate and common reservoirs, and compared the Markovian and non- Markovian scenarios. Their solutions apply in the weak coupling regime only. As in the cases of previous studies, ESD was observed in the separate bath scenario, however, the disentanglement time could be delayed by tuning the oscillators frequencies appropriately. In the common reservoir case, entanglement persists for long times when the two oscillators are at resonance, however, as we move away from the resonance condition, entanglement is abruptly degraded and its dynamics resembles that seen in the separate bath scenario. However, a mathematical condition was derived showing that even in the case of non-resonant oscillators, there may be asymptotic entanglement in the system provided the coupling strengths of the two oscillators to the heat bath vary. The main difference between the Markovian and non- Markovian studies was found to appear only in the initial stage of the system s dynamical evolution. In particular, it was found that for the non-identical oscillators, the initial entanglement persists for longer times in the non-markovian setting. The opposite was observed in the case of resonant oscillators: here the non-markovian effects were found to accelerate decoherence. Based on the above studies, it can be said that the asymptotic entanglement dynamics for Gaussian states has been studied quite extensively, however, only the weak coupling limit has been analysed in detail. It would be interesting to see what the changes in dynamics are for a general system environment coupling, however, this case is mathematically formidable. The studies also reveal that entanglement dynamics in CV systems is influenced by a variety of factors which must be carefully investigated and taken into account when implementing the developments of QIT. A new type of non-classical correlation Separable states, as opposed to entangled states, can be prepared classically, or more precisely, by local operations and classical communication. 9 Unsurprisingly, for a long time classicality was synonymous to separability [57]. The systems whose density matrix can be written in the form given by (2) cannot possibly possess entanglement. Yet, there is a class of states which despite fulfilling the separability criterion still display non-classical correlations in the sense that we cannot extract maximum information about them: an attempt to learn about the properties of the system would involve changing its state. As an example, let us consider a quantum state of a bipartite system described by the following density matrix: ρ = 1 2 { 0 0 A + + B + A 1 1 B } (5) with ± = 1 2 ( 0 ± 1 ). It is clear that the above density matrix is of the form given in (2), and yet the state is not classical due to the indistinguishability of the local states of the subsystems A and B. Phrased differently, the states 0(1) and (+) of the subsystem A(B) are not orthogonal, and hence not fully distinguishable. For the state of the system to be fully classical, we require that the state be left unchanged upon an attempt to learn about its properties. It is therefore clear that some non-classical correlations more general than entanglement are present in the state described by (5). The first attempt to account for this phenomenon resulted in defining a new type of quantum correlation, namely quantum discord. Discord in discrete systems 9 This is obviously in contrast to entangled states, which cannot be created in this way. Quantum discord was originally defined as a non-zero difference between two classically equivalent expressions for mutual information. In classical information theory, mutual information J(A : B) is a measure of classical correlations between two random variables A and B: how much can we infer about A knowing B? It is defined in terms of Shannon entropy H(X) = x p X=x log 2 p X=x, where the sum is taken over all members of the set X with the elements of the set denoted by x, and p X=x is the probability of obtaining a particular outcome x from X. Shannon entropy quantifies our uncertainty about these variables: the larger its value, the less we know about the system. Classical mutual information is defined as: J(A : B) = H(A) H(A B) (6) where H(A B) = b p B=bH(A B = b) is the conditional entropy of A knowing B it measures the average uncertainty that remains about A when B is given (here the probability distribution for the system is used to derive the corresponding probabilities for the subsystems, e.g. p A = b p A,B=b). Equation (6) states that the amount of correlations between the variables A and B is equivalent to the uncertainty 9

10 about A given B being subtracted from the uncertainty about A exclusively. Bayes rule can be used to show that H(A B) = H(A, B) H(B), where H(A, B) is the joint entropy of the system having the property H(A, B) = H(A) + H(B A) = H(B) + H(A B). Substitution of the former into (6) results in an alternative expression for mutual information, which reads: I(A : B) = H(A) + H(B) H(A, B) (7) Classically, J(A : B) = I(A : B). However, it should be expected that this will not be true in the quantum case due to the non-commutativity incorporated in the terms H(A B) (and H(B A)). In classical physics, a measurement on the subsystem B can be made without the overall state of the system, and in particular that of the subsystem A, being changed (or disturbed). However, in quantum mechanics a measurement of one quantity in general disturbs the system and changes its state, thereby influencing the outcome of a measurement of the second quantity of interest. The value of the conditional entropy in such a case is going to depend on the outcomes of different measurements done on the subsystem B followed by the measurements on the subsystem A, as different measurements performed on B can disturb A to a different extent. It is straightforward to generalise (7) to the quantum case: Shannon entropy is replaced with von Neumann entropy, and the probability distributions with the corresponding density matrices so that H(A(B)) H(ρ A(B) ) = Trρ A(B) log 2 ρ A(B), and H(A, B) H(ρ A,B ) = Trρ A,B log 2 ρ A,B. The expression for quantum mutual information is therefore: I(ρ A,B ) = H(ρ A ) + H(ρ B ) H(ρ A,B ) (8) A similar extension of the expression (6) involves the generalisation of the term H(A B). For simplicity, let us consider only perfect measurements [58] defined by the complete set of orthonormal projection operators {Π B k }, where the subscript k keeps track of different possible outcomes of the measurement. Therefore, after the measurement on B is performed, the density matrix for the whole system reads ρ A Π B = Π B k k ρ A,BΠ B k /p k, where p k = Tr A,B Π B k ρ A,B is the corresponding value of the probability. The conditional entropy is then H(ρ A Π B), which for the k complete measurement takes the form H(ρ A {Π B k } ) = k p kh(ρ A Π B). Therefore the quantum analogue of k J(A : B) can be written as: J(ρ A,B ) = H(ρ A ) H(ρ A {Π B k } ) (9) The value of J(ρ A,B ) will depend on the set of projection operators {Π B k } chosen for the measurement. It can be shown that J(ρ A,B ) maximised over all possible sets of projection operators is a measure of the total classical correlations C(ρ A,B ) of the given state (to be more precise, classical correlations C(ρ A,B ) are defined as the maximum information that can be gained about the subsystem A upon the measurement of the subsystem B) [59]: C(ρ A,B ) max {Π B k } [J(A : B) {Π B k } ] (10) Knowing the total amount of correlations in the system measured by the quantum mutual information in (9) and the total amount of classical correlations in (10), the total amount of purely quantum correlations can be defined as: D(ρ A,B ) I(ρ A,B ) C(ρ A,B ) (11) The quantity D(ρ A,B ) is known as quantum discord. The set of projection operators which minimizes D(ρ A,B ) corresponds to the measurements which disturb the state of the system to the smallest extent, and allow for the extraction of the maximum information about the system. Extension to continuous variable systems In order to evaluate quantum discord, an optimisation procedure over all of the possible sets of measurements for a given subsystem needs to be performed. It is clear that such a task is in general very difficult to do, which is why discord was originally defined in terms of orthogonal projective measurements for finite dimensional systems. Both the discord and the total amount of classical correlations as defined in [59] have been extended to continuous variable systems recently [60, 61], where the orthogonal projective measurements are discarded in favour of the generalised Gaussian positive operator valued measures (POVMs). The generalisation of discord to CV systems would involve introducing elements of measurement theory in quantum mechanics [62], and is beyond the scope of this review. Dynamics of discord in discrete variable systems The field of discord dynamics in discrete bipartite systems coupled to an external environment has been studied quite extensively, however, interesting developments in the topic were done in the last two years. 10

11 There are currently very few studies on discord dynamics in CV systems, which I have described in detail after introducing the interesting observations seen in DV systems. Robustness of discord against decoherence In general, discord is found to be more robust (or resilient) against decoherence than entanglement [63, 64, 65, 66]. A few comparative studies of the time evolution of these two correlations were performed, revealing that in the case of two independent qubits in quite a large number of different separate reservoirs obeying Markovian dynamics, discord decays asymptotically [63], while entanglement undergoes sudden death [33]. From these examples it is clear that discord is a more general type of quantum correlation, which is present even when the entanglement has decayed to zero. Discord sudden birth When two qubits prepared in an initially classical state are placed in a common reservoir [66], a phenomenon termed discord sudden birth can be observed. Interestingly, while the discord is non-zero for t 0, in exactly the same conditions the qubits never become entangled (Fig. 6). Similar effects were also shown to arise in other cases of Markovian dynamics [64]. A generalisation of this (and alike) studies on the influence of Markovianity on quantum discord indicated that a general Markovian heat bath can never cause the discord to vanish permanently for any state with non-zero discord (the obvious exception is the case of t ) [67]. It was also proven that (a) nearly all quantum states have non-zero discord, (b) the states which are classical, and therefore lack discord, are negligible in Hilbert space, and (c) any state with zero discord can be turned into a non-classical state provided a small perturbation is introduced. This study, while showing that discord is expected to be present in most cases, also exhausts the question of discord dynamics in a Markovian setting. It also implies that it is much easier to create discord than entanglement in discrete bipartite systems. Instantaneous disappearance versus sudden death Non-Markovian studies showed that quantum discord between two independent qubits in separate reservoirs at T = 0K exhibits a phenomenon termed instantaneous disappearance [65]. The authors avoid using the term discord sudden death, since discord Figure 6: Discord dynamics as a function of the rescaled time t for the initial state of the qubits being Ψ = Adapted from [66] Figure 7: Discord and entanglement measured by concurrence (blue dashed and solid black lines) for a specific initial state of the two qubits. The points of zero discord are circled. The revival of discord at later points in time is due to the memory effects of the environment. Adapted from [65]. vanishes only instantaneously at certain points in time (Fig. 7). However, a more realistic study [66] indicated that instantaneous disappearance really takes the more familiar form of sudden death for T 0K. The study considered two independent qubits placed in separate reservoirs obeying non-markovian dynamics, and showed that discord (like entanglement) decays to zero only to increase again at a certain later point in time. The periods of zero discord were shown to be shorter than those of zero entanglement, thereby implying once more that discord is a more general type of non-classical correlation. Sudden change dynamics and frozen discord Yet another interesting result was presented in the work of Maziero et al., who were the first ones to show that under suitable conditions both quantum and classical correlations undergo sudden changes in dynamics [7]. Inspired by their work, Mazolla et al. investigated the decoherence of two qubits in separate depolarizing environments. In the Markovian scenario, a transition time t > 0 was discovered, with the property that for any time t < t discord does not change and classical correlations undergo a decay, while for t > t the opposite is seen [68]. 11

12 Figure 8: Discord and entanglement dynamics measured with concurrence (blue solid and dashed black lines). It is clear that even when entanglement has decayed completely, under certain conditions discord remains constant. Adapted from [68]. Figure 10: Discord dynamics in the case of the common reservoir for a fixed photon number N and various values of the squeezing parameter r. r = 0, 0.2, 0.4 and r = 1 correspond to solid blue, dashed purple, dashed dotted citrine, and dotted green lines, respectively. Adapted from [69]. More interestingly, discord was found to be constant and non-zero while entanglement decayed to zero this phenomenon was termed frozen discord (Fig. 8). Due to the finite memory effects of the depolarizing channel in the non-markovian case, either (a) a series of transition times was observed, or (b) the so-called sudden change dynamics takes place [70], whereby the correlations are never constant, but undergo abrupt changes instead (Fig. 9). This is quite a remarkable result, as it shows that not only is discord more frequently encountered in the quantum world, but it can also be kept constant for certain periods of time, perhaps allowing for its implementation in quantum information protocols without the need to worry about its degradation by environment. Dynamics of discord in continuous variable systems Figure 9: Correlation dynamics of the system in question with mutual information, classical correlations and quantum discord depicted as orange dotted, red dashed, and blue solid lines, respectively. Both (a) the frozen and (b) sudden change dynamics are shown. Arrows emphasise the sudden changes. Adapted from [68]. Due to the definition of discord having been extended to continuous variable systems only recently [60, 61], there are currently only a few case studies focusing on discord dynamics of two mode Gaussian states in the literature [69]. Here the time evolution of discord was investigated for two identical non-interacting harmonic oscillators placed in both separate and common reservoirs. Weak non-markovian coupling between the oscillators and the environments was considered. For initially uncorrelated states in the separate reservoir scenario, the rate of increase of discord falls off for increasing average photon number N (set to be the same for both oscillators). At the same time, for the initially correlated states the discord decays exponentially and its decrease can be slowed down 12