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Environment-Induced Decoherence and the Transition From Quantum to Classical

We study dynamics of quantum open systems, paying special attention to those aspects of their evolution which are relevant to the transition from quantum to classical. We begin with a discussion of the conditional dynamics of simple systems. The resulting models are straightforward but suffice to illustrate basic physical ideas behind quantum measurements and decoherence. To discuss decoherence and environment-induced superselection einselection in a more general setting, we sketch perturbative as well as exact derivations of several master equations valid for various systems. Using these equations we study einselection employing the general strategy of the predictability sieve. Assumptions that are usually made in the discussion of decoherence are critically reexamined along with the ``standard lore'' to which they lead. Restoration of quantum-classical correspondence in systems that are classically chaotic is discussed. The dynamical second law -it is shown- can be traced to the same phenomena that allow for the restoration of the correspondence principle in decohering chaotic systems (where it is otherwise lost on a very short time-scale). Quantum error correction is discussed as an example of an anti-decoherence strategy. Implications of decoherence and einselection for the interpretation of quantum theory are briefly pointed out.


Introduction and overview
The quantum origin of the classical world was so difficult to imagine for the forefathers of quantum theory that they were often willing to either postulate its independent existence (Bohr), or even to give up quantum theory and look for something with more fundamental classical underpinnings (de Broglie, and, to a lesser extent, also Einstein). The source of the problem is the quantum principle of superposition, which, in effect, exponentially expands the set of available states to all of the conceivable superpositions. Thus, coherent superpositions of dead and alive cats have -in the light of the quantum theory-the same right to exist as either of the two classical alternatives. Within the Hilbert space describing a given system "classically legal" states are exceptional. The set of all states in the Hilbert space is enormous as compared with the size of the set of states where one finds classical systems. Yet, it is a fact of life that classical objects are only found in a very small subset of all possible (and in principle, allowed) states. So, one has to explain the origin of this apparent "super-selection" rule that prevents the existence of most states in the Hilbert space of some physical systems. Decoherence and its principal consequence -environment-induced super-selection or einselection-account for this experimental fact of life.
Decoherence is caused by the interaction between the system and its environment. Under a variety of conditions, which are particularly easy to satisfy for macroscopic objects, it leads to the einselection of a small subset of quasi-classical states from within the enormous Hilbert space. The classicality is then an emergent property, induced in the system by its interaction with the environment. Arbitrary superpositions are dismissed, and a preferred set of "pointer states" emerges. These preferred states are the candidate classical states. They correspond to the definite readings of the apparatus pointer in quantum measurements, as well as to the points in the phase space of a classical dynamical system.
The role of the process of decoherence in inducing classicality has become clear only relatively recently -within the past two decades. The key idea is relatively simple: An environment of a quantum system can, in effect, monitor its states through continuous interaction. The imprint of the system left on the environment will contain information about selected states of the system. The states that leave the imprint without getting perturbed in the process are the preferred states. Thus, the key property of quasi-classical pointer states is their insensitivity to monitoring by -and consequently their resistance to the entanglement caused by-interaction with the environment: states that entangle least are most stable. They are also, almost by definition, the only states that remain an accurate description of the the system alone: All other states evolve into joint system-environment states, preserving their purity (and, consequently, the information the observer has about them) only when both the system and the environment are included in a larger "super system".
The fact that the interaction between quantum systems produces entanglement was well known almost since the beginning of quantum theory. Indeed, because the ideas of decoherence and einselection rely on quantum theory, and on quantum theory alone, it may be useful to ask why it took so long to arrive at a natural explanation of the quantum origins of classicality. There are several possible explanations for this delay. We shall return to them later in the paper. But, for the moment, it is useful to note that the ability of environment-induced decoherence to result in the same set of preferred states, essentially independently of the initial state of the system and the environment, is crucial. This was not appreciated until relatively recently [1,2]. It is precisely this stability of the set of preferred states that allows them to be regarded as good candidates for the quantum counterparts of classical reality. Indeed, only still more recent research on the predictability sieve has allowed for more fundamental and general understanding of the emergent classicality (see [3,4] and also [5]).
The prejudice that seems to have delayed serious study of the role of the "openness" of a quantum system in the emergence of classicality is itself rooted in the classical way of thinking about the Universe. Within the context of classical physics, all fundamental questions were always settled in the context of closed systems. The standard strategy to ensure isolation involved enlarging a system -i. e., by including the immediate environment. The expectation was that in this manner one can always reduce any open system to a larger closed system. This strategy does indeed work in classical physics, where the enlargement can help in satisfying conservation laws for quantities such as energy or momentum. It fails in the quantum case under discussion, because now it is the information (about the state of the system) that must be prevented from spreading. Information is much harder to contain when the system in question becomes larger. Thus, in the end, the only truly isolated macroscopic system is the Universe as a whole. And we, the observers, are certainly not in a position to study it from the outside.
In what sense is the preferred set of states preferred? It is clear that generic superposition of the members of this preferred set will decay into mixtures. On the other hand, if the initial state is just one of the members of the preferred set, the temporal evolution will minimally affect the state, which will resist becoming entangled with the environment. Einselection can thus be thought of as a process by which a "record" of the state of the system is created dynamically (through interaction) in the state of the environment. It is this ongoing process by which the system is being continuously monitored by the environment that leads to the emergence of a natural set of preferred states that are the least affected by the interaction.
As sketched above, the physical principles of decoherence and einselection appear, in retrospect, rather straightforward. How much can be accomplished by exploring their consequences? There are several interesting and important questions that naturally arise in this context and that have been asked (and answered, in most cases) over the last two decades. First, one naturally asks how much can we explain with these ideas (i.e., is it consistent to think that all objects that are known to behave classically are doing so because of decoherence?). A closely related question is the one concerning natural time-scales associated with decoherence. How fast does decoherence take place? This is a very important question because a first look at the decoherence process may leave us wondering if decoherence may be consistent with the existence of a "reversible" classical world. Thus, if one believes that classicality is really an emergent property of quantum open systems one may be tempted to conclude that the existence of emergent classicality will always be accompanied by other manifestations of openness such as dissipation of energy into the environment (this would be a problem because, as we know, there are many systems that behave classically while conserving energy). Second, one also wonders how, in detail, is the preferred set of states dynamically selected through the interaction with the environment. In particular, it is interesting to know how this pointer basis is determined by the structure of the interaction Hamiltonian between system and environment and/or to the other details of the physics involved. Third, a related question arises in this context: are there observable manifestations of decoherence other than einselection?
A remarkable characteristic of the current debates on the nature of the quantum to classical transition and on the problem of quantum measurement is that for the first time in history there have been actual experiments probing the boundary between the quantum and the classical domains in a controlled way [6][7][8][9][10]. Controlled decoherence experiments (which are very difficult because nature provides us with classical or quantum systems but not with objects whose interaction with the environment can be controlled at will) were recently carried on for the first time and help us in understanding the nature of this process. Some of the most notable experiments in this area were performed at the Ecole Normale Superieure in Paris and are be part of Dr. Brune's lectures.
Our lectures start with an introduction to quantum conditional dynamics using two-state systems. Conditional dynamics is responsible both for setting up the problem of measurement, and for the decoherence and einselection that solve it. The resulting models are straightforward and can serve in the idealized studies of the measurement process. However, they are clearly too simple to be realistic -classicality is, after all, a property of essentially every sufficiently macroscopic object. To discuss decoherence and einselection in this more general setting, we shall therefore study dynamics of quantum open systems. Section 3 is devoted to the derivation of the key tool -a master equation for the reduced density matrix. This basic tool is immediately used in section 4, where environment-induced superselection is studied, including, in particular, the predictability sieve. Section 5 analyzes some "loose ends" -that is, essentially technical issues that are usually omitted in the derivations of the master equations. We show there that although the qualitative conclusions arrived at on the basis of the "naive" master equation approach are essentially unaffected by the detailed examination of some of the idealized assumptions that go into its derivation, quantitative estimates can change quite significantly when a more realistic approach is adopted. Section 6 is devoted to the study of the effect of decoherence on the quantum-classical correspondence in systems that are classically chaotic. We show there that decoherence not only explains the origin of classical dynamics, but that it may be responsible for the loss of information that accounts for the second law of thermodynamics as well. Section 7 is devoted to quantum error correction -to the strategies which can be used to suppress decoherence. The summary and conclusions are briefly stated in Section 8.

Quantum Measurements
In this section we shall introduce the measurement problem -the issue that has dominated the discussion of the relation between quantum and classical for a very long time. This will afford us the opportunity to study conditional dynamics that will be employed in one form or another throughout this review. Such interactions are necessary to achieve entanglement between quantum systems that set up the measurement problem. They are necessary for accomplishing decoherence, which leads to environment-induced superselection (or einselection), and thus resolves many of the problems arising on the border between quantum and classical. Last, but not least, quantum conditional dynamics and entanglement underlie quantum logic and will be of importance in the latter part of the paper devoted to quantum error correction.
Predictability is rightly regarded as one of the key attributes of classical dynamics. On the other hand, the defining feature of quantum mechanics is thought to be its probabilistic nature, which manifests itself in measurements. This discord between classical determinism and quantum randomness is often blamed for the difficulties with interpretation of quantum theory. Yet, the fundamental equations of either classical or quantum theory allow them -indeed, demand of them-to be perfectly predictable: It is just that what can be predicted with certainty, especially in the quantum case, cannot be often accessed by measurements. And, conversely, what can be measured in an evolving quantum system cannot usually be predicted, except in the probabilistic sense.
The Schrödinger equation allows one to predict the state of an isolated system at any subsequent moment of time. In an isolated quantum system, dynamical evolution is strictly deterministic. This perfect quantum predictability could be of use only if one were to measure observables that have the resulting evolving state as one of its eigenstates. These observables are generally inaccessible to reasonable measuring devices, and therefore are of no interest.
Quantum determinism is of little use for an observer who is only a part of the whole system. The overall quantum determinism could have predictive power only for someone who is (i) monitoring quantum systems from the outside. Moreover, it would help if the observer was endowed with (ii) enough memory to measure and store data, and (iii) sufficient ability to compute and to model deterministic evolution of the system of interest. For an observer trapped inside of the quantum universe, this is obviously not the case.
The universe is all there is. Therefore, by definition, it is a closed quantum system. Given the deterministic nature of the Schrödinger evolution, one may be surprised that there is a problem with the interpretation of quantum theory. After all, the interpretational ideal often mentioned in such discussions is deterministic Newtonian dynamics. However, the interpretation problem stems from the fact that deterministic unitary evolution of quantum theory is incompatible with classical determinism. Indeed, as the studies of chaotic systems demonstrate, classical dynamics has more room for randomness than quantum physics.
States of the quantum systems are perturbed by the very act of monitoring them. The elemental unpredictability associated with the act of observation cannot be avoided unless the observer knows in advance which observables can be measured with impunity. This feature of quantum information is essential to guarantee the security of quantum cryptography (see lectures of A. Ekert; also [11]) -the state of a quantum system cannot be found out by the eavesdropper unless the observation is carried out on the same basis as the one used by the intended recipient of the message. The "no cloning theorem" [12,13] prevents duplication of quantum information -amplification is associated with breaking the symmetry associated with the superposition principle.
Environment-induced superselection rules allow the observer to be a succesfull eavesdropper, and to extract useful information from the quantum systems without the environment getting in the way because (in contrast to the strategies employed in quantum cryptography) the measurements carried out by the environment are restricted to few observables. The state of the system is therefore of necessity "precollapsed" and commutes with these observables. Further measurements carried out by the observer will only reveal (rather than perturb) the pre-existing state of affairs. Thus, environment-induced decoherence supplies a justification for the persistent impression of "reality". In contrast to the observables encountered in the microscopic realm, macroscopic quantum systems can appear only in one of the preselected (pointer basis) set of quantum states. The "collapse of the wavepacket" viewed in this way is just a familiar classical process of finding out which of the possible outcomes has actually occurred. The danger of interference between the alternatives was eradicated by decoherence long before the observer became involved.
How can one ever hope for a resolution that would allow for the familiar combination of classical determinism and classical randomness to emerge? At the risk of anticipating results that will be justified in detail only later, we note that quantum determinism may be relevant only for an observer who knows the initial state of an isolated quantum system. For a quantum observer immersed in a quantum universe this is a very rare exception, attainable only in carefully controlled laboratory experiments, and only for rather small quantum systems. The information capacity, memory, and information processing abilities of an observer that is a (macroscopic, yet comparatively small) subsystem of the universe are miniscule compared to the task of simulating even a small quantum system, let alone the universe as a whole. And as soon as the idea of the observer knowing the entire state of the universe is recognized as not feasible, "environmental monitoring" of both the state of the observer and of the observables he recorded begins to matter. An observer with decohering memory can keep reliable records only in the einselected states of his/her memory bits [3,[14][15][16][17]. Records will have predictive power only when they correlate with the einselected observables in the rest of the universe.

Bit-by-bit measurement and quantum entanglement
This problem of transition from quantum determinism to classical definiteness is illustrated most vividly by the analysis of quantum measurements. An answer to a "generic" question about the state of a quantum system (and the outcome of a measurement of the corresponding observable) is not deterministic. In the usual textbook discussions, this random element is blamed on the "collapse of the wavepacket" that is invoked whenever a quantum system comes into contact with a classical apparatus. In a fully quantum discussion of the problem, this issue still arises, in spite (or rather because) of the overall deterministic quantum evolution of the state vector of the universe. Indeed, as carefully pointed out by von Neumann [18] in his quantum analysis of measurements, there seems to be no room for "real collapse" in purely unitary models of measurements.
To illustrate the ensuing difficulties, we consider with von Neumann a quantum system S initially in a state |ψ interacting with a quantum apparatus A initially in a state |A 0 . The interaction will generally result in an entangled final state, (2.1) Here {|A i } and {|s i } are states in the Hilbert spaces of the apparatus and of the system, respectively, and a i are complex coefficients. This transition can be accomplished by means of a unitary Schrödinger evolution. It leads to an uncomfortable conclusion. All that an appropriate interaction between A and S can achieve is putting the measuring apparatus (or an observer) in an EPR-like entangled state of all the possible outcomes consistent with the initial state [1]. Operationally, this EPR-like nature of the state emerging from the pre-measurement (as the step achieved by Eq. (2.1) is often called) can be made more explicit by rewriting the sum in a different basis All we have done is use an alternative basis for both the apparatus and the system, exploiting the freedom of choice guaranteed by the quantum principle of superposition. Therefore, if one were to associate the state of the apparatus (observer) with a state in the decomposition of |Ψ t , then even before one could start enquiring about the specific outcome of the measurement one would have to decide what decomposition of |Ψ t is to be used, because the change of the basis corresponds to a redefinition of the measured quantity. One could make the clash between quantum and classical even more dramatic by making an additional measurement on the same quantum system after the premeasurement correlation is established. In accord with Eq. (2.2), such additional measurement would have a power to select an arbitrary observable of the system S and would single out the corresponding states of the apparatus A. Yet, given the freedom to rewrite |Ψ t in an infinite number of ways, this state of A would be for almost any choice of the decomposition of the sum of Eq. (2.2) completely "nonclassical" in any reasonable sense, and it would depend on the initial state of the quantum system.
In a quantum domain, such an entanglement must be commonplace, along with its disturbing consequences. Indeed, a "Schrödinger kitten" state recently implemented by means of an atomic physics experiment ( [19] is an excellent illustration of the distinction between the quantum entanglement and the classical correlation in the context of quantum measurements). The NIST group in Boulder has managed -manipulating a single ion inside a trap with lasers-to establish a correlation between its internal state (designated here by {| ↑ , | ↓ }, respectively, for "excited" and "ground") and its location (|L or |R for "left" or "right"). The final correlated wavefunction has a premeasurement, EPR-like form, are superpositions of the ground and excited states. This very same |Ψ A can be written therefore as Thus, the same correlated state of the "atom cat" can be expressed in two very different-looking ways, implying the potential for still more kinds of ambiguous correlations. Expressed in the first way (see Eq. (2.3)), the atom can be in one of the two alternative locations, depending on its internal state that is defined as a superposition of ground and excited states. In the second way (given by Eq. (2.5)) the natural internal states of the atom are correlated with a very nonclassical state -a superposition of an atom in two locations. Monroe et al. [19] measure the internal state of the atom in the basis corresponding to the decomposition of Eq. (2.5) and verify that it is indeed in a superposition of |L and |R with either a positive or a negative sign (an "even" or and "odd" Schrödinger cat). Given the atomic size of this "kitten", its ability to appear in a superposition of two different widely separated locations may or may not be a surprise. But the point this recent experiment allows us to make is at the heart of the interpretation problem. If the quantum laws are universally valid, very nonclassical Schrödinger cat-like states should be commonplace for an apparatus that measures a quantum system and, indeed, for run-of-the-mill macroscopic systems in general. One should be able to prepare such nonclassical states at will, by entangling arbitrarily large objects with quantum states of microscopic systems and then measuring these quantum objects in some arbitrary basis. If such sequences of events were common, classical objects would almost always be in very nonlocal superposition states.
Quantum theory mandates this pandemonium. Yet, we never seem to encounter it, least of all in the course of measurements. The task of the interpretation of quantum theory is to understand why. In the Copenhagen interpretation, this problem never arises, because the apparatus is by definition classical. However, if one insists on the universality of quantum theory, the difficulty described above is inevitable. It arises, for instance, in Everett's Many Worlds Interpretation, which was in fact originally called "the Relative State Interpretation" [20]. Everett and other followers of the MWI philosophy tried to occasionally bypass this question by insisting that one should only discuss correlations. Correlations are indeed at the heart of the problem, but it is not enough to explain how to compute them; for that, quantum formalism is straightforward enough. What is needed instead is an explanation of why some states retain correlations, but most of them do not, in spite of the arbitrariness in basis selection that is implied by Eq. (2.2). Or, equivalently, what is needed is an explanation of the loss of general quantum entanglement, but a selective retention of classical correlations -correlations that are also quantum in their origin, but which consistently single out the same basis of the quantum states violating the spirit of the superposition principle.

Interactions and the information transfer in quantum measurements
The interaction required to accomplish the correlation between the measured system and the apparatus, Eq. (2.1), can be regarded as a generalization of the basic logical operation known as a "controlled not" or a c-not. Classically, c-not changes the state of the target bit when the control bit is in a state 1, and does nothing otherwise: Quantum c-not is a straightforward quantum version of Eq. (2.6). It differs from the classical case only because arbitrary superpositions of the control bit and of the target bit are allowed Above a "negation" of a state |¬a t is a basis-dependent operation defined by It suffices to identify |A 0 = |0 t , and |A 1 = |1 t to have an obvious correspondence between the c-not and a premeasurement.
In the classical c-not the direction of the information transfer is always consistent with the designations of the two participating bits. The state of the control bit remains unchanged while it controls the state of the target bit, Eq. (2.6). Written in terms of the logical {|0 , |1 } basis, the truth table of the quantum c-not is essentially -that is, save for the possibility of superpositions-the same as Eq. (2.6). One might therefore anticipate that the direction of information transfer and the designations ("control/system" and "target/apparatus") of the two qubits will also be unambiguous, as they are in the classical case. This expectation however is incorrect, as can be readily demonstrated by expressing the process in the conjugate basis {|+ , |− } that, for either control or target bit, is obtained through the Hadamard transform: The truth That is, in the complementary basis {|+ , |− } the roles of the control and the target bit are reversed. The state of the former target -represented by the second ket in Eq. (2.10)-remains unaffected in the new basis, and the state of the former control is conditionally "flipped".
In the above c-not (or bit-by-bit measurement), the appropriate interaction Hamiltonian is Above, g is a coupling constant, σ i are Pauli matrices, and the two operators refer to the system (i.e., to the former control), and to the apparatus pointer (the former target), respectively. It is easy to see that the states {|0 , |1 } S of the system are unaffected by H int , because Thus, the measured (control) observableǫ = e 0 |0 0| + e 1 |1 1| is a constant of motion under the evolution generated by H int . The states {|+ , |− } A of the apparatus (which encode the information about the phase between the logical states) have exactly the same "immunity" [ Hence, when the apparatus is prepared in a definite phase state (rather than in a definite pointer/logical state), it will pass its phase onto the system, as the truth  14) which, in comparison with Eq. (2.11), makes this "immunity" obvious. This basis-dependent direction of the information flow in a quantum c-not (or in a premeasurement) is a direct consequence of complementarity. It can be summed up by stating that although the information about the observable with the eigenstates {|0 , |1 } travels from the measured system to the apparatus, in the complementary {|+ , |− } basis it seems to be the apparatus that is being measured by the system. This observation also clarifies the sense in which phases are inevitably "disturbed" in measurements. They are not really destroyed, but, rather, as the apparatus measures a certain observable of the system, the system "measures" the phases between the possible outcome states of the apparatus. These phases in a macroscopic apparatus coupled to the environment are fluctuating rapidly and uncontrollably, thus leading to the destruction of phase coherence. However, even if this consequence of decoherence were somehow prevented (i.e., by perfectly isolating the apparatus pointer from the environment), preexisting phases between the outcome states of the apparatus would have to be known while, simultaneously, A is in |A 0 , the "ready-to-measure-state". This would require a simultaneous knowledge of the two non-commuting observables, and is therefore impossible because of Heisenberg indeterminacy.
It appears that even the question "which of the two interacting systems is a measuring device?" (which should be decided by the direction of the flow of information) depends on the initial states. In "classical practice" this ambiguity does not arise because the initial state of the apparatus can never be selected at the whim of the observer. Einselection limits the set of possible states of the apparatus to a small subset of all the states available in Hilbert space.

Monitoring by the environment and decoherence
In this section, we shall see how the quantum-classical correspondence can be reestablished by decoherence and einselection, caused by the monitoring of the to-be-classical observables by the environment. The environment is defined as any set of degrees of freedom that are coupled to the system of interest, and which can therefore 'monitor' -become entangled with-its states. Environments can be external (such as particles of air or photons that scatter off, say, the apparatus pointer) or internal (e.g., collections of phonons or other excitations in the materials from which an apparatus is constructed). Often, environmental degrees of freedom emerge from the split of the original set of degrees of freedom into the "system of interest" that is some collective observable (order parameter in a phase transition), and the "microscopic remainder".
The superposition principle applies only when the quantum system is closed. When the system is open, interaction with the environment will inevitably result in an incessant "monitoring" of some of the observables by the environmental degrees of freedom. This will result in the degradation of the pure states into mixtures. These mixtures will often -remarkably often-turn out to be diagonal in the same set of "preferred states" that are nearly independent of the initial state of the system and of the environment, but which are selected with the crucial help of the interaction Hamiltonian. This decoherence process determines the relative "fitness" of all the possible superpositions that exist in the Hilbert space. The resulting "natural selection" is responsible for the emergence of classical reality. Its consequence is known as environment-induced superselection [2], or einselection.
The set of habitually decohering states is often called "the pointer basis", in recognition of its role in the measurement problem. The criterion for the selection of pointer states goes well beyond the often-repeated characterizations based solely on the instantaneous eigenstates of the density matrix. What is of the essence is the ability of the einselected states to survive monitoring by the external degrees of freedom. This heuristic criterion can be made rigorous by quantifying the predictability of the evolution of the candidate classical states or of the associated pointer observables. To put it succinctly, measurement of the pointer observables yields an optimal initial condition. In spite of the openness of the system, its results can be employed for the purpose of prediction better than the other Hilbert space alternatives.
The contrast between the resilience of the states associated with the preferred (pointer) observables and the fragility of their superpositions can be analyzed in terms of Heisenberg's principle of indeterminacy. The environment monitors observables with the accuracy dictated by the interaction Hamiltonian. Thus, only a measurement that happens to commute with the observables monitored by the environment will result in a useful record that can be successfully employed for the purpose of prediction. In contrast, a system prepared by the measurement in an arbitrary superposition will also be monitored by the environment, which will tend to correlate with the pointer observable. When the initial superposition prepared by the observer does not commute with the observables monitored by the environment, Heisenberg's indeterminacy implies that the records of the observer are of no use for the purpose of prediction. The monitoring continuously carried out by the environment on the pointer observables makes anything except for the pointer states a poor choice.
Three quantum systems -the measured system S, the apparatus pointer or the memory of the observer A, and the environment E-and the correlations between them will be the subject of the discussion below. In quantum measurements, S and A will be coupled. Their quantum entanglement will be converted into an effectively classical correlation as a result of the interaction between A and E. In measurements of classical systems, both S and A will interact with E and decohere. In either case, states einselected by the environment will be the focus of attention. In A, they will be the repository of information, serving as pointer states of the apparatus or memory states of the observer. The system S can also look effectively classical when it is subject to einselection, and when A keeps records of its einselected states. This SAE triangle (or a triangle much like it) is necessary for careful study of decoherence and its consequences. By keeping all three corners of this triangle in mind, one can avoid the confusion about the relation of the instantaneous eigenstates of the density matrix (see, for example, the discussion following [21]). This three-system context is necessary to keep track of the correlations between the memory of the observer and the state of the measured system. The evolution from a quantum entanglement to the classical correlation may be the easiest relevant theme to define operationally. In spite of this focus on the correlation, I shall often suppress one of the corners of the above triangle to simplify the equations. All three parts of the triangle will however play a role in formulation of the questions we shall pose and in motivating of the criteria for classicality that we shall devise.

One-bit environment for a bit-by-bit measurement
The simplest discussion of a single act of decoherence involves just three onebit systems [1,22]. They are denoted by S, A, and E in an obvious reference to their designated roles. The measurement starts with the interaction of a measured system with the apparatus, where A 0 |A 1 = 0. For a general state, These formulae are an example of a c-not-like premeasurement that has already been discussed. As was noted previously, a correlated state of this form is not enough to claim that a measurement has taken place. The biggest problem with |Φ is the basis ambiguity. Equation (2.16) represents only an entanglement, the same as in Bohm's version of the EPR state [23]. The ambiguity of the basis selection in this simple example can be settled with the help of one additional system, E, which performs a premeasurement on the apparatus. As a result, It may seem that very little can be accomplished by repeating the step that has led to the S − A correlation and the associated problems. But this is not the case. A collection of three correlated quantum systems is no longer subject to the basis ambiguity we have pointed out in connection with the EPR-like state |Φ , Eq. (2.16). This is especially true when the states of the environment are correlated with the simple products of the states of the apparatus-system combination [1]. In Eq. (2.17) above, this can be guaranteed (irrespective of the value of α and β) providing that: When this condition is satisfied, the description of the A − S pair can be readily obtained in terms of a reduced-density-matrix: This reduced-density-matrix contains only terms corresponding to classical correlations.
If the condition of Eq. (2.18) did not hold -that is, if the orthogonal states of the environment were not correlated with the apparatus in the basis in which the original premeasurement was carried out-then the terms on the diagonal of the reduced density matrix ρ AS would be the sum of products rather than simply products of states of S and A. An extreme example of that situation is the pre-decoherence density matrix of the pure state: Its eigenstate is simply |Φ . When expanded, |Φ Φ| contains terms that are off-diagonal when expressed in the natural basis consisting of tensor products of states in the two subspaces. Their disappearance as a result of tracing over the environment signals the disappearance of the basis ambiguity. There is of course a conceptual difference with the classical case. In classical mechanics, it was in principle possible to imagine that the outcome was predetermined. In quantum mechanics this is usually impossible even in principle. However, that distinction can be made only with a more complete knowledge than the one typically available to the observer. The pointer observable that emerges from this simple case is easy to characterize. The interaction Hamiltonian between the apparatus and the environment, H AE , should have the same structure as for the c-not. It should be a function of the pointer observablê of the apparatus. Consequently, the states of the environment will bear an imprint of the pointer states {|A 1 , |A 0 }. As was also noted in the discussion of c-nots, [H AE ,Â] = 0 immediately implies thatÂ is a control, and its eigenstates will be preserved. Disappearance of quantum coherence because of a "one-bit" measurement has been verified experimentally in neutron and, more recently, in atomic interferometry [24][25][26]. A single act of quantum measurement we have discussed here should be regarded as an elementary discrete instance of continuous monitoring, which is required to bring about the appearance of classicality.

Decoherence of a single (qu)bit
Another example of decoherence is afforded by a two-state apparatus A interacting with an environment of N other spins [2]. We can think of it as just another two-state system, and, in that spirit, we shall identify in this section the two apparatus states as {| ⇑ , | ⇓ }. The process of decoherence is definitely not limited to states of the apparatus pointers, so these two generic candidate pointer states can belong to any system.
The simplest, yet already quite illustrative example of this situation occurs when the self-Hamiltonian of the apparatus disappears, H A = 0, and the interaction Hamiltonian has the form: Under the influence of this Hamiltonian, the initial state Here: The reduced density matrix is then The coefficient r(t) determines the relative size of the off-diagonal terms. It is given by Unless k'th spin of the environment is initially in an eigenstate of the interaction Hamiltonian, its contribution to the product will be less than unity. Consequently, for large environments consisting of many (N ) spins and at large times the off-diagonal terms are typically small, This effect can be illustrated with the help of the Bloch sphere. The density matrix of any two-state system can be represented by a point in the 3-D space. In terms of the coefficients a, b, and r(t) that we have previously used, the coordinates of the point representing ρ(t) are: z = (|a| 2 − |b| 2 ), x = ℜ(ab * r), and y = ℑ(ab * r). When the state is pure, x 2 + y 2 + z 2 = 1 -pure states lie on the surface of the Bloch sphere ( Figure 1). When the state is mixed, the point representing it lies inside that sphere. Any conceivable (unitary or non-unitary) quantum evolution of the two-state system can be thought of as a transformation of the surface of the pure states into the ellipsoid contained inside the Bloch sphere. Deformation of the Bloch sphere caused by decoherence is a special case of such general evolutions. The decoherence process does not affect a or b. Hence, evolution caused by decoherence alone occurs in a constant-z plane. Such a "slice" through the Bloch sphere would show the point representing the state at a fraction |r(t)| of its maximum distance. The complex number r(t) can be expressed as the sum of the complex phase factors rotating with the frequencies given by differences ∆ω j between the energy eigenvalues of the interaction Hamiltonian, weighted with the probabilities of finding these energy eigenstates in the initial state, The index j now denotes partial energy eigenstates of the environment of the interaction Hamiltonian (tensor products of ↑ and ↓ states of the environmental spins). The corresponding eigenvalue differences between the two complete energy eigenstates | ⇑ |j and | ⇓ |j are There are 2 N distinct states |j , and, barring degeneracies, the same number of different ∆ω j 's. The probabilities p j are given by which, in turn, is easily expressed in terms of the appropriate squares of the products of α k and β k . The evolution of r(t) given by Eq. (2.29) is a consequence of the rotations of the complex vectors p k exp(−i∆ω j t) with different frequencies. The resultant r(t) will then start with the amplitude 1 and quickly "crumble" to a value approximately equal to (2.32) Fig. 1. Decoherence can be seen in the Bloch sphere as the process that induce the states to "move towards the vertical axis", which is defined by the two pointer states on the poles. Classical domain consists of just two pointer states. The classical core is the set of all mixtures of pointer states.
In this sense, decoherence is exponentially effective -the expected magnitude of the off-diagonal terms decreases exponentially fast with the physical size N of the environment-with the number of systems (spins in our example). In effect, any initial state asymptotically approaches the z-axis as a result of decoherence. We note that the effectiveness of einselection depends on the initial state of the environment: When E is in the k'th eigenstate of H AE , and p j = δ jk , the coherence in the system will be retained because the environment is now in an eigenstate of the "control". This situation is, however, unlikely in realistic circumstances because the self-Hamiltonian of the environment H E will not commute, in general, with H AE . Moreover, even when H E = 0, finding an environment in an energy eigenstate of the Hamiltonian seems extremely unlikely -the eigenvalues of such eigenstates are bound to be dense in large systems, and therefore they will be easily perturbed by the interaction with their environments. Furthermore, the 2 N partial eigenstates of the interaction Hamiltonian are exponentially rare among arbitrary superpositions.
The geometry of the flows induced by decoherence inside the Bloch sphere exhibits characteristics that are encountered in more general physical situations, involving decoherence in bigger Hilbert spaces as follows: (i) Domain of pure quasi-classical states consisting of all the einselected pointer states ({| ⇑ , | ⇓ } in our case). Pointer states are the pure states least affected (here, unaffected) by decoherence.
(ii) Classical core of probability distributions, i.e., all the mixtures of pointer states. In Figure 1 it corresponds to the section [-1,+1] of the zaxis.
(iii) The rest of the space -the rest of the volume of the Bloch sphereconsists of more general density matrices. As a result of decoherence, that part of the Hilbert space is "ruled out" by einselection.
Visualization of this decoherence-induced decomposition of the Hilbert space is still possible in the simple two-dimensional case studied here, but the existence of the elements (i)-(iii) is a general feature. It characterizes the emergence of classicality under all circumstances. We shall therefore appeal to the intuitions developed in the course of this discussion later. However, it may be useful to anticipate a few of the phenomena that can take place when decoherence combines with the evolution induced by the self-Hamiltonian of the system or when it is caused by more complicated couplings to the environment.
(a) Approach to equilibrium would affect elements diagonal in the pointer basis, so that the density matrix would asymptotically approach a timeindependent distribution (such as ρ ∼ 1 for a thermal equilibrium at infinite temperature or ρ ∼ | ⇓ ⇓ | for decay). This corresponds to a flow towards some specific point (i.e., the center or the "south pole" in the above two examples) within the Bloch sphere. However, when decoherence dominates, the flow would start somewhere within the Bloch sphere, and quickly (on the decoherence timescale) converge towards a point on the z-axis (the classical core). This would be followed by a much slower relaxation, a flow more or less along the z-axis (and therefore essentially within the classical core) on a relaxation timescale.
(b) Approximately reversible classical dynamics can coexist with decoherence when the self-Hamiltonian of the system can generate motions within the surfaces of constant entropy inside the classical core. In the case considered here, the core is one-dimensional and the subspaces of constant entropy within it are zero-dimensional. Therefore, it is impossible to generate continuous isentropic motion within them. In multidimensional Hilbert spaces with richer dynamics that are nearly isentropic, approximately reversible evolution is often possible and allows for the idealization of trajectories in the classical limit.
(c) A sharp distinction between the classical core and the rest of the Hilbert space is possible only in idealized situations (or in an even more idealized "mathematical classical limit", in whichh → 0, mass→ ∞, etc.). In realistic situations, all that will be required is a clear contrast between the rates of the entropy production between the inside and the outside of the classical core. We shall refine such criteria in the discussion of the predictability sieve -a criterion for the selection of the preferred pointer states, which in effect demands that the entropy production rate should be minimized for the einselected states. In the case discussed here, pointer states obviously satisfy this criterion, and the entropy production vanishes in the classical domain. In more general situations, we shall not be equally lucky. For instance, in the case of chaotic systems, entropy will also be produced in a classical core, but at a rate set by the classical dynamics (i.e., by the self-hamiltonian rather than by the coupling with the environment) and much more slowly than outside of the classical core.

Decoherence, einselection, and controlled shifts
The above discussion of decoherence can be straightforwardly generalized to the situation where the system, the apparatus, and the environment have many states, and where the interactions between them are much more complicated. Here we assume that the system is isolated and that it interacts with the apparatus only briefly. As a result of that interaction, the state of the apparatus becomes entangled with the state of the system, ( i α i |s i )|A 0 → i α i |s i |A i . By analogy with a c-not, we shall refer to this conditional operation as a c-shift. This quantum correlation suffers from the basis ambiguity we have discussed previously: The S − A entanglement implies that for any state of either of the two systems there exists a corresponding pure state of the other. Indeed, when the initial state of S is chosen to be one of the eigenstates of the conjugate basis, |r l = N − 1 2 N −1 k=0 exp(2πikl/N )|s k , this c-shift would equally well represent a measurement of the apparatus state (in the basis conjugate to {|A k }) by the system [27]. Thus, it is not just the basis that is ambiguous, but also the roles of the control (system) and of the target (apparatus) can be reversed when the conjugate basis is selected. These ambiguities that exist for the SA pair can be removed by recognizing the role of the environment.
Decoherence is represented schematically in Figure 2 by a sequence of c-nots (or c-shifts) which, in some fixed basis, 'measure' the state of the apparatus and record the outcome of the measurement in the environment. The requirement for a good apparatus is to retain correlations between the measured observable of the system and some "pointer observable". This will happen when the c-shift between S and A correlates the state of the system with the observable of the apparatus that is itself monitored (but not perturbed) by the environment. That is, in an idealized measurement, the measured observable of the system is playing the role of the control with respect to the S − A c-shift. In a well-designed apparatus, the pointer observable is a target of the S − A c-shift, but a control of the A − E c-shifts. Eigenstates of the pointer observable of the apparatus play the role of an alphabet of a communication channel. They encode a state of the system and retain the correlation in spite of the interaction with the environment.
The graph in Figure 2 captures the essence of the idealized decoherence process, which yields -in spite of the interaction with the environment-a noiseless classical communication channel [28,29]. This is possible because in the pointer basis, the A − E c-shifts operate without disturbing the pointer observable, which is the constant of motion of the A − E interaction Hamiltonian.
The advantage of the graphical representation of the decoherence process as a sequence of c-shifts lies in its simplicity and suggestiveness. However, the actual process of decoherence is usually caused by a continuous interaction (so that it can be only approximately broken up into discrete c-shifts). Moreover, in contrast to the c-nots used in quantum logic circuits, the record inscribed in the environment is more often than not distributed over many degrees of freedom. Last but not least, the observable of the apparatus (or any other open system) may be a subject to noise (and not just decoherence) or it may evolve in a manner that will rotate pointer states into their superpositions.
The basic physics of decoherence is a simple premeasurement-like process carried out by the environment E as a result of the interaction with the apparatus, Decoherence leads to the einselection when the states of the environment |ε j corresponding to different pointer states become orthogonal, When this orthogonality condition is satisfied, Schmidt decomposition of the state vector |Φ SAE into a composite subsystem SA and E yields product states |s j |A j as partners of the orthogonal environment states. The density matrix describing the correlated but decohered SA pair is then: The reduced density matrix of the SA pair is diagonal in the product states. For notational simplicity, we shall often discard reference to the object that does not interact with the environment (here, the system S). Nevertheless, it is useful to keep in mind that the preservation of the SA correlations is the criterion used to define the pointer basis. The density matrix of a single object evolving in contact with the environment will be always diagonal in the same (instantaneous) Schmidt basis. This instantaneous diagonality should not be used as a sole criterion for classicality (although see [31,32]; as well as [33,34]). Rather, the ability of certain sets of states to retain correlations in spite of the coupling to the environment is decisive in the emergence of "classical reality". This is especially obvious in quantum measurements.
When the interaction with the apparatus has the form the basis {|A k } is left unperturbed. Then, any correlation with the states {|A k } will be preserved. And, by definition, the states that preserve correlations will be the pointer states. Any observable A co-diagonal with the interaction Hamiltonian will be an effective pointer observable. For, when the Hamiltonian depends on A, it will commute with A, Moreover, the dependence of the interaction Hamiltonian on the observable is an obvious precondition for the monitoring of that observable by the environment.

Dynamics of quantum open systems: master equations
One of the most practical tools for analyzing the dynamics of a quantum open system is the evolution equation for the reduced density matrix, known as the "master equation". In this section we will review some of the most common techniques to obtain such an equation. As usual, we divide our universe into a system of interest S that interacts with an environment E.
The reduced density matrix of the system is the operator that allows us to answer all physical questions that concern the system S only. We will denote the reduced density matrix as ρ, which is obtained from the total density matrix of the universe by tracing over the environment Hilbert space. Thus, where the total density matrix is denoted as ρ T . In principle, the evolution equation for ρ could be obtained by solving Schrödinger (or von Neumann) equation for the total density matrix and then taking the trace. However, this task can be analytically completed in very few cases, and the study of the evolution of the reduced density matrix should be done by using some approximations.
This section is divided in two parts. First we review some of the standard techniques used to obtain approximate master equations. Our plan is not to give a complete review of master equation techniques but to present some useful tools to be applied later in studying decoherence. We do this not only to ensure that the paper is self-contained but also because we think it might be useful to present some simple and helpful results that are not so well known. We focus on the simplest approximation scheme, obtaining master equations valid to a second order in a perturbative expansion in the system-environment coupling strength. We first review the general perturbative scheme and apply it to two physically interesting examples: (1) The Brownian motion of a particle coupled to an environment of independent oscillators and (2) A quantum particle locally coupled to an environment formed by a quantum scalar field. As a further illustration of the way in which perturbative master equations can be obtained, we find the corresponding equations for a two-level system coupled to a bosonic bath in two physically relevant cases (the decay of a two-level atom and the spin boson model).
In the second part of this section we review the properties of an important model that is amenable to an exact solution. Thus, we concentrate on the linear quantum Brownian motion model analyzing the properties of its exact master equation. In particular, we stress the fact that in this simple but physically relevant model, the exact master equation has the same functional form as the one obtained using perturbation theory and can always be cast in terms of a local differential equation with time-dependent coefficients.

Master equation: Perturbative evaluation
Here we present the general procedure that can be used to derive the master equation, assuming that the system-environment coupling is small. Thus, we sketch a textbook derivation of the master equation using perturbation theory. We think it is convenient to present this derivation just to stress the fact that perturbative master equations can always be shown to be local in time. The calculation we follow is closely related to the one presented, for example, in [35] and can be seen to be a variant of the time-convolutionless method discussed in [36].
Let us consider the total Hamiltonian to be where H S and H E are respectively the self-Hamiltonian of the system and the environment and V is the interaction term. The equation for the complete density matrix ρ T , in the interaction picture, reads (we use a tilde to denote operators in the interaction picture), where the interaction potential and density matrix areṼ Solving equation (3.1) perturbatively is rather straightforward and leads to the Dyson series, We can use this to compute the reduced density matrix to second order. To obtain the master equation we compute the time derivative of the resulting expression and perform the trace over the environment. We geṫ So far, the only assumption we made was the validity of a perturbative expansion up to second order. Now we will assume that the initial state is not entangled, i.e., that the total density matrix is a tensor product of the To finish the derivation, we make a rather trivial observation that enables us to rewrite the master equation in a very simple way: The initial state ρ(0) that appears in the right-hand-side of equation (3.4) could again be expressed in terms ofρ(t) using the same perturbative expansion that enabled us to obtain (3.4). By doing this we can rewrite the right-hand-side of the master equation entirely in terms of the reduced density matrix evaluated at time t. The resulting equation iṡ This, when rewritten in the Schrödinger picture, is the basic master equation we will use in this section. It is important to keep in mind that to derive it, we only made two important assumptions: (a) we used a perturbative expansion up to second order in the system-environment coupling constant and (b) we assumed uncorrelated initial conditions.
Below, we will apply this equation to study three interesting examples. Before doing that, let us stress that the master equation is local in time even though to obtain it, no Markovian assumption was made (see below). Moreover, this rather simple form can be simplified further by assuming that the system-environment coupling is of the form where S n (E n ) are operators acting on the Hilbert space of the system (environment) only. In such case, the master equation in the Schrödinger picture can be written aṡ where the bracket notation indicates the expectation value over the initial state of the environment and the kernels K (i) nm are simply determined by the two time correlation functions of the environment as follows: At this point, it is interesting to consider another important approximation that is usually employed in this context, i.e., the Markovian approximation that we have refrained from using so far. The Markovian approximation corresponds to considering cases for which the kernels K (i) are strongly peaked about t = t 1 . When this is the case, i.e. when the environment has a very short correlation time, one can transform the temporal integrals into integrals over the variable τ = t − t 1 , which can then be extended over the entire interval [0, ∞). As we mentioned above, so far, we have not used the Markovian assumption and therefore the above equations are valid even if the environment has a long correlation time and the kernels K (i) are not strongly peaked. In the examples below, we will mention some cases where this happens and use the above equation to study decoherence produced by a non Markovian environment.
It is also worth mentioning that to go one step beyond equation (3.7), one needs to know the temporal dependence of the free Heisenberg operators of the system (i.e., S n (t)) which obviously depend on the Hamiltonian H S that we have not specified so far. We will do so in some concrete examples below.

Example 1: Perturbative master equation in Quantum Brownian Motion
The system of interest is a quantum particle, which moves in a one dimensional space (generalization to higher dimensions is immediate). The environment is an ensemble of harmonic oscillators interacting bilinearly through position with the system. Thus, the complete Hamiltonian is and V = n λ n q n x. The Hamiltonian of the system will be left unspecified for the moment (we will concentrate later on the case of a harmonic oscillator). The initial state of the environment will be assumed to be a thermal equilibrium state at temperature T = 1/k B β. Under these assumptions the first-order term in the master equation disappears because T r E (Ṽ (t)ρ E ) = 0. Therefore, the master equation in the Schröedinger picture iṡ (3.10) The two kernels appearing here are respectively called the noise and the dissipation kernel and are defined as where J(ω) = n λ 2 n δ(ω − ω n )/2m n ω n is the spectral density of the environment and N (ω) is the mean occupation number of the environmental oscillators (i.e., 1 + 2N (ω) = coth(βhω/2)).
Equation (3.10) is already very simple but it can be further simplified if one assumes that the system is a harmonic oscillator. Thus, if we consider the Hamiltonian of the system to be H S = p 2 /2M + M Ω 2 x 2 /2, we can explicitly solve the Heisenberg equations for the system and determine the operator x(t) to be x(t) = x cos(Ωt) + 1 MΩ p sin(Ωt). Inserting this into (3.10), we get the final expression for the master equation, Here the time-dependent coefficients (the frequency renormalizationΩ(t), the damping coefficient γ(t), and the two diffusion coefficients D(t) and f (t)) arẽ From this equation it is possible to have a qualitative idea of the effects the environment produces on the system. First we observe that there is a frequency renormalization. Thus, the "bare" frequency of the oscillator is renormalized byΩ 2 . This term does not affect the unitarity of the evolution. The terms proportional to γ(t), D(t) and f (t) bring about non-unitary effects. Thus, one can easily see that the second term is responsible for producing friction (γ(t) plays the role of a time-dependent relaxation rate). The last two are diffusion terms. The one proportional to D(t) is the main cause for decoherence.
Of course, the explicit time dependence of the coefficients can only be computed once we specify the spectral density of the environment. To illustrate their qualitative behavior, we will consider a typical ohmic environment characterized by a spectral density of the form where Λ plays the role of a high-frequency cutoff and γ 0 is a constant characterizing the strength of the interaction. For this environment, it is rather straightforward to find the following exact expressions for the coefficients Ω(t) and γ(t): From these equations we see that these coefficients are initially zero and grow to asymptotic values on a timescale that is fixed by the high-frequency cutoff Λ. Thus, we see the relation between this result and the one we would obtain by using a Markovian approximation simply corresponds to taking the limit Λ → ∞. In such a case both coefficients are not continuous at t = 0 and jump into constant values (the frequency renormalization diverges as it is proportional to the product γ 0 Λ). The time dependence of the diffusion coefficients can also be studied for the above environment. However, the form of the coefficients for arbitrary temperature is quite complicated. To analyze the qualitative behavior, it is convenient to evaluate them numerically. In Figure 3 one can see the dependence of the coefficients (for both the long and short timescales) for several temperatures (high and low). We observe that both coefficients have an initial transient where they exhibit a behavior that is essentially temperature independent (over periods of time comparable with the one fixed by the cutoff). The direct diffusion coefficient D(t) after the initial transient rapidly settles into the asymptotic value given by D ∞ = M γ 0 Ω coth(βhΩ/2)Λ 2 /h(Λ 2 + Ω 2 ). The anomalous diffusion coefficient f (t) also approaches an asymptotic value (which for high temperatures is suppressed with respect to D ∞ by a factor of Λ), but the approach is algebraic rather than exponential. More general environments can be studied using our equation. In fact, the behavior of the coefficients is rather different for environments with different spectral content. This has been analyzed in the literature, in particular in relation to decoherence [37].
It is interesting to mention that the master equation (3.12) (although it has been derived perturbatively) can be shown to be very similar to its exact counterpart whose derivation we will discuss later in this section.

Example 2: Perturbative master equation for a two-level system coupled to a bosonic heat bath
As a second example we obtain the perturbative master equation for a two-level system coupled to an oscillator environment. We consider two different models characterized by different interaction Hamiltonians. First, we discuss the model describing the physics of the decay of a two-level atom (in the rotating wave approximation), where a n and a † n are annihillation and creation operators of the environment oscillators, and σ ± are the raising and lowering operators of the two-level system. The perturbative master equation obtained following the procedure described above iṡ where the kernel k(t) is defined as Using the solution of the free Heisenberg equations for the spin operator (i.e., σ ± (t) = σ ± exp(±i∆t)), we can deduce that the master equation iṡ where the time-dependent coefficients are We recognize in this equation similar features to those present in the one for quantum Brownian motion (QBM). The interaction with the environment on the one hand renormalizes the Hamiltonian of the particle through the term c(t) (including thermal fluctuations, we could verify that c(t) is generally temperature dependent, as opposed to the QBM case). The nonhermitian part has a zero temperature contribution that is responsible for the spontaneous decay of the two-level system. The decay rate is determined by b(t) and has a time dependence that is essentially the same as the one found for the diffusion coefficient in the zero temperature QBM case analyzed above. The finite temperature contributions can be shown to be responsible not only for the changes in the value of the decay rate b(t) (which in that case would account also for the induced decay) but also for adding new terms to the master equation that take into account the induced absorption.
Finally we obtain the perturbative master equation for the spin boson Hamiltonian, which is also widely used in various condensed-matter physics problems (and was thoroughly studied in the nonperturbative regime in [38]) where q n are the coordinates of the environmental oscillators. The master equation can be shown to bė (3.23) where the two kernels are the same as defined above in the QBM case (3.11). Using the free Heisenberg operator σ z (t) = σ z cos ∆t + σ y sin ∆t we obtain the master equation, where the effective Hamiltonian and the time-dependent coefficients are now given by As before, the interpretation is quite straightforward. The effect of the environment is to renormalize the frequency as well as to introduce the decay of the system. This effect takes place only if the bare frequency ∆ is nonzero (otherwise z(t) vanishes). The other effect of the environment is to destroy the nondiagonal terms in the density matrix, a task that is carried out by the term proportional toD, which is present even when the bare driving vanishes. As before, the expression for the time-dependent coefficients is qualitatively similar to the one observed in the QBM model.

Example 3: Perturbative master equation for a particle interacting with a quantum field
We consider the following simple model: The system is a particle with position x (moving in a 3-dimensional space) and the environment is a quantum scalar field φ. The interaction between them is local as described by the , where e is the coupling constant (the "charge" of the particle). Expanding the scalar field in normal modes, the Hamiltonian can be written as where the Fourier components h k are proportional to annihillation operators of the quantum field (i.e., h k = e a k /(2π) /2 (2ω k ) 1/2 ). More generally, we could consider models in which the particle-field interaction is slightly nonlocal taking into account the finite extent of the particle (thus, a nonrelativistic treatment of the quantum particle would only give consistent results if we do not attempt to localize it beyond its Compton wavelength). In this case, the interaction HamiltonianH int = e d yW ( x − y)φ( y) depends upon the window function W ( r) whose support lies inside a sphere of radius R (the Compton radius of the particle) centered around the origin. This nonlocal interaction corresponds to a Hamiltonian whose Fourier components h k are multiplied bŷ W ( k) (the Fourier transform of W ( r)). As we can see, the net effect of taking into account the finite size of the particle is to introduce an ultraviolet cutoff in the scalar field (the particle does not interact with the field modes with frequencies higher that its rest mass). It is interesting to note that for this class of models we can also derive a master equation for the reduced density matrix of the particle. Thus, using the perturbative approach described above, we simply obtain (assuming the initial state of the quantum field is thermal equilibrium) the master equation as follows: Here, x(t) is the Heisenberg position operator for the particle (evolved with the free Hamiltonian H) and G R,H ( k, t) are the Fourier transforms of the retarded and symmetric two-point functions of the scalar field (multiplied by the appropriate window function if the interaction is nonlocal). When the environment is a free field, we have 26) where N k is the number density of particles in the initial state of the quantum field (the above result is valid if the field is not free, in which case the propagators are appropriately dressed). This master equation is extremely rich. Here, we will use it for two main purposes. On the one hand, we can see that the Quantum Brownian Motion case is a special limit of this particle-field model that arises in the so-called dipole approximation. This is the most widely used approximation in this context and is valid whenever the dominant wavelengths in the environment are much larger than the lengthscale over which the position of the particle varies. If this is the case, we can expand the exponentials up to second order ( k x ≪ 1) and obtain: . Thus, our first example of a linear Brownian particle coupled to an oscillator environment arises as the dipole approximation of the particle field model. With this in mind, we will use the particle field model as an example to show that some of the results obtained in the QBM case are just artifacts of the dipole approximation. In particular, this will be the case with the dependence of the decoherence rate on distance. Using the master equation of our particle-field model we will easily show that the decoherence rate does not indefinitely grow with distance but exhibits saturation.

Exact master equation for Quantum Brownian Motion
After presenting some simple perturbative master equations one may wonder under what circumstances are they a reasonable approximation. To partially address this issue, it is interesting to compare these equations with the ones that can be obtained for exactly solvable problems. In particular, we describe the master equation for a model that has been thoroughly studied in connection with decoherence, i.e., the linear quantum Brownian motion. Thus, because the Hamiltonian is quadratic both in the coordinates of the system and the environment, it is not surprising that it can be exactly solved. In this subsection, we will describe a simple derivation of the exact master equation, discuss its main features, and show that its functional form is the same as the one obtained by using perturbation theory. Indeed, the exact master equation has the same functional form as (3.12), the only difference being that the time dependence of the coefficients is different in general, as expected.
It is interesting to note that the exact master equation for QBM has only been found recently in spite of the simplicity of the model (in particular, the fact that it can always be written as an equation that is local in time was not appreciated until very recently [37]). Unfortunately, the derivation of the exact master equation is not so simple and, to say the least, the original one presented in [37] is indeed rather complicated. Here we will present the simplest derivation of the exact master equation that we know of, which is done following the method proposed first in [39]. Previous studies of the master equation for QBM, obtained under various approximations, include the celebrated paper by Caldeira and Legget [40] among others (see also [41,42]).
The derivation will focus on properties of the evolution operator for the reduced density matrix. This operator will be denoted as J and is defined as the one that enables us to find the reduced density matrix at some arbitrary time from the initial one. Thus, by definition, this operator satisfies: The derivation of the exact master equation has two essential steps. The first step is to find an explicit form for the evolution operator of the reduced density matrix. The second step is to use this explicit form to obtain the master equation satisfied by the reduced density matrix. To make our presentation simpler, we postpone the proof of the first step, which will be done below using path integral techniques. Here, we first want to demonstrate how to obtain the master equation once we know the explicit form of the evolution operator. So, let us show what the evolution operator for the reduced density matrix looks like. For linear QBM we will show later that it can always be written as where for notational convenience we are using sum and difference coordinates (i.e., X = x + x ′ , Y = x − x ′ , etc) and the coefficients b i and a jl are time-dependent functions whose explicit form will be given below (and depend on the properties of the environment). Thus, the evolution operator (3.28) is simply a Gaussian function of its arguments with time-dependent coefficients. This comes as no surprise because the problem is linear. Knowing the propagator for the reduced density matrix, it is easy to obtain the master equation following the simple method described in [39]. This is the second step of the derivation of the master equation and is done as follows. We compute the temporal derivative of the propagator J noting that the only time dependence is through the coefficients b i and a jl . Thus, we obtainJ Using this equation, we can try to find the master equation through multiplying by the initial density matrix and integrating this over the initial coordinates. The master equation would be trivially obtained in this way if, after multiplying by the initial density matrix, we could integrate over all the initial coordinates. This is straightforward, with some of the terms appearing in (3.29) but it is not so obvious how to handle terms that explicitly depend upon the initial coordinates X 0 and Y 0 . Fortunately, there is a simple trick that we can use: because we know that the propagator (3.28) is Gaussian, we can make use of this fact to obtain the following simple relations: These two equations can be used in (3.29) and in this way we can express the right hand side of this equation entirely in terms of the reduced density matrix. The resulting master equation iṡ The coefficients appearing in this equation are determined by b i and a jl as follows: Thus, we showed that the exact master equation is a simple consequence of the Gaussian form of the evolution operator (3.28). To complete our derivation of this equation we need to explicitly show how to obtain equation (3.28) and also find the explicit form of the time-dependent coefficients (which is also required to simplify the expressions leading to the master equation (3.30).
To obtain the explicit form of the evolution operator we will follow a derivation based on the use of path integral techniques (see [37,39,[43][44][45]). To understand it, very little previous knowledge of path integrals is required. The main ingredient is the path integral expression for the evolution operator of the complete wave function. Thus, if the action of the combined system is S T [x, q], the matrix elements of the evolution operator U can be written as where the integration is over all paths that satisfy the boundary conditions, In the above and following equations, to avoid the proliferation of subindices we use q to collectively denote all the coordinates of the oscillators q n (we will not write the subscript n that should be implicitly assumed). Using this equation, one can obtain a path integral representation of the evolution operator of the complete density matrix and, after taking the final trace over the environment, we find a path integral representation of the propagator for the reduced density matrix. It is clear that the resulting expression will involve a double path integral (one to evolve kets and another one to evolve bras). For a generic initial state ρ T , the propagator is a somewhat complicated-looking expression. To simplify our presentation, we will only consider here factorizable initial states (and refer the reader to [45] for the most general situation where initial correlations are included). Thus, if the initial state can be factored we can express the reduced density matrix at arbitrary times as a function of the reduced density matrix at initial time using a (state-independent) propagator that has the following path integral representation: where the integral is over paths satisfying the above boundary conditions, S[x] is the action for the system only, and F [x, x ′ ] is the so-called "Influence Functional" first introduced by Feynman and Vernon [46]. This functional is responsible for carrying all the physical effects produced by the environment on the evolution of the system. In fact, if there is no coupling between the system and the environment, the Influence Functional is equal to the identity, and the above expression reduces to the one corresponding to the free Schrödinger evolution for the isolated system. The Influence Functional is defined as

35)
where ρ E is the initial state of the environment and S SE [q, x] is the action of the environment (including the interaction term with the system). It is easy to see that if there is no interaction (or if the two systems trajectories are the same, i.e., x = x ′ ), then the influence functional is equal to one.
Calculating the Influence Functional for an environment formed by a set of independent oscillators coupled linearly to the system is a rather straightforward task (and, to our knowledge, was first done by Feynman and Vernon in [46]). Assuming the initial state of the environment is thermal equilibrium at temperature T = 1/k B β, the result is where X = x + x ′ , Y = x − x ′ , and the two kernels ν(s) and η(s) are the so-called noise and dissipation kernels that were defined above in (3.11). Thus, all the influence of the environment on the evolution of the system is encoded in the noise and dissipation kernels (two different environments that produce the same kernels would be equivalent as to the impact they have on the system). To obtain the above expression is a simple exercise in path integrals. However, the calculation can also be done by a more straightforward procedure that makes no reference to path integrals. Indeed, one can notice that the influence functional can always be expressed in operator language as where T (T ) denotes the time ordered (antitime ordered) product of the corresponding Heisenberg operators, and V int is the interaction term between the system and the environment. If the interaction is bilinear and the initial state of the environment is thermal, one can easily realize that the result should be a Gaussian functional of both x and x ′ . Therefore, one can just write down such most general Gaussian functionals in terms of unknown kernels. These kernels could be identified by using the above expression, taking functional derivatives with respect to x and x ′ and evaluating the result when x = x ′ . In this way, one realizes that the result is given by (3.36), where the noise and dissipation kernels are given by expectation of symmetric and antisymmetric two-time correlation functions of the environment oscillators, exactly as in (3.11).
Knowing the Influence Functional enables us to compute the exact expression for the evolution operator of the reduced density matrix. In fact, all we need is to perform the path integral in (3.34). If the system is linear we see that the integrand is Gaussian and, therefore, the integral can also be explicitly computed. To perform this integral is not so trivial because the integrand is not separable into a product of functions of x and x ′ . However, the integral can be calculated simply by changing variables. First we should integrate over sum and difference coordinates X and Y . Then, we should change variables writing X = X c +X and Y = Y c +Ỹ where X c and Y c satisfy the equations obtained by varying the phase of the integrand and imposing the corresponding boundary conditions. In this way, we show that the result of the path integral is simply the integrand evaluated in the trajectories X c , Y c , multiplied by a time-dependent function that can be determined by normalization. The only nontrivial part of this derivation is to realize that the trajectories X c and Y c can be chosen as the ones extremizing only the phase of the integrand, (and not the entire exponent that, as we saw, has a real part coming from the noise). For more details on this derivation the interested reader can look in [37,44,45]. Therefore, the final result is given in equation (3.28) where the coefficients b i and a jl are time-dependent functions that are determined in the following way. Let the functions u 1 2 be two solutions of the equation, satisfying the boundary conditions u 1 (0) = u 2 (t) = 1 and u 1 (t) = u 2 (0) = 0. Then, the coefficients appearing in (3.28) are simply given by The time dependence of the coefficients of the master equation can be investigated after specifying the spectral density and the temperature of the environment. This has been done in great detail in a series of papers [37,39,43,47]. We will not review these results in detail but would just like to mention that for the case that is most interesting for studying decoherence, which is the underdamped (i.e., weakly coupled) harmonic oscillator, the time dependence of the exact coefficients is very similar to the one obtained by analyzing the coefficients appearing in the perturbative master equation. Indeed, the perturbative coefficients obtained above can be recovered by solving the equation for the functions u 1 perturbatively and replacing these equations inside (3.38) and (3.31). Thus, to get a qualitative idea about the behavior of the coefficients, we restrict ourselves to the analysis already made for the perturbative ones (see Figure 3).
It will be useful to analyze decoherence not only using the reduced density matrix but also the Wigner function that is the phase space distribution function that can be obtained from the density matrix as [48] W It is simple to show that for the case of the harmonic oscillator, the evolution equation for the Wigner function can be obtained from the master equation and has the form of a Fokker Planck equatioṅ The form of the evolution equation for the Wigner function for more general (nonlinear) systems will be discussed in Section 6. As a final remark, it is worth pointing out that the exact master equation does not have the so-called "Lindblad form". A master equation is of the Lindblad form [49] if it can be written aṡ for some operators L n and some (positive) constants γ n . As shown by Lindblad, this is the most general master equation with the property of being Markovian and preserving the positivity of the density matrix. The fact that the exact master equation does not have the Lindblad form may be puzzling but after some thinking becomes natural. Of course, the exact evolution also preserves positivity of the density matrix, but it does so in a more subtle way than through a Lindblad master equation. The true evolution is not Markovian (but in a very weak sense). The only memory effect relies on the fact that the system remembers the initial time when the (factorizable) initial conditions were imposed. This effect appears in the time dependence of the coefficients that is responsible also for enforcing positivity in an interesting way (see [45,47] for some discussion on the way positivity follows from the exact master equation). As a final comment, we would like to mention the fact that exact master equations are rather rare, but the above equation for QBM is not the only interesting exact master equation known. For example, it is possible to derive an exact master equation that has strong similarities with the one for QBM (i.e., an equation that is local in time and has time-dependent coefficients) for the model of a two-level system coupled to a bosonic bath through the Hamiltonian (3.17) (this equation was derived first in [50] and rediscovered by other means in [51]).

Decoherence of a superposition of two coherent states
We will analyze here the decoherence process in a simple example: the linear quantum Brownian motion model whose exact master equation is given in (3.30). For this we will first set up an initial state that is delocalized in position (or momentum) space and examine its temporal evolution, paying special attention to the fate of interference effects. Thus, we will consider a state of the form [47,52] Ψ(x, where Note that we assumed (just for simplicity) that the two wave packets are symmetrically located in phase space. The above expression allows us to study two extreme cases: the coherent states are separated in position or in momentum. In both cases, as a consequence of quantum interference, the Wigner function oscillates and becomes negative in some regions of phase space (and therefore cannot be interpreted as a probability distribution). When the coherent states are separated in position (momentum), the fringes are aligned along the p (x) axis.
To evolve this initial state, we should solve the master equation (3.30). Rather than doing this, one can use the explicit form of the evolution operator (3.28) and obtain the exact form of the reduced density matrix or the Wigner function at any time. We will adopt this strategy but will use the master equation (3.30) and the equation for the Wigner function (3.40) as a guide to interpret our results and to obtain simple estimates for the most important effects that take place as a result of the interaction between the system and the environment. The exact evolution of the above initial state is such that the Wigner function can be written always as the sum of two Gaussian peaks and an interference term, where All the coefficients appearing in these expressions are somewhat complicated functions of time that are determined by the coefficients that appear in the propagator (3.28) and the initial state (in the same way, they also depend on temperature and on the spectral density of the environment). Their explicit form can be found in [47]. The initial state is such that From the form of the exact solution, it is clear what the qualitative behavior of the quantum state is. The two Gaussian peaks follow the two classical trajectories (which get distorted by the interaction with the environment) and change their width along their evolution. On top of this, the interference fringes change their wavelength and also rotate somewhat following the rotation of the two wavepackets. The effect of decoherence is clearly manifested in the damping of the interference fringes that, in the above formulae, is produced by the exponential term exp(−A int ). Thus, we will look carefully at this term, which can be seen to be the "fringe visibility factor" defined as A close analysis of the definition of A int shows that it vanishes initially and is always bounded from above, i.e., The value of A int cannot grow to infinity as a consequence of the fact that the two Gaussian initial states have a finite overlap that is proportional to exp (−A int | max ).
To understand qualitatively and quantitatively the time dependence of the fringe visibility factor, it is interesting to obtain an evolution equation for A int . Using its definition, we know thaṫ This, after using the form of the Wigner function together with the evolution equation, can be transformed intȯ This equation enables us to obtain a clear picture of the time evolution of the fringe visibility function. Thus, we can see that the first term on the right-hand side is always positive and corresponds to the effect of normal diffusion. The normal diffusion will tend to wash out interference. The initial rate at which A int grows is determined by the diffusion coefficient and by the initial wavelength of the fringes in the momentum direction (remember that initially we have κ p = L 0 /h. As time goes by, we see that the effect of this term will be less important as the effective wavelength of the fringes grows (making κ p decrease).
Various simple estimates of the temporal behavior of the fringe visibility factor can be obtained from this equation. The most naive one is to neglect the time dependence of the diffusion coefficient and assume that the fringes always stay more or less frozen, as in the initial state. In such a case, we have A int ≈ 4L 2 0 Dt/h 2 . Thus, if we use the asymptotic expression of the diffusion coefficient, we obtain (at high temperatures) A int ≈ γt4L 2 0 /λ 2

DB
where λ DB is the thermal de Broglie wavelength. Consequently, we find that decoherence takes place at a rate which is the relaxation rate multiplied by a factor that could be very large in the macroscopic domain (this is the result originally obtained by one of us, see [52] where it is shown that for typical macroscopic parameters, i.e., room temperature, cm-scale distances and masses on the order of a gram, the factor 4L 2 0 /λ 2 DB can be as large as 10 40 ). By analyzing the temporal behavior of A int obtained using the exact solution, we can check that this naive estimate is an excellent approximation in many important situations. However, it may fail in other important cases. Here, we want to stress a message that we believe is very important (see [55]): It may be rather dangerous to draw conclusions that are too general from the theoretical analysis of simple models of decoherence (like the one of linear QBM). The reason is that simple estimates like the one corresponding to the decoherence timescale (4.10) are just that: simple estimates that apply to specific situations. They do not apply in other circumstances, some of which we will describe here (and in the next section).
For example, the above simple estimate of the decoherence timescale fails in the simple case of "ultrafast" decoherence. For, in the high-temperature approximation of the master equation we neglected (among other things) initial transients occurring in the timescale fixed by the cutoff. Nothing (not even decoherence) can happen faster than the cutoff timescale since only after such timescale the diffusion coefficient reaches a sizable value. Thus, studying the initial time behavior of the normal diffusion coefficient one realizes that for very short times, A int always grows quadratically (and not linearly). In fact, we have From this expression one sees that in this case A int is smaller than the one obtained under the assumption of a constant diffusion coefficient (at least for times t ≤ Λ −1 ). In this case, the decoherence timescale may be longer than the one corresponding to the high temperature approximation, On the other hand, the above estimate for A int also fails to take into account the fact that A int does not grow forever because it finally saturates to the value fixed by equation (4.7). Saturation is achieved in a timescale that can be estimated to be t sat ≈ γ −1 0 (hΩ/k B T ). At approximately this time the saturation of A int takes place (it is clear that this is a very short time, much shorter than any dynamical timescale).
The high-temperature approximation to the behavior of A int will clearly fail at very low temperatures (however, it is quite remarkable how robust an approximation this is; see [47] for a detailed analysis). We will comment in the next section about the effects arising at low temperatures giving more accurate estimates for A int in such a domain.

Predictability sieve and preferred states for QBM
The most important consequence of the decoherence process is the dynamical selection of a set of stable, preferred states. These are, by definition, the least affected by the interaction with the environment in the sense that they are the ones that become less entangled with it. To obtain these states, a systematic ("predictability sieve") criterion has been proposed [3,53]. The basic idea is the following: To find the pointer states, one should consider all possible pure initial states for the system and compute the entropy associated with its reduced density matrix after some time t. The pointer states are the ones that minimize the entropy production for a dynamic range of times.
The predictability sieve can be applied to the simplest models of a quantum measurement, for which the Hamiltonian of the system can be completely neglected. In such a case, the pointer states are directly associated with the eigenstates of the interaction Hamiltonian (actually, to its eigensubspaces that may be degenerate). In other more realistic situations where the self-Hamiltonian of the system is not negligible the pointer states are not going to be picked only by the interaction Hamiltonian but by the interplay between it and the evolution produced by the systems own Hamiltonian. The best example where we can explicitly compute these pointer states is the QBM model we have been studying in this section. To do this, the master equation is, as we will see, a very convenient tool.
To find pointer states, we should minimize the entropy production at some time (varying over times to find a stable answer). However, to make our task simpler, instead of using von Neumann entropy, we will simply study the evolution of the purity of the system as measured by ς = T rρ 2 . This quantity is equal to one for a pure state and decreases when the state of the system gets mixed because entanglement is generated by the evolution. The master equation directly enables us to write down an evolution equation for the purity ς. Thus, using the definition of ς and the equation (3.12) we obtain [3]: To simplify our treatment, we will once again use a perturbative approximation and substitute in the right hand side of this equation the expression for the free Heisenberg operators: x(t) = x cos Ωt + p/M Ω sin Ωt and p(t) = p cos Ωt − M Ωx sin Ωt. Moreover, we will average over one period of the harmonic motion, assuming that the coefficients of the master equation do not vary during that time (clearly, this is a crude approximation, and we will comment later about what happens when we relax it). We also assume that the initial state is pure (and use the fact that in that case ρ 2 = ρ). Moreover, we neglect the effect of the friction term because, as we see, this term will always try to increase the purity in a way that is not sensitive to the state itself (thus, friction always tries to localize the state competing against diffusion that has the opposite effect). Doing this, we find out that the change in purity over one period is simply given by (4.14) where ∆x and ∆p are respectively the position and momentum dispersion of the initial state. The anomalous diffusion term does not produce any net entropy increase (or purity decrease) because its effect averages out over one oscillation. The term responsible for purity decrease is simply coming from diffusion, and to minimize it, we should vary over all possible initial states. This can easily be implemented by varying over all values of the initial dispersion in position and momentum in such a way that the righthand side of (4.14) is minimized. Because ∆x∆p ≥h/2 must always be satisfied, it is clear that the minimum is obtained when the state saturates uncertainty relations. From the resulting equation we obtain the pointer states as having ∆x 2 =h/2M Ω and ∆p 2 =hM Ω/2. Therefore, the pointer states are simply given by coherent states with minimum uncertainty. This result is simple and satisfying. In fact, coherent states are the closest we can get to points in phase space. They are preferred states in QBM because they turn out to be the most robust and most effectively resist the combined effect of the system and the environment. They are also well localized in position and, therefore, are not significantly perturbed by the environment monitoring their position. Moreover, because of their symmetry, they are also not drastically altered by the evolution induced by the Hamiltonian of the system.

Energy eigenstates can also be selected by the environment!
So far, we have discussed two regimes in which the predictability sieve can be successfully applied. We first mentioned the case of a measurement (where the Hamiltonian of the system is negligible), and we just studied the case where both the system and the environment induce nontrivial evolution. There is a third regime that is interesting to study and is one in which the evolution of the environment is very slow as compared with the dynamic timescales of the system. If this is the case, it is possible to show [54] that the preferred states are simply the eigenstates of the Hamiltonian of the system. However, it is interesting to note that to find out this result, it is not possible to use a model like the linear QBM we described before. In fact, in such a model we can see that if we consider a very slow environment (with frequencies much smaller than the one belonging to the system) the master equation (3.12), which is still applicable, has time-dependent coefficients that are oscillatory functions of time with no well-defined sign. Therefore, the predictability sieve criterion does not give a robust set of states in this case. However, the third regime of einselection can be examined using a simple argument based on an adiabatic solution of the full Schrödinger equation. The main ingredient we need is, as will be shown below, a slow environment that couples to the system through an interaction Hamiltonian that has a nonzero expectation value in the energy eigenstates of the system. To see this, we will solve the full Schrödinger equation treating the environment adiabatically. Suppose that the initial state of the universe given as |Ψ(0) = n c n |φ n |ǫ 0 where the states |φ n are nondegenerate eigenstates of the Hamiltonian of the system (with distinct energies E n ), and |ǫ 0 is a state of the environment that, for simplicity, we will consider as a coherent state (the vacuum, for example). We can solve the full Schrödinger equa-tion in the adiabatic approximation and show that this state evolves into |Ψ(t) = n c n exp(−iE n t/h)|φ n |ǫ n (t) where the state |ǫ n (t) , that gets correlated with the nth energy eigenstate of the system, obeys the following Schrödinger equation Note that in this equation the operator φ n |H int |φ n acts on the Hilbert space of the environment and depends parametrically on the energy eigenstates of the system. We will assume that the interaction is such that the Hamiltonian is of the form H int = S ⊗ Π E , where the operator S acts on the system Hilbert space, and the environment operator Π E acts on the environment as a translation generator (it could be the momentum operator, for example, but from our discussion it will be clear that the choice of momentum here is not crucial). The decoherence in energy eigenbasis can easily be established as follows. Because Π E is a momentum operator and the initial state of the environment is a coherent state, the evolution turns out to be simply such that |ǫ n (t) = |ǫ 0 + S nn t , where S nn = φ n |S|φ n . Therefore, the overlap between the two states that correlate with different energy eigenstates can be estimated as ǫ n (t)|ǫ m (t) ≈ exp(−t 2 (S nn − S mm ) 2h2 ). Consequently, in this case, we see einselection of energy eigenstates (superpositions of energy eigenstates are degraded while pure energy eigenstates are not affected). For this reason, pointer states are energy eigenstates. This result has a rather natural interpretation. It just tells us that the environment is not able to react before the system has time to evolve and therefore only probes time-averaged quantities of the system. Energy, being the only observable that does not average out to zero is therefore the preferred observable. The conditions for energy eigenstates to become the pointer basis are the following: the environment must behave adiabatically (and be slow as compared with the dynamics of the system), and the interaction with the system must be through an observable with a nonvanishing expectation value in energy eigenstates.

Deconstructing decoherence: landscape beyond the standard models
Simple models of decoherence, like the one we discussed so far (linear quantum Brownian motion) are important to illustrate the simplicity and high efficiency of the decoherence process (two characteristics that may be interpreted as indicating its generality). However, it is important to keep in mind that no generic conclusions should be drawn from simple estimates. This is especially important in view of the possibility of carrying out experiments to test decoherence in a controlled manner. In such cases, it is essential to study specific models of the decoherence process in the correct context. Estimates of the decoherence timescale, nature of pointer states, and other characteristics of decoherence obtained in models like QBM should be taken as indications rather than as strong predictions.
In this section, we would like to stress the fact that some of the simple features that have became identified as "standard lore" in the decoherence process for the simplest case of linear QBM are not generic by showing explicitly how they fail in two specific examples. We will address basically two issues. First we will consider the status of one of the simplest predictions arising from studying decoherence in linear QBM: the "decoherence rate grows quadratically with distance". We will show that this is not the case for more realistic models where local interactions between particles and fields (rather than oscillators) are taken into account. Second, we will consider the status of predictions of the decoherence timescale like the ones in Eq. (4.10) at low temperature. In this case, by analyzing the same linear QBM at low temperatures, we will show that the decoherence process may be more complicated, allowing even for nonmonotonic behavior.

Saturation of the decoherence rate at large distances
One of the results obtained studying the decoherence process in linear QBM models is that the decoherence rate grows quadratically with the separation between different pieces of the system wavefunction. This result is natural (delocalized wavepackets decohere faster) but would certainly not be physical if it held for arbitrarily large separations. Apart from any arguments involving cutoff (see discussion following Eq. (4.10), it is clear that the environment should have a coherence length so that separations that are bigger than this natural lengthscale should be equivalent and therefore induce the saturation of the decoherence rate.
However, saturation is not present in the linear QBM model, as is clear from the discussion above. One therefore asks what kinds of models predict saturation. We will describe here the simplest of such models. The environment is formed by a quantum scalar field; the system is a quantum particle, and the interaction between them is local. This is the model whose perturbative master equation we derived in section 3.4. It is important to stress once more that the linear QBM model is obtained from the particlefield model by means of the dipole approximation. Thus, saturation in this context arises only if we do not make the dipole approximation (which is certainly not well justified for large separations). The issue of the saturation of the decoherence rate was analyzed first in [56] and also discussed in [55]. In this review we present a simpler discussion than the one of [55] that captures the main ingredients necessary for saturation and enables us to obtain the principal results without complicated calculations (some experimental results related to these issues were reported in [7]).
As we discussed in 3.4, the reduced density matrix of the particle obeys the perturbative master equation (3.25). In this equation, the Heisenberg operator of the particle x(t 1 ) appears. To simplify our argument, we will consider the system that is a free and very massive particle and therefore replace x(t) = x(0) in (3.25) (corrections to this approximation can be computed also). In the simplest example, we will consider as environment a massless scalar field (and replace the corresponding expressions for the Fourier transform of the two point functions; see Eq. (3.7)). In this case, we can express the master equation in the position representation aṡ where only the term producing decoherence has been written out, and the function Γ(x, x′) is defined as where r = |x − x′| (and, as before, W (k) is the Fourier transform of the window function that introduces a natural high-frequency cutoff). It is simplest to analyze the high-frequency limit of the above expression. In that case, the integral can be exactly computed and turns out to be From this expression we clearly see the saturation. Thus, the solution of the master equation in the "decoherence-dominated" approximation (neglecting all terms except the one producing decoherence) is simply ρ(x, x′) ≈ exp(− t 0 dt 1 Γ(x− x′, t 1 ))ρ(x, x′). The dependence of Γ(r) for long distances is given by the second instance in equation (5.3) that approaches a constant as r grows larger than 1/Λ and t. On the other hand, the quadratic dependence of the decoherence rate is recovered for small distances: by expanding the function Γ(r) around r ≈ 0, we obtain a quadratic behavior.

Decoherence at zero temperature
A simple estimate for the low temperature behavior of the fringe visibility function can be obtained as follows. Use the asymptotic form of the diffusion coefficient for low temperatures given by perturbation theory and integrate the equation for A int , neglecting both its time dependence as well as the temporal evolution of the wavelength of the fringes. In this way, we obtain A int ≈ γ 0 t(4L 2 0 /∆x 2 ) coth βΩ/2. However, this is not always a good approximation. On the one hand, if this behavior were correct, we could estimate the saturation time (the time for which A int would approach its maximum value) to be on the order of t sat ≈ γ −1 0 tanh βΩ/2, which for very low temperatures can be very close to, or even larger than, a dynamical timescale. Note that this does not imply that decoherence occurs in a dynamical timescale: For that, the important fact is the actual value of A int and not how close to the maximum value we are. The decoherence timescale at low temperatures is on the order of t dec ≈ γ −1 0 (∆x/2L 0 ) 2 , which is still much shorter than γ −1 0 for macroscopic parameters). The fact that the naive estimate for the saturation timescale becomes larger than typical dynamical times means that A int does not have a monotonic behavior in time. In fact, it turns out that at very low temperatures, the role of the anomalous diffusion term in the master equation starts to be relevant (its value is of the same order of magnitude as the normal diffusion coefficient). The contribution of this term to the evolution of A int is clearly seen in equation (4.9) where we see that the second term (associated with anomalous diffusion) does not have a well-defined sign (its sign changes as the interference fringes rotate in phase space). From this observation, one expects that if at low temperatures the fringe visibility factor does not saturate, its time dependence should exhibit some oscillatory behavior (modulating an overall increase dictated by normal diffusion). The periods of slower decoherence coincide with the moments when fringes get oriented along the position axis (this coincides with the instant when the two wavepackets are most separated in momentum). This qualitative prediction concerning the behavior of A int is confirmed by the exact numerical calculations shown in Figure 5. In this figure, the oscillations are clearly seen.
A very simple and interesting expression for A int can be obtained for the QBM model. Thus, in [45] it has been shown that the fringe visibility factor can always be written as follows: where ∆x 2 is the position dispersion in thermal equilibrium (i.e., ∆x 2 = h coth(βΩ/2)/M Ω) and S is the normalized position autocorrelation function defined as This equation enables us to obtain very simple qualitative estimates of the efficiency of decoherence. More interestingly, it clearly shows that decoherence has the same physical origin as other dissipative effects (and is closely related to the decay of the autocorrelation function through Eq. (5.4)). However, in spite of their common origin, the decay of correlations and the decoherence process have very different timescales. In fact, from the above equation we can estimate how much the correlation functions have to decay in order for the system to decohere. Thus, at the time for which A int approaches unity, the spatial correlations in the system should have decayed by a factor S(t dec )/S(0) = 1 − ∆x 2 /4L 2 0 , which is indeed very small (note that ∆x approaches the thermal de Broglie wavelength at high temperatures and the spread of the ground state at zero temperature).
On the other hand, the above formula (5.4) can also allow us to estimate correctly A int both at high and low temperatures for the underdamped Brownian motion model. In fact, we just need to obtain a reasonable approximation for the position correlation function. For example, assuming a simple exponential decay would lead us to conclude that This is a crude but very reasonable approximation that is, for example, not only very good at high temperatures and very early times but also exhibits the correct saturation behavior for long times. It can be further improved by better approximating the position correlation function. For example, computing S(t) in the highly underdamped regime we obtain which is a very good approximation for the low-temperature (low-damping) behavior exhibited in Figure 5.

Preexisting correlations between the system and the environment.
Almost all papers concerning decoherence assume that the initial state has no correlations between the system and the environment (i.e., that the state can be factored). In this section we will analyze what happens if we consider more general initial conditions. In particular, we are interested in analyzing initial conditions that are closer to what we encounter experimentally. Thus, we consider a situation in which the system and the environment are initially in a thermal equilibrium state at some temperature (which could be zero) and at the initial time we perform a measurement on the system to prepare an initial state. This measurement could be imperfect (i.e., may be characterized not by a projection operator, but by a POVM). After this measurement, we consider the evolution of the system coupled to the environment in the usual way. Under these circumstances, the initial state of the universe is generally not a product. Moreover, in the case when the initial state is a product (i.e., when the measurement performed on the system is perfect), the state of the environment depends functionally on the state of the system. This type of initial states can generally be written as where A j and A ′ j are Krauss operators (not necessarily projectors) acting on the Hilbert space of the system (see [29] for a good review).
We are not going to present any details of the calculations leading to the (exact) solution of this model. Our presentation follows closely the one in [45] where the influence of initial correlations on decoherence was examined. Here we present a summary of the results obtained in that paper.
First, it is worth stressing the fact that it is still possible to find a relatively simple master equation for the reduced density matrix of the system. However, the existence of initial correlations prevents us from expressing this equation entirely in terms of the reduced density matrix. Thus, the evolution of ρ not only depends on ρ itself but also on initial correlations between the system and the environment. Interestingly enough, for the case of the linear QBM model, an exact master equation that is very similar to (3.12) can be obtained. It reads as follows: It is important to stress that this equation is exact and valid for all spectral densities and initial temperatures. The time-dependent coefficients appearing in (5.8) are functions of time and temperature (and of the spectral density of the environment, of course). Explicit formulae are given in [45]. The interpretation of the first three lines of this equation is identical to the ordinary case where no correlations are present. The initial correlations appear in the time dependence of the coefficients but, for realistic environments, this dependence is very weak (thus, these coefficients are qualitatively the same as before). The last two lines make this equation nonhomogeneous. In fact, these terms are present because of the correlated nature of the initial state. Thus, in that case, the master equation cannot be entirely written in terms of the reduced density matrix. It can be shown that the two density matrices ρ 11 and ρ 12 are obtained by propagating two different initial states given by the "density matrices" ρ 11 = {q, ρ} and ρ 12 = i[q, ρ]. The evolution of ρ 1i can also be studied with this formalism because (apart from not being normalized) they belong to the class of initial conditions defined by (5.7). Therefore, the evolution equation obeyed by these operators is also (5.8), with new inhomogeneous terms. Thus, a hierarchy of equations, which are coupled because of the initial correlations, can be derived in this way (see [45] for more details).
The time dependence of all the coefficients has been studied in detail in [45] and the conclusion is that, for an ohmic environment at arbitrary temperatures, the coefficientsC 1 andC 2 , entering in the inhomogeneous terms of the master equation are exceedingly small and become negligible after a time that is on the order of the cutoff timescale. After this short initial transient, the impact of the initial correlations on the future evolution of the system can be entirely neglected. Of course, in less-realistic situations, it is possible to show that these coefficients have an important effect. For example, the formalism we described could be applied to the case of two coupled oscillators in which one considers one of them as the system and the other one as the environment. In this case, when the size of the system and environment are comparable, initial correlations play an important role. The time dependence of the other coefficients of the master equation is also affected by the correlations but they all behave qualitatively in a similar way as in the absence of such initial correlations (see [45] for a detailed study of these coefficients).
It is interesting to analyze the evolution of a delocalized initial state to see how decoherence takes place in this model, which includes the effect of initial correlations. For this, we consider the initial condition (5.7 with the operators associated to a projection onto a Schrödinger cat state (say, a superposition of two coherent states separated in position). Thus, we take whereP is a projector onto a pure state of the systemP = |Ψ Ψ| and the state |Ψ is itself a Schrödinger's cat state (i.e., a superposition of two Gaussian packets), where |Ψ ± are such that The decoherence process for this initial state has been analyzed in the previous section in the absence of initial correlations. The fate of this state is not very different from the behavior we described before but there are some subtle differences. Thus, initial correlations distort the Gaussian peaks in the initial Wigner function as well as the intermediate interference fringes. An exact solution of the problem is possible (see [45]) and it turns out that it is no longer true that the Wigner function can be written as the sum of two Gaussian peaks plus interference fringes. In fact, it turns out that each Gaussian peak is distorted in such a way that it can be written as the sum of two nearby Gaussians with a term between them. The same is true for the interference fringes, which get distorted and split into several (actually ten) terms. However, for realistic (ohmic) environments, this effect is very small (as discussed in [45]), and the decoherence process goes qualitatively in the same way as described in the previous section (in fact, in Figure  5, the two curves for the decoherence factor are almost indistinguishable from each other: one corresponds to an initially uncorrelated state while the other to the case described in this section).

Decoherence and chaos
Here we investigate environment induced superselection in the context of quantum chaos (i.e., quantum dynamics of systems that are classically chaotic). We first argue [60] that the evolution of a chaotic macroscopic (but, ultimately, quantum) system is not just difficult to predict (requiring accuracy exponentially increasing with time) but quickly ceases to be deterministic in principle as a result of the Heisenberg indeterminacy (which limits the resolution available in the initial conditions). This happens after a time th, which is only logarithmic in the Planck constant. A definitely macroscopic (if somewhat outrageous) example [61] is afforded by various components of the solar system that are chaotic, with the Lyapunov timescales ranging from a bit more then a month (Hyperion, a prolate moon of Saturn [57]) to millions of years (planetary system as a whole [58,59]). On the timescale th the initial minimum uncertainty wavepackets corresponding to celestial bodies would be smeared over distances of the order of the radii of their orbits into "Schrödinger cat-like" states, and the concept of a trajectory would cease to apply. In reality, such paradoxical states are eliminated by decoherence that helps restore quantum-classical correspondence. We shall also see that the price for the recovery of classicality is the loss of predictability. In the classical limit (associated with effective decoherence, and not just with the smallness ofh) the rate of increase of the von Neumann entropy of the decohering system is independent of the strength of the coupling to the environment and equal to the sum of the positive Lyapunov exponents.

Quantum predictability horizon: How the correspondence is lost.
As a result of chaotic evolution, a patch in the phase space that corresponds to some regular (and classically "reasonable") initial condition becomes drastically deformed. Classical chaotic dynamics is characterized by the exponential divergence of trajectories. Moreover, conservation of the volume in the phase space in the course of Hamiltonian evolution (which is initially a good approximation for sufficiently regular initial conditions, even in cases that are ultimately quantum) implies that the exponential divergence in some of the directions must be balanced by the exponential squeezing -convergence of trajectories-in other directions. It is that squeezing that forces a chaotic system to explore the quantum regime. As the wavepacket becomes narrow in the direction corresponding to momentum, (where ∆p 0 is its initial extent in momentum, and λ is the relevant Lyapunov exponent) the position becomes delocalized: The wavepacket becomes coherent over the distance ℓ(t) that can be inferred from Heisenberg's principle, Coherent spreading of the wavepacket over large domains of space is disturbing in its own right. Moreover, it may lead to a breakdown of the correspondence principle at an even more serious level. Predictions of the classical and quantum dynamics concerning some of the expectation values no longer coincide after a time th when the wave-packet coherence length ℓ(t) reaches the scale on which the potential is nonlinear. Such a scale χ can usually be defined by comparing the classical force (given by the gradient of the potential ∂ x V ) with the leading order nonlinear contribution ∼ ∂ 3 x V , For instance, for the gravitational potential χ ≃ R/ √ 2, where R is a size of the system (i.e., a size of the orbit of the planet). The reason for the breakdown of the correspondence is that when the coherence length of the wavepacket reaches the scale of nonlinearity, the effect of the potential energy on the motion can no longer be represented by the classical expression for the force [60], F (x) = ∂ x V (x), because it is not even clear where the gradient is to be evaluated for a delocalized wavepacket. As a consequence, after a time given by the expectation value of some of the observables of the system may even begin to exhibit noticeable deviations from the classical evolution [64]. This is also close to the time beyond which the combination of classical chaos and Heisenberg's indeterminacy makes it impossible in principle to employ the concept of a trajectory. Over the time ∼ th a chaotic system will spread from a regular Planck-sized volume in the phase space into a (possibly quite complicated) wavepacket with the dimensions of its envelope comparable to the range of the system. This timescale defines the quantum predictability horizon -a time beyond which the combination of classical chaos and quantum indeterminacy makes predictions not just exponentially difficult, but impossible in principle. The shift of the origin of the loss of predictability from classical deterministic chaos to quantum indeterminacy amplified by exponential instabilities is just one of the symptoms of the inability of classical evolution to track the underlying quantum dynamics.
This breakdown of correspondence can be investigated more rigorously by following the evolution of the Wigner function (defined in (3.39)) for the possibly macroscopic, yet ultimately quantum system. Dynamics of the Wigner function is generated by the Moyal bracket (that is simply the Wigner transform of the right-hand side of von Neumann equation for the density matrix). This Moyal bracket can be expressed through the familiar classical Poisson bracket: Above, H is the Hamiltonian of the system, and W is the Wigner transform of the density matrix. When the potential V in H is analytic, the Moyal bracket can be expanded in powers of the Planck constant. Consequently, the evolution of the Wigner function is given bẏ Correction terms above will be negligible when W (x, p) is a reasonably smooth function of p, that is, when the higher derivatives of W with respect to momentum are small. However, the Poisson bracket alone predicts that, in the chaotic system, they will increase exponentially quickly as a result of the "squeezing" of W in momentum, Eq. (6.1). Hence, after th, quantum "corrections" will become comparable to the first classical term on the right-hand side of Eq. (6.7). At that point, the Poisson bracket will no longer suffice as an approximate generator of evolution. The phase space distribution will be coherently extended over macroscopic distances, and interference between the fragments of W will play a crucial role. The timescale on which the quantum-classical correspondence is lost in a chaotic system can also be estimated (or rather, bounded from above) by the formula [62,63] t r = λ −1 ln(I/h) , (6.8) where I is the action.

Exponential instability vs. decoherence
In a quantum chaotic system weakly coupled to the environment, the process of decoherence briefly sketched above will compete with the tendency for coherent delocalization, which occurs on the characteristic timescale given by the Lyapunov exponent λ. Exponential instability would spread the wavepacket to the "paradoxical" size, but monitoring by the environment will attempt to limit its coherent extent by smoothing out interference fringes. The two processes shall reach status quo when their rates are comparable, Because the decoherence rate depends on δx, this equation can be solved for the critical, steady state coherence length, which yields ℓ c ∼ Λ dB (T ) × λ/γ.
A more careful analysis can be based on the combination of the Moyal bracket and the master equation approach to decoherence we have just sketched. In many cases, (including the situation of large bodies immersed in the typical environment of photons, rarefied gases, etc.) an effective approximate equation can be derived and translated into the phase space by performing a Wigner transform of the master equation. Then: As before, we are interested in the regime where we can neglet the term that causes relaxation, which, in the macroscopic limit, can be made very small without decreasing the effect of decoherence caused by the last, diffusive term. As we saw in the previous Section, the role of this decoherence term is to destroy the quantum coherence of the fragments of the wavefunction between spatially separated regions. Thus, in effect, this decoherence term can esure that the Poisson bracket is always reasonably accurate. Diffusion prevents the wavepacket from becoming too finely structured in momentum, which would have caused the failure of the correspondence principle. In the case of the thermal environment, the diffusion coefficient D = ηk B T , where η is viscosity. The competition between the squeezing resulting from the chaotic instability and spreading resulting from diffusion leads to a standoff when the Wigner function becomes coherently spread over This translates into the critical (spatial) momentum scale of σ c = 2D λ , (6.12) which nearly coincides with the quick estimate given by equation (6.9). Returning to an outrageous example of the solar system, for a planet of the size of Jupiter a chaotic instability on the four-million-year timescale and the consequent delocalization would be easily halted even by a very rarefied medium (0.1 atoms/cm 3 , comparable to the density of interplanetary gas in the vicinity of massive outer planets) at a temperature of 100K (comparable to the surface temperature of major planets): The resulting ℓ c is on the order of 10 −29 cm! Thus, decoherence is exceedingly effective in preventing the packet from spreading; ℓ c << χ, by an enormous margin. Hence, the paradox we have described in the first part of the paper has no chance of materializing.
The example of quantum chaos in the solar system is a dramatic illustration of the effectiveness of decoherence, but its consequences are, obviously, not restricted to celestial bodies: Schrödinger cats, Wigners friends, and, generally, all of the systems that are in principle quantum but sufficiently macroscopic will be forced to behave in accordance with classical mechanics as a result of the environment-induced superselection [1,2]. This will be the case whenever ℓ c ≪ χ , (6.13) because ℓ c is a measure of the resolution of "measurements" carried out by the environment. This incredible efficiency of the environment in monitoring (and, therefore, localizing) states of quantum objects is actually not all that surprising. We know (through direct experience) that photons are capable of maintaining an excellent record of the location of Jupiter (or any other macroscopic body). This must be the case, because we obtain our visual information about the universe by intercepting a minute fraction of the reflected (or emitted) radiation with our eyes.
Our discussion extends and complements developments that go back more than a decade [65]. We have established a simple criterion for the recovery of the correspondence, Eq. (6.13), which is generously met in the macroscopic examples discussed above. And, above all, we have demonstrated that the very same process of decoherence that delivers "pointer basis" in the measuring apparatus can guard against violation of the quantumclassical correspondence in dynamics.

The arrow of time: a price of classicality?
Decoherence is caused by the continuous measurement-like interactions between the system and the environment. Measurements involve the transfer of information, and decoherence is no exception: The state of the environment acquires information about the system. For an observer who has measured the state of the system at some initial instant the information he will still have at some later time will be influenced (and, in general, diminished) by the subsequent interaction between the system and the environment. When the observer and the environment monitor the same set of observables, information losses will be minimized. This is in fact the idea behind the predictability sieve [3,4] -an information-based tool which allows one to look for the einselected, effectively classical states under quite general circumstances. When, however, the state implied by the information acquired by the observer either differs right away from the preferred basis selected by the environment, or -as will be the case here-evolves dynamically into such a "discordant" state, the environment will proceed to measure it in the preferred basis, and, from the observer's point of view, information loss will ensue.
This information loss can be analyzed in several ways. The simplest is to compute the (von Neumann) entropy increase in the system. This will be our objective in this section. However, it is enlightening to complement this "external" view by looking at the consequences of decoherence from the point of view of the observer, who is repeatedly monitoring the system and updating his records. [66] The loss of information can be quantified by the increase of the von Neumann entropy, H = −T r ρ ln ρ , (6.14) where ρ is the reduced density matrix of the system. We shall now focus on the rate of increase of the von Neumann entropy in a dynamically evolving system subject to decoherence. As we have seen before, decoherence restricts the spatial extent of the quantum-coherent patches to the critical coherence length ℓ c , Eq. (6.11). A coherent wavepacket that overlaps a region larger than ℓ c will decohere rapidly, on a time-scale τ D shorter than the one associated with the classical predictability loss rate given by the Lyapunov exponent λ. Such a wavepacket will deteriorate into a mixture of states, each of which is coherent over a scale of dimension ℓ c by σ c =h/ℓ c . Consequently, the density matrix can be approximated by an incoherent sum of reasonably localized and approximately pure states. When N such states contribute more or less equally to the density matrix, the resulting entropy is H ≃ ln N.
The coherence length ℓ c determines the resolution with which the environment is monitoring the position of the state of a chaotic quantum system. That is, by making an appropriate measurement on the environment, one could in principle localize the system to within ℓ c . As time goes on, the initial phase space patch characterizing the observer's information about the state of the system will be smeared over an exponentially increasing range of the coordinate, Eq. (6.2). When the evolution is reversible, such stretching does not matter, at least in principle: It is matched by the squeezing of the probability density in the complementary directions (corresponding to negative Lyapunov exponents). Moreover, in the quantum case folding will result in the interference fringes -telltale signature of the long range quantum coherence, best visible in the structure of the Wigner functions.
Narrow wavepackets, and, especially, small-scale interference fringes are exceedingly susceptible to monitoring by the environment. Thus, the situation changes dramatically as a result of decoherence. In a chaotic quantum system, the number of independent eigenstates of the density matrix will increase as Consequently, the von Neumann entropy will grow at the rate: This equation emerged as a "corollary" of our discussion, but perhaps it is even its key result: Decoherence will help restore the quantum-classical correspondence. But we have now seen that this will happen at a price. Loss of information is an inevitable consequence of the eradication of the "Schrödinger cat" states that were otherwise induced by the chaotic dynamics. They disappear because the environment is "keeping an eye" on the phase space, monitoring the location of the system with an accuracy set by ℓ c . Throughout this section we have "saved" on notation, using "λ" to denote (somewhat vaguely) the rate of divergence of the trajectories of the hypothetical chaotic system. It is now useful to become a bit more precise. A Hamiltonian system with D degrees of freedom will have in general many (D) pairs of Lyapunov exponents with the same absolute value but with opposite signs. These global Lyapunov exponents are obtained by averaging local Lyapunov exponents, which are the eigenvalues of the Jacobian of the local transformation, and which describe the rates at which a small patch centered on a trajectory passing through a certain location in the phase space is being deformed.
The evolution of the Wigner function in the phase space is governed by the local dynamics. However, over the long haul, and in the macroscopic case, the patch that supports the probability density of the system will be exponentially stretched. This stretching and folding will produce a phase-space structure that differs from the classical probability distribution because of the presence of the interference fringes, with the fine structure whose typical scale is on the order ofh/ℓ (i) (t). In an isolated system, this fine structure will saturate only when the envelope of the Wigner function fills in the available phase space volume. Monitoring by the environment destroys these small-scale interference fringes and keeps W from becoming narrower than σ c in momentum. As a result -and in accord with Eq. (6.16) above-the entropy production will asymptotically approach the rate given by the sum of the positive Lyapunov exponents, This result [60] is at the same time familiar and quite surprising. It is familiar because it coincides with the Kolmogorov-Sinai formula for the entropy production rate for a classical chaotic system. Here we have seen underpinnings of its more fundamental quantum counterpart. All the same,  Fig. 6. Entropy production resulting from decoherence for a classically chaotic system becomes, after an initial transient, independent of the value of the diffusion constant and set by the Lyapunov exponent. See [71] it is surprising because it is independent of the strength of the coupling between the system and the environment, even though the process of decoherence (caused by the coupling to the environment) is the ultimate source of entropy increase. Over the last few years, the argument we presented above has been investigated and confirmed, using numerical simulations (see [67][68][69][70][71]). Figure 6 presents clear evidence showing that in the chaotic regime the entropy production rate approaches the value set by the Lyapunov exponent (data correspond to studies of a quantum particle moving in a harmonically driven double well potential [71]). This independence is indeed remarkable, and leads one to suspect that the cause of the arrow of time may be traced to the same phenomena that are responsible for the emergence of classicality in chaotic dynamics, and elsewhere (i.e., in quantum measurements). In a sense, this is of course not a complete surprise: Von Neumann knew that the measurements are irreversible [18]. And Zeh [32] emphasized the close kinship between the irreversibility of the "collapse" in quantum measurements and in the second law, cautioning against circularity of using one to solve the other. However, what is surprising is that both the classical-looking result ultimately has quantum roots, and that these roots are so well hidden from view that the entropy production rate depends solely on the classical Lyapunov exponents.
Environment may not enter explicitly into the entropy production rate, Eq. (6.17), but it will help determine when this asymptotic formula becomes valid. The Lyapunov exponents will "kick in" as the dimensions of the patch begin to exceed the critical sizes in the corresponding directions, ℓ (i) (t)/ℓ (i) c 1. The instant when that happens will be set by the strength of the interaction with the environment, which determines ℓ c . This "border territory" may be ultimately the best place to test the transition from quantum to classical. One may, for example, imagine a situation where the above inequality is comfortably satisfied in some directions in the phase space, but not in the others. In that case, the rate of the entropy production will be lowered to include only these Lyapunov exponents for which decoherence is effective.

Decoherence, einselection, and the entropy production.
The significance of the efficiency of decoherence goes beyond the example of the solar system or the task of reconciling quantum and classical predictions for classically chaotic systems. Every degree of freedom coupled to the environment will suffer loss of quantum coherence. Objects that are more macroscopic are generally more susceptible. In particular, the "hardware" responsible for our perceptions of the external universe and for keeping records of the information acquired in the course of our observations is obviously very susceptible to decoherence. Neurons are strongly coupled to the environment and are definitely macroscopic enough to behave in an effectively classical fashion. That is, they have a decoherence timescale many orders of magnitude smaller than the relatively sluggish timescale on which they can exchange and process information. As a result, in spite of the undeniably quantum nature of the fundamental physics involved, perception and memory have to rely on the information stored in the decohered (and, therefore, effectively classical) degrees of freedom.
An excellent illustration of the constraint imposed on information processing by decoherence comes from the recent discussions of the possibility of implementing real quantum computers. Decoherence is viewed as perhaps the most serious threat to the ability of a quantum information processing system to carry out a superposition of computations [11,30]. Yet, precisely such an ability to "compute" in an arbitrary superposition would be necessary for an observer to be able to "perceive" an arbitrary quantum state. Moreover, in the external universe only those observables that are resistant to decoherence and which correspond to "pointer states" are worth recording. Records are valuable because they allow for predictions, and resistance to decoherence is a precondition to predictability [3,14].
It is too early to claim that all the issues arising in the context of the transition from quantum to classical have been settled with the help of decoherence. Decoherence and einselection are, however, rapidly becoming a part of a standard lore [72,73]. Where expected, they deliver classical states, and -as we have seen above-guard against violations of the correspondence principle. The answers that emerge may not be to everyone's liking, and do not really discriminate between the Copenhagen Interpretation and the Many Worlds approach. Rather, they fit within either mold, effectively providing the missing elements -delineating the quantum-classical border postulated by Bohr (decoherence time fast or slow compared to the dynamical timescales on the two sides of the "border"), and supplying the scheme for defining distinct branches required by Everett (overlap of the branches is eliminated by decoherence).
7 How to fight against decoherence: Quantum error correcting codes.
It is clear that decoherence is a process that has a crucial role in the quantum-to-classical transition. But in many cases, physicists are interested in understanding the specific causes of decoherence just because we want to get rid of it. Thus, decoherence is responsible for washing out the quantum interference effects we would very much like to see as a signal in some experiments. This is the type of situation one is clearly facing in quantum computation (and in the physics of quantum information in general). A quantum computer is a gigantic interferometer whose wave function explores an exponential number of classical computations simultaneously. Coherence between branches of the computer wave function should be maintained because the existence of quantum interference between these branches is the basic reason why these computers can outperform their classical counterparts. Thus, decoherence in this context is a major problem. An obvious way of try to prevent decoherence from damaging quantum states is to reduce the strength of the coupling between the system and its environment. However, it is never possible to reduce this coupling to zero and eliminate decoherence in this way. Remarkably, in recent years new techniques that enable the active protection of the information stored in quantum states from the degrading effect of the interaction with the environment have emerged. They come under the name of "Quantum Error Correcting Codes" (QECC) and were invented by people working on quantum computation [74,75]. They are based on remarkably simple and beautiful ideas and could be found to be useful in other areas of physics. For this reason, we believe it could be interesting to include this final main theme to give a simple-minded presentation of the methods that could enable us in principle to "fight against decoherence" preserving quantum states.

How to protect a classical bit
To introduce the basic idea of Quantum Error Correcting Codes it is better to start with a short discussion of the simplest ways in which one can protect classical information. Suppose that we have a single qubit b that lives in a noisy environment. Because of the effect of the noise we will assume that the bit has a probability p to flip after some time. Therefore, if we look at the bit after this time, the probability of the bit being unaltered by the noise is 1−p and therefore, the information is degraded. Can we protect this classical bit? The answer is "yes" and the way to do it is by using an error correcting code. The simplest such procedure is based on the brute force use of redundancy as follows. We can "encode" this one bit of information using more carriers, mapping the state of the bit into many identical copies (i.i. b → (b, b, . . . , b)). If we do this, we can recover the initial information after the noise occurred by voting on and adopting as our result the one that gets the majority of votes. In this way we also discover which carriers were altered by the noise (i.e., the minority) and recuperate the information. Of course, this works if the error probability is small enough. To be precise, let us assume that we encode the information in three carrier bits (this is the simplest repetition code). The probability that no flip occurs is P (no f lip) = (1 − p) 3 , and the others are simply P (one f lip) = 3p(1 − p) 2 , P (two f lips) = 3p 2 (1 − p), and P (three f lips) = p 3 . Thus, the above error-correcting strategy (encoding one into three bits and voting at the end) increases the probability of keeping the information intact from 1 − p to 1 − 3p 2 + 2p 3 = 1 − O(p 2 ), which is close to unity, provided p is small enough. This example illustrates the simplest classical error correction code. Of course, much more sophisticated codes exist, and we are probably not doing justice to the beautiful theory of classical error-correcting codes (see [76]) by using this naive code as an example. However, we think it is enough for the purpose of our discussion.

How to protect a quantum bit
The basic question then becomes if it is possible to generalize this simple procedure to quantum mechanics. One may be tempted to guess that this task is impossible because a quantum version of the naive repetition code described above could never work as a consequence of the nonclonability of quantum states. Also, the fact that measurements drastically affect the state of quantum systems [12,13] is somehow suggestive of the difficulties of implementing an error-correcting quantum strategy naively translating the classical error-correcting ideas. However, these expectations were proven to be incorrect when in 1995 Peter Shor created the first quantum errorcorrecting code [75]. His work, once again triggered a lot of activity and over the last four years the theory of Quantum Error Correcting Codes was fully developed. So far, there have been some experimental demonstrations showing the workings of these codes (only in NMR experiments) but in our view, the interesting ideas of QECC still are waiting for phycisists to give a definite answer to whether or not they will be useful for other purposes than the ones that originally motivated them. For this reason, we find it interesting to bring these issues to this review.
Let us now describe how it is possible to create QECC. For this, we consider a quantum bit prepared in an arbitrary quantum state Ψ = α|0 + β|1 . To be precise, we will first describe how noise affects the state of the qubit whose state we want to protect. We will first consider the simplest case of a noise that just produces "dephasing". We assume that the noise introduces a random phase with a probability p or leaves the state intact with a probability 1 − p. Although this is not the most general kind of operation that a noisy environment can produce on a quantum system, we will later show that this is not a restrictive assumption and that the treatment we present here can be generalized to include all of the effects that the noise can produce. So, for the moment we will just consider this "dephasing" noise. The dephasing can be simply described by the action of a σ z operator on the state of the system. In this chapter, we will adopt the following notation. The Pauli matrices σ x,y,z are simply denoted as X, Y, Z. Thus, if the initial state of the system is Ψ 0 = α|0 + β|1 the final state (after the noise has occurred) is described by a density matrix as, where ρ in = |Ψ 0 Ψ 0 |. It is easy to see that the interaction with the noise degrades the quantum state, causing the loss of quantum coherence. As a measure of this degradation, we can compute the "fidelity" of the process that is simply given by the overlap between the ideal state and the actual state. Using the above form for the density matrix, we find out that fidelity is reduced to F = T r(ρ out ρ in ) = 1 − 4p|αβ| 2 . Thus, fidelity is reduced by an amount that is linear in the error probability p. Another measure of the degradation is given by the loss of purity of the final state that can be measured, for example, by T r(ρ 2 out ) = 1 − 8p(1 − p)|αβ| 2 . In what follows, we will present a method that enables us to protect the quantum state in such a way that the fidelity (or the loss of purity) does not decay linearly with the error probability but it does so quadratically.
So, let us present a way to protect the state of our qubit from the effect of a dephasing environment. As in the classical case, we will use many carriers to protect one qubit of information (in our example, we use three qubits to protect one). But the use of redundancy has to be more subtle in the the quantum case. The key idea is to encode the logical states into entangled states of the three qubits in such a way that when an error occurs, the logical states are mapped into other orthogonal subspaces (one subspace for each error we want to correct). If this is the case, we can learn about the error by measuring an observable that just tells us in what two-dimensional subspace the state is in. In this way, we learn what the error was without getting any information about the state itself. Once we know the error, we can correct it and start the process all over again. This idea is illustrated clearly (we hope!) by the three-qubit example. In this case, we can use the following encodinf for the logical states: (the subscript L is used to denote the logical states). The "encoding" process is simply the mapping of the physical states of the three independent carriers onto the above entangled logical states. This task is the first one that one has to do to protect the information and is represented by a unitary operator (the encoding operator E). One takes the qubit whose quantum state is to be protected and applies an operation to it together with the other two carriers we use. This operation maps the initial state into the encoded state, i.e., E|α|0 + β|1 )|00 = α|||0 L + β|1 L . Later in this section we will describe ways in which the encoding operation can be implemented. The reason why (7.2) is a good encoding can be seen as follows. It is a simple exercise to show that when we apply an error operator to any of the two logical states (i.e., when we act with a Z operator on any one of the qubits) we obtain mutually orthogonal states. Thus, one can show that |0 L ⊥ Z i |0 L ⊥ |1 L ⊥ Z i |1 L for i = 1, 2, 3, i.e., that the two logical states and their "erroneous descendants" are a set of eight mutually orthogonal states that constitute a basis of the complete Hilbert space of the three qubits. Therefore, the total Hilbert space can be decomposed in the direct sum of four two-dimensional subspaces. The "logical subspace" H L , which is generated by the two vectors {|0 L , |1 L }, has three "erroneous descendents" which are simply Z i H L , and the total Hilbert space is the direct sum of H L and Z i H L (i = 1, 2, 3). As a consequence, there is an observable that we could measure to determine in which one of the four subspaces the state is in. In so doing, we discover the error and can correct it trivially.
To complete our description, we just have to exhibit what this observable whose measurement reveals the error is. To do this, it is interesting to look at the symmetries of the logical states (7.2). It is clear that these states are eigenstates of the operators M 1 = X 1 X 2 and M 2 = X 2 X 3 with eigenvalue +1 (thus, |0 L is an homogeneous superposition of all states with an even number of ones and |1 L contains all states with an odd number of ones; therefore these states are invariant when we flip any two states, which is precisely what the X i X j operators do). Moreover, it is easy to show that M 1 and M 2 are two commuting hermitian operators whose eigenvalues are ±1 (this follows from the fact that these operators square to the identity, i.e., M 2 i = 1). Moreover, it is simple to show that all the "erroneous descendents" of the logical subspace are also eigenspaces of M i . For example, the subspace Z 1 H L is formed by linear superpositions of the vectors {z 1 |0 L , |1 L } that are eigenstates of M 1 and M 2 with eigenvalues equal to −1. This follows from the fact that as the error-operator Z 1 anticommutes with M 1 and M 2 , it transforms eigenstates of these operators into eigenstates with a different eigenvalue (i.e., if M i |φ = |φ , then M i Z 1 |φ = −Z 1 |φ ). Therefore, if our goal is to find out in which of the four two-dimensional subspaces the state is in, we just have to measure the two operators M 1 and M 2 . The result of this measurement is always represented by a set of two numbers that are ±1 (the two eigenvalues of M i ) and each of the four possible alternatives (that are known as the error syndromes) identify uniquely one of the four subspaces (H L corresponds to the syndrome (+1, +1), Z 1 H L to (−1, −1), Z 2 H L to (−1, +1) and Z 3 H L to (+1, −1).
It is also interesting to think about what kind of physical procedure we should follow to perform this kind of measurement. As discussed, we need to measure the operators M i that are constructed as tensor products of Pauli matrices. However, it is very important to realize that we must do this without measuring individually the factors appearing in these products! Thus, in our case, we need to measure only M 1 = X 1 X 2 and M 2 = X 1 X 3 , but we cannot do this by measuring the three operators X i individually. If we were to do this, we would be measuring a complete set of commuting observables and causing the system to collapse into a particular state. Instead, quantum error correction needs measuring, not a complete set of observables but only enough observables to gain information about the error without destroying the coherence in the state of the system (thus, we want our measurement to project the state into a two-dimensional subspace and not to collapse it into one ray).
It is not hard to find a systematic way to devise a strategy that will enable us to measure any operator that is the tensor product of Pauli matrices without measuring the individual factors. To do this, it is clear that because the observables we measure are collective, we should induce an interaction between the qbits in such a way that after the interaction, the result of the measurement is "written" on only one particle. For example, suppose that we have two particles and we want to measure the operator M = X 1 X 2 . Suppose also that we find a unitary operator D satisfying the condition Z 2 D = DM . This condition implies that the operator D will transform an eigenstate of M with eigenvalue m (that can only be ±1) into an eigenstate of Z with eigenvalue m. Therefore, if we want to measure M , we can first apply the unitary operation D and then measure Z 2 (in other words, D is the operator that changes basis from M to Z 2 eigenstates). Thus, now we just need to construct this operator. This can be done by using a simple quantum circuit. In fact, the quantum circuit for the operator D is shown in Figure 7. We just have to apply a Hadammard rotation to each qubit and then do a c-not using the first qbit as the control and the second one as the target. To show that this is the correct circuit for D, we just have to show that the relation Z 2 D = DM is satisfied. For this purpose, we apply M = X 1 X 2 to the left of the circuit and start moving the X i operators to the right. As these operators satisfy that RX = ZR, they transform into Z operators when they pass through the Hadammard rotations. Then, the Z operator in the control goes through the end of the circuit but the one acting on the target generates an extra Z in the control qubit that cancels the first one. Therefore, this implies that the circuit satisfies the required identity. Using this simple idea, is possible to design simple quantum circuits that can be used to measure any collective observables built as tensor products of Pauli matrices. Moreover, this can be generalized to any number of qubits. For example, the circuit to measure M 1 = X 1 X 2 and M 2 = X 1 X 3 is given in Figure 6 and consists of three Hadammard rotations (one in each qubit) followed by two c-not gates with the first qubit acting as the control. It is easy to see that if we measure the second and third qubits after the circuit, we learn about the syndrome and therefore find out what the error was.
To recover from the error, we just have to apply a simple operation to the remaining qubit that we do not measure (the first one in our example). This qubit contains the quantum state up to some unitary transformation that we can undo. To find out how to recover from the error, the idea is simply to see what the circuit does to the errors themselves. In fact, it is easy to show that the operator D associated with the decoding circuit appearing in Figure 7 satisfies that Z 1 D = DX 1 X 2 X 3 , and that Z 2 D = DX 2 , Z 3 D = DX 3 . Therefore, this means that if the encoded state is affected by a Z 1 -type error, the resulting state after decoding will have the last two qubits set to one (we already knew that this was the syndrome corresponding to this error), and the first qubit will be affected by an X rotation that we should undo. On the contrary, the other two errors (Z 2 and Z 3 ) do not require any corrective action.
So, to summarize, the error-correcting procedure is the following: (1) We encode the qubit in three carriers applying the encoding circuit shown in Figure 7. (2) After the errors act on the system, we decode the state, detect the syndromes, and apply corrective operations. (3) We refresh the syndrome qubits (resetting them to the zero state) and encoding again. It is clear that measurement of the syndrome is not really necessary because it can always be replaced by a corrective operation performed by means of a quantum circuit (in our case a c-c-not that is controlled by the second and third qubits). The essential part of this method is the refreshing of the syndrome qubits that is the part responsible for taking away the "entropy" generated by errors. Two final comments are worth making before giving a more formal presentation. First, we should remark that our discussion so far assumed that errors were applied by some agent that acted on a single (unknown) qubit. However, we can extend this method to consider a situation in which there is a probability p for any one qubit to be affected by a Z-type error. In this case, the state of the three qubits before the decoding and corrective circuit is applied is given by the following density matrix: After we apply the decoding and corrective procedure to this density matrix, it is clear that the first two terms will now be simply proportional to ρ in . Thus, in this way we have completely eliminated the term that is linear in the error probability p. The final state differs from the ideal one only through terms that are quadratic in the error probability. Therefore, the fidelity of the whole process will be given by F = 1 − O(p 2 ). On this linearto-quadratic change in the dependence of F on p relies the whole power of quantum error correction (which clearly only has a good chance of working at this level, without concatenation, if p is small enough).
Finally, we could worry about not having considered more general classes of errors. However, it should be clear by now that the general idea described so far could be generalized to include more general operations. It is important to realize that to take into account all possible effects the environment could cause on a qubit, we should protect not only against phase errors (associated with Z operators) but also against bit flips (associated with X operators) and on a combination of both (associated with Y operators). It is clear that if we are able to protect against three types of independent errors, we could also fight efficiently against arbitrary unitary (or nonunitary) errors that can always be written in terms of operators that are linear combinations of these three elements and of the identity. So, the question is how to invent codes that protect against arbitrary errors affecting any one of the carrier qubits. A code like this was first presented by Peter Shor [75] and can be constructed using our previous three qubit QECC as a building block. In fact, Shor encodes one qubit using nine carriers organized in three blocks of three qubits each. The logical states are a product of three factors like the ones shown in (7.2). This code has the following eight symmetry operations (the previous one had the two symmetries, M 1 and M 2 ): First, we easily find six symmetry operators that generalize the previous M 1 and M 2 in the three blocks of three qubits. Second, we find two other independent symmetries corresponding to the fact that the three blocks are repeated: It is easy to show that each of the 27 different errors that can affect the nine carriers corresponds to a different syndrome (and therefore maps the logical states into orthogonal subspaces). The decoding should be done by measuring the above eight operators that reveal the syndrome and allow us to know the error that took place enabling us to correct it. A decoding circuit for this code can be easily constructed following the same discussion presented above for the three qubit. It is interesting to note that the code presented by Shor is by no means the most efficient way to correct errors. In fact, we notice that we are using an enormous Hilbert space of dimension 2 9 = 512, but we would only need a space with enough room to accommodate for all the subspaces where we would map independent errors (in this case we require for this purpose only 2(1 + 3 × 9) = 56). Smaller codes have been developed, and the smallest one that corrects general onequbit errors requires five qubits [77], because n = 5 saturates the identity 2 n = 2(3n + 1). This is the so-called "perfect" QECC and has the following symmetry operators M 1 = Z 2 Z 3 Z 4 Z 5 , M 2 = Y 1 Z 3 X 4 Y 5 , M 3 = Z 1 X 2 Z 3 X 5 , and M 4 = Z 1 Y 3 Y 4 Z 5 . To show that these symmetry operators constitute a good QECC requires showing that all independent errors produce a different anticommutation pattern with the M i operator (this is left as an exercise). The construction of an encoding-decoding circuit for this code can also be done by generalizing the ideas we have described before.

Stabilizer quantum error-correcting codes
A more formal description of the principles underlying the theory of quantum error-correcting codes can be given (following the presentation of [78] we restrict ourselves to discuss a rather wide class of codes known as stabilizer codes (for more general codes and for a more thorough discussion of QECC we refer the reader to [79][80][81]). We can consider codes that protect k qubits by encoding them into n carriers. Here, the code space H k (or logical space) is a 2 k dimensional subspace of the total Hilbert space of the n carriers. H n is a tensor product of n two-dimensional factors and has a natural basis whose elements are product states of the individual carriers. This is the "physical basis" that can be formed with the common eigenstates of the operators {Z 1 , . . . , Z n } (for convenience, we label states of this basis not by the eigenvalues of the corresponding operators, which are ±1, but by the eigenvalues of the projectors onto the −1 subspace, which are 0 or 1: thus, the label z j = 0 (z j = 1) corresponds to a +1 (−1) eigenvalue of the operator Z j ). Furthermore, we order the n carriers in such a way that the last k qubits are the ones whose state we encode, and the first n − k are the ancillary carriers. Therefore, states of the physical basis are of the form |s, z P = |s P ⊗ |z P (where the strings s = (s 1 , . . . , s n−k ), z = (z 1 , . . . , z k ) store the corresponding eigenvalues and the subscript P is used to identify the states of the physical basis).
An error-correcting code is a mapping from the physical product states |0 P ⊗ |Ψ P onto the code space H k , which is formed by entangled states of n carriers. A rather general class of codes can be described in terms of their stabilizer group (see [80]). The stabilizer of the code is an Abelian group formed by all operators that are tensor products of Pauli matrices and have H k as an eigenspace with an eigenvalue equal to +1. Every element of the stabilizer, which is a finite group with 2 n−k elements, can be obtained by appropriately multiplying n − k generators, which will be denoted as M 1 , . . . , M n−k . The elements of the stabilizer are completely degenerate in the code space H k (since all states in H k are eigenstates with eigenvalue +1 of all M j ). To define a basis in the code space, we choose k extra operators L 1 , . . . , L k , which being tensor products of Pauli matrices commute with all elements of the stabilizer. These operators L j ′ , j ′ = 1, . . . , k are the "logical pointers" because they define the directions in H k associated with the logical states |0 L , . . . , |2 k − 1 L (logical pointers belong to the group of operators that commute with the stabilizer, known as the normalizer).
The n − k generators of the stabilizer together with the k logical pointers are a Complete Set of Commuting Operators (CSCO) whose common eigenstates form a complete basis of the Hilbert space H n . Elements of this "logical basis", labeled by n quantum numbers, are denoted as |m, l L , where the bit strings m = (m 1 , . . . , m n−k ), and l = (l 1 , . . . , l k ) identify the corresponding eigenvalues, and the subscript L refers to logical states. The CSCO formed by the generators of the stabilizer and the logical pointers defines a prescription for decomposing the original Hilbert space of the ncarriers into a tensor product of a 2 k -dimensional logical space L and a 2 n−k -dimensional syndrome space Y. In fact, elements of the logical basis (which are entangled states of the n-carriers) are tensor products of states belonging to L and Y: |m, l L = |m L ⊗ |l L . Encoded states, which belong to H k , are also product states of the form |Ψ = |0 L ⊗ l c l |l L .
The code protects quantum states against any error E a whose action on states of the logical basis is to change the logical syndrome and, eventually, rotate the logical state in L in a syndrome-dependent way, E a |m L ⊗ |l L = e iφma |m + c a L ⊗ U a |l L . (7.4) Here, U a is a unitary operator acting on the collective logical space L, and φ ma is a phase that may depend on the syndrome and the error. The error E a changes the syndrome from m to m + c a where c a is the bit string storing the commutation pattern between the error and the generators of the stabilizer (the jth bit of this string is one if the error anticommutes with M j and is zero otherwise). The reason for this is that when acting on a logical state, the error E a changes the eigenvalue of the operator M j only if {M j , E a } = 0. The label a used to identify errors is arbitrary and, for the case of nondegenerate codes (which are the only ones we will consider here) it is always possible to label errors E a using simply the commutation pattern c a (i.e., we can choose a = c a ).
To correct against the action of any of the errors E a (or against any linear superposition of them) one can first detect the error by measuring the collective syndrome (i.e., measuring the observables M j , j = 1, . . . , n − k) and later recover from the error by applying the corresponding operator U † a . This detection-recovery process can be conveniently described as a quantum operation defined by the following mapping from the erroneous density matrix ρ in into the corrected one ρ out , where the sum runs over all syndromes (N = 2 n−k − 1), and the recovery operator for each syndrome is By construction, these operators satisfy the identity N m=0 R † m R m = I. Because our description of error detection-recovery process is entirely formulated on the logical basis, it does not involve so far any reference to encoding or decoding operations that can be simply defined as a change of basis. The encoding operator C is a unitary operator mapping the physical basis, formed by product states of the n carriers, onto the logical basis, formed by entangled states. Accordingly, C transforms the operators Z i (whose eigenvalues define states on the physical basis) into the operators M j , L j ′ (that label states on the logical basis). Thus, the encoding operator C is such that Z j = C † M j C, j = 1, . . . , n − k, and Z n−k+j ′ = C † L j ′ C, j ′ = 1, . . . , k. Taking this into account, the action of the operator R m can be described, in the physical basis, as the following sequence of operations: i) decode the state, ii) measure the syndrome in the physical basis by measuring Z j in the first (n − k) carriers, iii) if the result of the measurement is the string s, apply the syndrome-dependent recovery operator U † s resetting the syndrome back to zero, and iv) encode the resulting state.
Finding a stabilizer code correcting a given set of errors is a rather hard task that involves designing generators having appropriate commutation patterns with the errors. Once the generators are found and the logical pointers are chosen, an encoding or decoding operator can be constructed (strategies for designing encoding or decoding circuits from the stabilizer are known; see [82,83]). The recovery operators depend on the encoding or decoding strategy and can be explicitly found from the encoding circuit by running errors through it.
As we mentioned above, the simplest code protecting k = 1 qubit using n = 3 carriers correcting against phase errors in any of the carriers can be understood as a particular example of this general stabilizer code class. In such a case, the basic errors to correct are E 1 = Z 1 , E 2 = Z 2 and E 3 = Z 3 . The stabilizer of the code can be chosen to be generated by M 1 = X 1 X 2 and M 2 = X 1 X 3 . The commutation pattern associated with each error is c 1 = 11 (because the error Z 1 anticommutes with both M 1 and M 2 ), c 2 = 10, c 3 = 01 (note that we could relabel the errors ordering them according to their commutation pattern). The decoding circuits exhibited in Figure 7 has the properties C † Z 1 C = X 1 X 2 X 3 , C † Z 2 C = X 2 , and C † Z 3 C = X 3 . (7.7) These properties entirely determine the action of the errors Z i in the logical basis. For example, the last identity implies that E 3 |m L |l L = |m+c 3 L |l+ 1 L . Thus, the error E 3 not only changes the syndrome but also modifies the logical state by flipping it. This means that the recovery operator for this error is U 3 = X. Analogously, we can find how the other errors act on the logical basis, showing that U 1 = U 2 = I.

Discussion
We have seen "decoherence in action" in a variety of settings. Our aim was not to review all of the studies of decoherence done in recent years. Thus, we left aside from our review the discussion of some very interesting physical problems where the role of environment-induced decoherence is relevant. For example, in cosmology, the way in which decoherence can account for the quantum to classical transition of density fluctuations (and of spacetime itself) has been -and still is-a matter of debate (see [84] for an incomplete list of relevant papers). Fortunately, there are also other areas where decoherence can be analyzed and tested in the laboratory. Among them, the use of systems of trapped and cold atoms (or ions) may offer the possibility of engeneering the environment (effectively choosing the pointer states) as proposed in [85]. Trapped atoms inside cavities were discussed [86] and the relation between decoherence and other cavity QED effects (such as Casimir effect) was analyzed [87]. On the mesoscopic scale, the nature of decoherence may receive increasing attention specially in the context of BEC both as a key ingredient in the phenomenological description [88] and as a threat to the longevity of BEC Schrödinger cats [89]. Moreover, the nature of decoherence is being studied experimentally in the context of condensed matter systems (see, for example, [90]). The aim of this section is to describe briefly what is (and point out what is not) accomplished by decoherence, and to show how it facilitates understanding the transition from quantum to classical. Environment-induced superselection is clearly the key interpretational benefit arising from decoherence. The quantum principle of superposition does not apply to open quantum systems. States in the Hilbert space are no longer "equal". Under a broad variety of realistic physical assumptions, one is now forced to conclude that for macroscopic objects only a small subset of states can ever contribute to the "familiar classical reality". Only the einselected pointer states will persist for long enough to retain useful (stable) correlations with -say-the memories of the observers, or, more generally, with other stable states. By contrast, their superpositions will degrade into mixtures that are diagonal in the pointer basis.
The precondition for "perception" (as in "perception of classical reality") is the ability of the state to persist, or to evolve in a more or less predictable manner during a time interval over which the observer is monitoring it. This time interval can occasionally be quite short, but it should not be as unreasonably short as the typical decoherence time for the macroscopic systems. Thus, the only states that have a chance of being perceived as "real" are the preferred (pointer) states. Indeed, given the limited accuracy of the observer's efforts, it may be more precise to say that broad superpositions of pointer states are definitely ruled out.
It is important to emphasize that the environment-induced superselection leads to a probability distribution that is diagonal in the preferred basis, and not to a single pointer state. Thus, the uniqueness of perceptions of the observer has its roots in the stability of the correlations between the states of the macroscopic objects in the outside world and the records in the observer's memory (which, incidentally, must also use preferred states of, say, neurons to store records of the observations).
The information possessed by the observer is not an abstract, esoteric entity. Rather, "information is physical" [30] and "there is no information without representation" [3]. In practice, this means that the state of the observer is in part determined by what he knows about the rest of the universe. Thus, the physical existence of long-lasting records underlies the essence of the process of perception. Observers will be aware of their own records, and of the external universe in a state consistent with these records.
This viewpoint known as the existential interpretation [3,17] accounts for the apparent collapse, but is consistent with either the Many Worlds or Copenhagen Interpretation.
The nature of the preferred states is dynamically negotiated in the course of the interaction between the system and the environment, but, as we have already seen, the self-Hamiltonian of the system plays an important role. Truly realistic models are difficult to treat, but lessons of the predictability sieve applied to simple models allow one to infer with some confidence that, in general, pointer states will be localized in position. After all, most interactions depend on distance. Thus, localization is an inescapable consequence [2,14]. Nevertheless, as we have already seen in perhaps the most relevant exactly solvable case of a decohering harmonic oscillator, preferred states tend to be localized in both position and momentum and can be regarded as quantum counterparts of classical points.
Investigation of the coexistence of decoherence with chaos is an example of a bit more complicated case. There, we have seen that localization is effectively enforced (even if such systems cannot be treated analytically, and extensive numerical studies are required).
An exciting "corollary" of decoherence in the setting of quantum chaos is the quantum derivation of the classically anticipated entropy production rate, given by the sum of positive Lyapunov exponents. This suggests a quantum origin of the second law of thermodynamics. Indeed, it seems that the resolution of the two outstanding puzzles of physics -the arrow of time and the apparent classicality-may originate from the same essentially quantum source, from decoherence and einselection.
The study of decoherence and einselection over the past two decades has yielded a new paradigm of emergent, effective classicality. It leads to a new understanding of the quantum origins of the classical. To be sure, not all of the interpretational questions have been settled, and much further work is required. Nevertheless, as a result of this paradigm shift, the quantumto-classical transition has become a subject of experimental investigations, while previously it was mostly a domain of philosophy.