Decoherence, Einselection, and the Quantum Origins of the Classical

REVIEWS OF MODERN PHYSICS, VOLUME 75, JULY 2003 Decoherence, einselection, and the quantum origins of the classical

Wojciech Hubert Zurek Theory Division, LANL, Mail Stop B210, Los Alamos, New Mexico 87545 (Published 22 May 2003)

The manner in which states of some quantum systems become effectively classical is of great significance for the foundations of quantum physics, as well as for problems of practical interest such as quantum engineering. In the past two decades it has become increasingly clear that many (perhaps all) of the symptoms of classicality can be induced in quantum systems by their environments. Thus decoherence is caused by the interaction in which the environment in effect monitors certain observables of the system, destroying coherence between the pointer states corresponding to their eigenvalues. This leads to environment-induced superselection or einselection, a quantum process associated with selective loss of information. Einselected pointer states are stable. They can retain correlations with the rest of the universe in spite of the environment. Einselection enforces classicality by imposing an effective ban on the vast majority of the Hilbert space, eliminating especially the flagrantly nonlocal ‘‘Schro¨ dinger-cat states.’’ The classical structure of phase space emerges from the quantum Hilbert space in the appropriate macroscopic limit. Combination of einselection with dynamics leads to the idealizations of a point and of a classical trajectory. In measurements, einselection replaces quantum entanglement between the apparatus and the measured system with the classical correlation. Only the preferred pointer observable of the apparatus can store information that has predictive power. When the measured quantum system is microscopic and isolated, this restriction on the predictive utility of its correlations with the macroscopic apparatus results in the effective ‘‘collapse of the wave packet.’’ The existential interpretation implied by einselection regards observers as open quantum systems, distinguished only by their ability to acquire, store, and process information. Spreading of the correlations with the effectively classical pointer states throughout the environment allows one to understand ‘‘classical reality’’ as a property based on the relatively objective existence of the einselected states. Effectively classical pointer states can be ‘‘found out’’ without being re-prepared, e.g, by intercepting the information already present in the environment. The redundancy of the records of pointer states in the environment (which can be thought of as their ‘‘fitness’’ in the Darwinian sense) is a measure of their classicality. A new symmetry appears in this setting. Environment-assisted invariance or envariance sheds new light on the nature of ignorance of the state of the system due to quantum correlations with the environment and leads to Born’s rules and to reduced density matrices, ultimately justifying basic principles of the program of decoherence and einselection.

CONTENTS A. Models of einselection 730 1. Decoherence of a single qubit 730 2. The classical domain and quantum halo 731 I. Introduction 716 3. Einselection and controlled shifts 732 A. The problem: Hilbert space is big 716 B. Einselection as the selective loss of information 733 1. Copenhagen interpretation 716 1. Conditional state, entropy, and purity 733 2. Many-worlds interpretation 717 2. Mutual information and discord 733 B. Decoherence and einselection 717 C. Decoherence, entanglement, dephasing, and C. The nature of the resolution and the role of noise 734 envariance 718 D. Predictability sieve and einselection 735 D. Existential interpretation and quantum V. Einselection in Phase Space 736 Darwinism 719 A. Quantum Brownian motion 737 II. Quantum Measurements 720 A. Quantum conditional dynamics 720 B. Decoherence in quantum Brownian motion 740 1. Decoherence time scale 740 1. Controlled NOT and bit-by-bit measurement 720 2. Measurements and controlled shifts 721 2. Phase-space view of decoherence 741 3. Amplification 722 C. Predictability sieve in phase space 742 B. Information transfer in measurements 723 D. Classical limit in phase space 744 1. Reduced density matrices and correlations 723 1. Mathematical approach (ប→0) 744 2. Action per bit 723 2. Physical approach: The macroscopic limit 744 C. ‘‘Collapse’’ analog in a classical measurement 724 3. Ignorance inspires confidence in classicality 745 III. Chaos and Loss of Correspondence 725 E. Decoherence, chaos, and the second law 745 A. Loss of quantum-classical correspondence 725 1. Restoration of correspondence 745 B. Moyal bracket and Liouville flow 727 2. Entropy production 746 C. Symptoms of correspondence loss 728 3. Quantum predictability horizon 749 1. Expectation values 728 VI. Einselection and Measurements 749 2. Structure saturation 729 A. Objective existence of einselected states 749 IV. Environment-Induced Superselection 729 B. Measurements and memories 750

C. Axioms of quantum measurement theory 751 with a focus on decoherence, einselection, and the emer- 1. Observables are Hermitian—axiom (iii a) 751 gence of classicality, and also attempts to offer a preview 2. Eigenvalues as outcomes—axiom (iii b) 751 of the future of this exciting and fundamental area. 3. Immediate repeatability, axiom (iv) 752 4. Probabilities, einselection, and records 752 D. Probabilities from envariance 753 A. The problem: Hilbert space is big 1. Envariance 754 2. Born’s rule from envariance 755 The interpretation problem stems from the vastness of 3. Relative frequencies from envariance 757 Hilbert space, which, by the principle of superposition, 4. Other approaches to probabilities 757 admits arbitrary linear combinations of any states as a VII. Environment as a Witness 758 possible quantum state. This law, thoroughly tested in A. Quantum Darwinism 759 the microscopic domain, bears consequences that defy 1. Consensus and algorithmic simplicity 759 classical intuition: It appears to imply that the familiar 2. Action distance 760 3. Redundancy and mutual information 760 classical states should be an exceedingly rare exception. 4. Redundancy ratio rate 762 And, naively, one may guess that the superposition prin- B. Observers and the existential interpretation 762 ciple should always apply literally: Everything is ulti- C. Events, records, and histories 763 mately made out of quantum ‘‘stuff.’’ Therefore there is VIII. Decoherence in the Laboratory 764 no a priori reason for macroscopic objects to have defi- A. Decoherence due to entangling interactions 765 nite position or momentum. As Einstein noted1 localiza- B. Simulating decoherence with classical noise 766 tion with respect to macrocoordinates is not just inde- 1. Decoherence, noise, and quantum chaos 766 pendent of, but incompatible with, quantum theory. 2. Analog of decoherence in a classical system 767 How, then, can one establish a correspondence between C. Taming decoherence 767 the quantum and the familiar classical reality? 1. Pointer states and noiseless subsystems 767 2. Environment engineering 768 1. Copenhagen interpretation 3. Error correction and resilient quantum computing 768 Bohr’s solution was to draw a border between the IX. Concluding Remarks 769 quantum and the classical and to keep certain objects— X. Acknowledgments 771 especially measuring devices and observers—on the References 771 classical side (Bohr, 1928, 1949). The principle of superposition was suspended ‘‘by decree’’ in the classical domain. The exact location of this border was difficult to I. INTRODUCTION pinpoint, but measurements ‘‘brought to a close’’ quantum events. Indeed, in Bohr’s view the classical domain The interpretation of quantum theory has been an is- was more fundamental. Its laws were self-contained sue ever since its inception. Its tone was set by the dis- (they could be confirmed from within) and established ¨ cussions of Schrodinger (1926, 1935a, 1935b), Heisen- the framework necessary to define the quantum. berg (1927), and Bohr (1928, 1949; see also Jammer, The first breach in the quantum-classical border ap- 1974; Wheeler and Zurek, 1983). Perhaps the most inci- peared early: In the famous Bohr-Einstein double-slit sive critique of the (then new) theory was that of Ein- debate, quantum Heisenberg uncertainty was invoked stein, who, searching for inconsistencies, distilled the es- by Bohr at the macroscopic level to preserve wave- sence of the conceptual difficulties of quantum particle duality. Indeed, since the ultimate components mechanics through ingenious gedanken experiments. We of classical objects are quantum, Bohr emphasized that owe to him and Bohr clarification of the significance of the boundary must be moveable, so that even the human quantum indeterminacy in the course of the Solvay Con- nervous system could be regarded as quantum, provided gress debates (see Bohr, 1949) and elucidation of the that suitable classical devices to detect its quantum fea- nature of quantum entanglement (Bohr, 1935; Einstein, tures were available. In the words of Wheeler (1978, ¨ Podolsky, and Rosen, 1935; Schrodinger, 1935a, 1935b). 1983), who has elucidated Bohr’s position and decisively The issues they identified then are still a part of the contributed to the revival of interest in these matters, subject. ‘‘No [quantum] phenomenon is a phenomenon until it is Within the past two decades, the focus of research on a recorded (observed) phenomenon.’’ the fundamental aspects of quantum theory has shifted from esoteric and philosophical to more down to earth as a result of three developments. To begin with, many 1 of the old gedanken experiments [such as the Einstein- In a letter dated 1954, Albert Einstein wrote to Max Born, ␺ ␺ ¨ Podolsky-Rosen (EPR) ‘‘paradox’’] became compelling ‘‘Let 1 and 2 be solutions of the same Schrodinger equation... . When the system is a macrosystem and when ␺ and ␺ demonstrations of quantum physics. More or less simul- 1 2 are ‘narrow’ with respect to the macrocoordinates, then in by taneously the role of decoherence began to be appreci- far the greater number of cases this is no longer true for ␺ ated and einselection was recognized as key to the emer- ϭ␺ ϩ␺ 1 2 . Narrowness with respect to macrocoordinates is not gence of classicality. Last but not least, various only independent of the principles of quantum mechanics, but, developments led to a new view of the role of informa- moreover, incompatible with them.’’ [The translation from tion in physics. This paper reviews progress in the field Born (1969) quoted here is due to Joos (1986), p. 7].

Rev. Mod. Phys., Vol. 75, No. 3, July 2003 Wojciech Hubert Zurek: Decoherence, einselection, and the quantum origins of the classical 717

This is a pithy summary of a point of view—known as very uncomfortable place to do physics. The many- the Copenhagen interpretation (CI)—that has kept worlds interpretation is incomplete: it does not explain many a physicist out of despair. On the other hand, as what is effectively classical and why. Nevertheless, it was long as no compelling reason for the quantum-classical a crucial conceptual breakthrough. Everett reinstated border could be found, the CI universe would be gov- quantum mechanics as a basic tool in the search for its erned by two sets of laws, with poorly defined domains interpretation. of jurisdiction. This fact has kept many students, not to mention their teachers, awake at night (Mermin 1990a, B. Decoherence and einselection 1990b, 1994). Environment can destroy coherence between the 2. Many-worlds interpretation states of a quantum system. This is decoherence. Accord- ing to quantum theory, every superposition of quantum The approach proposed by Everett (1957a, 1957b) states is a legal quantum state. This egalitarian quantum and elucidated by Wheeler (1957), DeWitt (1970), and principle of superposition applies in isolated systems. others (see Zeh, 1970, 1973; DeWitt and Graham, 1973; However, not all quantum superpositions are treated Geroch, 1984; Deutsch, 1985, 1997; Deutsch et al., 2001) equally by decoherence. Interaction with the environ- was to enlarge the quantum domain. Everything is now ment will typically single out a preferred set of states. represented by a unitarily evolving state vector, a gigan- These pointer states remain untouched in spite of the tic superposition splitting to accommodate all the alter- environment, while their superpositions lose phase co- natives consistent with the initial conditions. This is the herence and decohere. Their name—pointer states— essence of the many-worlds interpretation (MWI). It originates from the context of quantum measurements, does not suffer from the dual nature of the Copenhagen where they were originally introduced (Zurek, 1981). interpretation. However, it also does not explain the They are the preferred states of the pointer of the appa- emergence of classical reality. ratus. They are stable and, hence, retain a faithful record The difficulty many have in accepting the many- of and remain correlated with the outcome of the mea- worlds interpretation stems from its violation of the in- surement in spite of decoherence. tuitively obvious ‘‘conservation law’’—that there is just Einselection is this decoherence-imposed selection of one universe, the one we perceive. But even after this the preferred set of pointer states that remain stable in question is dealt with, many a convert from the Copen- the presence of the environment. As we shall see, einse- hagen interpretation (which claims the allegiance of a lected pointer states turn out to have many classical majority of physicists) to the many-worlds interpretation properties. Einselection is an accepted nickname for (which has steadily gained popularity; see Tegmark and environment-induced superselection (Zurek, 1982). Wheeler, 2001, for an assessment) eventually realizes Decoherence and einselection are two complementary that the original many-worlds interpretation does not views of the consequences of the same process of envi- address the preferred-basis question posed by Einstein ronmental monitoring. Decoherence is the destruction (see footnote 1) (see Bell, 1981, 1987; Wheeler, 1983; of quantum coherence between preferred states associ- Stein, 1984; Kent, 1990, for critical assessments of the ated with the observables monitored by the environ- many-worlds interpretation). And as long as it is unclear ment. Einselection is its consequence—the de facto ex- what singles out preferred states, perception of a unique clusion of all but a small set, a classical domain outcome of a measurement and, hence, of a single uni- consisting of pointer states—from within a much larger 2 verse cannot be explained either. Hilbert space. Einselected states are distinguished by In essence, the many-worlds interpretation does not their resilience—stability in spite of the monitoring en- address, but only postpones, the key question. The vironment. quantum-classical boundary is pushed all the way to- The idea that the ‘‘openness’’ of quantum systems wards the observer, right against the border between the might have anything to do with the transition from material universe and the consciousness, leaving it at a quantum to classical was ignored for a very long time, probably because in classical physics problems of fundamental importance were always settled in isolated sys- 2DeWitt, in the many-worlds reanalysis of quantum measure- tems. In the context of measurements, Gottfried (1966) ments, makes this clear: in DeWitt and Graham (1973), the last anticipated some of the later developments. The fragility paragraph of p. 189, he writes about the key ‘‘remaining prob- of energy levels of quantum systems was emphasized by lem.’’ ‘‘Why is it so easy to find apparata in states [with a well the seminal papers of Zeh (1970, 1973), who argued [in- defined value of the pointer observable]? In the case of mac- spired by remarks relevant to what would be called to- roscopic apparata it is well known that a small value for the day ‘‘deterministic chaos’’ (Borel, 1914)] that macro- mean square deviation of a macroscopic observable is a fairly scopic quantum systems are in effect impossible to stable property of the apparatus. But how does the mean square deviation become so small in the first place? Why is a isolate. large value of the mean-square deviation of a macroscopic ob- The understanding of how the environment distills the servable virtually never, in fact, encountered in practice? ... a classical essence from quantum systems is more recent proof of this does not yet exist. It remains a program for the (Zurek, 1981, 1982, 1993a). It combines two observa- future.’’ tions: (1) In quantum physics, ‘‘reality’’ can be attributed

Rev. Mod. Phys., Vol. 75, No. 3, July 2003 718 Wojciech Hubert Zurek: Decoherence, einselection, and the quantum origins of the classical to the measured states. (2) Information transfer usually ͉⌿͑t͒͘ϭexp͑ϪiHt/ប͉͒⌿͑0͒͘, (1.1) associated with measurements is a common result of al- and, in the absence of systems, the problem of interpre- most any interaction of a system with its environment. tation seems to disappear. Some quantum states are resilient to decoherence. There is simply no need for ‘‘collapse’’ in a universe This is the basis of einselection. Using Darwinian anal- with no systems. Our experience of the classical reality ogy, one might say that pointer states are the most ‘‘fit.’’ does not apply to the universe as a whole, seen from the They survive monitoring by the environment to leave outside, but to the systems within it. Yet, the division ‘‘descendants’’ that inherit their properties. The classical into systems is imperfect. As a consequence, the uni- domain of pointer states offers a static summary of the verse is a collection of open (interacting) quantum sys- result of quantum decoherence. Save for classical dy- tems. Since the interpretation problem does not arise in namics, (almost) nothing happens to these einselected quantum theory unless interacting systems exist, we states, even though they are immersed in the environ- shall also feel free to assume that an environment exists ment. when looking for a resolution. It is difficult to catch einselection in action. Environ- Decoherence and einselection fit comfortably in the ment has little effect on the pointer states, since they are context of the many-worlds interpretation in which they already classical. Therefore it is easy to miss the define the ‘‘branches’’ of the universal state vector. De- decoherence-driven dynamics of einselection by taking coherence makes the many-worlds interpretation com- for granted its result—existence of the classical domain plete: It allows one to analyze the universe as it is seen and a ban on arbitrary quantum superpositions. Macro- by an observer, who is also subject to decoherence. Ein- scopic superpositions of einselected states disappear selection justifies elements of Bohr’s Copenhagen inter- rapidly. Einselection creates effective superselection pretation by drawing the border between the quantum rules (Wick, Wightman, and Wigner, 1952, 1970; Wight- and the classical. This natural boundary can sometimes man, 1995). However, in the microscopic domain, deco- be shifted. Its effectiveness depends on the degree of herence can be slow in comparison with the dynamics. isolation and on the manner in which the system is Einselection is a quantum phenomenon. Its essence probed, but it is a very effective quantum-classical bor- cannot even be motivated classically. In classical physics, der nevertheless. arbitrarily accurate measurements (also by the environ- Einselection fits either the MWI or the CI framework. ment) can, in principle, be carried out without disturbing It sets limits on the extent of the quantum jurisdiction, the system. Only in quantum mechanics acquisition of delineating how much of the universe will appear classi- information inevitably brings the risk of altering—of re- cal to observers who monitor it from within, using their preparation of the state of the system. limited capacity to acquire, store, and process informa- The quantum nature of decoherence and the absence tion. It allows one to understand classicality as an ideali- of classical analogs are a source of misconceptions. For zation that holds in the limit of macroscopic open quan- instance, decoherence is sometimes equated with relax- tum systems. ation or classical noise that can be introduced by the The environment imposes superselection rules by pre- environment. Indeed, all of these effects often appear serving part of the information that resides in the corre- together and as a consequence of ‘‘openness.’’ The dis- lations between the system and the measuring apparatus tinction between them can be briefly summed up: Relax- (Zurek, 1981, 1982). The observer and the environment ation and noise are caused by the environment perturb- compete for information about the system. The ing the system, while decoherence and einselection are environment—because of its size and its incessant inter- caused by the system perturbing the environment. action with the system—wins that competition, acquiring Within the past few years decoherence and einselec- information faster and more completely than the ob- tion have become familiar to many. This does not mean server. Thus a record useful for the purpose of predic- that their implications are universally accepted (see tion must be restricted to observables that are already comments in the April 1993 issue of Physics Today; monitored by the environment. In that case, the ob- d’Espagnat, 1989, 1995; Bub, 1997; Leggett, 1998, 2002; server and the environment no longer compete and de- the exchange of views between Anderson, 2001, and coherence becomes unnoticeable. Indeed, typically ob- Adler, 2003; Stapp, 2002). In a field where controversy servers use the environment as a communication has reigned for so long this resistance to a new paradigm channel, and monitor it to find out about the system. is no surprise. The spreading of information about the system through the environment is ultimately responsible for the emergence of ‘‘objective reality.’’ The objectivity of a C. The nature of the resolution and the role of envariance state can be quantified by the redundancy with which it is recorded throughout the universe. Intercepting frag- Our aim is to explain why the quantum universe ap- ments of the environment allows observers to identify pears classical when it is seen ‘‘from within.’’ This ques- (pointer) states of the system without perturbing it tion can be motivated only in the context of a universe (Zurek, 1993a, 1998a, 2000; see especially Sec. VII of divided into systems, and must be phrased in the lan- this paper for a preview of this new ‘‘environment as a guage of the correlations between systems. The Schro¨ - witness’’ approach to the interpretation of quantum dinger equation dictates deterministic evolution; theory).

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When an effect of a transformation acting on a system There is usually no need to trace the collapse of the can be undone by a suitable transformation acting on wave packet all the way to the observer’s memory. It the environment, so that the joint state of the two re- suffices that the states of a decohering system quickly mains unchanged, the transformed property of the sys- evolve into mixtures of preferred (pointer) states. All tem is said to exhibit ‘‘environment-assisted invariance’’ that can be known, in principle, about a system (or even, or envariance (Zurek, 2003b). The observer must obvi- introspectively, about an observer by himself) is its ously be ignorant of the envariant properties of the sys- decoherence-resistant identity tag—a description of its tem. Pure entangled states exhibit envariance. Thus, in einselected state. quantum physics, perfect information about the joint Apart from this essentially negative function as a cen- state of the system-environment pair can be used to sor, the environment also plays a very different role as a prove ignorance of the state of the system. broadcasting agent, relentlessly cloning the information Envariance offers a new fundamental insight into about the einselected pointer states. This role of the en- what is information and what is ignorance in the quan- vironment as a witness in determining what exists was tum world. It leads to Born’s rule for the probabilities not appreciated until very recently. Over the past two and justifies the use of reduced density matrices as a decades, the study of decoherence has focused on the description of a part of a larger combined system. Deco- effect of the environment on the system. This led to a herence and einselection rely on reduced density matri- multitude of technical advances, which we shall review, ces. Envariance provides a fundamental resolution of but it also missed one crucial point of paramount con- many of the interpretational issues. It will be discussed ceptual importance: observers monitor systems indi- in Sec. VI.D. rectly, by intercepting small fractions of their environments (e.g., a fraction of the photons that have been D. Existential interpretation and quantum Darwinism reflected or emitted by the object of interest). Thus, if an understanding of why we perceive the quantum universe What the observer knows is inseparable from what as classical is the principal aim, our study should focus the observer is: the physical state of his memory implies on the information spread throughout the environment. his information about the universe. The reliability of this This leads one away from the models of measurement information depends on the stability of its correlation inspired by the von Neumann chain (von Neumann, with external observables. In this very immediate sense 1932) to studies of information transfer involving condi- decoherence brings about the apparent collapse of the tional dynamics and the resulting branching and ‘‘fan- wave packet: after a decoherence time scale, only the out’’ of information throughout the environment einselected memory states will exist and retain useful (Zurek, 1983, 1998a, 2000). This view of the role of the correlations (Zurek, 1991, 1998a, 1998b; Tegmark, 2000). environment, known as ‘‘quantum Darwinism’’ because The observer described by some specific einselected of the analogy between the selective amplification of the state (including a configuration of memory bits) will be information concerning pointer observables and the re- able to access (‘‘recall’’) only that state. The collapse is a production which is key to natural selection, is comple- consequence of einselection and of the one-to-one cor- mentary to the usual image of the environment as a respondence between the state of the observer’s memory source of perturbations that destroy the quantum coher- and of the information encoded in it. Memory is simul- ence of the system. It suggests that the redundancy of taneously a description of the recorded information and the imprint of a system in the environment may be a part of an ‘‘identity tag,’’ defining the observer as a quantitative measure of its relative objectivity and hence physical system. It is as inconsistent to imagine the ob- of the classicality of quantum states. Quantum Darwin- server perceiving something other than what is implied ism is discussed in Sec. VII of this review. by the stable (einselected) records in his possession as it The benefits of recognizing the role of environment is impossible to imagine the same person with a different include not just an operational definition of the objec- DNA. Both cases involve information encoded in a state tive existence of the einselected states, but—as is also of a system inextricably linked with the physical identity detailed in Sec. VI—a clarification of the connection be- of an individual. tween quantum amplitudes and probabilities. Einselec- Distinct memory/identity states of the observer tion converts arbitrary states into mixtures of well- (which are also his ‘‘states of knowledge’’) cannot be defined possibilities. Phases are envariant. Appreciation superposed. This censorship is strictly enforced by deco- of envariance as a symmetry tied to ignorance about the herence and the resulting einselection. Distinct memory state of the system was the missing ingredient in the states label and inhabit different branches of Everett’s attempts of no-collapse derivation of Born’s rule and its many-worlds universe. The persistence of correlations probability interpretation. While both envariance and between the records (data in the possession of the ob- quantum Darwinism are only beginning to be investi- servers) and the recorded states of macroscopic systems gated, the extension of the program of einselection they is all that is needed to recover ‘‘familiar reality.’’ In this offer allows one to understand the emergence of ‘‘clas- manner, the distinction between ontology and sical reality’’ from the quantum substrate as a funda- epistemology—between what is and what is known to mental consequence of quantum laws and goes far be- be—is dissolved. In short (Zurek, 1994), there can be no yond the ‘‘for all practical purposes’’ only view of the information without representation. role of the environment.

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II. QUANTUM MEASUREMENTS → → ¬ 0cat 0cat ;1cat 1c at . (2.3) The need for a transition from the quantum determin- The quantum c-NOT is a straightforward quantum ver- ism of the global state vector to the classical definiteness sion of Eq. (2.3). It was known as a ‘‘bit-by-bit measure- of states of individual systems is traditionally illustrated ment’’ (Zurek, 1981, 1983) and already used to elucidate by the example of quantum measurements. The out- the connection between entanglement and premeasure- come of a generic measurement of the state of a quan- ment before it acquired its present name and significance in the context of quantum computation (see, for tum system is not deterministic. In the textbook discus- example, Nielsen and Chuang, 2000). Arbitrary superpo- sions, this random element is blamed on the ‘‘collapse of sitions of the control bit and of the target bit states are the wave packet,’’ invoked whenever a quantum system allowed: comes into contact with a classical apparatus. In a fully ͑␣͉ ϩ␤͉ ͉ →␣͉ ͉ ϩ␤͉ ͉¬ quantum discussion this issue still arises, in spite of (or 0c͘ 1c͘) at͘ 0c͘ at͘ 1c͘ at͘. (2.4) rather because of) the overall deterministic quantum ͉¬ Here negation, at͘, of a state is basis dependent: evolution of the state vector of the universe. As pointed ¬͑␥͉ ͘ϩ␦͉ ͘ ϭ␥͉ ͘ϩ␦͉ ͘ out by von Neumann (1932), there is no room for a real 0t 1t ) 1t 0t . (2.5) collapse in the purely unitary models of measurements. ͉ ϭ͉ ͉ ϭ͉ With A0͘ 0t͘, A1͘ 1t͘ we have an obvious analogy between a c-NOT and a premeasurement. A. Quantum conditional dynamics In the classical controlled NOT, the direction of information transfer is consistent with the designations of the To illustrate the difficulties, consider a quantum sys- two bits. The state of the control remains unchanged tem S initially in a state ͉␺͘ interacting with a quantum while it influences the target, Eq. (2.3). Classical mea- apparatus A initially in a state ͉A ͘: 0 surement need not influence the system. Written in the logical basis ͕͉0͘,͉1͖͘, the truth table of the quantum ͉⌿ ͘ϭ͉␺͉͘A ͘ϭͩ ͚ a ͉s ͪ͘ ͉A ͘ 0 0 i i 0 i c-NOT is essentially—save for the possibility of superpositions—the same as Eq. (2.3). One might have → ͉ ͉ ϭ͉⌿ anticipated that the direction of information transfer ͚ ai si͘ Ai͘ t͘. (2.1) i and the designations (control/system and target/ apparatus) of the two qubits would also be unambigu- ͕͉ ͖͘ ͕͉ ͖͘ Above, Ai and si are states in the Hilbert spaces ous, as in the classical case. This expectation is incorrect. of the apparatus and of the system, respectively, and ai In the conjugate basis ͕͉ϩ͘,͉Ϫ͖͘ defined by are complex coefficients. The conditional dynamics of such premeasurement, as the step achieved by Eq. (2.1) ͉Ϯ͘ϭ͉͑0͘Ϯ͉1͘)/ͱ2, (2.6) is often called, can be accomplished by means of a uni- the truth table, Eq. (2.3) (as such equations providing a ¨ tary Schrodinger evolution. Yet it is not enough to claim map from the inputs to the outputs of the logic gates are that a measurement has been achieved. Equation (2.1) known), along with Eq. (2.6), lead to a new complemen- ͉⌿ ͘ leads to an uncomfortable conclusion: t is an EPR- tary truth table: like entangled state. Operationally, this EPR nature of the state emerging from the premeasurement can be ͉Ϯ͉͘ϩ͘→͉Ϯ͉͘ϩ͘, (2.7) made more explicit by rewriting the sum in a different ͉Ϯ͉͘Ϫ͘→͉ϯ͉͘Ϫ͘. (2.8) basis: In the complementary basis ͕͉ϩ͘,͉Ϫ͖͘, the roles of the ͉⌿ ϭ ͉ ͉ ϭ ͉ ͉ control and of the target are reversed. The former target t͘ ͚ ai si͘ Ai͘ ͚ bi ri͘ Bi͘. (2.2) i i (basis ͕͉0͘, ͉1͖͘)—represented by the second ket above— This freedom of basis choice—basis ambiguity—is guar- remains unaffected, while the state of the former control anteed by the principle of superposition. Therefore, if (the first ket) is conditionally flipped. one were to associate states of the apparatus (or the In the bit-by-bit case the measurement interaction is ͉⌿ ͘ ϭ ͉ ͉ ͉Ϫ Ϫ͉ observer) with decompositions of t , then even before Hint g 1͗͘1 S ͗͘ A inquiring about the specific outcome of the measure- g ment one would have to decide on the decomposition of ϭ ͉ ͗͘ ͉ ͓ Ϫ͉͑ ͗͘ ͉ϩ͉ ͗͘ ͉͔͒ ͉⌿ 1 1 S 1 0 1 1 0 A . (2.9) t͘; a change of the basis redefines the measured quan- 2 tity. Here g is a coupling constant, and the two operators refer to the system (i.e., to the former control) and to the apparatus pointer (the former target), respectively. It 1. Controlled NOT and bit-by-bit measurement is easy to see that the states ͕͉0͘,͉1͖͘S of the system are The interaction required to entangle a measured sys- unaffected by Hint , since tem and the measuring apparatus, Eq. (2.1), is a gener- ͓H , e ͉0͗͘0͉Sϩe ͉1͗͘1͉S͔ϭ0. (2.10) alization of the basic logical operation known as a ‘‘con- int 0 1 ⑀ϭ ͉ ͉ϩ ͉ ͉ trolled NOT’’orac-NOT. A classical c-NOT changes the The measured observable ˆ e0 0͗͘0 e1 1͗͘1 is a state at of the target when the control is 1, and does constant of motion under Hint . The c-NOT requires in- nothing otherwise: teraction time t such that gtϭ␲/2.

Rev. Mod. Phys., Vol. 75, No. 3, July 2003 Wojciech Hubert Zurek: Decoherence, einselection, and the quantum origins of the classical 721

͕͉ϩ͘ ͉Ϫ͖͘ NϪ1 The states , A of the apparatus encode infor- 2␲i ͉ ͘ϭ Ϫ1/2 ͩ Ϫ ͪ ͉ ͘ mation about phases between the logical states. They Ak N ͚ exp kl Bl , (2.14b) ϭ N have exactly the same ‘‘immunity:’’ l 0 ͓ ͉ϩ ϩ͉ ϩ ͉Ϫ Ϫ͉ ͔ϭ the inverse of the transformation of Eq. (2.14a). Hence Hint , fϩ ͗͘ A fϪ ͗͘ A 0, (2.11) where fϮ are the eigenvalues of this ‘‘phase observable.’’ ͉␺ ϭ ␣ ͉ ϭ ␤ ͉ ͘ ͚ l Al͘ ͚ k Bk͘, (2.16) Hence, when the apparatus is prepared in a definite l k phase state (rather than in a definite pointer/logical ␤ state), it will pass its phase on to the system, as Eqs. where the coefficients k are NϪ1 (2.7) and (2.8) show. Indeed, Hint can be written as 2␲i ␤ ϭ Ϫ1/2 ͩ Ϫ ͪ ␣ ϭ ͉ ͉ ͉Ϫ Ϫ͉ k N ͚ exp kl l . (2.17) Hint g 1͗͘1 S ͗͘ A lϭ0 N g The Hadamard transform of Eq. (2.6) is a special case of ϭ ͓1Ϫ͉͑Ϫ͗͘ϩ͉ϩ͉ϩ͗͘Ϫ͉͔͒S ͉Ϫ͗͘Ϫ͉A (2.12) 2 the more general transformation considered here. To implement the truth tables involved in premea- making this immunity obvious. surements, we define observable Aˆ and its conjugate: This basis-dependent direction of information flow in NϪ1 a quantum c-NOT (or in a premeasurement) is a conse- ˆ ϭ ͉ ͗͘ ͉ quence of complementarity. While the information A ͚ k Ak Ak , (2.18a) kϭ0 about the observable with the eigenstates ͕͉0͘,͉1͖͘ travels from the system to the measuring apparatus, in the NϪ1 ˆ ϭ ͉ ͉ complementary ͕͉ϩ͘,͉Ϫ͖͘ basis it seems that the appara- B ͚ l Bl͗͘Bl . (2.18b) ϭ tus is measured by the system. This observation (Zurek l 0 1998a, 1998b; see also Beckman et al., 2001) clarifies the The interaction Hamiltonian sense in which phases are inevitably ‘‘disturbed’’ in mea- ϭ ˆ ˆ surements. They are not really destroyed, but rather, as Hint gsB (2.19) the apparatus measures a certain observable of the sys- is an obvious generalization of Eqs. (2.9) and (2.12), tem, the system simultaneously ‘‘measures’’ phases be- with g the coupling strength and tween the possible outcome states of the apparatus. This NϪ1 leads to loss of phase coherence. Phases become ‘‘shared ϭ ͉ ͉ sˆ ͚ l sl͗͘sl . (2.20) property,’’ as we shall see in more detail in the discus- lϭ0 sion of envariance. ͕͉ ͖͘ ˆ The question ‘‘what measures what?’’ (decided by the In the Ak basis, B is a shift operator, ץ direction of the information flow) depends on the initial iN states. In the classical practice this ambiguity does not Bˆ ϭ . (2.21) Âץ 2␲ arise. Einselection limits the set of possible states of the apparatus to a small subset. To show how Hint works, we compute exp͑ϪiH t/ប͉͒s ͉͘A ͘ 2. Measurements and controlled shifts int j k NϪ1 The truth table of a whole class of c-NOT-like transfor- ϭ͉ Ϫ1/2 ͓Ϫ ͑ បϩ ␲ ͒ ͔͉ sj͘N ͚ exp i jgt/ 2 k/N l Bl͘. mations that includes general premeasurement, Eq. lϭ0 (2.1), can be written as (2.22) ͉ ͉͘ ͘→͉ ͉͘ ͘ sj Ak sj Akϩj . (2.13) We now adjust the coupling g and the duration of the Equation (2.1) follows when kϭ0. One can therefore interaction t so that the action ␫ expressed in Planck model measurements as controlled shifts—c-shifts—or units 2␲ប is a multiple of 1/N: ͕͉ ͖͘ generalizations of the c-NOT. In the bases sj and ␫ϭgt/បϭG 2␲/N. (2.23a) ͉ * ͕ Ak͖͘, the direction of information flow appears to be unambiguous—from the system S to the apparatus A. For an integer G, Eq. (2.22) can be readily evaluated: However, a complementary basis can be readily defined ͑Ϫ ប͉͒ ͉ ϭ͉ ͉ exp iHintt/ sj͘ Ak͘ sj͘ A͕kϩG j͖ ͘. (2.24) (Ivanovic, 1981; Wootters and Fields, 1989): * N NϪ1 This is a shift of the apparatus state by an amount G*j 2␲i ͉ ͘ϭ Ϫ1/2 ͚ ͩ ͪ ͉ ͘ proportional to the eigenvalue j of the state of the sys- Bk N exp kl Al . (2.14a) ϩ lϭ0 N tem. G plays the role of gain. The index ͕k G*j͖N is evaluated modN, where N is the number of possible out- Above, N is the dimensionality of the Hilbert space. An comes, that is, the dimensionality of the Hilbert space of analogous transformation can be carried out on the basis the apparatus pointer A. When G jϾN, the pointer will ͕͉s ͖͘ of the system, yielding states ͕͉r ͖͘. * i j just rotate through the initial zero. The truth table for Orthogonality of ͕͉A ͖͘ implies k Gϭ1 defines a c-shift, Eq. (2.13), and with kϭ0 leads to ͉ ϭ␦ ͗Bl Bm͘ lm , (2.15) a premeasurement, Eq. (2.1).

Rev. Mod. Phys., Vol. 75, No. 3, July 2003 722 Wojciech Hubert Zurek: Decoherence, einselection, and the quantum origins of the classical

The form of the interaction, Eq. (2.19), in conjunction and the simplifying assumption about the coefficients, with the initial state, decides the direction of informa- ␣ ͑ ͒ϭ␣͑ Ϫ ͒ tion transfer. Note that—as was the case with the l j j l , (2.31) c-NOT’s—the observable that commutes with the interac- has been made. The aim of this simplification is to focus tion Hamiltonian will not be perturbed: on the case when the apparatus states are peaked ͓ ͔ϭ ␣ ϳ Ϫ Hint ,sˆ 0. (2.25) around a certain value l [e.g., l(j) exp͕ (j Ϫl)2/2⌬2͖], and where the form of their distribution sˆ commutes with H and is therefore a nondemolition int over ͕͉A ͖͘ does not depend on l. observable (Braginsky, Vorontsov, and Thorne, 1980; k A good measurement allows one to distinguish states Caves et al., 1980; Bocko and Onofrio, 1996). of the system. Hence it must satisfy 3. Amplification ͉ ͉ ͉2ϭ͉ ␣͓ ϩ ͑ Ϫ ͔͒␣ ͑ ͉͒2 ͗alϩGk alϩGkЈ͘ ͚ j G k kЈ * j Amplification has often been regarded as the process j forcing quantum potentialities to become classical real- Ϸ␦ ity. An example of it is the extension of the measure- kЈ,k . (2.32) ment model described above. States of the system that need to be distinguished should Assume the Hilbert space of the apparatus pointer is rotate the pointer of the apparatus to the correlated out- large compared with the space spanned by the eigen- come states that are approximately orthogonal. When ˆ states of the measured observable s: the coefficients ␣(k) are peaked around kϭ0 with dis- ⌬ Nϭdim͑HA͒ӷdim͑HS͒ϭn. (2.26) persion , this implies Then one can increase ␫ to an integer multiple G of ⌬ӶG. (2.33) 2␲/N, Eqs. (2.23a) and (2.24). However, larger ␫ will lead to redundancy only when the Hilbert space of the In the general case of an initial mixture, Eq. (2.29), apparatus has many more dimensions than possible out- one can evaluate the dispersion of the expectation value comes. Otherwise, only ‘‘wrapping’’ of the same record of the record observable Aˆ as will ensue. The simplest example of such wrapping, 2 ˆ 2 ˆ 2 0 ˆ 2 0 ˆ 2 (c-NOT) , is the identity operation. For Nӷn, however, ͗A ͘Ϫ͗A͘ ϭTr␳AA Ϫ͑Tr␳AA͒ . (2.34) one can attain gain: The outcomes are distinguishable when GϭNgt/2␲ប. (2.23b) ͗ ˆ 2͘Ϫ͗ ˆ ͘2Ӷ The outcomes are now separated by GϪ1 empty eigen- A A G. (2.35) ӷ states of the record observable. In this sense, G 1 Interaction with the environment yields a mixture of the achieves redundancy, providing that wrapping of the form of Eq. (2.29). Amplification can protect measure- record is avoided. This is guaranteed when ment outcomes from noise through redundancy.3 nGϽN. (2.27) Amplification is useful in the presence of noise. For 3 example, it may be difficult to initiate the apparatus in The above model of amplification is unitary. Yet it contains ͉A ͘, so the initial state may be a superposition: seeds of irreversibility. The reversibility of a c-shift is evident: 0 as the interaction continues, the two systems will eventually disentangle. For instance, it takes t ϭ2␲ប/(gN) [see Eq. ͉ ͘ϭ͚ ␣ ͑ ͉͒ ͘ e al l k Ak . (2.28a) (2.23b) with Gϭ1] to entangle S ͓dim(HS)ϭn͔ with an A k with dim(HA)ϭNуn pointer states. However, as the interac- Indeed, typically a mixture of such superpositions, tion continues, A and S disentangle. For a c-shift, this recur- ϭ ϭ ␲ប rence time scale is tRec Nte 2 /g. It corresponds to a gain 0 ϭ ϭ ␳ ϭ͚ w ͉a ͗͘a ͉, (2.28b) G N. Thus, for an instant of less than te at t tRec , the ap- A i i i ϩ ϭ i paratus disentangles from the system, as ͕k N*j͖N k.Re- versibility results in recurrences of the initial state, but for N may be the starting point for a premeasurement. Then ӷ1, they are rare. For less regular interactions (e.g, involving the environment) ͉ ͗͘ ͉␳ ϭ͉ ͗͘ ͉ ͉ ͗͘ ͉ sk skЈ A sk skЈ ͚ wl al al the recurrence time is much longer. In that case, tRec is, in l ϳ Ϸ effect, a Poincare´ time: tRec tPoincare´ N!te . In any case tRec ӷ te for large N. Undoing entanglement in this manner would →͉ ͉ ͉ ϩ ͉ sk͗͘skЈ ͚ wl al Gk͗͘alϩGkЈ , be exceedingly difficult because one would need to know pre- l cisely when to look and because one would need to isolate the (2.29) apparatus or the immediate environment from other degrees of freedom—their environments. ͉ ͉ where alϩGk͘ obtains from al͘, Eq. (2.28a), through The price of letting the entanglement undo itself by waiting for an appropriate time interval is at the very least given by the ͉ ϭ ␣ ͑ ͉͒ cost of storing the information based on how long it is neces- alϩGk͘ ͚ l j AjϩGk͘, (2.30) j sary to wait. In the special c-shift case this is proportional to

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B. Information transfer in measurements later when in Sec. VI we consider the relation between decoherence and probabilities. In order to derive Born’s 1. Reduced density matrices and correlations rule it will be important not to assume it in some guise. Following premeasurement, the information about Information transfer is the objective of the measure- the subsystems available to the observer locally de- ment process. Yet quantum measurements have only creases. This is quantiﬁed by the increase of the entro- rarely been analyzed from that point of view. As a result pies: S A of the interaction of the system with the apparatus , Ϫ ͉⌿ ͘ N 1 their joint state is still pure t , Eq. (2.1), but each of ϭϪ ␳ ␳ ϭϪ ͉ ͉2 ͉ ͉2 HS Tr S ln S ͚ ai ln ai the subsystems is in a mixture: iϭ0 Ϫ N 1 ϭϪ ␳ ␳ ϭ ␳ ϭ ͉⌿ ⌿ ͉ϭ ͉ ͉2͉ ͉ Tr A ln A HA . (2.37) S TrA t͗͘ t ͚ ai si͗͘si , (2.36a) iϭ0 Since the evolution of the whole SA is unitary, the in- NϪ1 crease of entropies in the subsystems is compensated for ␳ ϭ ͉⌿ ⌿ ͉ϭ ͉ ͉2͉ ͉ by the buildup of correlations, and the resulting increase A TrS t͗͘ t ͚ ai Ai͗͘Ai . (2.36b) iϭ0 in mutual information: The partial trace leads to reduced density matrices, here NϪ1 ␳ ␳ I͑S A͒ϭ ϩ Ϫ ϭϪ ͉ ͉2 ͉ ͉2 S and A , which are important for what follows. They : HS HA HSA 2 ͚ ai ln ai . describe subsystems to the observer who, before the pre- iϭ0 measurement, knew pure states of the system and of the (2.38) apparatus, but who has access to only one of them after- This has been used in quantum theory as a measure of wards. entanglement (Zurek, 1983; Barnett and Phoenix, 1989). The reduced density matrix is a technical tool of paramount importance. It was introduced by Landau (1927) 2. Action per bit as the only density matrix that gives rise to the correct measurement statistics given the usual formalism that An often raised question concerns the price of infor- includes Born’s rule for calculating probabilities (see, for mation in units of some other ‘‘physical currency’’ (Bril- example, p. 107 of Nielsen and Chuang, 2000, for an louin, 1962, 1964; Landauer, 1991). Here we shall estab- insightful discussion). This remark will come to haunt us lish that the least action necessary to transfer one bit is of the order of a fraction of ប for quantum systems with two-dimensional Hilbert spaces. Information transfer can be made cheaper on the ‘‘wholesale’’ level, when the log N memory bits. In situations when eigenvalues of the interaction Hamiltonian are not commensurate, it will be more like systems involved have large Hilbert spaces. ϳlog N!ϷN log N, since the entanglement will get undone only Consider Eq. (2.1). It evolves the initial product state after a Poincare´ time. Both classical and quantum cases can be of the two subsystems into a superposition of product ͚ ␣ ͉ ͉ →͚ ␣ ͉ ͉ analyzed using algorithmic information. For related discus- states, ( j j sj͘) A0͘ j j sj͘ Aj͘. The expectation sions see Zurek (1989, 1998b), Caves (1994), and Schack and value of the action involved is no less than Caves (1996). NϪ1 Ampliﬁed correlations are hard to contain. The return to ϭ ͉␣ ͉2 ͉ ͉ ͉ I ͚ j arccos ͗A0 Aj͘ . (2.39) purity after tRec in the manner described above can be hoped jϭ0 for only when the apparatus or the immediate environment E ͉ (i.e., the environment directly interacting with the system) can- When ͕ Aj͖͘ are mutually orthogonal, the action is not ‘‘pass on’’ the information to their more remote environ- Iϭ␲/2 (2.40) ments EЈ. The degree of isolation required puts a stringent limit on the coupling gEEЈ between the two environments. Re- in Planck (ប) units. This estimate can be lowered by ͉ turn to purity can be accomplished in this manner only if tRec using as the initial A0͘ a superposition of the outcomes Ͻt ϭ2␲ប/(NЈgEE ), where NЈ is the dimension of the Hil- ͉ eЈ Ј Aj͘. In general, an interaction of the form bert space of the environment EЈ. Hence the two estimates of NϪ1 NϪ1 t translate into gEE Ͻg/NЈ for the regular spectrum and the Rec Ј ϭ ͉ ͗͘ ͉ ͉͑ ͗͘ ͉Ϫ ͒ much tighter gEE Ͻg/N!NЈ for the random case more relevant HSA ig ͚ sk sk ͚ Ak Al H.c. , (2.41) Ј kϭ0 lϭ0 for decoherence. In short, once information ‘‘leaks’’ into the correlations be- where H.c. is the Hermitian conjugate, saturates the tween the system and the apparatus or the environment, keep- lower bound given by ing it from spreading further ranges between very hard and next to impossible. With the exception of very special cases Iϭarcsin ͱ1Ϫ1/N. (2.42) (small N, regular spectrum), the strategy of enlarging the sys- For a two-dimensional Hilbert space the average action tem, so that it includes the environment—occasionally men- ␲ប tioned as an argument against decoherence—is doomed to fail, can be thus brought down to /4 (Zurek, 1981, 1983). unless the universe as a whole is included. This is a question- As the size of the Hilbert space increases, the action able setting (since the observers are inside this ‘‘isolated’’ sys- involved approaches the asymptotic estimate of Eq. tem) and in any case makes the relevant Poincare´ time ab- (2.40). The entropy of entanglement can be as large as surdly long. log N where N is the dimension of the Hilbert space of

Rev. Mod. Phys., Vol. 75, No. 3, July 2003 724 Wojciech Hubert Zurek: Decoherence, einselection, and the quantum origins of the classical

͉␴ the smaller of the two systems. Thus the least action per ways be convinced that the system was in a state i͘ bit of information decreases with the increase of N: already before he has measured it in accord with Eq. I ␲ (2.44b). ␫ϭ Ϸ . (2.43) This sequence of events as seen by the discoverer log N 2 log N 2 2 looks like a collapse (see Zurek, 1998a, 1998b). For in- This may be one reason why information appears ‘‘free’’ stance, an insider who knew the state of the system be- in the macroscopic domain, but expensive (close to fore the discoverer carried out his measurement need ប/bit) in the quantum case of small Hilbert spaces. not notice any change of that state when he makes further confirmatory measurements. This property is the C. ‘‘Collapse’’ analog in a classical measurement cornerstone of the ‘‘reality’’ of classical states—they need not ever change as a consequence of measure- Definite outcomes that we perceive appear to be at ments. We emphasize, however, that while the state of odds with the principle of superposition. They can nev- the system may remain unchanged, the state of the ob- ertheless also occur in quantum physics when the initial server must change to reflect the acquired information. state of the measured system is—already before the Last but not least, an outsider—someone who knows measurement—in one of the eigenstates of the mea- about the measurement, but (in contrast to the insider) sured observable. Then Eq. (2.1) will deterministically not about the initial state of the system nor (in contrast rotate the pointer of the apparatus to the appropriate to both the insider and the discoverer) about the out- record state. The result of such a measurement can be come of the measurement—will describe the same pro- predicted by an insider—an observer aware of the initial cess still differently: state of the system. This a priori knowledge can be rep- ͉ resented by the preexisting record Ai͘, which is only ͉ ͗͘ ͉͉ ͗͘ ͉ ͉␴ ͗͘␴ ͉ A␳S A␳S A0 A0 ͚ pi i i corroborated by an additional measurement: i ͉ ͉ ͉␴ →͉ ͉ ͉␴ Ai͘ A0͘ i͘ Ai͘ Ai͘ i͘. (2.44a) →͉A␳ ͗͘A␳ ͉ͩ ͚ p ͉A ͗͘A ͉͉␴ ͗͘␴ ͉ͪ . (2.44c) S S i i i i i In classical physics complete information about the i initial state of an isolated system always allows for an This view of the outsider, Eq. (2.44c), combines a one- exact prediction of its future state. A well-informed ob- to-one classical correlation of the states of the system server will even be able to predict the future of the classical universe as a whole (‘‘Laplace’s demon’’). Any ele- and the records with the indefiniteness of the outcome. ment of surprise (any use of probabilities) must We have just seen three distinct quantum-looking de- therefore be blamed on partial ignorance. Thus, when scriptions of the very same classical process (see Zurek, the information available initially does not include the 1989 and Caves, 1994 for previous studies of the insider- exact initial state of the system, the observer can use an outsider theme). They differ only in the information ensemble described by ␳S—by a list of possible initial available ab initio to the observer. The information in ͉␴ the possession of the observer prior to the measurement states ͕ i͖͘ and their probabilities pi . This is the ignorance interpretation of probabilities. We shall see in Sec. determines in turn whether—to the observer—the evo- VI that—using envariance—one can justify ignorance lution appears to be (a) a confirmation of the preexisting about a part of the system by relying on perfect knowl- data, Eq. (2.44a); (b) a collapse associated with the in- edge of the whole. formation gain, Eq. (2.44b), and with the entropy de- Through measurement the observer finds out which of crease translated into algorithmic randomness of the ac- the potential outcomes consistent with his prior (incom- quired data (Zurek, 1989, 1998b); or (c) an entropy- plete) information actually happens. This act of informa- preserving establishment of a correlation, Eq. (2.44c). tion acquisition changes the physical state of the ob- All three descriptions are classically compatible, and can server, the state of his memory. The initial memory state be implemented by the same (deterministic and revers- containing a description A␳ of an ensemble and a ible) dynamics. S In classical physics the insider view always exists, in ‘‘blank’’ A , ͉A␳ ͗͘A␳ ͉͉A ͗͘A ͉, is transformed into a 0 S S 0 0 principle. In quantum physics it does not. Every ob- ͉ ͗͘ ͉͉ ͗͘ ͉ record of a specific outcome: A␳S A␳S Ai Ai .In server in a classical universe could, in principle, aspire to quantum notation this process will be described by such be an ultimate insider. The fundamental contradiction a discoverer as a random ‘‘collapse:’’ between every observer’s knowing precisely the state of the rest of the Universe (including the other observers) ͉ ͗͘ ͉͉ ͗͘ ͉ ͉␴ ͗͘␴ ͉ A␳S A␳S A0 A0 ͚ pi i i can be swept under the rug (if not really resolved) in a i universe where the states are infinitely precisely determined and the observer’s records (as a consequence of →͉A␳ ͗͘A␳ ͉͉A ͗͘A ͉͉␴ ͗͘␴ ͉. (2.44b) S S i i i i the ប→0 limit) may have an infinite capacity for infor- This is only the description of what happens as reported mation storage. However, given a set value of ប, the by the discoverer. Deterministic representation of this information storage resources of any finite physical sys- very same process by Eq. (2.44a) is still possible. In tem are finite. Hence, in quantum physics, observers re- other words, in classical physics the discoverer can al- main largely ignorant of the detailed state of the uni-

Rev. Mod. Phys., Vol. 75, No. 3, July 2003 Wojciech Hubert Zurek: Decoherence, einselection, and the quantum origins of the classical 725

verse, since there can be no information without exponentially compressing in stable directions. The rates representation (Zurek, 1994). of stretching and compression are given by positive and ⌳ Classical collapse is described by Eq. (2.44b). The ob- negative Lyapunov exponents i . Hamiltonian evolu- server discovers the state of the system. From then on, tion demands that the sum of all the Lyapunov expo- the state of the system will remain correlated with his Ϯ⌳ nents be zero. In fact, they appear in i pairs. record, so that all future outcomes can be predicted, in Loss of correspondence in chaotic systems is a conse- effect by iterating Eq. (2.44a). This disappearance of all quence of the exponential stretching of the effective the potential alternatives save for one that becomes a support of the probability distribution in the unstable ‘‘reality’’ is the essence of the collapse. There need not direction (say, x) and its exponential narrowing in the be anything quantum about it. complementary direction (Zurek and Paz, 1994; Zurek, Einselection in the observer’s memory provides many 1998b). As a consequence, the classical probability dis- of the ingredients of classical collapse in the quantum tribution will develop structures on the scale context. In the presence of einselection, a one-to-one correspondence between the state of the observer and ⌬ ϳ⌬ ͑Ϫ⌳ ͒ p p0 exp t . (3.1) his knowledge about the rest of the universe can be ⌬ firmly established, and (at least, in principle) operation- Above, p0 is the measure of the initial momentum ally verified. One could measure bits in the observer’s spread and ⌳ is the net rate of contraction in the direc- memory or even the ‘‘imprint’’ of their state on the en- tion of momentum given by the Lyapunov exponents vironment and determine what he knows without alter- (but see Boccaletti, Farini, and Arecchi, 1997). In a real ing his records—without altering his state. After all, one chaotic system, stretching and narrowing of the prob- can do so with a classical computer. The existential in- ability distribution in both x and p occur simultaneously, terpretation recognizes that the information possessed as the initial patch is rotated and folded. Eventually, the by the observer is reflected in his einselected state, ex- envelope of its effective support will swell to fill in the plaining his perception of a single branch—‘‘his’’ classi- available phase space, resulting in a wave packet that is cal universe. coherently spread over a spatial region of no less than ⌬ ϳ͑ប ⌬ ͒ ͑⌳ ͒ x / p0 exp t . (3.2) III. CHAOS AND LOSS OF CORRESPONDENCE until it becomes confined by the potential, while the The study of the relationship between the quantum small-scale structure will continue to descend to ever and the classical has been, for a long time, focused al- smaller scales (Fig. 1). Breakdown of the quantum- most entirely on measurements. However, the problem classical correspondence can be understood in two of measurement is difficult to discuss without observers. complementary ways, either as a consequence of small ⌬ And once the observer enters, it is often hard to avoid p (see the discussion of the Moyal bracket below) or as ⌬ its ill-understood anthropic attributes such as conscious- a result of large x. ness, awareness, and the ability to perceive. Coherent exponential spreading of the wave packet— ⌬ We shall sidestep these ‘‘metaphysical’’ problems and large x—must cause problems with correspondence. focus on the information-processing underpinnings of This is inevitable, since classical evolution appeals to the observership. It is nevertheless fortunate that there is idealization of a point in phase space acted upon by a ץ another problem with the quantum-classical correspon- force given by the gradient xV of the potential V(x) dence that leads to interesting questions not motivated evaluated at that point. But the quantum wave function by measurements. As was anticipated by Einstein (1917) can be coherent over a region larger than the nonlinear- ␹ before the advent of modern quantum theory, chaotic ity scale over which the gradient of the potential ␹ motion presents such a challenge. The full implications changes significantly. can usually be estimated by of classical dynamical chaos were understood much (V, (3.3 ץ/V ץ␹Ӎͱ later. The concern about the quantum-classical corre- x xxx spondence in this modern context dates to Berman and and is typically of the order of the size L of the system: Zaslavsky (1978) and Berry and Balazs (1979) (see ϳ␹ Haake, 1991 and Casati and Chirikov, 1995a, for refer- L . (3.4) ences). It has even led some to question the validity of An initially localized state evolving in accord with quantum theory (Ford and Mantica, 1992). Eqs. (3.1) and (3.2) will spread over such scales after

A. Loss of quantum-classical correspondence ⌬p ␹ Ӎ⌳Ϫ1 0 tប ln ប . (3.5) The interplay between quantum interference and chaotic exponential instability leads to the rapid loss of It is then impossible to tell what force is acting upon the quantum-classical correspondence. Chaos in dynamics is system, since it is not located in any speciﬁc x. This es- characterized by the exponential divergence of the clas- timate of what can be thought of as Ehrenfest time, the sical trajectories. As a consequence, a small patch rep- time over which a quantum system that has started in a resenting the probability density in phase space is expo- localized state will continue to be sufﬁciently localized nentially stretching in unstable directions and for the quantum corrections to the equations of motion

Rev. Mod. Phys., Vol. 75, No. 3, July 2003 726 Wojciech Hubert Zurek: Decoherence, einselection, and the quantum origins of the classical

FIG. 1. Chaotic evolution generated from the same initial Gaussian by the Hamiltonian Hϭp2/2mϪ␬ cos(xϪl sin t)ϩax2/2. (a)– (c) Snapshots of the quantum (បϭ0.16) Wigner function; (d) classical probability distribution in phase space. For mϭ1, ␬ϭ0.36, lϭ3, and aϭ0 – 0.01 the Hamiltonian exhibits chaos with the Lyapunov exponent ⌳ϭ0.2 (Karkuszewski, Zakrzewski, and Zurek, 2002). Quantum (a) and classical (d) are obtained at the same instant, tϭ20. They exhibit some similarities [i.e., the shape of the regions of signiﬁcant probability density, ‘‘ridges’’ in the topographical maps of (a) and (d)], but the difference—the presence of the interference patterns with W(x,p) assuming negative values (marked with blue)—is striking. Saturation of the size of the smallest patches is anticipated already at this early time, and indeed the ridges of the classical probability density are narrower than in the corresponding quantum features. Saturation is even more visible in (b) taken at tϭ60 and (c), tϭ100 [note change of scale from (a) and (d)]. Sharpness of the classical features makes simulations going beyond tϭ20 unreliable, but quantum simulations can be effectively carried out much further, since the necessary resolution can be anticipated in advance from Eqs. (3.14)–(3.16) (Color).

obeyed by its expectation values to be negligible (Gott- Another way of describing the root cause of a break- fried, 1966), is valid for chaotic systems. Logarithmic de- down of correspondence is to note that after a time scale pendence is the result of inverting the exponential sen- of the order of tប , the quantum wave function of the sitivity. In the absence of exponential instability (⌳ϭ0), system would have spread over all of the available space divergence of trajectories is typically polynomial and and would be forced to fold onto itself. Fragments of the ␣ leads to a power-law dependence, tបϳ(I/ប) , where I is wave packet arrive at the same location (although with the classical action. Thus macroscopic (large-I) inte- different momenta, and having followed different grable systems can follow classical dynamics for a very paths). The ensuing evolution depends critically on long time, providing they were initiated in a localized whether they have retained phase coherence. When co- state. For chaotic systems tប also becomes inﬁnite in the herence persists, a complicated interference event de- limit ប→0, but that happens only logarithmically slowly. cides the subsequent evolution. And, as can be antici- As we shall see below, in the context of quantum- pated from the double-slit experiment, there is a big classical correspondence this is too slow for comfort. difference between coherent and incoherent folding in

Rev. Mod. Phys., Vol. 75, No. 3, July 2003 Wojciech Hubert Zurek: Decoherence, einselection, and the quantum origins of the classical 727 the configuration space. This translates into a loss of cor- was a complete description of planetary dynamics, plan- respondence, which sets in surprisingly quickly, at tប . ets would be delocalized along their orbits. To find out how quickly, we estimate tប for an obviously macroscopic object, Hyperion, a chaotically tumbling moon of Saturn (Wisdom, 1985). Hyperion has the B. Moyal bracket and Liouville flow prolate shape of a potato and moves on an eccentric ϭ orbit with a period tO 21 days. Interaction between its Heuristic arguments about the breakdown of gravitational quadrupole and the tidal field of Saturn quantum-classical correspondence can be made more leads to chaotic tumbling with Lyapunov time ⌳Ϫ1 rigorous with the help of the Wigner function. We start Ӎ42 days. with the von Neumann equation To estimate the time over which the orientation of iប␳˙ ϭ͓H,␳͔. (3.9) Hyperion becomes delocalized, we use a formula (Ber- This can be transformed into the equation for the man and Zaslavsky, 1978; Berry and Balazs, 1979): Wigner function W, which is defined in phase space as LP I t ϭ⌳Ϫ1 ln ϭ⌳Ϫ1 ln . (3.6) 1 ipy y y r ប ប W͑x,p͒ϭ ͵ expͩ ͪ ␳ͩ xϪ ,xϩ ͪ dy. 2␲ប ប 2 2 Above L and P give the range of values of the coordi- (3.10) nates and momenta in phase space of the system. Since The result is LӍ␹ and PϾ⌬p , it follows that t уtប . On the other 0 r ˙ ϭ͕ ͖ hand, LPӍI, the classical action of the system. W H,W MB . (3.11) The advantage of Eq. (3.6) is its insensitivity to initial Here ͕...,...͖MB stands for the Moyal bracket, the conditions and the ease with which the estimate can be Wigner transform of the von Neumann bracket (Moyal, obtained. For Hyperion, a generous overestimate of the 1949). classical action I can be obtained from its binding energy The Moyal bracket can be expressed in terms of the EB and its orbital time tO : Poisson bracket ͕...,...͖, which generates Liouville flow in បӍ បӍ 77 classical phase space, by the formula I/ EBtO / 10 . (3.7) iប͕...,...͖ ϭsin͑iប͕...,...͖͒. (3.12) The above estimate (Zurek, 1998b) is ‘‘astronomically’’ MB large. However, in the calculation of the loss of corre- When the potential V(x) is analytic, the Moyal bracket spondence, Eq. (3.6), only the logarithm of I enters. can be expanded (Hillery et al., 1984) in powers of ប: Thus ប2n͑Ϫ͒n 2nϩ1ץ 2nϩ1ץ ϭ͕ ͖ϩ ˙ HyperӍ ͓ ͔ 77Ӎ ͓ ͔ W H,W ͚ 2n x V p W. tr 42 days ln 10 20 yr . (3.8) nу1 2 ͑2nϩ1͒! After approximately 20 yr Hyperion would be in a co- (3.13) herent superposition of orientations that differ by 2␲. The first term is just the Poisson bracket. Alone, it We conclude that after a relatively short time an ob- would generate classical motion in phase space. How- viously macroscopic chaotic system becomes forced into ever, when the evolution is chaotic, quantum corrections a flagrantly nonlocal ‘‘Schro¨ dinger-cat’’ state. In the (proportional to the odd-order momentum derivatives original discussion (Schro¨ dinger, 1935a, 1935b) an inter- of the Wigner function) will eventually dominate the mediate step in which the decay products of the nucleus right-hand side of Eq. (3.10). This is because the expo- were measured to determine the fate of the cat was es- nential squeezing of the initially regular patch in phase sential. Thus it was possible to maintain that the prepos- space (which begins its evolution in the classical regime, terous superposition of the dead and live cat could be where the Poisson bracket dominates) leads to an expo- avoided, providing that quantum measurement (with the nential explosion of the momentum derivatives. Conse- collapse it presumably induces) was properly under- quently, after a time logarithmic in ប [Eqs. (3.5) and stood. (3.6)], the Poisson bracket will cease to be a good esti- This cannot be the resolution for chaotic quantum sys- mate of the right-hand side of Eq. (3.13). tems. They can evolve, as the example of Hyperion dem- The physical reason for the ensuing breakdown of the onstrates, into states that are nonlocal and, therefore, quantum-classical correspondence has already been ex- extravagantly quantum, simply as a result of exponen- plained: exponential instability of the chaotic evolution tially unstable dynamics. Moreover, this happens surpris- delocalizes the wave packet. As a result, the force acting ingly quickly, even for very macroscopic examples. Hy- on the system is no longer given by the gradient of the perion is not the only chaotic system. There are potential evaluated at the location of the system. It is asteroids that have chaotically unstable orbits (e.g., Chi- not even possible to say where the system is, since it is in ron), and even indications that the solar system as a a superposition of many distinct locations. Conse- whole is chaotic (Laskar, 1989; Sussman and Wisdom, quently, the phase-space distribution and even the ex- 1992). In all of these cases straightforward estimates of pectation values of the observables of the system differ tប yield answers much smaller than the age of the solar noticeably when evaluated classically and quantum me- system. Thus, if unitary evolution of closed subsystems chanically (Haake, Kus´, and Sharf, 1987; Habib, Shi-

Rev. Mod. Phys., Vol. 75, No. 3, July 2003 728 Wojciech Hubert Zurek: Decoherence, einselection, and the quantum origins of the classical

FIG. 2. Difference between the classical and quantum averages of the dispersion of momentum ⌬2ϭ͗p2͘Ϫ͗p͘2 plotted for (a) same initial condition, but three different values of ប in the model defined in Fig. 1, with the parameter aϭ0. The instant when the difference between the classical and quantum averages becomes significant varies with ប in a manner anticipated from Eqs. (3.5) and (3.6) for the Ehrenfest time, as can be seen in the inset; (b) same value of ប, but for four different initial conditions. Inset appears to indicate that the typical variance difference ␦ varies only logarithmically with decreasing ប, although the large error bars (tied to the large systematic changes of behavior for different initial conditions) preclude one from arriving at a firmer conclusion. (See Karkuszewski, Zakrzewski, and Zurek, 2002, for further details and discussion.) (Color). zume, and Zurek, 1998; Karkuszewski, Zakrzewski, and through measurements that are subject to Heisenberg Zurek, 2002). Moreover, this will happen after an un- indeterminacy. Still, it should be fair to compare aver- comfortably short time tប . ages over an evolving Wigner function with an initially identical classical probability distribution (Haake, Kus´, and Sharf, 1987; Ballentine, Yang, and Zibin, 1994; Fox C. Symptoms of correspondence loss and Elston, 1994a, 1994b; Miller, Sarkar, and Zarum, 1998). These are shown in Fig. 2 for an example of a The wave packet becomes rapidly delocalized in a driven chaotic system. Clearly, there is reason for con- chaotic system, and the correspondence between classi- cern. Figure 2 (corroborated by other studies—see cal and quantum is quickly lost. Flagrantly nonlocal Karkuszewski, Zakrzewski, and Zurek, 2002, for refer- Schro¨ dinger-cat states appear no later than tប , and this ences) demonstrates that not just the phase-space por- is the overarching interpretational as well as physical trait but also the averages diverge at a time ϳtប . problem. In the familiar real world we never seem to In integrable systems, the rapid loss of correspon- encounter such smearing of the wave function even in dence between the quantum and the classical expecta- the examples of chaotic dynamics where it is predicted tion values may still occur, but only for very special ini- by quantum theory. tial conditions, due to the local instability in phase space. 1. Expectation values Indeed, a double-slit experiment is an example of a regular system in which a local instability (splitting of Measurements usually average out fine phase-space the paths) leads to correspondence loss, but only for ju- interference structures, which may be a striking, but ex- diciously selected initial conditions. Thus one may dis- perimentally inaccessible symptom of the breakdown of miss such a breakdown as a consequence of a rare correspondence. Thus one might hope that when inter- pathological starting point and argue that the conditions ference patterns in the Wigner function are ignored by that lead to discrepancies between classical and quan- looking at the coarse-grained distribution, the quantum tum behavior exist, but are of measure zero in the clas- results should be in accord with the classical. This would sical limit. not exorcise the ‘‘chaotic cat’’ problem. Moreover, the In the chaotic case the loss of correspondence is typi- breakdown of correspondence can also be seen in the cal. As shown in Fig. 2, it happens after a disturbingly expectation values of quantities that are smooth in short tប for generic initial conditions. The time at which phase space. the quantum and classical expectation values diverge in Trajectories diverge exponentially in a chaotic system. the example studied here is consistent with the estimates A comparison between expectation values for a single of tប , Eq. (3.5), but exhibits a significant scatter. This is trajectory and for a delocalized quantum state (which is not too surprising—exponents characterizing local insta- how the Ehrenfest theorem mentioned above is usually bility vary noticeably with location in phase space. stated) would clearly lead to a rapid loss of correspon- Hence stretching and contraction in phase space will oc- dence. However, one may rightly object to the use of a cur at a rate that depends on the specific trajectory. The single trajectory and argue that both the quantum and dependence of a typical magnitude on ប is still not clear. the classical state should be prepared and accessed only Emerson and Ballentine (2001a, 2001b) studied coupled

Rev. Mod. Phys., Vol. 75, No. 3, July 2003 Wojciech Hubert Zurek: Decoherence, einselection, and the quantum origins of the classical 729 spins and argued that it is of the order of ប, but Fig. 2 small perturbations of the potential (Karkuszewski, suggests it decreases more slowly than that, and that it Zakrzewski, and Zurek, 2002). may be only logarithmic in ប (Karkuszewski et al., 2002). IV. ENVIRONMENT-INDUCED SUPERSELECTION

2. Structure saturation The principle of superposition applies only when the quantum system is closed. When the system is open, in- Evolution of the Wigner function leads to rapid teraction with the environment results in an incessant buildup of interference fringes. These fringes become monitoring of some of its observables. As a result, pure progressively smaller, until saturation, when the wave states turn into mixtures that become rapidly diagonal in packet is spread over the available phase space. At that einselected states. These pointer states are chosen with time their scales in momentum and position are typically the help of the interaction Hamiltonian and are inde- given by pendent of the initial state of the system. Their predict- dpϭប/L, (3.14) ability is key to the effective classicality (Zurek, 1993a; ϭប Zurek, Habib, and Paz, 1993). dx /P, (3.15) Environments can be external (such as particles of the where L(P) defines the range of positions (momenta) of air or photons that scatter off, say, the apparatus the effective support of W in phase space. pointer) or internal (collections of phonons or other in- Hence the smallest structures in the Wigner function ternal excitations). Often, environmental degrees of occur (Zurek, 2001) on scales corresponding to an action freedom emerge from a split of the original set of degrees of freedom into a ‘‘system of interest,’’ which may aϭdxdpϭបϫប/LPϭប2/I, (3.16) be a collective observable (e.g., an order parameter in a where IӍLP is the classical action of the system. Action phase transition) and a ‘‘microscopic remainder.’’ aӶប for macroscopic I. The set of einselected states is called the pointer basis Sub-Planck structure is a kinematic property of quan- (Zurek, 1981) in recognition of its role in measurements. tum states. It helps determine their sensitivity to pertur- The criterion for the einselection of states goes well be- bations and has applications outside quantum chaos or yond the often repeated characterizations based on the decoherence. For instance, a Schro¨ dinger-cat state can instantaneous eigenstates of the density matrix. What is be used as a weak force detector (Zurek, 2001), and its of the essence is the ability of the einselected states to sensitivity is determined by Eqs. (3.14)–(3.16). survive monitoring by the environment. This heuristic Structure saturation on scale a is an important distinc- criterion can be made rigorous by quantifying the pre- tion between the quantum and the classical. In chaotic dictability of the evolution of the candidate states, or of systems, the smallest structures in classical probability the associated observables. Einselected states provide density exponentially shrink with time, in accord with optimal initial conditions. They can be employed for the Eq. (3.1) (see Fig. 1). Equation (3.16) has implications purpose of prediction better than other Hilbert-space for decoherence, since a controls the sensitivity of sys- alternatives—they retain correlations in spite of their tems as well as of environments (Zurek, 2001; Karkus- immersion in the environment. zewski, Jarzynski, and Zurek, 2002). As a result of the Three quantum systems—the measured system S, the smallness of a, Eq. (3.16), and as anticipated by Peres apparatus A, and the environment E—and the correla- (1993), quantum systems are more sensitive to perturbations between them are the subject of our study. In pre- tions when their classical counterparts are chaotic (see measurements S and A interact. Their resulting en- also Jalabert and Pastawski, 2001). But in contrast to tanglement transforms into an effectively classical classical chaotic systems they are not exponentially sen- correlation as a result of the interaction between A and sitive to infinitesimally small perturbations. Rather, the E. smallest perturbations that can be effective are set by This SAE triangle helps us to analyze decoherence Eq. (3.16). and study its consequences. By keeping all three corners The emergence of Schro¨ dinger-cat states through dy- of this triangle in mind, one can avoid confusion and namics is a challenge to quantum-classical correspon- maintain focus on the correlations between, for ex- dence. It is not yet clear to what extent one should be ample, the memory of the observer and the state of the concerned about the discrepancies between quantum measured system. The evolution from a quantum en- and classical averages. The size of this discrepancy may tanglement to a classical correlation we are about to dis- or may not be negligible. But in the original cuss may be the easiest relevant aspect of the quantum- Schro¨ dinger-cat problem, quantum and classical expec- to-classical transition to define operationally. In the tation values (for the survival of the cat) were in accord. language of the last part of Sec. II, we are about to In both cases it is ultimately the state of the cat that is justify the ‘‘outsider’’ point of view, Eq. (2.44c), before most worrisome. considering the measurement from the vantage point of Note that we have not dealt with dynamical localiza- the ‘‘discoverer,’’ Eq. (2.44b), and before tackling the tion (Casati and Chirikov, 1995a). This is because it ap- issue of collapse. In spite of this focus on correlations, pears after too long a time (ϳបϪ1) to be a primary con- we shall often suppress one of the corners of the SAE cern in the macroscopic limit and is quite sensitive to triangle to simplify notation. All three parts will, how-

Rev. Mod. Phys., Vol. 75, No. 3, July 2003 730 Wojciech Hubert Zurek: Decoherence, einselection, and the quantum origins of the classical

ever, play a role in formulating questions and in moti- The single eigenstate of this density matrix is ͉⌽͘. When vating the criteria for classicality. expanded, ͉⌽͗͘⌽͉ contains terms that are off diagonal when expressed in any of the natural bases consisting of the tensor products of states in the two systems. Their disappearance as a result of tracing over the environ- A. Models of einselection ment removes the basis ambiguity. Thus, for example, the reduced density matrix ␳AS , Eq. (4.5a), has the same The simplest case of a single act of decoherence in- form as the outsider description of the classical measure- volves just three one-bit systems (Zurek, 1981, 1983). ment, Eq. (2.44c). They are denoted by S, A, and E in an obvious reference In our simple model, pointer states are easy to char- to their roles. The measurement starts with the interac- acterize. To leave pointer states untouched, the Hamil- tion of the measured system with the apparatus tonian of interaction HAE should have the same struc- ͉↑ ͉ →͉↑ ͉ ture as that for the c-NOT, Eqs. (2.9) and (2.10). It should ͘ A0͘ ͘ A1͘, (4.1a) be a function of the pointer observable, Aˆ ͉↓͉͘ ͘→͉↓͉͘ ͘ ϭ ͉ ͉ϩ ͉ ͉ A0 A0 , (4.1b) a0 A0͗͘A0 a1 A1͗͘A1 of the apparatus. Then the ͉ ϭ states of the environment will bear an imprint of the where ͗A0 A1͘ 0. For a general state, pointer states ͕͉A ͘,͉A ͖͘. As noted in Sec. II, ͑␣͉↑ ϩ␤͉↓ ͉ →␣͉↑ ͉ ϩ␤͉↓ ͉ ϭ͉⌽ 0 1 ͘ ͘) A0͘ ͘ A1͘ ͘ A0͘ ͘. (4.2) ͓HAE ,Aˆ ͔ϭ0 (4.6) These formulas represents a c-NOT implementation of the premeasurement discussed in Sec. II. immediately implies that Aˆ is a control, and its eigen- The basis ambiguity [that is, the ability to rewrite ͉⌽͘, states will be preserved. Eq. (4.2), in any basis of, say, the system, with the principle of superposition guaranteeing the existence of the 1. Decoherence of a single qubit corresponding pure states of the apparatus] disappears An example of continuous decoherence is afforded by E when an additional system, , performs a premeasure- two-state apparatus A interacting with an environment A ment on : of N other spins (Zurek, 1982). The two apparatus states ͑␣͉↑ ͉ ϩ␤͉↓ ͉ ͉␧ ͕͉⇑͘ ͉⇓͖͘ ͘ A1͘ ͘ A0͘) 0͘ are , . For the simplest, yet already interesting example, the self-Hamiltonian of the apparatus disappears, →␣͉↑͉͘ ͉͘␧ ͘ϩ␤͉↓͉͘ ͉͘␧ ͘ϭ͉⌿͘ A1 1 A0 0 . (4.3) HAϭ0, and the interaction has the form A collection of three correlated quantum systems is no ϭ͉͑⇑ ⇑͉Ϫ͉⇓ ⇓͉͒ ͉͑↑ ↑͉Ϫ͉↓ ↓͉͒ longer subject to the basis ambiguity we have pointed HAE ͗͘ ͗͘ ͚ gk ͗͘ ͗͘ k . out in connection with the EPR-like state ͉⌽͘, Eq. (4.2). k This is especially true when states of the environment (4.7) are correlated with the simple products of the states of Under the inﬂuence of this Hamiltonian the initial state, the apparatus-system combination (Zurek, 1981; Elby N and Bub, 1994). In Eq. (4.3) this can be guaranteed (ir- ͉⌽͑ ͒ ϭ͑ ͉⇑ ϩ ͉⇓ ͑␣ ͉↑ ϩ␤ ͉↓ 0 ͘ a ͘ b ͘) ͟ k ͘k k ͘k), (4.8) respective of the values of ␣ and ␤) providing that kϭ1 ␧ ͉␧ ϭ ͗ 0 1͘ 0. (4.4) evolves into

When this orthogonality condition is satisﬁed, the state ͉⌽͑t͒͘ϭa͉⇑͉͘E⇑͑t͒͘ϩb͉⇓͉͘E⇓͑t͒͘; (4.9) A S of the - pair is given by a reduced density matrix N Ϫ ͉E ͑ ͒ ϭ ͑␣ igkt͉↑ ϩ␤ igkt͉↓ ϭ͉E ͑Ϫ ͒ ␳ASϭTrE͉⌿͗͘⌿͉ ⇑ t ͘ ͟ ke ͘k ke ͘k) ⇓ t ͘. kϭ1 ϭ͉␣͉2͉↑ ↑͉͉ ͉ϩ͉␤͉2͉↓ ↓͉͉ ͉ ͗͘ A1͗͘A1 ͗͘ A0͗͘A0 (4.10) (4.5a) The reduced density matrix is 2 containing only classical correlations. ␳Aϭ͉a͉ ͉⇑͗͘⇑͉ϩab*r͑t͉͒⇑͗͘⇓͉ϩa*br*͑t͉͒⇓͗͘⇑͉ If the condition of Eq. (4.4) did not hold, that is, if the ϩ͉ ͉2͉⇓͗͘⇓͉ orthogonal states of the environment were not corre- b . (4.11) lated with the apparatus in the basis in which the origi- The coefﬁcient r(t)ϭ͗E⇑͉E⇓͘ determines the relative size nal premeasurement was carried out, then the eigen- of the off-diagonal terms. It is given by ␳ states of the reduced density matrix AS would be sums N S of products rather than simply products of states of ͑ ͒ϭ ͓ ϩ ͉͑␣ ͉2Ϫ͉␤ ͉2͒ ͔ r t ͟ cos 2gkt i k k sin 2gkt . (4.12) and A. An extreme example of this situation is the pre- kϭ1 decoherence density matrix of the pure state For large environments consisting of many (N) spins at ͉⌽ ⌽͉ϭ͉␣͉2͉↑ ↑͉͉ ͉ϩ␣␤ ͉↑ ↓͉͉ ͉ ͗͘ ͗͘ A1͗͘A1 * ͗͘ A1͗͘A0 large times the off-diagonal terms are typically small: ϩ␣ ␤͉↓ ↑͉͉ ͉ϩ͉␤͉2͉↓ ↓͉͉ ͉ N * ͗͘ A0͗͘A1 ͗͘ A0͗͘A0 . ͉ ͑ ͉͒2Ӎ ϪN ͓ ϩ͉͑␣ ͉2Ϫ͉␤ ͉2͒2͔ r t 2 ͟ 1 k k . (4.13) (4.5b) kϭ1

Rev. Mod. Phys., Vol. 75, No. 3, July 2003 Wojciech Hubert Zurek: Decoherence, einselection, and the quantum origins of the classical 731

͉⇓ ͉ ⌬␻ ϭ ⇑͉ ͉ ͉ ͉⇓ N and ͘ j͘ are j ͗ ͗j HAE j͘ ͘. There are 2 distinct ͉j͘’s, and, barring degeneracies, the same number ⌬␻ of different j’s. Probabilities pj are given by ϭ͉ ͉E͑ ϭ ͒ ͉2 pj ͗j t 0 ͘ , (4.15) which is in turn easily expressed in terms of the appro- ␣ ␤ priate squares of k and k . The evolution of r(t), Eq. (4.14), is a consequence of Ϫ ⌬␻ the rotations of the complex vectors pk exp( i jt) with different frequencies. The resultant r(t) will then start with the amplitude 1 and, as is anticipated by Eq. (4.13), quickly ‘‘crumble’’ to

2N ͉͗ ͑ ͉͒2͘ϳ 2ϳ ϪN r t ͚ pj 2 . (4.16) jϭ1 In this sense, decoherence is exponentially effective. The magnitude of the off-diagonal terms decreases exponentially fast, with the physical size N of the environment effectively coupled to the state of the system. We note that the effectiveness of einselection depends on the initial state of the environment. When E is in the ϭ␦ FIG. 3. Schematic representation of the effect of decoherence kth eigenstate of HAE , pj jk , the coherence in the on a Bloch sphere. When interaction with the environment system will be retained. This special environment state singles out pointer states located at the poles of the Bloch is, however, unlikely in realistic circumstances. sphere, pure states (which lie on its surface) will evolve towards the vertical axis. This classical core is a set of all the mixtures of the pointer states. 2. The classical domain and quantum halo The geometry of flows induced by decoherence in a The density matrix of any two-state system can be Bloch sphere exhibits characteristics encountered in represented by a point in three-dimensional space. In general: terms of the coefficients a, b, and r(t), the coordinates of the point representing it are zϭ(͉a͉2Ϫ͉b͉2), x (i) A classical set of the einselected pointer states ϭRe(ab*r), and yϭIm(ab*r), the real and imaginary (͕͉⇑͘,͉⇓͖͘ in our case). Pointer states are the pure parts of the complex ab*r. When the state is pure, x2 states least affected by decoherence. ϩy2ϩz2ϭ1. Pure states lie on the surface of the Bloch (ii) A classical domain consisting of all the pointer sphere (Fig. 3). states and their mixtures. In Fig. 3 this corre- Ϫ ϩ Any conceivable (unitary or nonunitary) quantum sponds to the section [ 1, 1] of the z axis. evolution can be thought of as a transformation of the (iii) The quantum domain, the rest of the volume of surface of the pure states into the ellipsoid contained the Bloch sphere, consisting of more general den- inside the Bloch sphere. Deformation of the Bloch sity matrices. sphere surface caused by decoherence is a special case of such general evolutions (Zurek, 1982, 1983; Berry, 1995). Visualizing the decoherence-induced decomposition Decoherence does not affect ͉a͉ or ͉b͉. Hence evolution of Hilbert space may be possible only in the simple case due to decoherence alone occurs in the zϭconst plane. studied here, but whenever decoherence leads to classi- Such a slice through the Bloch sphere would show the cality, the emergence of generalized and often approxi- point representing the state at a fraction ͉r(t)͉ of its mate versions of the elements (i)–(iii) is expected. maximum distance. The complex r(t) can be expressed As a result of decoherence the part of Hilbert space as the sum of the complex phase factors rotating with outside the classical domain is ruled out by einselection. ⌬␻ The severity of the prohibition on its states varies. One the frequencies given by the differences j between the energy eigenvalues of the interaction Hamiltonian, may measure the nonclassicality of (pure or mixed) weighted with the probabilities of finding them in the states by quantifying their distance from the classical do- initial state: main with the rate of entropy production and comparing it to the much lower rate in the classical domain. Classi- 2N cal pointer states would then be enveloped by a ‘‘quan- ͑ ͒ϭ ͑Ϫ ⌬␻ ͒ r t ͚ pj exp i jt . (4.14) tum halo’’ (Anglin and Zurek, 1996) of nearby, relatively ϭ j 1 decoherence-resistant but still somewhat quantum The index j denotes the environment part of the energy states, with more flagrantly quantum (and more fragile) eigenstates of the interaction Hamiltonian, Eq. (4.7), for Schro¨ dinger-cat states further away. ͉ ͘ϭ͉↑͘ ͉↓͘ ͉↑͘ By the same token, one can define an einselection- example: j 1 2 ¯ N . The corresponding differences between the energies of the eigenstates ͉⇑͉͘j͘ induced metric in the classical domain, with the distance

Rev. Mod. Phys., Vol. 75, No. 3, July 2003 732 Wojciech Hubert Zurek: Decoherence, einselection, and the quantum origins of the classical

NϪ1 ͉ ϭ Ϫ1/2 ͑ ␲ ͉͒ rl͘ N ͚ exp 2 ikl/N sk͘, (4.17) kϭ0 the c-shift could equally well represent a measurement ͉ of the apparatus (in the basis conjugate to ͕ Ak͖͘)by the system. Thus it is not just the basis that is ambiguous, but also the roles of the control (system) and of the target (apparatus), which can be reversed when the conjugate basis is selected. These ambiguities can be re- moved by recognizing the role of the environment. Figure 4 captures the essence of an idealized decoherence process that allows the apparatus to be—in spite of its interaction with the environment—a noiseless classical communication channel (Schumacher, 1996; Lloyd, 1997). This is possible because the A-E c-shifts do not disturb the pointer states. The advantage of this idealization of the decoherence process as a sequence of c-shifts lies in its simplicity. However, the actual process of decoherence is usually FIG. 4. Information transfer in a c-NOT or a c-shift ‘‘carica- continuous (so that it can only be approximately broken ture’’ of measurement, decoherence, and decoherence with up into discrete c-shifts). Moreover, in contrast to the noise. Bit-by-bit measurement is shown on the top. This dia- c-NOT’s used in quantum logic circuits, the record in- gram is the fundamental logic circuit used to represent deco- scribed in the environment is usually distributed over herence affecting the measuring apparatus. Note that the di- many degrees of freedom. Last but not least, the observ- rection of the information flow in decoherence, from the able of the apparatus (or any other open system) may be decohering apparatus and to the environment, differs from the information flow associated with noise. In short, as a result of subject to noise (and not just decoherence), or its self- decoherence, environment is perturbed by the state of the sys- Hamiltonian may rotate instantaneous pointer states tem. Noise is, by contrast, perturbation inflicted by the envi- into their superpositions. These very likely complica- ronment. Preferred pointer states are selected so as to mini- tions will be investigated in specific models below. mize the effect of the environment—to minimize the number Decoherence is caused by a premeasurementlike proof c-NOT’s pointing from the environment at the expense of cess carried out by the environment E: those pointing towards it. ͉⌿SA͉͘␧ ͘ϭͩ ͚ ␣ ͉s ͉͘A ͪ͘ ͉␧ ͘→͚ ␣ ͉s ͉͘A ͉͘␧ ͘ 0 j j j 0 j j j j j j between two pointer states given by the rate of entropy ϭ͉⌽ ͘ production of their superposition. This is not the only SAE . (4.18) way to define a distance. As we shall see in Sec. VII, the Decoherence leads to einselection when the states of the ͉␧ redundancy of the record of a state imprinted on the environment j͘ corresponding to different pointer environment is a very natural measure of its classicality. states become orthogonal: In the course of decoherence, pointer states tend to be ͗␧ ͉␧ ͘ϭ␦ recorded redundantly and can be deduced by intercept- i j ij . (4.19) ing a very small fraction of the environment (Zurek, Then the Schmidt decomposition of the state vector 2000; Dalvit, Dziarmaga, and Zurek, 2001; Ollivier, Pou- ͉⌽SAE͘ into composite subsystems SA and E yields prod- ͉ ͉ lin, and Zurek, 2002). uct states sj͘ Aj͘ as partners of the orthogonal environment states. The decohered density matrix describing the SA pair is then diagonal in product states: 3. Einselection and controlled shifts ␳D ϭ ͉␣ ͉2͉ ͉͉ ͉ϭ ͉⌽ ⌽ ͉ SA ͚ j sj͗͘sj Aj͗͘Aj TrE SAE͗͘ SAE . Discussion of decoherence can be generalized to the j situation in which the system, the apparatus, and the (4.20) environment have many states, and their interactions For simplicity we shall often omit reference to the object are complicated. Here we assume that the system is iso- that does not interact with the environment (here, the lated, and that it interacts with the apparatus in the system S). Nevertheless, preservation of the SA correla- c-shift manner discussed in Sec. II. As a result of that tions is the criterion defining the pointer basis. Invoking interaction the state of the apparatus becomes entangled it would eliminate much confusion (see, for example, ͚ ␣ ͉ ͉ with the state of the system: ( i i si͘) A0͘ discussions in Halliwell, Perez-Mercader, and Zurek, →͚ ␣ ͉ ͉ i i si͘ Ai͘. This state suffers from basis ambiguity: 1994; Venugopalan, 1994, 2000). The density matrix of a the entanglement of S and A implies that for any state of single object in contact with the environment will always either there exists a corresponding pure state of its part- be diagonal in an (instantaneous) Schmidt basis. This ner. Indeed, when the initial state of S is chosen to be instantaneous diagonality should not be used as the sole one of the eigenstates of the conjugate basis, criterion for classicality (although see Zeh, 1973, 1990;

Rev. Mod. Phys., Vol. 75, No. 3, July 2003 Wojciech Hubert Zurek: Decoherence, einselection, and the quantum origins of the classical 733

Albrecht, 1992, 1993). Rather, the ability of certain Normalized ␳S͉⌸ can be obtained by using the prob- j states to retain correlations in spite of coupling to the ability of the outcome: environment is decisive. Ϫ1 ␳S͉⌸ ϭp ˜␳S͉⌸ , (4.26) When the interaction with the apparatus has the form j j j

p ϭTr˜␳S͉⌸ . ϭ AE ͉ ͗͘ ͉͉␧ ͗͘␧ ͉ϩ j j HAE ͚ gklm Ak Ak l m H.c., (4.21) k,l,m The conditional density matrix represents the descrip- S the basis ͕͉A ͖͘ is left unperturbed and any correlation tion of the system available to the observer who knows k A ⌸ ͉ that the apparatus is in a subspace defined by j . with the states ͕ Ak͖͘ is preserved. But, by definition, pointer states preserve correlations in spite of decoherence, so that any observable Aˆ codiagonal with the interaction Hamiltonian will be pointer observable. For 1. Conditional state, entropy, and purity ˆ ␳P ͉ ͘ when the interaction is a function of A, it can be ex- Before decoherence, S͉C ͘ is pure for any state Cj , ˆ ˆ j panded in A as a power series, so it commutes with A: P 2 P ͑␳S͉⌸ ͒ ϭ␳S͉⌸ ᭙͉C ͘; (4.27a) j j j ͓HAE͑Aˆ ͒,Aˆ ͔ϭ0. (4.22) providing the initial premeasurement state, Eq. (4.23), The dependence of the interaction Hamiltonian on the was pure as well. It follows that observable is an obvious precondition for the monitor- P H͑␳SA͉ ͒͘ϭ0 ᭙͉C ͘. (4.28a) ing of that observable by the environment. This admits Cj j ˆ P the existence of degenerate pointer eigenspaces of A. For this same case given by the initially pure ␳SA of Eq. (4.23), conditional density matrices obtained from the B. Einselection as the selective loss of information D decohered ␳SA will be pure if and only if they are con- ͕͉ ͖͘ The establishment of a measurementlike correlation ditioned upon the pointer states Ak : between the apparatus and the environment changes the ͑␳D ͒2ϭ␳P ϭ͉ ͉⇔͉ ϭ͉ S͉ ͘ S͉ ͘ sk͗͘sk Cj͘ Aj͘; (4.27b) P Cj Cj density matrix from the premeasurement ␳SA to the de- D D P cohered ␳SA , Eq. (4.20). For the initially pure ͉⌿SA͘, H͑␳S͉ ͒͘ϭH͑␳S͉ ͒͘. (4.28b) Aj Aj Eq. (4.18), this transition is represented by This last equation is valid even when the initial states of ␳P ϭ ␣ ␣ ͉ ͗͘ ͉͉ ͗͘ ͉ the system and of the apparatus are not pure. Thus only SA ͚ i j* si sj Ai Aj i,j in the pointer basis will the predecoherence strength of the SA correlation be maintained. In all other bases →͚ ͉␣ ͉2͉ ͗͘ ͉͉ ͗͘ ͉ϭ␳D ͑␳D ͒2Ͻ ␳D ͉ ͘¹͕͉ ͖͘ i si si Ai Ai SA . (4.23) Tr S͉C ͘ Tr S͉C ͘ ; Cj Aj , (4.27c) i j j P D H͑␳S͉ ͒͘ϽH͑␳S͉ ͒͘; ͉C ͘¹͕͉A ͖͘. (4.28c) Einselection is accompanied by an increase of entropy, Cj Cj j j ⌬ ͑␳ ͒ϭ ͑␳D ͒Ϫ ͑␳P ͒у ͉ H SA H SA H SA 0, (4.24) In particular, in the basis ͕ Bj͖͘ conjugate to the pointer ͕͉ ͖͘ and by the disappearance of the ambiguity in what was states Aj , Eq. (2.14), there is no correlation left with measured (Zurek, 1981, 1993a). Thus, before decoher- the state of the system. That is, ence, the conditional density matrices of the system, ␳D ϭ Ϫ1 ͉ ͗͘ ͉ϭ ␳S͉ ͘ , are pure for any state ͉C ͘ of the apparatus S͉B ͘ N ͚ sk sk 1/N, (4.29) Cj j j k pointer. They are defined using the unnormalized where 1 is a unit density matrix. Consequently ˜␳S͉⌸ ϭTrA⌸ ␳SA , (4.25) j j ͑␳D ͒2ϭ␳ S͉B ͘ S͉B ͘ /N, (4.27d) ⌸ ϭ͉ ͉ j j where in the simplest case j Cj͗͘Cj projects onto a 4 D P pure state of the apparatus. H͑␳S͉ ͒͘ϭH͑␳S͉ ͒͘Ϫln NϭϪln N. (4.28d) Bj Bj Note that, initially, the conditional density matrices were

4 also pure in the conjugate (and any other) basis, pro- This can be generalized to projections onto multidimen- vided that the initial state was the pure entangled pro- sional subspaces of the apparatus. In that case, the purity of P jection operator ␳ ϭ͉⌿SA͗͘⌿SA͉, Eq. (4.23). the conditional density matrix will usually be lost during the SA trace over the states of the pointer. This is not surprising. When the observer reads off the pointer of the apparatus only in a coarse-grained manner, he will forgo part of the informa- 2. Mutual information and discord tion about the system. The ampliﬁcation we have considered before can prevent some of this loss of resolution due to coarse Selective loss of information everywhere except in the graining in the apparatus. Generalizations to density matrices pointer states is the essence of einselection. It is re- that are conditioned upon projection-operator-valued mea- ﬂected in the change of the mutual information which sures (Kraus, 1983) are also possible. starts from

Rev. Mod. Phys., Vol. 75, No. 3, July 2003 734 Wojciech Hubert Zurek: Decoherence, einselection, and the quantum origins of the classical

IP͑S A͒ϭ ͑␳P͒ϩ ͑␳P ͒Ϫ ͑␳P ͒ Prior to decoherence, the use of probabilities would not : H S H A H S,A have been legal. ϭϪ ͉␣ ͉2 ͉␣ ͉2 For the case considered here, Eq. (4.18), the condi- 2͚ i ln i . (4.30a) ␳ ϭ i tional entropy H( S͉A) 0. In the pointer basis there is a perfect correlation between the system and the appara- As a result of einselection, for initially pure cases, this tus, providing that the premeasurement Schmidt basis decreases to at most, half its initial value: and the pointer basis coincide. Indeed, it is tempting to D D D D define a good apparatus or a classical correlation by in- I ͑S:A͒ϭH͑␳S ͒ϩH͑␳A ͒ϪH͑␳S A͒ , sisting on such a coincidence. The difficulties with conditional entropy and mutual ϭϪ ͉␣ ͉2 ͉␣ ͉2 ͚ i ln i . (4.30b) i information are a symptom of the quantum nature of the problem. The trouble with H(␳S͉A) arises for states that This level is reached when the pointer basis coincides exhibit quantum correlations—entanglement of ͉⌿SA͘ with the Schmidt basis of ͉⌿SA͘. The decrease in mutual being an extreme example—and thus do not admit an information is due to the increase of the joint entropy interpretation based on probabilities. A useful sufficient ␳ H( S,A): condition for the classicality of correlations is then the existence of an apparatus basis that allows quantum ver- ⌬I͑S:A͒ϭIP͑S:A͒ϪID͑S:A͒ sions of the two classically identical expressions for the ϭ ͑␳D ͒Ϫ ͑␳P ͒ϭ⌬ ͑␳ ͒ mutual information to coincide: I(S:A)ϭJA(S:A) H S,A H S,A H S,A . (4.31) (Zurek, 2000, 2003a; Ollivier and Zurek, 2002; Vedral, Classically, an equivalent definition of the mutual infor- 2003). Equivalently, the discord mation obtains from the asymmetric formula ␦IA͑S͉A͒ϭI͑S:A͒ϪJA͑S:A͒ (4.36) JA͑S:A͒ϭH͑␳S͒ϪH͑␳S͉A͒, (4.32) must vanish. Unless ␦IA(S͉A)ϭ0, probabilities for the with the help of the conditional entropy H(␳S͉A). distinct apparatus pointer states cannot exist. Above, the subscript A indicates the member of the cor- We end this subsection with part summary, part antici- related pair that will be the source of the information patory remarks. Pointer states retain undiminished cor- about its partner. A symmetric counterpart of the above relations with the measured system S, or with any other equation, JS(S:A)ϭH(␳A)ϪH(␳A͉S), can also be writ- system, including observers. The loss of information ten. caused by decoherence is given by Eq. (4.31). This loss is In the quantum case, the definition of Eq. (4.32) is so precisely such as to lift conditional information from the far incomplete, since a quantum analog of the classical paradoxical (negative) values, Eq. (4.33), to the classi- conditional information has not yet been specified. In- cally allowed level. This is equal to the information deed, Eqs. (4.30a) and (4.32) jointly imply that in the gained by the observer when he consults the apparatus case of entanglement a quantum conditional entropy pointer. This is no accident—the environment has ‘‘measured’’ (become correlated with) the apparatus in the H(␳S͉A) would have to be negative. For in that case, very same pointer basis at which observers have to ac- A ͑␳ ͒ϭ ͉␣ ͉2 ͉␣ ͉2Ͻ cess to take advantage of the remaining (classical) H S͉A ͚ i ln i 0 (4.33) i correlation between the pointer and the system. Only when observers and the environment monitor codiago- I S A ϭJ S A would be required to allow for ( : ) A( : ). Vari- nal observables do they not get in each other’s way. I S A ␳ ous quantum redefinitions of ( : )orH( S͉A) have In the idealized case, the preferred basis was distin- been proposed to address this (Lieb, 1975; Schumacher guished by its ability to retain perfect correlations with and Nielsen, 1996; Cerf and Adami, 1997; Lloyd, 1997). the system in spite of decoherence. This remark will We shall simply regard this fact as an illustration of the serve as a guide in other situations. It will lead to a strength of the quantum correlations (i.e., entangle- criterion—the predictability sieve—used to identify pre- ment) that allow I(S:A) to violate the inequality ferred states in less idealized circumstances. For example, when the self-Hamiltonian of the system is non- I͑S:A͒рmin͑HS ,HA͒. (4.34) trivial, or when the commutation relation, Eq. (4.22), This inequality follows directly from Eq. (4.32) and the does not hold exactly for any observable of S, we shall non-negativity of classical conditional entropy (see, for seek states that are best in retaining correlations with example, Cover and Thomas, 1991). the other systems. Decoherence decreases I(S:A) to this allowed level (Zurek, 1983). Moreover, now the conditional entropy C. Decoherence, entanglement, dephasing, and noise can be defined in the classical pointer basis as the average of partial entropies computed from the conditional In the symbolic representation of Fig. 4, noise is the D ␳S͉ ͘ over the probabilities of different outcomes: process in which the environment acts as a control, in- Ai scribing information about its state on the state of the ͑␳ ͒ϭ ͑␳ ͒ system, which assumes the role of the target. However, H S͉A ͚ p͉A ͘H S͉A ͘D . (4.35) i i i the direction of the information flow in c-NOT’s and

Rev. Mod. Phys., Vol. 75, No. 3, July 2003 Wojciech Hubert Zurek: Decoherence, einselection, and the quantum origins of the classical 735 c-shifts depends on the choice of initial states. Control Nevertheless, each member of the ensemble may exist in and target switch roles when, for a given interaction a state as pure as it was before dephasing. Nuclear mag- Hamiltonian, one prepares the input of the c-NOT in the netic resonance (NMR) offers examples of dephasing basis conjugate to the logical pointer states. Einselected (which can be reversed using spin echo). Dephasing is a states correspond to the set of states that, when used in loss of phase coherence between members of the en- c-NOT’s or c-shifts, minimizes the effect of interactions semble due to differences in the noise in phases each directed from the environment to the system. member experiences. It does not result in an informa- Einselection is caused by the premeasurement carried tion transfer to the environment. out by the environment on the pointer states. Decoher- Dephasing cannot be used to justify the existence of a ence follows from Heisenberg’s indeterminacy. Pointer preferred basis in individual quantum systems. Never- theless, the ensemble as a whole may obey the same observable is measured by the environment. Therefore master equation as do individual systems entangling the complementary observable must become at least as with the environment. Indeed, many of the symptoms indeterminate as is demanded by Heisenberg’s principle. exhibited by, e.g., the expectation values for a single de- As the environment and the systems entangle through cohering system can be reproduced by ensemble aver- an interaction that favors a set of pointer states, their ages in this setting. In spite of the light shed on this issue phases become indeterminate [see Eq. (4.29) and the by the discussion of simple cases (Wootters and Zurek, discussion of envariance in Sec. VI]. Decoherence can 1979; Stern, Aharonov, and Imry, 1989), more remains to be thought of as the resulting loss of phase relations. be understood, perhaps by considering the implications Observers can be ignorant of phases for reasons that of envariance (see Sec. VI). do not lead to an imprint of the state of the system on Noise is an even more familiar and less subtle effect the environment. Classical noise can cause such dephas- represented by transitions that break the one-to-one cor- ing when the observer does not know the time- respondence in Eq. (4.39). Noise in the apparatus would dependent classical perturbation Hamiltonian respon- ͉ cause a random rotation of states Aj͘. It could be mod- sible for this unitary, but unknown, evolution. For (n) eled by a collection of Hamiltonians similar to Hd but example, in the predecoherence state vector, Eq. (4.18), not codiagonal with the observable of interest. Then, random-phase noise will cause a transition: after an ensemble average similar to Eq. (4.39), the one- to-one correspondence between S and A would be lost. ͑n͒ ͉⌿SA͘ϭͩ ͚ ␣ ͉s ͉͘A ͪ͘ →͚ ␣ exp͑i␾ ͉͒s ͉͘A ͘ j j j j j j j However, as before, the evolution is unitary for each n, j j and the unperturbed state could be reconstructed from ͑n͒ ϭ͉⌿SA͘. (4.37) information the observer could have in advance. Hence, in the case of dephasing or noise, information A dephasing Hamiltonian acting either on the system or about the cause obtained either in advance, or after- on the apparatus can lead to such an effect. In this sec- wards, sufﬁces to undo the effect. Decoherence relies on ond case its form could be entangling interactions [although, strictly speaking, it need not involve entanglement (Eisert and Plenio, ͑n͒ϭ ␾˙ ͑n͒͑ ͉͒ ͗͘ ͉ Hd ͚ j t Aj Aj . (4.38) 2002)]. Thus neither prior nor posterior knowledge of j the state of the environment is enough. Transfer of in- In contrast to interactions causing premeasurements, en- formation about a decohering system to the environ- tanglement, and decoherence, Hd cannot inﬂuence the ment is essential, and plays a key role in the interpreta- SA nature or the degree of the correlations. Hd does not tion. imprint the states of S or A anywhere else in the uni- We note that, while the nomenclature used here verse. For each individual realization n of the phase seems the most sensible to this author and is widely ͕␾(n) ͖ noise [each selection of j (t) in Eq. (4.37)] the state used, it is unfortunately not universal. For example, in ͉⌽(n)͘ ͕␾(n)͖ SA remains pure. Given only j one could re- the context of quantum computation ‘‘decoherence’’ is store the predephasing state on a case-by-case basis. sometimes used to describe any process that can cause However, in the absence of such detailed information, errors (but see related discussion in Nielsen and one is often forced to represent SA by the density ma- Chuang, 2000). trix averaged over the ensemble of noise realizations:

¯␳SAϭ͉͗⌿SA͗͘⌿SA͉͘ D. Predictability sieve and einselection

ϭ ͉␣ ͉2͉ ͉͉ ͉ The evolution of a quantum system prepared in a clas- ͚ j sj͗͘sj Aj͗͘Aj j sical state should emulate classical evolution that can be idealized as a ‘‘trajectory’’—a predictable sequence of ͑ ͒ ͑ ͒ ϩ i͓␾ n Ϫ␾ n ͔␣ ␣ ͉ ͉͉ ͉ objectively existing states. For a purely unitary evolu- ͚ ͚ e j k j k sj͗͘sk Aj͗͘Ak . j,k n tion, all of the states in the Hilbert space retain their purity and are therefore equally predictable. However, (4.39) in the presence of an interaction with the environment, a In this phase-averaged density matrix off-diagonal terms generic superposition representing correlated states of will be suppressed and may even completely disappear. the system and of the apparatus will decay into a mix-

Rev. Mod. Phys., Vol. 75, No. 3, July 2003 736 Wojciech Hubert Zurek: Decoherence, einselection, and the quantum origins of the classical ture diagonal in pointer states, Eq. (4.23). Only when Viola, 2000; Zanardi, 2001). However, calculations are in the predecoherence state of SA is a product of a single general quite difficult even for the initial pure-state ͉ apparatus pointer state Ai͘ with the corresponding out- cases. come state of the system (or a mixture of such product The idea of the sieve selecting candidates for the clas- states) does decoherence have no effect: sical states is one decade old, but still only partly explored. We shall see it in action below. We have outlined ␳P ϭ͉ ͉͉ ͉ϭ␳D SA si͗͘si Ai͗͘Ai SA . (4.40) two criteria for sifting through the Hilbert space in search of classicality; von Neumann entropy and purity A correlation of a pointer state with any state of an define, after all, two distinct functionals. Entropy is ar- isolated system is untouched by the environment. By the guably an obvious information-theoretic measure of pre- A same token, when the observer prepares in the dictability loss. Purity is much easier to compute. It is ͉ ͘ pointer state Ai , he can count on its remaining pure. often used as a ‘‘cheap substitute’’ and has a physical ͉ One can even think of si͘ as the record of the pointer significance of its own. It seems unlikely that pointer state of A. Einselected states are predictable: they pre- states selected by the predictability and purity sieves serve correlations and hence are effectively classical. could differ substantially. After all, In the above idealized cases, the predictability of ϪTr␳ ln ␳ϭTr␳͕͑1Ϫ␳͒Ϫ͑1Ϫ␳͒2/2ϩ , (4.43) some states follows directly from the structure of the ¯ relevant Hamiltonians (Zurek, 1981). A correlation with so that one can expect the most predictable states to also a subspace associated with a projection operator PA will remain the purest (Zurek, 1993a). However, the expan- be immune to decoherence providing that sion, Eq. (4.43), is very slowly convergent. Therefore a more mathematically satisfying treatment of the differ- ͓ ϩ ͔ϭ HA HAE ,PA 0. (4.41) ences between the states selected by these two criteria In more realistic cases it is difficult to demand the exact would be desirable, especially in cases where (as we conservation guaranteed by such a commutation condi- shall see in the next section for the harmonic oscillator) tion. Looking for approximate conservation may still be the preferred states are coherent (Zurek, Habib, and a good strategy. The various densities used in hydrody- Paz, 1993), and hence the classical domain forms a rela- namics are one obvious choice (see, e.g., Gell-Mann and tively broad ‘‘mesa’’ in Hilbert space. The possible discrepancy between the states selected Hartle, 1990, 1993). by sieves based on predictability and those based on pu- In general, it is useful to invoke a more fundamental rity raises a more general question. Will all the sensible predictability criterion (Zurek, 1993a). One can measure criteria yield identical answers? After all, one can imag- the loss of predictability caused by evolution for every ͉⌿͘ ine other reasonable criteria for classicality, such as the pure state by von Neumann entropy or some other yet-to-be-explored ‘‘distinguishability sieve’’ of Schuma- measure of predictability such as the purity: cher (1999), which picks out states whose descendants 2 ␵⌿͑t͒ϭTr␳⌿͑t͒. (4.42) are most distinguishable in spite of decoherence. More- over, as we shall see in Sec. VII (also, Zurek, 2000), one In either case, predictability is a function of time and a can ascribe classicality to the states that are most redun- functional of the initial state as ␳⌿(0)ϭ͉⌿͗͘⌿͉. Pointer dantly recorded by the environment. The menu of vari- states are obtained by maximizing the predictability ous classicality criteria already contains several posi- functional over ͉⌿͘. When decoherence leads to classi- tions, and more may be added in the future. There is no cality, good pointer states exist, and the answer is robust. a priori reason to expect that all of these criteria will A predictability sieve sifts all of Hilbert space, order- lead to identical sets of preferred states. It is neverthe- ing states according to their predictability. The top of the less reasonable to hope that, in the macroscopic limit in list will be the most classical. This point of view allows which classicality is indeed expected, differences be- for unification of the simple definition of the pointer tween various sieves should be negligible. The same sta- states in terms of the commutation relation, Eq. (4.41), bility in the selection of the classical domain is expected with the more general criteria required to discuss classi- with respect to changes of, say, the time of evolution cality in other situations. The eigenstates of the exact from the initial pure state. Reasonable changes of such pointer observable are selected by the sieve. Equation details within the time interval in which einselection is (4.41) guarantees that they will retain their purity in expected to be effective should lead to more or less simi- spite of the environment and are (somewhat trivially) lar preferred states, and certainly to preferred states predictable. contained within each other’s ‘‘quantum halo’’ (Anglin and Zurek, 1996). As noted above, this seems to be the The predictability sieve can be generalized to situa- case in the examples explored to date. It remains to be tions where the initial states are mixed (Paraoanu and seen whether all criteria will agree in other situations of Scutera, 1998; Paraoanu, 2002). Often whole subspaces interest. emerge from the predictability sieve, naturally leading to decoherence-free subspaces (see, for example, Lidar V. EINSELECTION IN PHASE SPACE et al., 1999) and can be adapted to yield ‘‘noiseless subsystems’’ (which are a non-Abelian generalization of Einselection in phase space is a special, yet very im- pointer states; see, for example, Knill, Laflamme, and portant, topic. It should lead to phase-space points and

Rev. Mod. Phys., Vol. 75, No. 3, July 2003 Wojciech Hubert Zurek: Decoherence, einselection, and the quantum origins of the classical 737

trajectories and to classical (Newtonian) dynamics. The 2 cn special role of position in classical physics can be traced C͑␻͒ϭ͚ ␦͑␻Ϫ␻ ͒. (5.4) 2m ␻ n to the nature of interactions that depend on distance n n n (Zurek, 1981, 1982, 1991) and therefore commute with The effect of the environment can be expressed position [see Eq. (4.22)]. Evolution of open systems in- through the propagator J acting on the reduced ␳S : cludes, however, the ﬂow in phase space induced by the ␳ ͑ Ј ͒ϭ ͵ Ј ͑ Ј ͉ Ј ͒␳ ͑ Ј ͒ self-Hamiltonian. Consequently a set of preferred states S x,x ,t dx0dx0J x,x ,t x0 ,x0 ,t0 S x0 ,x0 ,t0 . turns out to be a compromise, localized in both position (5.5) and momentum, localized in phase space. Einselection is responsible for the classical structure We focus on the case in which the system and the envi- of phase space. States selected by the predictability sieve ronment are initially statistically independent, so that become phase-space ‘‘points,’’ and their time-ordered their density matrices start from a product state:

sequences turn into trajectories. In underdamped, clas- ␳SEϭ␳S␳E . (5.6) sically regular systems one can recover this phase-space structure along with (almost) reversible evolution. In This is a restrictive assumption. One can try to justify it S chaotic systems there is a price to be paid for classicality: as an idealization of a measurement that correlates S E combination of decoherence with the exponential diver- with the observer and destroys correlations of with , gence of classical trajectories (which is the deﬁning fea- but that is only an approximation, since realistic mea- ture of chaos) leads to entropy production at a rate surements leave partial correlations with the environ- given—in the classical limit—by the sum of positive ment intact. Fortunately, such preexisting correlations Lyapunov exponents. Thus the second law of thermody- lead to only minor differences in the salient features of namics can emerge from the interplay of classical dy- the subsequent evolution of the system (Anglin, Paz, namics and quantum decoherence, with entropy produc- and Zurek, 1997; Romero and Paz, 1997). ␳ tion caused by information ‘‘leaking’’ into the The evolution of the whole SE can be represented as environment (Zurek and Paz, 1994, 1995a; Zurek, ␳SE͑x,q,xЈ,qЈ,t͒ 1998b; Paz and Zurek, 2001). ϭ ͵ Ј Ј␳ ͑ Ј Ј ͒ A. Quantum Brownian motion dx0dx0dq0dq0 SE x0 ,q0 ,x0 ,q0 ,t0

ϫ ͑ ͒ ͑ Ј Ј Ј Ј͒ The quantum Brownian motion model consists of an K x,q,t,x0 ,q0 K* x ,q ,t,x0 ,q0 . (5.7) E environment —a collection of harmonic oscillators (co- Above, we suppress the sum over the indices of the in- ␻ ordinates qn , masses mn , frequencies n , and coupling dividual environment oscillators. The evolution operator constants c )—interacting with the system S (coordinate n K(x,q,t,x ,q ) can be expressed as a path integral x), with a mass M and a potential V(x). We shall often 0 0 consider harmonic V(x)ϭM⍀2x2/2 so that the whole i ͑ ͒ϭ ͵ ͩ ͓ ͔ͪ SE is linear and one can obtain an exact solution. This K x,q,t,x0 ,q0 DxDq exp ប I x,q , (5.8) assumption will be relaxed later. ͓ ͔ The Lagrangian of the system-environment entity is where I x,q is the action functional that depends on the trajectories x and q. The integration must satisfy the ͑ ͒ϭ ͑ ͒ϩ ͑ ͕ ͖͒ L x,qn LS x LSE x, qn ; (5.1) boundary conditions the system alone has the Lagrangian ͑ ͒ϭ ͑ ͒ϭ ͑ ͒ϭ ͑ ͒ϭ x 0 x0 ; x t x; q 0 q0 ; q t q. (5.9) M M The expression for the propagator of the density matrix L ͑x͒ϭ x˙ 2ϪV͑x͒ϭ ͑x˙ 2Ϫ⍀2x2͒. (5.2) S 2 2 can now be written in terms of actions corresponding to the two Lagrangians, Eqs. (5.1)–(5.3): The effect of the environment is modeled by the sum of ͑ Ј ͉ Ј ͒ the Lagrangians of individual oscillators and of the J x,x ,t x0 ,x0 ,t0 system-environment interaction terms: i m c x 2 ϭ ͵ DxDxЈ expͩ ͑IS͓x͔ϪIS͓xЈ͔͒ͪ ϭ͚ n ͫ ˙ 2 Ϫ␻2 ͩ Ϫ n ͪ ͬ ប LSE qn n qn ␻2 . (5.3) n 2 mn n ϫ ͵ Ј␳ ͑ Ј͒ This Lagrangian takes into account the renormalization dqdq0dq0 E q0 ,q0 of the potential energy of the Brownian particle. The interaction depends (linearly) on the position x of the i ϫ ͵ Ј ͩ ͑ ͓ ͔Ϫ ͓ Ј Ј͔͒ͪ DqDq exp ISE x,q ISE x ,q . harmonic oscillator. Hence we expect x to be an instan- h taneous pointer observable. In combination with the harmonic evolution this leads to Gaussian pointer states, (5.10) well localized in both x and p. An important character- The separability of the initial conditions, Eq. (5.6), was istic of the model is the spectral density of the environ- used to make the propagator depend only on the initial ment: conditions of the environment. Collecting all terms con-

Rev. Mod. Phys., Vol. 75, No. 3, July 2003 738 Wojciech Hubert Zurek: Decoherence, einselection, and the quantum origins of the classical

E taining integrals over in the above expression leads to The time-dependent coefficients bk and aij are com- the influence functional (Feynman and Vernon, 1963) puted from the noise and dissipation kernels, which reflect properties of the environment. They obtain from ͑ Ј͒ϭ ͵ Ј␳ ͑ Ј͒ the solutions of the equation F x,x dqdq0dq0 E q0 ,q0 s i u¨ ͑s͒ϩ⍀2u͑s͒ϩ2 ͵ ds␩͑sϪsЈ͒u͑sЈ͒ϭ0, (5.17) ϫ ͵ Ј ͑ ͓ ͔Ϫ ͓ Ј Ј͔͒ 0 DqDq expͩ ប ISE x,q ISE x ,q ͪ . where ⍀ is the ‘‘bare frequency’’ of the oscillator. Two (5.11) such solutions that satisfy the boundary conditions ϭ ϭ ϭ ϭ Influence functional can be evaluated explicitly for u1(0) u2(t) 1 and u1(t) u2(0) 0 can be used for specific models of the initial density matrix of the envi- this purpose. They yield the coefficients of the Gaussian ronment. An environment in thermal equilibrium pro- propagator through vides a useful and tractable model for the initial state. ͑ ͒ϭ ˙ ͑ ͒ ͑ ͒ϭ ˙ ͑ ͒ The density matrix of the nth mode of the thermal en- b1͑2͒ t u2͑1͒ t /2, b3͑4͒ t u2͑1͒ 0 /2, (5.18a) vironment is 1 t t ͑ ͒ϭ ͵ ͵ Ј ͑ ͒ ͑ Ј͒␯͑ Ϫ Ј͒ ␻ aij t ϩ␦ ds ds ui s uj s s s . mn n 1 ij 0 0 ␳E ͑q,qЈ͒ϭ n ប␻ (5.18b) n 2␲ប sinhͩ ͪ kBT The master equation can now be obtained by taking the time derivative of Eq. (5.5), which in effect reduces m ␻ ϫ Ϫ n n to the computation of the derivative of the propagator, exp ប␻ ͭ n Eq. (5.16), above: 2␲ប sinhͩ ͪ kBT ˙ ϭ ˙ ϩ ˙ ϩ ˙ Ϫ ˙ Ϫ ˙ J ͕b3 /b3 ib1XY ib2X0Y ib3XY0 ib4X0Y0 ប␻ ϫͫ͑ 2 ϩ Ј2͒ ͩ n ͪ Ϫ Јͬ Ϫa˙ Y2Ϫa˙ YY Ϫa˙ Y2͖J. (5.19) qn qn cosh 2qnqn . 11 12 0 22 0 kBT ͮ The time derivative of ␳S can be obtained by multiplying the operator on the right-hand side by an initial density (5.12) matrix and integrating over the initial coordinates The influence functional F can be written as (Grabert, X0 ,Y0 . Given the form of Eq. (5.19), one may expect Schramm, and Ingold, 1988) that this procedure will yield an integro-differential (nonlocal in time) evolution operator for ␳S . However, t s i ln F͑x,xЈ͒ϭ ͵ ds͑xϪxЈ͒͑s͒ ͵ du͓␩͑sϪsЈ͒͑xϩxЈ͒͑sЈ͒ the time dependence of the evolution operator disap- 0 0 pears as a result of the two identities satisfied by the propagator: Ϫi␯͑sϪsЈ͒͑xϪxЈ͒͑sЈ͔͒, (5.13) b i ͪ ץ where ␯(s) and ␩(s) are known as the dissipation and ϭͩ 1 ϩ Y0J Y X J, (5.20a) noise kernels, respectively, and are defined in terms of b3 b3 the spectral density: b1 i 2a11 a12b1 Ϫiͩ ϩ ͪ Y ץ X JϭͫϪ XϪ ϱ 0 b b Y b b b ␯͑s͒ϭ ͵ d␻C͑␻͒coth͑ប␻␤/2͒cos͑␻s͒; (5.14) 2 2 2 2 3 0 a ͬ ץ ϩ 12 ϱ X J. (5.20b) b2b3 ␩͑s͒ϭ ͵ d␻C͑␻͒sin͑␻s͒. (5.15) 0 After the appropriate substitutions, the resulting equation with renormalized Hamiltonian Hren has the form With the assumption of thermal equilibrium at kBT ϭ1/␤, and in the harmonic-oscillator case V(x) i 2 2 ␳˙ ͑ Ј ͒ϭϪ ͗ ͉͓ ͑ ͒ ␳ ͔͉ Ј͘Ϫ͓␥͑ ͒͑ Ϫ Ј͒ ϭM⍀ x /2, the integrand of Eq. (5.10) for the propaga- S x,x ,t ប x Hren t , S x t x x tor is Gaussian. The integral can be computed exactly ͒ ͑ Ϫ ͑ ͒͑ Ϫ ͒2͔␳͒ ץϩ ץϫ͑ and should also have a Gaussian form. The result can be x xЈ D t x xЈ S x,xЈ,t conveniently written in terms of the diagonal and off- .␳S͑x,xЈ,t͒͒ ץϩ ץϪif͑t͒͑xϪxЈ͒͑ diagonal coordinates of the density matrix in the posi- x xЈ tion representation, XϭxϩxЈ,YϭxϪxЈ: (5.21) ͑ ͉ ͒ The calculations leading to this master equation are J X,Y,t X0 ,Y0 ,t0 nontrivial. They involve the use of relations between the b exp͓i͑b XYϩb X YϪb XY Ϫb X Y ͔͒ ϭ 3 1 2 0 3 0 4 0 0 coefficients bk and aij . The final result leads to explicit ␲ ͑ 2ϩ ϩ 2͒ . 2 exp a11Y 2a12YY0 a22Y0 formulae for these coefficients: ⍀˜ 2͑ ͒ ϭ ˙ Ϫ ˙ (5.16) t /2 b1b2 /b2 b1 , (5.22a)

Rev. Mod. Phys., Vol. 75, No. 3, July 2003 Wojciech Hubert Zurek: Decoherence, einselection, and the quantum origins of the classical 739

FIG. 5. Time-dependent coefficients of the perturbative master equation for quantum Brownian motion. The parameters used in these plots (where the time is measured in Ϫ units of ⍀ 1) are ␥/⍀ϭ0.05, ⌫/⍀ϭ100, ប⍀ϭ kBT/ 10, 1, and 0.1. Plots on the right show the initial portion of the plots on the left—the initial transient—illustrating its independence of temperature (although higher temperatures produce higher final values of the coefficients). Plots on the right show that the final values of the coefficients strongly depend on temperature, and that anomalous diffusion is of importance only for very low temperatures.

␥͑ ͒ϭϪ Ϫ ˙ 2 t t b1 b2/2b2 , (5.23a) ␥͑ ͒ϭ ͵ ͑⍀ ͒␩͑ ͒ t ⍀ ds sin s s , (5.23b) ͑ ͒ϭ Ϫ ϩ M 0 D t a˙ 11 4a11b1 a˙ 12b1 /b3 Ϫ ˙ ͑ ϩ ͒ 1 t b2 2a11 a12b1 /b3 /b2 , (5.24a) D͑t͒ϭ ͵ ds cos͑⍀s͒␯͑s͒, (5.24b) ប 0 ͑ ͒ϭ Ϫ ˙ ͑ ͒Ϫ 2f t a˙ 12 /b3 b2a12 / b2b3 4a11 . (5.25a) 1 t The fact that the exact master equation, Eq. (5.21), is f͑t͒ϭϪ ͵ ds sin͑⍀s͒␩͑s͒. (5.25b) M⍀ local in time for an arbitrary spectrum of the environ- 0 ment is remarkable. This was demonstrated by Hu, Paz, These coefficients can be made even more explicit and Zhang (1992) following discussions carried out un- when a convenient specific model is adopted for the der more restrictive assumptions by Caldeira and Leg- spectral density: gett (1983); Haake and Reibold (1985); Grabert, ␻ ⌫2 Schramm, and Ingold (1988); and Unruh and Zurek ͑␻͒ϭ C 2M␥ 2 2 . (5.27) (1989). It is the linearity of the problem that allows one 0 ␲ ⌫ ϩ␻ to anticipate the (Gaussian) form of the propagator. ␥ Above, 0 characterizes the strength of the interaction, The above derivation of the exact master equation and ⌫ is the high-frequency cutoff. Then used the method of Paz (1994; see also Paz and Zurek, ␥ ⌫3 ⍀ 2001). Explicit formulas for the time-dependent coeffi- 2 0 Ϫ⌫ ⍀˜ 2ϭϪ ͫ1Ϫͩ cos ⍀tϪ sin ⍀tͪ e tͬ; cients can be obtained when one focuses on the pertur- ⌫2ϩ⍀2 ⌫ bative master equation. The formulas can be derived ab (5.22c) initio (see Paz and Zurek, 2001) but can also be obtained ␥ ⌫2 ⌫ from the above results by finding a perturbative solution ␥͑ ͒ϭ 0 ͫ Ϫͩ ⍀ Ϫ ⍀ ͪ Ϫ⌫tͬ t ⌫2ϩ⍀2 1 cos t ⍀ sin t e . to Eq. (5.17) and then substituting it in Eqs. (5.22a)– (5.25a). The resulting master equation in the operator (5.23c) form is Note that both of these coefficients are initially zero. i i␥͑t͒ They grow to their asymptotic values on a time scale set ␳˙ ϭϪ ͓ ϩ ⍀˜ ͑ ͒2 2 ␳ ͔Ϫ ͓ ͕ ␳ ͖͔ by the inverse of the cutoff frequency ⌫. S ប HS M t x /2, S ប x, p, S The two diffusion coefficients can also be studied, but f͑t͒ it is more convenient to evaluate them numerically. In Ϫ ͑ ͓͒ ͓ ␳ ͔͔Ϫ ͓ ͓ ␳ ͔͔ D t x, x, S ប x, p, S . (5.26) Fig. 5 we show their behavior. The normal diffusion coefficient quickly settles into its long-time asymptotic Coefficients such as the frequency renormalization ⍀˜ , value: the relaxation coefficient ␥(t), and the normal and Ϫ1 2 2 2 DϱϭM␥ ⍀ប coth͑ប⍀␤/2͒⌫ /͑⌫ ϩ⍀ ͒. (5.28) anomalous diffusion coefficients D(t) and f(t) are given 0 by The anomalous diffusion coefficient f(t) also approaches its asymptotic value. For high temperature it is 2 t ⌫ ⍀˜ 2͑t͒ϭϪ ͵ ds cos͑⍀s͒␩͑s͒, (5.22b) suppressed by a cutoff with respect to Dϱ , but the M 0 approach to fϱ is more gradual, algebraic rather than

Rev. Mod. Phys., Vol. 75, No. 3, July 2003 740 Wojciech Hubert Zurek: Decoherence, einselection, and the quantum origins of the classical

exponential. Environments with different spectral con- term of Heff . After evaluating x˙ and x˙ Ј in terms of p and tent exhibit different behavior (Hu, Paz, and Zhang, pЈ in the expression for Heff one obtains 1992; Paz, Habib, and Zurek, 1993; Paz, 1994; Anglin, ϭ͓ ˙ Ϫ␥͑ Ϫ Ј͔͒2 Ϫ͓ ЈϪ␥͑ Ϫ Ј͔͒2 Paz, and Zurek, 1997). Heff p x x /2M p x x /2M ϩ ͑ ͒Ϫ ͑ ͒Ϫ ␥ ͑ Ϫ ͒2 ប2 V x V xЈ i2M kBT xЈ x / . B. Decoherence in quantum Brownian motion (5.33) This expression yields the operator that generates the The coefficients of the master equation we have just evolution of the density matrix, Eq. (5.29). derived can be computed under a variety of different The coefficients of Eq. (5.21) approach their high- assumptions. The two obvious characteristics of the en- temperature values quickly (see Fig. 5). Already for T vironment that one can change are its temperature T well below what the rigorous derivation would demand, ␻ and its spectral density C( ). In the case of high tem- the high-temperature limit appears to be an excellent peratures, D(t) tends to a temperature-dependent con- approximation. The discrepancy is manifested by symp- stant and dominates over f(t). Indeed, in this case all of toms such as some of the diagonal terms of ␳S(xЈ,x) the coefficients settle to asymptotic values after an initial assuming negative values when the evolution starts from transient. Thus an initial state that is so sharply localized in position as i to have kinetic energy in excess of the values allowed by ␳͒ ץϪ ץ␳˙ ϭϪ ͓ ␳ ͔Ϫ␥͑ Ϫ Ј͒͑ S ប Hren , S x x x xЈ S the high-temperature approximation. However, this is limited to the initial instant of order 1/⌫, and is known to 2M␥k T be essentially unphysical for other reasons (Unruh and Ϫ B ͑ Ϫ Ј͒2␳ ប2 x x S . (5.29) Zurek, 1989; Ambegoakar, 1991; Anglin, Paz, and Zurek, 1997; Romero and Paz, 1997). This short-time ␳ This master equation for (x,xЈ) obtains in the unreal- anomaly is closely tied to the fact that Eq. (5.33) (and, istic but convenient limit known as the high-temperature indeed, many of the exact or approximate master equa- approximation, which is valid when kBT is much higher tions derived to date) does not have the Lindblad form than all the other relevant energy scales, including the (Kossakowski, 1973; Lindblad, 1976; see also Gorini, energy content of the initial state and the frequency cut- Kossakowski, and Sudarshan, 1976; Alicki and Lendi, ␻ off in C( ) (see Caldeira and Leggett, 1983). However, 1987) of a dynamical semigroup. when these restrictive conditions hold, Eq. (5.29) can be The high-temperature master equation (5.29) is a written for an arbitrary V(x). To see why, we give a good approximation in a wider range of circumstances derivation patterned on that of Hu, Paz, and Zhang than the one for which it was derived (Feynman and (1993). Vernon, 1963; Dekker, 1977; Caldeira and Leggett, ␳ We start with the propagator, Eq. (5.5), S(x,xЈ,t) 1983). Moreover, our key qualitative conclusion—rapid ϭ Ј ͉ Ј ␳ Ј J(x,x ,t x0 ,x0 ,t0) S(x0 ,x0 ,t0), which we shall treat decoherence in the macroscopic limit—does not cru- as if it were an equation for a state vector of the two- cially depend on the approximations leading to Eq. dimensional system with coordinates x,xЈ. The propaga- (5.29). We shall therefore use it in our further studies. tor is then given by the high-temperature version of Eq. (5.10): 1. Decoherence time scale ͑ Ј ͉ Ј ͒ ប J x,x ,t x0 ,x0 ,t0 In the macroscopic limit [that is, when is small compared to other quantities with dimensions of action, such i ͱ ͗ Ϫ Ј 2͘ ϭ ͵ DxDxЈ expͩ ͕I ͑x͒ϪI ͑xЈ͖͒ͪ as 2MkBT (x x ) in the last term] the high- ប R R temperature master equation is dominated by

Ϫ ␥͑͐t ͓ ˙ Ϫ ˙ ϩ ˙ Ϫ ˙ ͔ϩ͓ ប2͔͓ Ϫ ͔2͒ Ϫ ͒ 2 ϫ M ds xx xЈxЈ xxЈ xЈx 2kBT/ x xЈ ͑x xЈ ͒ ␳ ͑ Ј ͒ϭϪ␥ͭ ͮ ␳ ͑ Ј ץ . e 0 t S x,x ,t ␭ S x,x ,t . (5.34) (5.30) T The term in the exponent can be interpreted as the ef- Above, fective Lagrangian of a two-dimensional system: ប ␭ ϭ (5.35) L ͑x,xЈ͒ϭMx˙ 2/2ϪV ͑x͒ϪMx˙ Ј2/2ϩV ͑xЈ͒ T ͱ eff R R 2MkBT ϩ␥͑xϪxЈ͒͑x˙ ϩx˙ Ј͒ is the thermal de Broglie wavelength. Thus the density matrix loses off-diagonal terms in position representa- 2M␥k T ϩ B ͑ Ϫ Ј͒2 tion: i ប2 x x . (5.31) Ϫ␥t͑xϪxЈ/␭ ͒2 ␳S͑x,xЈ,t͒ϭ␳S͑x,xЈ,0͒e T , (5.36) One can readily obtain the corresponding Hamiltonian, while the diagonal (xϭxЈ) remains untouched. (x˙ ЈϪL . (5.32ץ/ Lץx˙ ϩx˙ Јץ/ Lץ ˙H ϭx eff eff eff eff Quantum coherence decays exponentially at a rate ϭ ϭ ϩ␥ Ϫ Conjugate momenta p px Mx˙ (x xЈ) and pЈ given by the relaxation rate times the square of the dis- ϭ ϭϪ ϩ␥ Ϫ pxЈ Mx˙ Ј (x xЈ) are used to express the kinetic tance, measured in units of thermal de Broglie wave-

Rev. Mod. Phys., Vol. 75, No. 3, July 2003 Wojciech Hubert Zurek: Decoherence, einselection, and the quantum origins of the classical 741

length (Zurek, 1984a). Position is the instantaneous ϩϱ 1 ͑ ប͒ y y pointer observable. If Eq. (5.36) was always valid, eigen- W͑x,p͒ϭ ͵ dye ipy/ ␳ͩ xϩ ,xϪ ͪ . (5.39) 2␲ប Ϫϱ 2 2 states of position would attain classical status. The importance of position can be traced to the na- The evolution equation followed by the Wigner function ture of the interaction Hamiltonian between the system obtains through the Wigner transform of the corre- and the environment. According to Eq. (5.3) sponding master equation. In the high-temperature limit, Eq. (5.29) (valid for general potentials) yields ϭ ץ ϩ͒ ͑ ץ␥ ϭ ϩ ץ (HSE x͚ cnqn . (5.37 n tW ͕Hren ,W͖MB 2 p pW D ppW. (5.40)

This form of HSE is motivated by physics (Zurek, 1982, The first term, the Moyal bracket, is the Wigner trans- 1991). Interactions depend on the distance. However, form of the von Neumann equation (see Sec. III). In the had we endeavored to find a situation in which a differ- linear case it reduces to the Poisson bracket. The second ent form of the interaction Hamiltonian—say, a term is responsible for relaxation. The last diffusive term momentum-dependent interaction—was justified, the is responsible for decoherence. form and consequently the predictions of the master Diffusion in momentum occurs at the rate set by D equation would have been analogous to Eq. (5.36), but ϭ ␥ 2M kBT. Its origin can be traced to the continuous with a substitution of the relevant observable monitored measurement of the position of the system by the envi- by the environment for x. Such situations may be experi- ronment. In accord with Heisenberg indeterminacy, mentally accessible (Poyatos, Cirac, and Zoller, 1996), measurement of the position results in an increase of the providing a test of one of the key ideas of einselection: uncertainty in the momentum (see Sec. IV). the relation between the form of interaction and the pre- Decoherence in phase space can be explained through ferred basis. the example of a superposition of two Gaussian wave The effect of the evolution, Eqs. (5.34)–(5.36), on the packets. The Wigner function in this case is given by density matrix in the position representation is easy to ͑ ͒ϭ ͑ ϩ ͒ϩ ͑ Ϫ ͒ϩ͑␲ប͒Ϫ1 envisage. Consider a superposition of two minimum- W x,p G x x0 ,p G x x0 ,p uncertainty Gaussians. Off-diagonal peaks represent co- 2 2 2 2 2 ␶ ϫexp͑Ϫp ␰ /ប Ϫx /␰ ͒cos͑⌬xp/ប͒, herence. They decay on a decoherence time scale D ,or with a decoherence rate (Zurek, 1984a, 1991) (5.41) Ϫ 2 Ϫ x xЈ where ␶ 1ϭ␥ͩ ͪ . (5.38) D ␭ Ϫ͑ ϯ ͒2 ␰2Ϫ͑ Ϫ ͒2␰2 ប2 T e x x0 / p p0 / ͑ Ϯ Ϫ ͒ϭ ␭ G x x0 ,p p0 . (5.42) The thermal de Broglie wavelength T is microscopic for ␲ប massive bodies and for the environment at reasonable temperatures. For a mass of1gatroom temperature We have assumed that the Gaussians are not moving ϭ and for the separation xЈϪxϭ1 cm, Eq. (5.38) predicts a (p0 0). decoherence rate approximately 1040 times faster than The oscillatory term in Eq. (5.41) is the signature of relaxation. Even the cosmic microwave background suf- superposition. The frequency of the oscillations is pro- fices to cause rapid loss of quantum coherence in objects portional to the distance between the peaks. When the as small as dust grains (Joos and Zeh, 1985). These esti- separation is only in position x, this frequency is ϭ⌬ បϭ ប mates for the rates of decoherence and relaxation f x/ 2x0 / . (5.43) should be taken with a grain of salt. Often the assumptions that have led to the simple high-temperature mas- Ridges and valleys of the interference pattern are parallel to the separation between the two peaks. This, and ter equation, Eq. (5.29), are not valid (Gallis and Flem- ប ing, 1990; Gallis, 1992; Anglin, Paz, and Zurek, 1997). the fact that appears in the interference term in W,is For example, the decoherence rate cannot be faster than important for the phase-space derivation of the decoher- the inverse of the spectral cutoff in Eq. (5.27), nor than ence time. We focus on the dominant effect and direct the rate with which the superposition is created. More- our attention to the last term of Eq. (5.40). Its effect on over, for large separations the quadratic dependence of a rapidly oscillating interference term will be very differ- the decoherence rate may saturate (Gallis and Fleming, ent from its effect on the two Gaussians. The interfer- 1990; Anglin, Paz, and Zurek, 1997), as seen in the simu- ence term is dominated by the cosine: lated decoherence experiments of Cheng and Raymer ⌬x ϳ ͩ ͪ (1999). Nevertheless, in the macroscopic domain deco- Wint cos ប p . (5.44) herence of widely delocalized Schro¨ dinger-cat states will occur very much faster than relaxation, which proceeds This is an eigenfunction of the diffusion operator. The at the rate given by ␥. decoherence time scale emerges (Zurek, 1991) from the corresponding eigenvalue ˙ ϷϪ ⌬ 2 ប2 ϫ 2. Phase-space view of decoherence Wint ͕D x / ͖ Wint . (5.45) ␶ A useful alternative way of illustrating decoherence is We have recovered the formula for D , Eq. (5.38), from afforded by the Wigner function representation a different-looking argument. Equation (5.40) has no ex-

Rev. Mod. Phys., Vol. 75, No. 3, July 2003 742 Wojciech Hubert Zurek: Decoherence, einselection, and the quantum origins of the classical

FIG. 6. Evolution of the Wigner function of a decohering harmonic oscillator. Note the difference between the rate at which the interference term disappears for the initial superposition of two minimal uncertainty Gaussians in position and in momentum.

plicit dependence on ប for linear potentials (in the non- perposition of momenta leads to a superposition of linear case ប enters through the Moyal bracket). Yet the positions, and hence to decoherence, albeit on a dynami- ប ប ␶ decoherence time scale contains explicitly. enters cal (rather than D) time scale. through Eq. (5.43), that is, through its role in determin- An intriguing example of a long-lived superposition of ing the frequency of the interference pattern Wint . two seemingly distant Gaussians was pointed out by The evolution of a pure initial state of the type con- Braun, Braun, and Haake (2000) in the context of super- sidered here is shown in Fig. 6. There we illustrate the radiance. As they note, the relevant decohering interac- evolution of the Wigner function for two initial pure tion cannot distinguish between some such superposi- states: superposition of two positions and superposition tions, leading to a Schro¨ dinger-cat pointer subspace. of two momenta. There is a noticeable difference between these two cases in the rate at which the interference term disappears. This was anticipated. The interaction in Eq. (5.3) is a function of x. Therefore x is C. Predictability sieve in phase space monitored by the environment directly, and the superposition of positions decoheres almost instantly. By Decoherence rapidly destroys nonlocal superposi- contrast, the superposition of momenta is initially in- tions. Obviously, states that survive must be localized. sensitive to monitoring by the environment—the corre- However, they cannot be localized to a point in x, since sponding initial state is already well localized in the ob- this would imply—by Heisenberg’s indeterminacy—an servable singled out by the interaction. However, a su- inﬁnite range of momenta and hence of velocities. As a

Rev. Mod. Phys., Vol. 75, No. 3, July 2003 Wojciech Hubert Zurek: Decoherence, einselection, and the quantum origins of the classical 743

result, a wave function localized too well at one instant would become very nonlocal a moment later. Einselected pointer states minimize the damage done by decoherence over the time scale of interest (usually associated with predictability or with dynamics). They can be found through the application of a predictability sieve as outlined at the end of Sec. IV. To implement it, we compute entropy increase or purity loss for all initially pure states in the Hilbert space of the system under the cumulative evolution caused by the self- Hamiltonian and by the interaction with the environment. It would be a tall order to carry out the requisite calculations for an arbitrary quantum system interacting with a general environment. We focus on an FIG. 7. Predictability sieve in action. The plot shows purity exactly solvable case. Tr␳2 for mixtures that have evolved from initial minimum- In the high-temperature limit the master equations uncertainty wave packets with different squeeze parameters s (5.26) and (5.29) can be expressed in the operator form in an underdamped harmonic oscillator with ␥/␻ϭ10Ϫ4. Co- ␥ ␩ herent states, which have the same spread in position as in 1 kBT ␳˙ ϭ ͓H ,␳͔ϩ ͓͕p,x͖,␳͔Ϫ ͓x,͓x,␳͔͔ momentum, sϭ1, are clearly most predictable. iប ren iប ប2 i␥ Ϫ ͓͑ ␳ ͔Ϫ͓ ␳ ͔͒ ប x, p p, x . (5.46) cillation period. For a harmonic oscillator with mass M and frequency ⍀, one can compute the purity loss aver- Above, ␩ϭ2M␥ is the viscosity. Only the last two terms aged over ␶ϭ2␲/⍀: can change entropy. Terms of the form ˆ ⌬␵͉2␲/⍀ϭϪ ͓⌬ 2ϩ⌬ 2 ͑ ⍀͒2͔ ␳˙ ϭ͓O,␳͔, (5.47) 0 2D x p / M . (5.51) where Oˆ is the Hermitian, leave the purity ␵ϭTr␳2 and the von Neumann entropy HϭϪTr␳ ln ␳ unaffected. Above, ⌬x and ⌬p are dispersions of the state at the This follows from the cyclic property of the trace: initial time. By Heisenberg indeterminacy, ⌬x⌬pуប/2. The loss of purity will be smallest when d N Tr␳Nϭ ͚ ͑Tr␳kϪ1͓Oˆ ,␳͔␳NϪk͒ϭ0. (5.48) dt ϭ k 1 ⌬x2ϭប/2M⍀, ⌬p2ϭបM⍀/2. (5.52) Constancy of Tr␳2 is obvious, while for Tr␳ ln ␳ it follows ␳ when the logarithm is expanded in powers of . Coherent quantum states are selected by the predictabil- Equation (5.46) leads to the loss of purity at the rate ity sieve in an underdamped harmonic oscillator (Zurek, (Zurek, 1993a) 1993a; Zurek, Habib, and Paz, 1993; Tegmark and Sha- ␩ d 4 kBT piro, 1994; Gallis, 1996; Wiseman and Vaccaro, 1998; Tr␳2ϭϪ Tr͓␳2x2Ϫ͑␳x͒2͔ϩ2␥Tr␳2. (5.49) dt ប2 Paraoanu, 1999, 2002). Rotation induced by the self- Hamiltonian turns preference for states localized in po- The second term increases purity—or decreases sition into preference for localization in phase space. entropy—as the system is damped from an initial highly This is illustrated in Fig. 7. mixed state. For the predictability sieve this term is usu- We conclude that for an underdamped harmonic os- ally unimportant, since for a vast majority of initially cillator coherent Gaussians are the best quantum theory pure states its effect will be negligible when compared to has to offer as an approximation to a classical point. the first decoherence-related term. Thus, in the case of Similar localization in phase space should be obtained in pure initial states, the reversible classical limit in which the familiar symp- ␩ toms of the openness of the system, such as the finite d 4 kBT Tr␳2ϭϪ ͑͗x2͘Ϫ͗x͘2͒. (5.50) relaxation rate ␥ϭ␩/2M, become vanishingly small. dt ប2 This limit can be attained for large mass M→ϱ, while Therefore the instantaneous loss of purity is minimized the viscosity ␩ remains fixed and sufficiently large to for perfectly localized states (Zurek, 1993a). The second assure localization (Zurek, 1991, 1993a). This is, of term of Eq. (5.49) allows for equilibrium. Nevertheless, course, not the only possible situation. Haake and Walls early on, and for very localized states, its presence (1987) discussed the overdamped case, in which pointer causes an (unphysical) increase of purity to above unity. states are still localized, but become relatively narrower This is a well-known artifact of the high-temperature in position. On the other hand, an ‘‘adiabatic’’ environ- approximation [see discussion following Eq. (5.33)]. ment with high-frequency cutoff large compared to the To find the most predictable states relevant for dy- level spacing in the system enforces einselection in en- namics, we consider the increase in entropy over an os- ergy eigenstates (Paz and Zurek, 1999).

Rev. Mod. Phys., Vol. 75, No. 3, July 2003 744 Wojciech Hubert Zurek: Decoherence, einselection, and the quantum origins of the classical

D. Classical limit in phase space point was no more than (2␲ប)d in the d-dimensional configuration space. In the presence of decoherence There are three strategies that allow one to simulta- arising from a coordinate-dependent interaction, evolu- neously recover the classical phase-space structure and tion of a general quantum superposition should be, after the classical equations of motion from quantum dynam- a few decoherence times, well approximated by a prob- ics and decoherence. ability distribution over such Gaussian points. 1. Mathematical approach (ប→0) 2. Physical approach: The macroscopic limit This mathematical classical limit could not be implemented without decoherence, since the oscillatory terms The possibility of the ប→0 classical limit in the pres- associated with interference do not have an analytic ence of decoherence is of interest. But បϭ1.054 59 Ϫ ប→0 limit (see, for example, Peres, 1993). However, in ϫ10 27 erg s. Therefore a physically more reasonable the presence of the environment, the relevant terms in approach increases the size of the object, and, hence, its the master equations increase as O(បϪ2) and make the susceptibility to decoherence. This strategy can be nonanalytic manifestations of interference disappear. implemented starting with Eq. (5.40). Reversible dy- ␥→ ϭ ␥ ϭ␩ Thus phase-space distributions can always be repre- namics obtains as 0 while D 2M kBT kBT in- sented by localized coherent state points, or by distribu- creases. tions over the basis consisting of such points. The decrease of ␥ and the simultaneous increase of ␩ This strategy is easiest to implement starting from the kBT can be anticipated with the increase of the size phase-space formulation. It follows from Eq. (5.45) that and mass. Assume that the density of the object is inde- the interference term in Eq. (5.41) will decay (Paz, pendent of its size R, and that the environment quanta Habib, and Zurek, 1993) over the time interval ⌬t as scatter from its surface (as would photons or air mol- ecules). Then MϳR3 and ␩ϳR2. Hence D⌬x2 ⌬x ϳ ͩ Ϫ⌬ ͪ ͩ ͪ 2 Wint exp t ប2 cos ប p . (5.53) ␩ϳO͑R ͒→ϱ, (5.56) As long as ⌬t is large compared to the decoherence time ␥ϭ␩/2MϳO͑1/R͒→0, (5.57) ␶ Ӎប2 ⌬ 2 scale D /D x , oscillatory contributions to the as R→ϱ. Localization in phase space and reversibility ប→ Wigner function W(x,p) should disappear as 0. Si- can be simultaneously achieved in a macroscopic limit. multaneously, Gaussians representing likely locations of The existence of a macroscopic classical limit in ␦ the system become narrower, approaching Dirac func- simple cases was pointed out some time ago (Zurek, tions in phase space. For instance, in Eq. (5.42), 1984a, 1991; Gell-Mann and Hartle, 1993). We shall ana- ͑ Ϫ Ϫ ͒ϭ␦͑ Ϫ Ϫ ͒ lim G x x0 ,p p0 x x0 ,p p0 , (5.54) lyze it in the next section in a more complicated chaotic ប→0 setting, where reversibility can no longer be taken for granted. In the harmonic-oscillator case, approximate providing that half-widths of the coherent states in x and reversibility is effectively guaranteed, since the action p decrease to zero as ប→0. This would be assured when, associated with the 1-␴ contour of the Gaussian state for instance, in Eqs. (5.41) and (5.42), increases with time at the rate (Zurek, Habib, and Paz, ␰2ϳប. (5.55) 1993) Thus individual coherent-state Gaussians approach k T ˙ϭ␥ B phase-space points. This behavior indicates that in a I ប⍀ . (5.58) macroscopic open system nothing but probability distributions over localized phase-space points can survive in Action I is a measure of the lack of information about the ប→0 limit for any time of dynamical or predictive phase-space location. Hence its rate of increase is a mea- significance. [Coherence between immediately adjacent sure of the rate of predictability loss. The trajectory is a points separated only by ϳ␰, Eq. (5.55), can last longer. limit of the ‘‘tube’’ swept through phase space by the This is no threat to the classical limit. Small-scale coher- moving contour representing the instantaneous uncer- ence is a part of a quantum halo of the classical pointer tainty of the observer about the state of the system. Evo- states (Anglin and Zurek, 1996).] lution is approximately deterministic when the area of The mathematical classical limit implemented by let- this contour is nearly constant. In accord with Eqs. ting ប→0 becomes possible in the presence of decoher- (5.56) and (5.57) I˙ tends to zero in the reversible mac- ence. It is tempting to carry this strategy to its logical roscopic limit: conclusion and represent every probability density in I˙ϳO͑1/R͒. (5.59) phase space in the point(er) basis of narrowing coherent states. Such a program is beyond the scope of this re- The existence of an approximately reversible trajectory- view, but the reader should by now be convinced that it like thin tube provides an assurance that, having local- is possible. Indeed, Perelomov (1986) showed that a ized the system within a regular phase-space volume at general quantum state could be represented in a sparse tϭ0, we can expect to find it later inside the Liouville- basis of coherent states that occupy the sites of a regular transported contour of nearly the same measure. Similar lattice, providing that the volume per coherent state conclusions follow for integrable systems.

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3. Ignorance inspires confidence in classicality ␹ϭͱ nent, while Vx /Vxxx characterizes the dominant ⌬ Dynamical reversibility can be achieved with einselec- scale of nonlinearities in the potential V(x), and p gives the coherence scale in the initial wave packet. The tion in the macroscopic limit. Moreover, I˙/I or other above estimate, Eq. (3.5), depends on the initial condi- measures of predictability loss decrease with the in- tions. It is smaller than, but typically close to, tr crease of I. This is especially dramatic when quantified Ϫ ϭ⌳ 1 ln I/ប, Eq. (3.6), where I is the characteristic ac- in terms of the von Neumann entropy, that, for Gaussian tion of the system. By contrast, phase-space patches of states, increases at the rate (Zurek, Habib, and Paz, regular systems undergo stretching with a power of time. 1993) Consequently, loss of correspondence occurs only over a ϩ ϳ ប ␣ I 1 much longer tr (I/ ) , which depends polynomially on H˙ ϭI˙ ln . (5.60a) ប IϪ1 . The resulting H˙ is infinite for pure coherent states (I ϭ1), but quickly decreases with increasing I. Similarly, 1. Restoration of correspondence the rate of purity loss for Gaussians is Exponential instability spreads the wave packet to a ␵˙ ϭI˙/I2. (5.60b) paradoxical extent at the rate given by the positive Lyapunov exponents ⌳(i) . Einselection attempts to en- Again, it tapers off for more mixed states. ϩ force localization in phase space by tapering off interfer- This behavior is reassuring. It leads us to conclude ence terms at a rate given by the inverse of the decoher- that irreversibility quantified through, say, von Neumann ␶ ϭ␥Ϫ1 ␭ ⌬ 2 ˙ ence time scale, D ( T / x) . The two processes entropy production, Eq. (5.60a), will approach H reach status quo when the coherence length ᐉ of the ˙ c Ϸ2I/I, vanishing in the limit of large I. When, in the wave packet makes their rates comparable, that is, spirit of the macroscopic limit, we do not insist on the ␶ ⌳ Ӎ maximal resolution allowed by quantum indeterminacy, D ϩ 1. (5.61) the subsequent predictability losses measured by the in- This yields an equation for the steady-state coherence crease of entropy or through the loss of purity will di- length and for the corresponding momentum dispersion: minish. Illusions of reversibility, determinism, and exact ᐉ Ӎ␭ ͱ⌳ ␥ classical predictability become easier to maintain in the c T ϩ/2 ; (5.62) presence of ignorance about the initial state! ␴ ϭប/ᐉ ϭͱ2D/⌳ϩ. (5.63) To think about phase-space points one may not even c c need to invoke a specific quantum state. Rather, a point Above, we have quoted results (Zurek and Paz, 1994) can be regarded as the limit of an abstract recursive pro- that follow from a more rigorous derivation of the co- ᐉ cedure in which the phase-space coordinates of the sys- herence length c than the rough and ready approach tem are determined better and better in a succession of that led to Eq. (5.61). They embody the same physical increasingly accurate measurements. One may be argument, but seek asymptotic behavior of the Wigner tempted to extrapolate this limiting process ad infinit- function that evolves according to the equation essimum, which would lead beyond Heisenberg’s inde- ប2n͑Ϫ͒n 2nϩ1ץ 2nϩ1ץ terminacy principle and to a false conclusion that ideal- ˙ ϭ͕ ͖ϩ W H,W ͚ 2n͑ ϩ ͒ x V p W ized points and trajectories exist objectively, and that the nу1 2 2n 1 ! (W. (5.64 2ץinsider view of Sec. II can always be justified. While in ϩD our quantum universe this conclusion is wrong, and the p extrapolation described above illegal, the presence, The classical Liouville evolution generated by the Pois- within Hilbert space, of localized wave packets near the son bracket ceases to be a good approximation of the minimum-uncertainty end of such imagined sequences decohering quantum evolution when the leading quan- of measurements is reassuring. Ultimately, the ability to tum correction becomes comparable to the classical represent motion in terms of points and their time- force: ordered sequences (trajectories) is the essence of classi- ប2 ប2 V W cal mechanics. Ϸ x p VxxxWppp ␹2 ␴2 . (5.65) 24 24 c

The term VxWp represents the classical force in the E. Decoherence, chaos, and the second law Poisson bracket. Quantum corrections are small when ␴ ␹ӷប. (5.66) The breakdown of correspondence in this chaotic set- c ting was described in Sec. III. It is anticipated to occur in Equivalently, the Moyal bracket generates approxi- all nonlinear systems, since stretching of the wave mately Liouville ﬂow when the coherence length satis- packet by the dynamics is a generic feature, absent only ﬁes from a harmonic oscillator. However, the exponential in- ᐉ Ӷ␹. (5.67) stability of chaotic dynamics implies rapid loss of c quantum-classical correspondence after the Ehrenfest This last inequality has an obvious interpretation: it is a ϭ⌳Ϫ1 ␹⌬ ប ⌳ ᐉ time, tប ln p/ . Here is the Lyapunov expo- condition for localization to within a region c that is

Rev. Mod. Phys., Vol. 75, No. 3, July 2003 746 Wojciech Hubert Zurek: Decoherence, einselection, and the quantum origins of the classical

FIG. 8. Snapshots of a chaotic system with a double-well potential: Hϭp2/2mϩAx4ϪBx2ϩCx cos(ft). In the example discussed here mϭ1, Aϭ0.5, Bϭ10, fϭ6.07 and Cϭ10, yielding the Lyapunov exponent ⌳Ϸ0.45 (see Habib, Shizume, and Zurek, 1998). All ﬁgures were obtained after approximately eight periods of the driving force. The evolution started from the same minimum- uncertainty Gaussian and proceeded according to (a) the quantum Moyal bracket, (b) the Poisson bracket, and (c) the Moyal bracket with decoherence [constant Dϭ0.025 in Eq. (5.64)]. In the quantum cases បϭ0.1, which corresponds to the area of the rectangle in the image of the Wigner function above. Interference fringes are clearly visible in (a), and the Wigner function shown there is only vaguely reminiscent of the classical probability distribution in (b). Even modest decoherence ͓Dϭ0.25 used to get (c) ᐉ ϭ corresponds to coherence length c 0.3] dramatically improves the correspondence between the quantum and the classical. The remaining interference fringes appear on relatively large scales, which implies small-scale quantum coherence (Color).

␹ (i) (i) (i) small compared to the scale of the nonlinearities of the Lyapunov exponents ⌳ϩ , with L ϳexp͓⌳ϩ t͔, the potential. When this condition holds, classical force will squeezing mandated by the Liouville theorem in the dominate over quantum corrections. complementary directions corresponding to ⌳(i) will halt Restoration of correspondence is illustrated in Fig. 8 Ϫ ␴(i) where Wigner functions are compared with classical at c , Eq. (5.63). In this limit, the number of pure probability distributions in a chaotic system. The differ- states needed to represent the resulting mixture in- ence between the classical and quantum expectation val- creases exponentially: ues in the same chaotic system is shown in Fig. 9. Even ͑i͒Ӎ ͑i͒ ᐉ͑i͒ relatively weak decoherence suppresses the discrepancy, N L / c (5.68) ϭ helping reestablish the correspondence: D 0.025 trans- in each dimension. The least number of pure states over- lates through Eq. (5.62) into coherence over ᐉ Ӎ0.3, not c lapped by W will then be Nϭ⌸ N(i). This implies much smaller than the nonlinearity scale ␹Ӎ1 for the i investigated Hamiltonian of Fig. 8. NӍ ⌳͑i͒ ץӍ ˙ H t ln ͚ ϩ . (5.69) 2. Entropy production i Irreversibility is the price for the restoration of This estimate for the entropy production rate be- quantum-classical correspondence in chaotic dynamics. comes accurate as the width of the Wigner function ␴(i) It can be quantiﬁed through the entropy production reaches saturation at c . When a patch in phase space rate. The simplest argument recognizes that decoher- corresponding to the initial W is regular and smooth on ᐉ ␴(i) ence restricts spatial coherence to c . Consequently, as scales large compared to c , evolution will start nearly the exponential instability stretches the size L(i) of the reversibly (Zurek and Paz, 1994). However, as squeezing distribution in directions corresponding to the positive brings the extent of the effective support of W close to

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FIG. 8. (Continued.)

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toring a decohering chaotic system, recording its state at time intervals small compared to ⌳Ϫ1, but large compared to the decoherence time scale. One can show (Zurek, 1998b) that the average increase in the size of the algorithmically compressed records of measurement of a decohering chaotic system (that is, the algorithmic randomness of the acquired data; see, for example, Cover and Thomas, 1991) is given by Eq. (5.69). This conclusion holds, providing that the effect of the col- lapses of the wave packet caused by the repeated measurements is negligible, i.e., the observer is ‘‘skillful.’’ A possible strategy that the skillful observer may adopt is that of indirect measurements, of monitoring a fraction of the environment responsible for decoherence to determine the state of the system. As we shall see in more detail in the following sections of the paper, this is a very natural strategy, often employed by observers. A classical analog of Eq. (5.69) was obtained by Kol- FIG. 9. Classical and quantum expectation values of position ͗x͘ as a function of time for an example of Fig. 8. Evolution mogorov (1960) and Sinai (1960) starting from very dif- started from a minimum-uncertainty Gaussian. Noticeable dis- ferent, mathematical arguments that in effect relied on crepancies between the quantum and classical averages appear an arbitrary but fixed coarse graining imposed on phase on a time scale consistent with the Ehrenfest time tប . Deco- space (see Wehrl, 1978). Decoherence leads to a similar- herence, even in modest doses, dramatically decreases differ- looking quantum result in a very different fashion: Ef- ences between the expectation values (Color). fective ‘‘coarse graining’’ is imposed by coupling to the environment, but only in the sense implied by einselec- ␴(i) tion. Its graininess (resolution) is set by the accuracy of c , diffusion bounds from below the size of the smallest features of W. Stretching in the unstable directions the monitoring environment. This is especially obvious continues unabated. As a consequence, the volume of when the indirect monitoring strategy mentioned imme- the support of W will grow exponentially, resulting in an diately above is adopted by the observers. Preferred entropy production rate set by Eq. (5.69), that is, by the states will be partly localized in x and p, but (in contrast sum of the classical Lyapunov exponents. Yet, it has an to the harmonic-oscillator case with its coherent states) obviously quantum origin in decoherence. This quantum details of this environment-imposed coarse graining will origin may be apparent initially, since the rate of Eq. likely depend on phase-space location, the precise na- (5.69) will be approached from above when the initial ture of the coupling to the environment, etc. Yet, in the state is nonlocal. On the other hand, in a multidimen- appropriate limit, Eqs. (5.66) and (5.67), the asymptotic sional system different Lyapunov exponents may begin entropy production rate defined with the help of the al- to contribute to entropy production at different instants gorithmic contribution discussed above [i.e., in the man- [since the saturation condition, Eq. (5.61), may not be ner of physical entropy, that is, the sum of the measure (i) met simultaneously for all ⌳ϩ ]. Then the entropy proof ignorance given by the von Newmann entropy and duction rate can accelerate, before subsiding as a conse- the algorithmic randomness of the records, Zurek quence of approaching equilibrium. (1989)] does not depend on the strength or nature of the The time scales on which this estimate of entropy pro- coupling, but is instead given by the sum of the positive duction applies are still subject to investigation (Zurek Lyapunov exponents. and Paz, 1995a; Zurek, 1998b; Monteoliva and Paz, von Neumann entropy production consistent with the 2000) and even controversy (Casati and Chirikov, 1995b; above discussion has now been seen in numerical studies Zurek and Paz, 1995b). The instant when Eq. (5.69) be- of decohering chaotic systems (Shiokawa and Hu, 1995; comes a good approximation corresponds to the mo- Furuya, Nemes, and Pellegrino, 1998; Schack, 1998; ment when the exponentially unstable evolution forces Miller and Sarkar, 1999; Monteoliva and Paz, 2000). Ex- the Wigner function to develop small phase-space struc- tensions to situations in which relaxation matters, as well tures on the scale of the effective decoherence-imposed as in the opposite direction to where decoherence is coarse graining, Eq. (5.63). Equation (5.69) will be a relatively gentle have also been discussed (Brun, Per- good approximation until the time tEQ at which equilib- cival, and Schack, 1996; Miller, Sarkar, and Zarum, 1998; rium sets in. Both tប and tEQ have a logarithmic depen- Pattanayak, 2000). A related development is the experi- dence on the corresponding (initial and equilibrium) mental study of the Loschmidt echo using NMR tech- phase-space volumes I0 and IEQ , so the validity of Eq. niques (Levstein, Usaj, and Pastawski, 1998; Levstein Ϫ Ӎ⌳Ϫ1 (5.69) will be limited to tEQ tប ln IEQ /I0 . et al., 2000; Jalabert and Pastawski, 2001), which sheds There is a simple and conceptually appealing way to new light on the irreversibility in decohering complex extend the interval over which entropy is produced at dynamical systems. We shall return briefly to this subject the rate given by Eq. (5.69). Imagine an observer moni- in Sec. VIII.

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3. Quantum predictability horizon state of an isolated quantum system is without prior in- The cross section I of the trajectorylike tube contain- formation about the observables used to prepare it. ing the state of the harmonic oscillator in phase space Measurements of observables that do not commute with increases only slowly, Eq. (5.58), at a rate which—once this original set will inevitably create a different state. the limiting Gaussian is reached—does not depend on I. The continuous monitoring of the einselected observ- By contrast, in chaotic quantum systems this rate is ables by the environment allows pointer states to exist in much the same way as do classical states. This ontological role of the einselected quantum states can be justi- ˙Ӎ ⌳͑i͒ I I͚ ϩ . (5.70) fied operationally, by showing that in the presence of i einselection one can find out what the quantum state is, A fixed rate of entropy production implies an exponen- without inevitably re-preparing it by the measurement. tial increase of the cross section of the tube of, say, the Thus einselected quantum states are no longer just epis- ␴ 1- contour containing points consistent with the initial temological. In a system monitored by the environment, conditions. Phase-space support expands exponentially. what is the einselected states coincides with what is This quantum view of chaotic evolution can be com- known to be—what is recorded by the environment pared with the classical deterministic chaos. In both (Zurek, 1993a, 1993b, 1998a). cases, in the appropriate classical limit, which may in- The conflict between the quantum and the classical volve either mathematical ប→0, or a macroscopic limit, was originally noted and discussed almost exclusively in the future state of the system can, in principle, be pre- the context of quantum measurements.5 Here I shall dicted to a set accuracy for an arbitrarily long time. consider measurements, and, more to the point, acquisi- However, such predictability can be accomplished only when the initial conditions are given with the resolution tion of information in quantum theory from the point of that increases exponentially with the time interval over view of decoherence and einselection. which the predictions are to be valid. Given the fixed A. Objective existence of einselected states value of ប, there is therefore a quantum predictability horizon after which the Wigner function of the system To demonstrate the objective existence of einselected starting from an initial minimum-uncertainty Gaussian states we now develop an operational definition of exis- becomes stretched to a size of the order of the charac- tence and show how, in an open system, one can find out teristic dimensions of the system (Zurek, 1998b). The what the state was and is, rather than just prepare it. ability to predict the location of the system in phase This point has been made before (Zurek, 1993a, 1998a), ϳ space is then lost after t tប , Eq. (3.5), regardless of but this is the first time I shall discuss it in more detail. whether evolution is generated by the Poisson or Moyal The objective existence of states can be defined op- bracket or, indeed, whether the system is closed or open. erationally by considering two observers. The first of The case of regular systems is closer to that of a har- them is the record keeper R. He prepares the states with monic oscillator. The rate at which the cross section of the original measurement and will use his records to de- the phase-space trajectory tube increases, consistent termine if they were disturbed by measurements carried with the initial patch in phase space, will asymptote to out by other observers, e.g., the spy S. The goal of S is to ˙Ӎ I const. discover the state of the system without perturbing it. H˙ ϭI˙/Iϳ1/t. (5.71) When an observer can consistently determine the state of a system without changing it, that state, by our opera- Thus initial conditions allow one to predict the future of tional definition, will be said to exist objectively. a regular system for time intervals that are exponentially In the absence of einselection the situation of S is longer than those in the chaotic case. The rate of en- hopeless: R prepares states by measuring sets of com- tropy production of an open quantum system is there- muting observables. Unless S picks, by sheer luck, the fore a very good indicator of its dynamics, as was con- same observables in the case of each state, his measure- jectured some time ago (Zurek and Paz, 1995a), and as ments will re-prepare the states of the system. Thus, seems borne out in the numerical simulations (Shiokawa when R remeasures using the original observables, he and Hu, 1995; Miller, Sarkar, and Zarum, 1998; Miller will likely find answers different from his records of pre- and Sarkar, 1999; Monteoliva and Paz, 2000). paratory measurements. The spy S will ‘‘get caught’’ because it is impossible to find out an initially unknown state of an isolated quantum system. VI. EINSELECTION AND MEASUREMENTS

It is often said that quantum states play only an epis- 5See, for example, Bohr, 1928; Mott, 1929; von Neumann, temological role, describing the observer’s knowledge 1932; Dirac, 1947; Zeh, 1971, 1973, 1993; d’Espagnat, 1976, about past measurement outcomes that have prepared 1995; Zurek, 1981, 1982, 1983, 1991, 1993a 1993b, 1998a; the system (Jammer, 1974; d’Espagnat, 1976, 1995; Fuchs Omne`s, 1992, 1994; Elby, 1993, 1998; Donald, 1995; Butterﬁeld, and Peres, 2000). In particular—and this is a key argu- 1996; Giulini et al., 1996; Bub, 1997; Bacciagaluppi and ment against their objective existence (against their on- Hemmo, 1998; Healey, 1998; Healey and Hellman, 1998; Saun- tological status)—it is impossible to determine what the ders, 1998.

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In the presence of environmental monitoring the na- eigenspaces Pi can be expressed (Bogolubov et al., 1990) ture of the game between R and S is dramatically al- through Eq. (6.1). Einselection attains this, guaranteeing tered. Now it is no longer possible for R to prepare an the diagonality of density matrices in the projectors Pi arbitrary pure state that will persist or predictably corresponding to pointer states. These are sometimes evolve without losing purity. Only einselected states that called decoherence-free subspaces when they are degen- are already monitored by the environment, that are se- erate (compare also the non-Abelian case of noiseless lected by the predictability sieve, will survive. By the subsystems discussed in quantum computation; see Za- same token, S is no longer clueless about the observ- nardi and Rasetti, 1997; Duan and Guo, 1998; Zanardi, ables he can measure without leaving incriminating evi- 1998, 2001; Lidar, Bacon, and Whaley, 1999; Blanchard dence. For example, he can independently prepare and and Olkiewicz, 2000; Knill, Laflamme, and Viola, 2000). test the survival of various states in the same environment to establish which states are einselected, and then measure appropriate pointer observables. Better yet, S B. Measurements and memories can forgo direct measurements of the system and gather information indirectly, by monitoring the environment. The memory of a measuring device or of an observer This last strategy may seem contrived, but indirect can be modeled as an open quantum apparatus A, inter- measurements—acquisition of information about the acting with S through a Hamiltonian explicitly propor- system by examining fragments of the environment that tional to the measured observable6 sˆ: ץ have interacted with it—is in fact more or less the only strategy employed by observers. Our eyes, for example, ϭϪ ˆ ϳ Hint gsˆB sˆ . (6.2) Âץ intercept only a small fraction of the photons that scatter from various objects. The rest of the photons constitute von Neumann (1932) considered an apparatus isolated the environment, which retains at least as complete a from the environment. At the instant of the interaction record of the same einselected observables as we can between the apparatus and the measured system this is a obtain (Zurek, 1993a, 1998a). E convenient assumption. For us it suffices to assume that, The environment acts as a persistent observer, domi- at that instant, the interaction Hamiltonian between the nating the game with frequent questions, always about system and the apparatus dominates. This can be accom- the same observables, compelling both R and S to focus E plished by taking the coupling g in Eq. (6.2) to be g(t) on the einselected states. Moreover, can be persuaded ϳ␦(tϪt ). Premeasurement happens at t : to share its records of the system. This accessibility of 0 0 the einselected states is not a violation of the basic te- ͩ ͚ ␣ ͉s ͪ͘ ͉A ͘→͚ ␣ ͉s ͉͘A ͘. (6.3) i i 0 i i i nets of quantum physics. Rather, it is a consequence of i i the fact that the data required to turn a quantum state into an ontological entity, an einselected pointer state, In practice the action is usually large enough to accom- are abundantly supplied by the environment. plish amplification. As we have seen in Sec. II, all this We emphasize the operational nature of this criterion can be done without an appeal to the environment. for existence. There may, in principle, be a pure state of For a real apparatus, interaction with the environment the universe including the environment, the observer, is inevitable. Idealized effectively classical memory will and the measured system. While this may matter to retain correlations, but will be subject to einselection. some (Zeh, 2000), real observers are forced to perceive Only the einselected memory states (rather than their the universe the way we do: We are a part of the uni- superpositions) will be useful for (or, for that matter, verse, observing it from within. Hence, for us, accessible to) the observer. The decoherence time scale environment-induced superselection specifies what exists. is very short compared to the time after which memory Predictability emerges as a key criterion of existence. states are typically consulted (i.e., copied or used in in- The only states R can rely on to store information are formation processing), which is in turn much shorter the pointer states. They are also the obvious choice for S than the relaxation time scale, on which memory ‘‘for- to measure. Such measurements can be accomplished gets.’’ without danger of re-preparation. Einselected states are Decoherence leads to classical correlation, insensitive to measurement of the pointer observables— they have already been measured by the environment. 6 Therefore additional projections Pi onto the einselected The observable sˆ of the system and Bˆ of the apparatus basis will not perturb the density matrix (Zurek, 1993a); memory need not be discrete with a simple spectrum as was it will be the same before and after the measurement: previously assumed. Even when sˆ has a complicated spectrum, the outcome of the measurement can be recorded in the eigen- ␳D ϭ ␳D states of the memory observable Aˆ , the conjugate of Bˆ ,Eq. after ͚ Pi beforePi . (6.1) i (2.21). For the case of discrete sˆ the necessary calculations that attain premeasurement—the quantum correlation that is the Correlations with the einselected states will be left intact first step in the creation of the record—were already carried (Zurek, 1981, 1982). out in Sec. II. For the other situations they are quite similar. In ˆ ϭ͚ ␭ Superselection for the observable A i iPi with es- either case, they follow the general outline of von Neumann’s ␭ sentially arbitrary nondegenerate eigenvalues i and (1932) discussion.

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be used as measuring devices and/or memories and as ␳P ϭ ␣ ␣ ͉ ͗͘ ͉͉ ͗͘ ͉ SA ͚ i j* si sj Ai Aj quantum environments that interact with either or both. i,j Axioms (iii)–(v) contain many idealizations. For instance, in real life or in laboratory practice measure- → ͉␣ ͉2͉ ͗͘ ͉͉ ͗͘ ͉ϭ␳D ͚ i si si Ai Ai SA , (6.4) ments have errors (and hence can yield outcomes other i than the eigenvalues oi). Moreover, only rarely do they following an entangling premeasurement. The left-hand prepare the system in the eigenstate of the observable side of Eq. (6.4) coincides with Eq. (2.44c), the outsider’s they are designed to measure. Furthermore, coherent view of the classical measurement. We shall see how and states—often an outcome of measurements, e.g., in to what extent its other aspects, including the insider’s quantum optics—form an overcomplete basis. Thus Eq. (2.44a) and the discoverer’s Eq. (2.44b), can be un- their detection does not correspond to a measurement derstood through einselection. of a Hermitian observable. Last but not least, the measured quantity may be inferred from some other quan- C. Axioms of quantum measurement theory tity (e.g., beam deflection in the Stern-Gerlach experiment). Yet, we shall not go beyond the idealizations of Our goal is to establish whether the above model can (i)–(v) above. Our goal is to describe measurements in a fulfill the requirements expected from measurement in quantum theory without collapse, to use axioms (ø), (i), textbooks (which are, essentially without exception, and (ii) to understand the origin of the other axioms. written in the spirit of the Copenhagen interpretation). Nonideal measurements are a fact of life incidental to There are several equivalent textbook formulations of this goal. the axioms of quantum theory. We shall (approximately) follow Farhi, Goldstone, and Gutmann (1989) and posit 1. Observables are Hermitian—axiom (iii a) them as follows: In the model of measurement considered here the ob- S (i) The states of a quantum system are associated servables are Hermitian as a consequence of an assumed ͉␺͘ with the vectors , which are the elements of premeasurement interaction, e.g., Eq. (2.24). In particu- H S the Hilbert space S that describes . lar, H is a product of the to-be-measured observable ˙ int (ii) The states evolve according to iប͉␺͘ϭH͉␺͘, of the system and of the ‘‘shift operator’’ in the pointer where H is Hermitian. of the apparatus or in the record state of the memory. (iii a) Every observable O is associated with a Hermit- Interactions involving non-Hermitian operators (e.g., ˆ ϳ † ϩ † ian operator O. Hint a b ab ) may, however, also be considered. (iii b) The only possible outcome of a measurement of It is tempting to speculate that one could dispose of ˆ O is an eigenvalue oi of O. the observables [and hence of the postulate (iii a)] alto- (iv) Immediately after a measurement that yields the gether in the formulation of the axioms of quantum ͉ ˆ theory. The only ingredients necessary to describe mea- value oi the system is in the eigenstate oi͘ of O. (v) If the system is in a normalized state ͉␺͘, then a surements are then the effectively classical, but ulti- ˆ mately quantum, apparatus and the measured system. measurement of O will yield the value oi with the probability p ϭ͉͗o ͉␺͉͘2. Observables emerge as a derived concept, as a useful i i idealization, ultimately based on the structure of the The first two axioms make no reference to measure- Hamiltonians. Their utility relies on the conservation ments. They state the formalism of the theory. Axioms laws, which relate the outcomes of several measure- (iii)–(v) are, on the other hand, at the heart of the ments. The most basic of these laws states that the sys- present discussion. In spirit, they go back to Bohr and tem that did not (have time to) evolve will be found in Born. In letter, they follow von Neumann (1932) and the same state when it is remeasured. This is the content Dirac (1947). The two key issues are the projection pos- of axiom (iv). Other conservation laws are also reflected tulate, implied by a combination of (iv) with (iii b), and in the patterns of correlation in the measurement the probability interpretation, axiom (v). records, which must in turn arise from the underlying To establish (iii b), (iv), and (v) we shall interpret in symmetries of the Hamiltonians. operational terms statements such as ‘‘the system is in Einselection should be included in this program, as it the eigenstate’’ and ‘‘measurement will yield value decides which observables are accessible and useful— . . . with the probability...’’by specifying what these which are effectively classical. It is conceivable that the statements mean for the sequences of records made and fundamental superselection may also emerge in this maintained by an idealized, but physical memory. manner (see Zeh, 1970 and Zurek, 1982, for early specu- We note that the above Copenhagen-like axioms pre- lations; see Giulini, Kiefer, and Zeh, 1995; Kiefer, 1996, sume the existence of quantum systems and of classical and Giulini, 2000, for the present status of this idea). measuring devices. This (unstated) axiom (ø) comple- ments axioms (i)–(v). Our version of axiom (ø) posits 2. Eigenvalues as outcomes—axiom (iii b) that the universe consist of quantum systems, and asserts that a composite system can be described by a tensor This axiom is the first part of the collapse postulate. product of the Hilbert spaces of the constituent systems. Given einselection, axiom (iii b) is easy to justify: we Some quantum systems can be measured, and others can need to show that only the records inscribed in the ein-

Rev. Mod. Phys., Vol. 75, No. 3, July 2003 752 Wojciech Hubert Zurek: Decoherence, einselection, and the quantum origins of the classical selected states of the apparatus pointer can be read off, Immediate repeatability for Hermitian observables and that, in a well-designed measurement, they correlate with discrete spectra is straightforward to justify on the with the eigenstates (and therefore, eigenvalues) of the basis of the Schro¨ dinger evolution generated by Hint of measured observable sˆ. Eq. (6.2) alone, although its implications depend on With Dirac (1947) and von Neumann (1932) we as- whether the premeasurement is followed by einselec- sume that the apparatus is built so that it satisfies the tion. Everett (1957a, 1957b) used the ‘‘no-decoherence’’ obvious truth table when the eigenstates of the mea- version as a foundation of his relative-state interpreta- sured observable are at the input: tion. On the other hand, without decoherence and einselection one could postpone the choice of what was ͉s ͉͘A ͘→͉s ͉͘A ͘. (6.5) i 0 i i actually recorded by taking advantage of the entangle- To assure this one can implement the interaction in ac- ment between the system and the apparatus and the record with Eq. (6.2) and the relevant discussion in Sec. II. sulting basis ambiguity, as is evident on the right-hand This is not to say that there are no other ways: Aha- side of Eq. (6.3). For instance, a measurement carried ͉ ronov, Anandan, and Vaidman (1993); Braginski and out on the apparatus in a basis different from ͕ Ai͖͘ Khalili (1996); and Unruh (1994) have all considered would also exhibit a one-to-one correlation with the sys- ͚ ␣ ͉ ͉ ϭ͚ ␤ ͉ ͉ ‘‘adiabatic measurements’’ that correlate the apparatus tem: i i si͘ Ai͘ k k rk͘ Bk͘. This flexibility to re- with the discrete energy eigenstates of the measured sys- write wave functions in different bases comes at the tem, nearly independently of the structure of Hint . price of relaxing the demand that the outcome states ͉ The truth table of Eq. (6.5) does not require collapse; ͕ rk͖͘ be orthogonal (so that there would be no associ- ͉ for any initial si͘ it represents a classical measurement ated Hermitian observable). However, as was already in quantum notation in the sense of Sec. II. However, noted, coherent states associated with a non-Hermitian Eq. (6.5) typically leads to a superposition of outcomes. annihilation operator can also be an outcome of a mea- This is the ‘‘measurement problem.’’ To address it, we surement. Therefore [and in spite of the strict interpre- ͉ assume that the record states ͕ Ai͖͘ are einselected. This tation of axiom (iii a)] this is not a very serious restric- has two related consequences: (i) Following the mea- tion. surement, the joint density matrix of the system and the In the presence of einselection the basis ambiguity apparatus decoheres, Eq. (6.3), so that it satisfies the disappears. Immediate repeatability would apply only to ϭ͉ ͉ superselection condition, Eq. (6.1), for Pi Ai͗͘Ai . (ii) the records made in the einselected states. Other appa- Einselection restricts states that can be read off as if ratus observables lose correlation with the state of the they were classical to pointer states. system on the decoherence time scale. In the effectively Indeed, following decoherence only the pointer states classical limit it is natural to demand repeatability ex- ͉ ͕ Ai͖͘ of the memory can be measured without dimin- tending beyond that very brief moment. This demand ishing the correlation with the states of the system. makes the role of einselection in establishing axiom (iv) Without decoherence, as we have seen in Sec. II, one evident. Indeed, such repeatability is, albeit in a more could use the entanglement between S and A to end up general context, the motivation for the predictability with almost arbitrary superposition states of either and sieve. hence to violate the letter and the spirit of axiom (iii b). Outcomes are restricted to the eigenvalues of measured observables because of einselection. Axiom (iii b) 4. Probabilities, einselection, and records is then a consequence of the effective classicality of the Density matrix alone, without the preferred set of pointer states, the only ones that can be found out with- states, does not suffice as a foundation for a probability out being disturbed. They can be consulted repeatedly interpretation. For, any mixed-state density matrix ␳S and remain unaffected under the joint scrutiny of the can be decomposed into sums of density matrices that observers and of the environment (Zurek, 1981, 1993a, add up to the same resultant ␳S , but need not share the a b 1998a). same eigenstates. For example, consider ␳S and ␳S , representing two different preparations (i.e., involving the measurement of two distinct, noncommuting observables) of two ensembles, each with multiple copies of a 3. Immediate repeatability, axiom (iv) system S. When they are randomly mixed in proportions This axiom supplies the second half of the collapse pa and pb, the resulting density matrix postulate. It asserts that in the absence of (the time for) a∨b a a b b ␳S ϭp ␳Sϩp ␳S evolution the quantum system will remain in the same state, and its remeasurement will lead to the same out- is the complete description of the unified ensemble (see come. Hence, once the system is found out to be in a Schro¨ dinger, 1936; Jaynes, 1957). a b a∨b certain state, it really is there. As in Eq. (2.44b) the Unless ͓␳S ,␳S͔ϭ0, the eigenstates of ␳S do not co- observer perceives potential options collapse to a single incide with the eigenstates of the components. This fea- actual outcome. [The association of axiom (iv) with the ture makes it difficult to regard any density matrix in collapse advocated here seems obvious, but it is not terms of probabilities and ignorance. Such ambiguity common. Rather, some form of our axiom (iii b) is usu- would be especially troubling if it arose in the descrip- ally regarded as the sole collapse postulate.] tion of an observer (or, for that matter, of any classical

Rev. Mod. Phys., Vol. 75, No. 3, July 2003 Wojciech Hubert Zurek: Decoherence, einselection, and the quantum origins of the classical 753 system). The ignorance interpretation, i.e., the idea that is admissible for an effectively classical A correlated probabilities are the observer’s way of dealing with un- with a quantum S. Now the discord ␦AI(S͉A)ϭ0. certainty about the outcome which we have briefly ex- Mixing of ensembles of pairs of correlated systems, plored in the discussion of the insider-outsider di- one of which is subject to einselection, does not lead to chotomy, Eqs. (2.44), requires at the very least that the the ambiguities discussed above. The discord ␦A(S͉A) set of events (‘‘the sample space’’) exists independently disappears in the einselected basis of A, and the eigen- of the information at hand, that is, independently of pa values of the density matrices can behave as classical and pb in the example above. Eigenstates of the density probabilities associated with events with the records. matrix do not supply such events, since the additional The menu of possible events, in the sample space, e.g., loss of information associated with mixing of the en- records in memory, is fixed by einselection. Whether one sembles alters the candidate events. can really justify this interpretation of the eigenvalues of Basis ambiguity would be disastrous for record states. the reduced density matrix is a separate question we are Density matrices describing a joint state of the memory about to address. A and of the system S ␳a∨bϭ ␳a ϩ ␳b D. Probabilities from envariance AS pa AS pb AS would have to be considered. In the absence of einselec- The view of the emergence of the classical based on a∨b tion the eigenstates of such ␳AS need not even be asso- environment-induced superselection has been occasion- ciated with a fixed set of record states of the presumably ally described as ‘‘for all practical purposes only’’ (see, a∨b classical A. Indeed, in general ␳AS has a nonzero for example, Bell, 1990), to be contrasted with the more discord,7 and its eigenstates are entangled (even when fundamental (if nonexistent) solutions of the problem a∨b the above ␳AS is separable, and can be expressed as a one could imagine (i.e., by modifying quantum theory; mixture of matrices that have no entangled eigenstates). see Bell, 1987, 1990). This attitude can be traced in part This would imply an ambiguity of what the record states to the reliance of einselection on reduced density matri- are precluding a probability interpretation of measure- ces. For even when explanations of all aspects of the ment outcomes. effectively classical behavior are accepted in the frame- The observer may nevertheless have records of a sys- work of, say, Everett’s many-worlds interpretation, and tem that is in the ambiguous situation described above. after the operational approach to the objectivity and Thus perception of unique outcomes based on the existential interpretation explained earlier is adopted, one major ␳a∨bϭ ͉ ͉͑ a ␳a ϩ b␳b ͒ gap remains: Born’s rule—axiom (v) connecting prob- AS ͚ wk Ak͗͘Ak p S p S k k k k ϭ͉␺ ͉2 k abilities with amplitudes, pk k —has to be postu- lated in addition to axioms (ø)–(ii). True, one can show that within the framework of einselection Born’s rule 7As we have seen in Sec. IV, Eqs. (4.30)–(4.36), discord emerges naturally (Zurek, 1998a). Decoherence is, how- ␦IA(S͉A)ϭI(S:A)ϪJA(S:A) is a measure of the ‘‘quantum- ever, based on reduced density matrices. Since their in- ness’’ of correlations. It should disappear as a result of the troduction by Landau (1927), it is known that a partial classical equivalence of two definitions of the mutual informa- trace leading to reduced density matrices is predicated tion, but is in general positive for quantum correlations, in- on Born’s rule (see Nielsen and Chuang, 2000, for a dis- cluding, in particular, predecoherence ␳SA . Discord is asym- cussion). Thus derivations of Born’s rule that employ ␦I S͉A Þ␦I A͉S ␦I S͉A metric, A( ) S( ). The vanishing of A( ) [i.e., reduced density matrices are open to the charge of cir- of the discord in the direction exploring the classicality of the cularity (Zeh, 1997). Moreover, repeated attempts to A ␳ S A states of , on which H( S͉A) in the asymmetric JA( : ), Eq. justify p ϭ͉␺ ͉2 within the no-collapse many-worlds in- (4.32), is conditioned] is necessary for the classicality of the k k terpretation (Everett, 1957a, 1957b; DeWitt, 1970; De measurement outcome (Ollivier and Zurek, 2002; Zurek, 2000, Witt and Graham, 1973; Geroch, 1984) have failed (see, 2003a). ␦IA(S͉A) can disappear as a result of decoherence in the einselected basis of the apparatus. Following einselection it for example, Stein, 1984; Kent, 1990; Squires, 1990). The is then possible to ascribe probabilities to the pointer states. In problem is their circularity. An appeal to the connection perfect measurements of Hermitian observables discord van- (especially in certain limiting procedures) between the ishes ‘‘both ways’’: ␦IA(S͉A)ϭ␦IS(A͉S)ϭ0 for the pointer smallness of the amplitude and the vanishing of the basis and for the eigenbasis of the measured observable corre- probabilities has to be made to establish that the relative lated with it. Nevertheless, it is possible to encounter situations frequencies of events averaged over branches of the uni- when vanishing of the discord in one direction is not accompa- versal state vector are consistent with Born’s rule. In nied by its vanishing ‘‘in reverse.’’ Such correlations are ‘‘clas- particular, one must a claim that ‘‘maverick’’ branches of sical one way’’ (Zurek, 2003a). the MWI state vector that have ‘‘wrong’’ relative fre- This asymmetry between classical A and quantum S arises quencies have a vanishing probability because their from the einselection. Classical record states are not arbitrary superpositions. The observer accesses his memory in the basis Hilbert-space measures are small. This is circular, as in which it is monitored by the environment. The information noted even by the proponents (DeWitt, 1970). stored is effectively classical because it is being widely dissemi- My aim here is to look at the origin of ignorance, nated. States of the observer’s memory exist objectively; they information, and, therefore, probabilities from a very can be determined through their imprints in the environment. quantum and fundamental perspective. Rather than fo-

Rev. Mod. Phys., Vol. 75, No. 3, July 2003 754 Wojciech Hubert Zurek: Decoherence, einselection, and the quantum origins of the classical

cus on probabilities for an individual isolated system I ͕͉s ͖͘ ␴ k ͉␺ ͘ϭ ␣ i k͉ ͉͘␧ ͘ shall—in the spirit of einselection, but without employ- uS SE ͚ ke sk k (6.9) k ing its usual tools such as trace or reduced density matrices—consider what the observer can (and cannot) can be undone by

know about a system entangled with its environment. ͕͉␧ ͖͘ ⑀ k ϭ i k͉␧ ␧ ͉ Within this context I shall demonstrate that Born’s rule uE ͚ e k͗͘ k , (6.10) k follows from the very quantum fact that one can know ⑀ ϭ ␲ Ϫ␴ precisely the state of the composite system and yet be providing that k 2 lk k for some integer lk . QED. provably ignorant of the state of its components. This is Thus phases associated with the Schmidt basis are en- due to environment-assisted invariance or envariance, a variant. We shall see below that they are the only envari- hitherto unrecognized symmetry I am about to describe. ant property of entangled states. The transformations Envariance of pure states is a symmetery conspicu- defined by Eq. (6.8) are rather specific—they share (Schmidt) eigenstates. Still, their existence leads us to ously missing from classical physics. It allows one to de- S fine ignorance as a consequence of invariance, and thus Theorem 6.1: A local description of the system entangled with a causally disconnected environment E to understand the origin of Born’s rule, the probabilities, must not depend on the phases of the coefficients ␣ in and ultimately the origin of information through argu- k the Schmidt decomposition of ͉␺SE͘. ments based on assumptions different from Gleason’s It follows that all the measurable properties of S are (1957) famous theorem. Rather, it is based on the Ma- ͉␣ ͉ ͉ completely specified by the list of pairs ͕ k ; sk͖͘.A chian idea of the relativity of quantum states, suggested different way of establishing this phase envariance theo- by this author two decades ago (see p. 772 of Wheeler rem appeals even more directly to causality. Phases of and Zurek, 1983), but not developed until now. Envari- ͉␺SE͘ can be arbitrarily changed by acting on E alone ance (Zurek, 2003b) addresses the question of the mean- ͉␧ [e.g, by the local Hamiltonian with eigenstates k͘, gen- ing of these probabilities by defining ‘‘ignorance’’ and erating evolution of the form of Eq. (6.9)]. But causality leads to correct relative frequencies. prevents faster-than-light communication. Hence no measurable property of S can be effected by acting on E. Clearly, there is an intimate connection between envariance and causality. Independence of the local state of S ␣ 1. Envariance from the phases of the Schmidt coefficients k follows from envariance alone, but it could be also established Environment-assisted invariance is a symmetry exhib- through an appeal to causality. The situation is similar as ited by a system S correlated with another system with ‘‘no cloning theorem.’’ It was proved using linearity (which we shall call the environment E). When a state of of quantum theory, but one could have also inferred im- the composite SE can be transformed by uS acting solely possibility of cloning an unknown quantum state from on the Hilbert space HS, but the effect of this transfor- special-relativistic causality. The proof based on linearity mation can be undone with an appropriate uE acting is ‘‘less expensive,’’ as it does not require ingredients only on HE, so that the joint state ͉␺SE͘ remains unal- that go beyond quantum theory. tered, so that Phase envariance theorem will turn out to be the crux of our argument. It relies on an input—entanglement uEuS͉␺SE͘ϭuE͉␩SE͘ϭ͉␺SE . (6.6) and envariance—which has not been employed to date in discussions of the origin of probabilities. In particular, Such a ͉␺SE͘ is envariant under uS . The generalization this input is different and more ‘‘physical’’ than that of to mixed ␳SE is obvious, but we shall find it easier to the successful derivation of Born’s rules by Gleason assume that SE has been purified in the usual fashion, (1957). i.e., by enlarging the environment. We also note that information contained in the ‘‘data- Envariance is best elucidated by considering an ex- base’’ ͕͉␣ ͉;͉s ͖͘ implied by Theorem 6.1 is the same as ample, an entangled state of S and E. It can be expressed k k in the reduced density matrix of the system ␳S . Al- in the Schmidt basis as though we do not yet know the probabilities of various ͉s ͘, the preferred basis of S has been singled out; ͉␺ ϭ ␣ ͉ ͉␧ k SE͘ ͚ k sk͘ k͘, (6.7) Schmidt states (sometimes regarded as instantaneous k pointer states; see, for example, Albrecht, 1992, 1993) where ␣ are complex, while ͕͉s ͖͘ and ͕͉␧ ͖͘ are ortho- play a special role as the eigenstates of envariant trans- k k k ͉␣ ͉ normal. For ͉␺SE͘ (and hence, given our above remark formations. Moreover, probabilities can depend on k about purification, for any system correlated with the (but not on the phases). We still do not know that pk ϭ͉␣ ͉2 environment) it is easy to demonstrate the following. k . Lemma 6.1: Unitary transformations codiagonal with The causality argument we could have used to estab- ͉␺ ͘ lish Theorem 6.1 applies of course to arbitrary transfor- the Schmidt basis of SE leave it envariant. E The proof relies on the form of such transformations: mations one could perform on . However, such transformations would in general not be envariant (i.e., could ͕͉s ͖͘ ␴ not be undone by acting on S alone). Indeed, this is one k ϭ i k͉ ͉ uS ͚ e sk͗͘sk , (6.8) k way to see that causality is a more potent ingredient than envariance. In particular, all envariant transforma- ␴ where k is a phase. Hence tion have a fairly restricted form as follows.

Rev. Mod. Phys., Vol. 75, No. 3, July 2003 Wojciech Hubert Zurek: Decoherence, einselection, and the quantum origins of the classical 755

Lemma 6.2: All of the unitary envariant transforma- N N ͉␺ ͉ ͉ ͉␧ → ͉ ͉ ͉␧ ϳ͉⌽ tions of SE͘ have Schmidt eigenstates. A0͘ ͚ sk͘ k͘ ͚ Ak͘ sk͘ k͘ SAE͘. (6.13) The proof relies on the fact that other unitary trans- kϭ1 kϭ1 ͉ →͉ formations would rotate the Schmidt basis, sk͘ ˜sk͘. Above, we have assumed that the absolute values of the The rotated basis becomes a new ‘‘Schmidt,’’ and this coefficients are equal (and omitted them for notational fact cannot be affected by unitary transformations of E, simplicity). We have also ignored phases (which need by state rotations in the environment. But a state that not be equal) since by the phase envariance theorem has a different Schmidt decomposition from the original they will not influence the state (and hence, the prob- ͉␺SE͘ is different. Hence a unitary transformation must abilities) associated with S. be codiagonal with the Schmidt basis of ␺SE to leave it Before the measurement the observer with access to S envariant. QED. cannot notice swaps in the states [such as Eq. (6.13)] with equal absolute values of the Schmidt coefficients. This follows from the envariance of the premeasurement ͉␺SE͘ under swaps, Eq. (6.11). One could argue this point in more detail by compar- 2. Born’s rule from envariance ing what happens for two very different input states: an When absolute values of some of the coefficients in entangled ͉␺SE͘ with equal absolute values of Schmidt Eq. (6.7) are equal, any orthonormal basis is Schmidt in coefficients and a product state: the corresponding subspace of HS. This implies envari- ͉␸SE͘ϭ͉s ͉͘␧ ͘. ance of a more general nature, e.g., under a swap: J J ␸ ␾ When the observer knows he is dealing with SE ,he ͑ ↔ ͒ϭ i kj͉ ͉ϩ uS k j e sk͗͘sj H.c. (6.11) ͉ knows the system is in the state sJ͘, and can predict the A swap can be generated by a phase rotation, Eq. (6.8), outcome of the corresponding measurement on S. The but in a basis complementary to the one swapped. Its Schro¨ dinger equation or just the resulting truth table, envariance does not contradict Lemma 6.2, as any ortho- Eq. (6.5), implies with certainty that his state—the fu- ͉␣ ͉ ͉ ͘ normal basis in this case is also Schmidt. So when k ture state of his memory—will be AJ . Moreover, swaps ϭ͉␣ ͉ ͉ ͘ ␸ j , the effect of a swap on the system can be undone involving sJ are not envariant for SE . They just swap by an obvious counterswap in the environment: the outcomes [i.e., when uS(J↔L) precedes the mea- ͉ Ϫ ͑␾ ϩ␾ Ϫ␾ ϩ ␲ ͒ surement, memory will end up in AL͘]. ͑ ↔ ͒ϭ i kj k j 2 lkj ͉␧ ͗͘␧ ͉ϩ uS k j e k j H.c. (6.12) By contrast, A swap can be applied to states that do not have equal N ␾ absolute values of the coefficients, but in that case it is ͉␺ ϳ i k͉ ͉␧ SE͘ ͚ e sk͘ k͘ no longer envariant. Partial swaps can also be generated, kϭ1 ͕͉ ͖͘ ri for example, by underrotating or by a uS , Eq. (6.8), is envariant under swaps. This allows the observer (who ͉ but with the eigenstates ͕ ri͖͘ intermediate between knows the joint state of SE exactly) to conclude that the those of the swapped and the complementary (Had- probabilities of all the envariantly swappable outcomes amard) basis. A swap followed by a counterswap ex- must be the same. The observer cannot predict his changes coefficients of the swapped states in the memory state after the measurement of S because he Schmidt expansion, Eq. (6.7). Hence, ␺SE is envariant knows too much: the exact combined state of SE. ↔ ͉␣ ͉ϭ͉␣ ͉ under swaps uS(j k) only when k j . For completeness, we note that when there are system States of correlated classical systems can also exhibit states that are absent from the above sum, i.e., states something akin to envariance under a classical version that appear with zero amplitude, they cannot be envari- of swaps. For instance, a correlated state of a system and antly swapped with the states present in the sum. Of ␳ ϳ͉ ͉͉ ͉ an apparatus described by SA sk͗͘sk Ak͗͘Ak course, the observer can predict with certainty that he ϩ͉ ͉͉ ͉ sj͗͘sj Aj͗͘Aj can be swapped and counterswapped. will not detect any of the corresponding zero-amplitude The corresponding transformations would be still given outcomes. For, following the measurement that corre- ͉ by, in effect, Eqs. (6.11) and (6.12), but without phases, lates memory of the observer with the basis ͕ sk͘} of the and swaps could no longer be generated by rotations system, there will be simply no terms describing ob- around the complementary basis. This situation corre- server with the record of such nonexistent states of S. sponds to the outsider’s view of the measurement pro- This argument about the ignorance of the observer con- cess, Eq. (2.44c). The outsider can be aware of the cor- cerning his future state, concerning the outcome of the relation between the system and the apparatus, but measurement he is about to perform, is based on his ignorant of their individual states. This connection be- perfect knowledge of a joint state of SE. tween ignorance and envariance will be exploited below. Probabilities refer to the guess the observer makes on Envariance based on ignorance may be found in the the basis of his information before the measurement classical setting, but envariance of pure states is purely about the state of his memory—the future outcome— quantum. Observers can know perfectly the quantum after the measurement. Since the left-hand side of Eq. joint state of SE, yet be provably ignorant of S. Consider (6.13) is envariant under swaps of the system states, the a measurement carried out on the state vector of SE probabilities of all the states must be equal. Thus, by from the point of view of envariance: normalization,

Rev. Mod. Phys., Vol. 75, No. 3, July 2003 756 Wojciech Hubert Zurek: Decoherence, einselection, and the quantum origins of the classical

ϭ ␣ ϳͱ pk 1/N. (6.14) where k mk and mk is a natural number (and, by Theorem 6.1, we drop the phases). To get an envariant Moreover, the probability of n mutually exclusive events state we increase the resolution, of A by assuming that that all appear in Eq. (6.13) with equal coefficients must be mk ͉ ϭ ͉ ͱ Ak͘ ͚ aj ͘/ mk . (6.18) ϭ k p ∨ ∨ ∨ ϭn/N. (6.15) jk 1 k1 k2 ¯ kn This concludes the discussion of the equal probability An increase of resolution is a standard trick, used in case. Our case rests on the independence of the state of classical probability theory ‘‘to even the odds.’’ Note ͉ ͘ S entangled with E from the phases of the coefficients in that we assume that basis states such as Ak are normal- the Schmidt representation, the phase envariance theo- ized (as they must be in a Hilbert space). This leads to rem 6.1, which in the case of equal coefficients, Eq. mk ͉ (6.13), allows envariant swapping, and yields Eqs. (6.14) N ͚ aj ͘ j ϭ1 k and (6.15). ͉⌽ ϳ ͱ k ͉ ͉␧ SAE͘ ͚ mk sk͘ k͘. (6.19) ϭ ͱ After a measurement the situation changes. In accord k 1 mk with our preceding discussion we interpret the presence We now assume that A and E interact (e.g., through a of the term ͉A ͘ in Eq. (6.13) as evidence that an out- k c-shift of Sec. II, with a truth table ͉a ͉͘␧ ͘→͉a ͉͘e ͘ ͉ jk k jk jk come sk͘ can be (or indeed has been—the language where ͕͉e ͖͘ are all orthonormal). After simplifying and here is somewhat dependent on the interpretation) re- jk ͉ rearranging terms we get a sum, over a new fine-grained corded. Conversely, the absence of some AkЈ͘ in the ͉ index, with the states of S that remain the same within sum above implies that the outcome skЈ͘ cannot occur. After a measurement the memory of the observer who coarse-grained cells, with the cell size measured by mk : ͉ ͉ N m M has detected sk͘ will contain the record Ak͘. Further k ͉⌽˜ ͘ϳ ͉ ͩ͘ ͉ ͉͘ ͪ͘ ϭ ͉ ͉͘ ͉͘ ͘ measurements of the same observable on the same sys- SAE ͚ sk ͚ aj ej ͚ sk͑j͒ aj ej . ϭ ϭ k k ϭ tem will confirm that S is in indeed in the state ͉s ͘. k 1 jk 1 j 1 k (6.20) This postmeasurement state is still envariant, but only ϭ͚N ϭ р ϭ under swaps that involve jointly the state of the system Above, M kϭ1mk , k(j) 1 for j m1 , k(j) 2 for Ͻ р ϩ and the correlated state of the memory: m1 j m1 m2 , etc. The above state is envariant under i␾ combined swaps: uAS͑k↔j͒ϭe kj͉s ,A ͗͘s ,A ͉ϩH.c. (6.16) k k j j ͑ ↔ ͒ϭ ͑ ␾ ͉͒ ͉ϩ uSA j jЈ exp i jjЈ sk͑j͒ ,aj͗͘ajЈ ,sk͑jЈ͒ h.c. Thus if another observer (‘‘Wigner’s friend’’) was getting SA ready to find out, either by direct measurement of S or Suppose that an additional observer measures in the by communicating with observer A, the outcome of A’s obviously swappable joint basis. By our equal- measurement, he would be (on the basis of envariance) coefficients argument, Eq. (6.14), we get ϭ provably ignorant of the outcome A has detected, but p(sk(j) ,aj) 1/M. could be certain of the AS correlation. We shall employ But the observer can ignore states aj . Then the prob- this joint envariance in the discussion of the case of un- ability of different Schmidt states of S is, by Eq. (6.15), equal probabilities immediately below. ͑ ͒ϭ ϭ͉␣ ͉2 Note that our reasoning does not really appeal to the p sk mk /M k . (6.21) information lost in the environment in the sense in This is Born’s rule. which this phrase is often used. Perfect knowledge of the The case with coefficients that do not lead to com- combined state of the system and the environment is the mensurate probabilities can be treated by assuming con- basis of the argument for the ignorance of S alone. For tinuity of probabilities as a function of the amplitudes, entangled SE, perfect knowledge of SE is incompatible and taking appropriate (and obvious) limits. This can be with perfect knowledge of S. This is really a conse- physically motivated: One would not expect probabili- quence of indeterminacy; joint observables with en- ties to change drastically depending on infinitesimal tangled eigenstates such as ␺SE simply do not commute changes of state. One can also extend the strategy out- (as the reader is invited to verify) with the observables lined above to deal with probabilities (and probability of the system alone. Hence ignorance associated with densities) in cases such as ͉s(x)͘, i.e., when the index of envariance is ultimately mandated by Heisenberg inde- the state vector changes continuously. This can be ac- terminacy. complished by discretizing it [so that the measurement The case of unequal coefficients can be reduced to the of Eq. (6.17) correlates different apparatus states with case of equal coefficients. This can be done in several small intervals of x] and then repeating the strategy of ways, of which we choose one that makes use of the Eqs. (6.17)–(6.21). The wave function s(x) should be preceding discussion of the envariance of the postmea- sufficiently smooth for this strategy to succeed. surement state. We start with We note that the increase of resolution we have ex- N ploited, Eqs. (6.18)–(6.21), need not be physically imple- ͉⌽ ϳ ␣ ͉ ͉ ͉␧ mented for the argument to proceed. The very possibil- SAE͘ ͚ k Ak͘ sk͘ k͘, (6.17) kϭ1 ity of carrying out these steps within the quantum

Rev. Mod. Phys., Vol. 75, No. 3, July 2003 Wojciech Hubert Zurek: Decoherence, einselection, and the quantum origins of the classical 757 formalism forces one to adopt Born’s rule. For example, This is the distribution one would expect from Born’s if the apparatus did not have the requisite extra resolu- rule. To establish the connection with relative frequen- tion, Eq. (6.18), the interaction of the environment with cies we appeal to the de Moivre–Laplace theorem a still different ‘‘counterweight’’ system C that yields (Gnedenko, 1982), which allows one to approximate N above pN(n) with a Gaussian: ͉⌿ ϭ ͱ ͉ ͉ ͉␧ ͉ ϪN͉␣͉2 2 SAEC͘ ͚ mk sk͘ Ak͘ k͘ Ck͘ (6.22) 1 1 n kϭ1 pN͑n͒Ӎ expͭ Ϫ ͫ ͬ ͮ . (6.26) ͱ2␲N͉␣␤͉ 2 ͱN͉␣␤͉ would lead one to Born’s rule through steps similar to ͉ N these that we have invoked before, providing that ͕ Ck͖͘ This last step requires large , but our previous discus- ͉ ϭ͚mk ͉ ͱ sion including Eq. (6.25) is valid for arbitrary N. Indeed, has the requisite resolution, Ck͘ ϭ cj ͘/ mk .An jk 1 k Eq. (6.21) can be regarded as the Nϭ1 case. interaction resulting in a correlation, Eq. (6.22), can oc- Nevertheless, for large N the relative frequency is E C cur between and , and happen far from the system of sharply peaked around the expected ͗n͘ϭN͉␣͉2. In- interest or from the apparatus. Thus it will not influence deed, in the limit N→ϱ the appropriately rescaled 2 the probabilities of the outcomes of measurements car- pN(n) tends to a Dirac ␦(vϪ͉␣͉ ) in the relative fre- ried out on S or of the records made by A. Yet, the fact quency vϭn/N. This justifies the relative-frequency in- that it can happen leads us to the desired conclusion. terpretation of the squares of amplitudes as probabilities in the MWI context. Maverick universes with different relative frequencies exist, but have a vanishing probabil- 3. Relative frequencies from envariance ity (and not just a vanishing Hilbert-space measure) for N Relative frequency is a common theme in studies that large . aim to elucidate the physical meaning of probabilities in Our derivation of the physical significance of the probabilities, while it led to relative frequencies, was quantum theory (Everett, 1957a, 1957b; Hartle, 1968; based on a very different set of assumptions than previ- DeWitt, 1970; Graham, 1970; Farhi, Goldstone, and ous derivations. The key idea behind it is the connection Gutmann, 1989; Aharonov and Reznik, 2002). In par- between symmetry (envariance) and ignorance (impos- ticular, in the context of the no-collapse many-worlds sibility of knowing something). The unusual feature of interpretation relative frequency seems to offer the best our argument is that this ignorance (for an individual hope of arriving at Born’s rule and elucidating its physi- system S) is demonstrated by appealing to the perfect cal significance. Yet, it is generally acknowledged that knowledge of the larger joint system that includes S as a the MWI derivations offered to date have failed to at- subsystem. tain this goal (Kent, 1990). We emphasize that one could not carry out the basic We postpone a brief discussion of these efforts to the step of our argument—the proof of the independence of next section, and describe an approach to relative fre- the likelihoods from the phases of the Schmidt expan- quencies based on envariance. Consider an ensemble of sion coefficients—for an equal-amplitude pure state of a many (N) distinguishable systems prepared in the same single, isolated system. The problem with ͉␺͘ ϭ Ϫ1/2͚N ␾ ͉ ͘ initial state: N k exp(i k) k is the accessibility of the phases. Consider, for instance, ͉␺͘ϳ͉0͘ϩ͉1͘Ϫ͉2͘ and ͉␴S͘ϭ␣͉0͘ϩ␤͉1͘. (6.23) ͉␺Ј͘ϳ͉2͘ϩ͉1͘Ϫ͉0͘. In the absence of decoherence the We focus on the two-state case to simplify the notation. swapping of k’s is detectable. Interference measure- We also assume that ͉␣͉2 and ͉␤͉2 are commensurate, so ments (i.e., measurements of the observables with that the state vector of the whole ensemble of correlated phase-dependent eigenstates ͉1͘ϩ͉2͘, ͉1͘Ϫ͉2͘, etc.) would triplets SAE after the requisite increases of resolution have revealed the difference between ͉␺͘ and ͉␺Ј͘. In- [see Eqs. (6.18)–(6.20) above] is given by deed, given an ensemble of identical pure states an ob- m M N server will simply determine what they are. Loss of ͉⌽N ͘ϳͩ ͉ ͉͘ ͉͘ ͘ϩ ͉ ͉͘ ͉͘ ͪ͘ phase coherence is essential to allow for the shuffling of SAE ͚ 0 aj ej ͚ 1 aj ej , jϭ1 jϭmϩ1 the states and coefficients. (6.24) Note that in our derivation the environment and ein- save for the obvious normalization. This state is envari- selection play an additional, more subtle role. Once a ͉ measurement has taken place, i.e., a correlation with the ant under swaps of the joint states s,aj͘, as they appear with the same (absolute value) of the amplitude in Eq. apparatus or with the memory of the observer has been (6.24). (By Theorem 6.1 we can omit phases.) established, one would hope that the records would re- After the exponentiation is carried out, and the result- tain validity over a long time, well beyond the decohering product states are sorted by the number of 0’s and 1’s ence time scale. This is a precondition for axiom (iv). in the records, we can calculate the number of terms Thus a collapse from a multitude of possibilities to a N n NϪn single reality can be confirmed by subsequent measure- with exactly n 0’s: ␯N(n)ϭ( )m (MϪm) . To get n ments only in the einselected pointer basis. the probability, we normalize: N mn͑MϪm͒NϪn N 4. Other approaches to probabilities ͑ ͒ϭͩ ͪ ϭͩ ͉ͪ␣͉2n͉␤͉2͑NϪn͒ pN n N . n M n Gnedenko (1982), in his classic textbook, lists three (6.25) classical approaches to probability:

Rev. Mod. Phys., Vol. 75, No. 3, July 2003 758 Wojciech Hubert Zurek: Decoherence, einselection, and the quantum origins of the classical

(a) Definitions that appeal to the relative frequency theorems are classical. One can adopt this approach of occurrence of events in a large number of trials. (Zurek, 1998a) to probabilities in quantum physics only (b) Definitions of probability as a measure of the cer- after decoherence intervenes, restoring the validity of tainty of the observer. the distributive law, which is not valid in quantum phys- (c) Definitions that reduce probability to the more ics (Birkhoff and von Neumann, 1936). primitive notion of equal likelihood. One can carry out the equal-likelihood approach in In the quantum setting, the relative-frequency ap- the context of decoherence (Zurek, 1998a). The prob- proach has been to date the most popular, especially in lems are, as pointed out before, the use of the trace and the context of the no-collapse many-worlds interpreta- the dangers of circularity. An attempt to pursue a strat- tion (Everett, 1957a, 1957b; DeWitt, 1970; Graham, egy akin to equal likelihood in the quantum setting at 1970). Counting the number of ‘‘clicks’’ seems most di- the level of the pure states of individual systems has also rectly tied to the experimental manifestations of prob- been made by Deutsch in his (unpublished) ‘‘signaling’’ ability. Yet, the Everett interpretation versions were approach to probabilities. The key idea is to consider a generally found lacking (Kent, 1990; Squires, 1990), source of pure states, and to find out when the permu- since they relied on circular reasoning, invoking, without tations of a set of basis states can be detected, and there- physical justification, an abstract measure of Hilbert fore, used for communication. When permutations are space to obtain a physical measure (frequency). Some of undetectable, the probabilities of the permuted set of the criticisms seem relevant also for the versions of this states are declared equal. The problem with this idea (or approach that allow for the measurement postulates (iii) with its more formal version described by DeWitt, 1998) and (iv) (Hartle, 1968; Farhi, Goldstone, and Guttmann, is that it works only for superpositions that have all the 1989). Nevertheless, for the infinite ensembles consid- coefficients identical, including their phases. Thus, as we ered in the above references (where, in effect, the have already noted, for closed systems, phases matter Hilbert-space measure of the many-worlds interpreta- and there is no invariance under swapping. In a recent tion branches that violate relative-frequency predictions paper Deutsch (1999) adopted a different approach is zero) the eigenvalues of the frequency operator acting based on decision theory. The basic argument focuses on a large or infinite ensemble of identical states will be again on individual states of quantum systems, but, as consistent with the (Born formula) prescription for noted in the critical comment by Barnum et al. (2000), probabilities. seems to appeal to some of the aspects of decision However, the infinite size of the ensemble necessary theory that depend on probabilities. In my view, it also to prove this point is troubling (and unphysical) and tak- leaves the problem of phase dependence of the coeffi- ing the limit starting from a finite case is difficult to jus- cients unaddressed. tify (Stein, 1984; Kent, 1990; Squires, 1990). Moreover, Among other approaches, the recent work of Gott- the frequency operator is a collective observable of the fried (2000) shows that in a discrete quantum system whole ensemble. It may be possible to relate observables coupled with a continuous quantum system Born’s for- defined for such an infinite ensemble supersystem to the mula follows from the demand that the continuous sys- states of individual subsystems, but the frequency opera- tem should follow classical mechanics in the appropriate tor does not do this. This is well illustrated by the gedan- limit. A somewhat different strategy, with a focus on the ken experiment envisaged by Farhi et al. (1989). To pro- coincidences of the expected magnitude of fluctuations, vide a physical implementation of the frequency was proposed by Aharonov and Reznik (2002). operator they consider a version of the Stern-Gerlach In comparison with all of the above strategies, ‘‘prob- experiment where all the spins are attached to a com- abilities from envariance’’ is the most radically quantum, mon lattice. Thus during the passage through the inho- in that it ultimately relies on entanglement (which is still mogeneity of the magnetic field, the center of mass of sometimes regarded as ‘‘a paradox,’’ and ‘‘to be ex- the whole lattice is deflected by an angle proportional to plained’’; I have used it as an explanation). This may be the projection of the net magnetic moment associated the reason why it has not been discovered until now. The with the spins on the direction defined by the field gra- insight offered by envariance into the nature of igno- dient. The deflection is proportional to the eigenvalue of rance and information sheds new light on probabilities the frequency operator, which is then a collective in physics. The (very quantum) ability to prove the ig- observable—states of individual spins remain in super- norance of a part of a system by appealing to perfect positions, uncorrelated with anything outside. This diffi- knowledge of the whole may resolve some of the difficulty can be addressed with the help of decoherence culties of the classical approaches. (Zurek, 1998a), but using decoherence without justifying Born’s formula first is fraught with the danger of circu- VII. ENVIRONMENT AS A WITNESS larity. The measure of certainty seems to be a rather vague The emergence of classicality can be viewed either as concept. Yet Cox (1946) has demonstrated that Boolean a consequence of the widespread dissemination of the logic leads, after the addition of a few reasonable as- information about the pointer states through the envi- sumptions, to the definition of probabilities that, in a ronment, or as a result of the censorship imposed by sense, appear as an extension of the logical truth values. decoherence. So far I have focused on this second view, However, the rules of symbolic logic that underlie Cox’s defining existence as persistence—predictability in spite

Rev. Mod. Phys., Vol. 75, No. 3, July 2003 Wojciech Hubert Zurek: Decoherence, einselection, and the quantum origins of the classical 759 of the environmental monitoring. The predictability broadcast information about themselves spreading out sieve is a way of discovering states that are classical in their ‘‘copies’’ throughout the rest of the universe. Am- this sense (Zurek, 1993a, 1993b; Zurek, Habib, and Paz, plified information is easiest to amplify. This leads to 1993; Gallis, 1996). analogies with ‘‘fitness’’ in the Darwinian sense, and sug- A complementary approach focuses not on the sys- gests looking at einselection as a sort of natural selec- tem, but on the records of its state spread throughout tion. the environment. Instead of seeking the least-perturbed states one can ask what states of the system are easiest A. Quantum Darwinism to discover by looking at the environment. Thus the environment is no longer just a source of decoherence, but Consider the ‘‘bit-by-byte’’ example of Sec. IV. Spin S acquires the role of a communication channel with basis- system is correlated with the environment: dependent noise that is minimized by the preferred ͉␺SE͘ϭa͉↑͉͘00...0͘ϩb͉↓͉͘11...1͘ pointer states. This approach can be motivated by the old dilemma: ϭa͉↑͉͘E↑͘ϩb͉↓͉͘E↓͘. (7.1) On one hand, quantum states of isolated systems are The basis ͕͉↑͘,͉↓͖͘ of S is singled out by the redundancy purely ‘‘epistemic’’ (see, for example, Peres, 1993; Fuchs of the record. This can be illustrated by rewriting the and Peres, 2000). Quantum cryptography (Bennett and same ͉␺SE͘, DiVincenzo, 2000; Nielsen and Chuang, 2000, and refer- ͱ ences therein) uses this impossibility of determining the ͉␺SE͘ϭ͉᭪͑͘a͉00...0͘ϩb͉11...1͘) 2 unknown state of an isolated quantum system. On the ϩ͉ ͑͘ ͉ ͘Ϫ ͉ ͘ ͱ other hand, classical reality seems to be made up of a 00...0 b 11...1 )/ 2 quantum building blocks: States of macroscopic systems ͱ ϭ͉͑᭪͉͘E᭪͘ϩ͉ ͉͘E ͘)/ 2, (7.2) exist objectively; they can be determined by many ob- ͕͉᭪͘ ͉ ͖͘ servers independently, without being destroyed or re- in terms of the Hadamard transformed , . S ͉↑͘ ͉↓͘ prepared. So the question arises: How can objective One can find out whether is or from a small existence—the ‘‘reality’’ of the classical states—emerge subset of the environment bits. By contrast, states ͕͉᭪͘ ͉ ͖͘ from purely epistemic wave functions? , cannot be easily inferred from the environ- ͕͉E ͘ ͉E ͖͘ There is not much one can do about this in the case of ment. States ᭪ , are typically not even orthogo- ͗E ͉E ͘ϭ͉ ͉2Ϫ͉ ͉2 ͉ ͉2Ϫ͉ ͉2ϭ a single state of an isolated quantum system. But open nal, ᭪ a b . And even when a b 0, systems are subject to einselection and can bridge the the record in the environment is fragile. Only one rela- ͉E ͘ ͉E ͘ chasm dividing their epistemic and ontic roles. The most tive phase distinguishes ᭪ from in that case, in direct way to see this arises from the recognition of the contrast with multiple records of the pointer states in fact that we never directly observe any system. Rather, ͉E↑͘ and ͉E↓͘. Remarks that elaborate this observation we discover states of macroscopic systems from the im- follow. They correspond to several distinct measures of prints they make on the environment: A small fraction the analogs of the Darwinian fitness of the states. of the photon environment intercepted by our eyes is 1. Consensus and algorithmic simplicity often all that is needed. States that are recorded most redundantly in the rest of the universe (Zurek, 1983, From the state vector ͉␺SE͘, Eqs. (7.1) and (7.2), the 1998a, 2000) are also the easiest to discover. They can be observer can find the state of the quantum system just found out indirectly, from multiple copies of the evi- by looking at the environment. To accomplish this, the dence imprinted in the environment, without a threat to total N of the environment bits can be divided into their existence. Such states exist and are real; they can samples of n bits each, with 1ӶnӶN. These samples be found out without being destroyed as if they were can then be measured using observables that are the really classical. same within each sample, but that differ between Environmental monitoring creates an ensemble of samples. They may correspond, for example, to different ‘‘witness states’’ in the subsystems of the environment antipodal points in the Bloch spheres of the environ- that allows one to invoke some of the methods of the ment bits. In the basis ͕͉0͘,͉1͖͘ (or bases closely aligned statistical interpretation (Ballentine, 1970) while sub- with it) the record inferred from the bits of information verting its ideology—to work with an ensemble of objec- scattered in the environment will be easiest to come by. tive evidence of a state of a single system. From this Thus, starting from the environment part of ͉␺SE͘, Eq. ensemble of witness states one can infer the state of the (7.1), the observer can find out, with no prior knowl- quantum system that has led to such ‘‘advertising.’’ This edge, the state of the system. Redundancy of the record can be done without disrupting the einselected states. in the environment allows for a trial-and-error indirect The predictability sieve selects states that entangle approach while leaving the system untouched. least with the environment. Questions about predictabil- In particular, measurement of n environment bits in a ity simultaneously lead to states that are most redun- Hadamard transform of the basis ͕͉0͘,͉1͖͘, Eq. (7.2), dantly recorded in the environment. Indeed, this idea is yields a random-looking sequence of outcomes (i.e., ͉ϩ ͉Ϫ ͉Ϫ the essence of the ‘‘quantum Darwinism’’ we alluded to ͕ ͘1 , ͘2 ,..., ͘n͖). This record is algorithmically in the Introduction. The einselected pointer states are random. Its algorithmic complexity is of the order of its not only best at surviving the environment, they also length (Li and Vita`nyi, 1993):

Rev. Mod. Phys., Vol. 75, No. 3, July 2003 760 Wojciech Hubert Zurek: Decoherence, einselection, and the quantum origins of the classical

͑ E ͉ϩ Ϫ ͒Ӎ K ͗ n , ͘ n. (7.3) Given a fixed division of the environment into subsystems the action distance is a metric on the Hilbert By contrast, the algorithmic complexity of the measure- space (Zurek, 1998a). That is, ment outcomes in the ͕͉0͘,͉1͖͘ basis will be small: ⌬͉͑␺͘,͉␺͘)ϭ0, (7.7) ͑ E ͉ ͒Ӷ K ͗ n 0,1͘ n, (7.4) ⌬͉͑␺͘,͉␸͘)ϭ⌬͉͑␸͘,͉␺͘)у0, (7.8) since the outcomes will be either 00...0or11...1.The observer seeking the preferred states of the system by and the triangle inequality looking at the environment should then search for the ⌬͉͑␺͘,͉␸͘)ϩ⌬͉͑␸͘,͉␥͘)у⌬͉͑␺͘,͉␥͘) (7.9) minimal record size and thus, for the maximum redundancy in the environmental record. States of the system are all satisfied. that are recorded redundantly in the environment must In defining ⌬ it is essential to restrict rotations to the have survived repeated instances of environment moni- subspaces of the subsystems of the whole Hilbert space, toring, and are obviously robust and predictable. and to insist that the unitary operations used in defining The predictability we have utilized before to devise a distance act on these subspaces. It is possible to relax sieve to select preferred states is used here again, but in constraints on such unitary operations by allowing, for a different guise. Rather than search for predictable sets example, pairwise or even more complex interactions of states of the system, we are now looking for the between subsystems. Clearly, in the absence of any re- records of the states of the system in the environment. strictions the action required to rotate any ͉␺͘ into any Sequences of states of environment subsystems corre- ͉␸͘ would be no more than (␲/2)ប. Thus the constraints lated with pointer states are mutually predictable and imposed by the natural division of the Hilbert space of hence, collectively algorithmically simple. States that are the environment into subsystems play an essential role. predictable in spite of interactions with the environment The preferred states of the system can be sought by ex- are also easiest to predict from their impact on its state. tremizing the action distance between the corresponding The state of the form of Eq. (7.1) can serve as an record states of the environment. In simple cases [e.g., example of amplification. The generation of redundancy see ‘‘bit-by-byte,’’ Eq. (4.7), and below] the action dis- through amplification brings about the objective exis- tance criterion for preferred states coincides with the tence of the otherwise subjective quantum states. States predictability sieve definition (Zurek, 1998a). ͉↑͘ and ͉↓͘ of the system can be determined reliably from a small fraction of the environment. By contrast, to de- 3. Redundancy and mutual information termine whether the system was in a state ͉᭪͘ or ͉͘ one would need to detect all of the environment. Objectivity The most direct measure of the reliability of the envi- can be defined as the ability of many observers to reach ronment as a witness is the information-theoretic redun- consensus independently. Such consensus concerning dancy of einselection itself. When the environment states ͉↑͘ and ͉↓͘ is easily established—many (ϳN/n) monitors the system (see Fig. 4), the information about observers can independently measure fragments of the its state will spread to more and more subsystems of the environment. environment. This can be represented by the state vector ͉␺SE͘, Eq. (7.1), with increasingly long sequences of 0’s and 1’s in the record states. The record size, the num- 2. Action distance ber N of the subsystems of the environment involved, One measure of the robustness of environmental does not affect the density matrix of the system S. Yet, it records is the action distance (Zurek, 1998a). It is given obviously changes the accessibility and robustness of the by the total action necessary to undo the distinction be- information analogs of the Darwinian fitness. As an il- tween the states of the environment corresponding to lustration, let us consider c-shifts. One subsystem of the E different states of the system, subject to the constraints environment (say, 1) with the dimension of the Hilbert arising from the fact that the environment consists of space no less than that of the system,

subsystems. Thus to obliterate the difference between dimHE уdimHS, 1 ͉E↑͘ and ͉E↓͘ in Eq. (7.1), one needs to ‘‘flip’’ one by one N subsystems of the environment. That implies an ac- suffices to eradicate the off-diagonal elements of ␳S in tion, i.e., the least total angle by which a state must be the control basis. On the other hand, when N subsystems rotated, see Sec. II.B, of of the environment correlate with the same set of states S ␲ of , the information about these states is simulta- ⌬͉͑E ͘ ͉E ͘ ϭ ͫ បͬ neously accessible more widely. While ␳S is no longer ↑ , ↓ ) N . (7.5) 2 • changing, spreading of the information makes the exis- S By contrast a flip of phase of just one bit will reverse the tence of the pointer states of more objective—they are correspondence between the states of the system and easier to discover without being perturbed. those of the environment superpositions that make up Information-theoretic redundancy is defined as the difference between the least number of bits needed to ͉E᭪͘ and ͉E ͘ in Eq. (7.2). Hence uniquely specify the message and the actual size of the ␲ ⌬͉͑E ͘ ͉E ͘ ϭ ͫ បͬ encoded message. Extra bits allow for detection and cor- ᭪ , ) 1 . (7.6) 2 • rection of errors (Cover and Thomas, 1991). In our case,

Rev. Mod. Phys., Vol. 75, No. 3, July 2003 Wojciech Hubert Zurek: Decoherence, einselection, and the quantum origins of the classical 761 the message is the state of the system, and the channel is of) S that exist in E. The maximal redundancy ratio R the environment. The information about the system will I max is of course basis independent. H I often spread over all of the Hilbert space E, which is The information defined through the symmetric k , enormous compared to HS. The redundancy of the Eq. (7.11), is in general inaccessible to observers who record of the pointer observables of selected systems interrogate the environment one subsystem at a time can also be huge. Moreover, typical environments con- (Zurek, 2003a). It therefore makes sense to consider the sist of obvious subsystems (i.e., photons, atoms, etc.). It basis-dependent locally accessible information and de- is then useful to define the redundancy of the record by fine the corresponding redundancy ratio RJ using the number of times the information about the system J ϭJ͑S E ͒ϭ ͑S͒ϩ ͑E ͒Ϫ͓ ͑S͒ϩ ͑E ͉S͔͒ k : k H H k H H k . has been copied, or by how many times it can be inde- (7.15) pendently extracted from the environment. The conditional entropy must be computed in a specific In the simple example of Eq. (7.1) such a redundancy basis of the system [see Eq. (4.32)]. All of the other ratio R for the ͕͉↑͘,͉↓͖͘ basis will be given by N, the num- steps that have led to the definition of RI can now be ber of environment bits perfectly correlated with the ob- max repeated using J . In the end, a basis dependent viously preferred basis of the system. More generally, k RJ͕͉͑s͖͘)ϭRJ͑ HE ͒ (7.16) but in the same case of perfect correlation, we obtain k

ln͑dimHE͒ is obtained. RJ(͕͉s͖͘) quantifies the mutual information Rϭ ϭlog H dimHEϭN, (7.10) ͑ H ͒ dim S between the collection of subsystems HE of the environ- ln dim S k ment and the basis ͕͉s͖͘ of the system. We note that the where HE is the Hilbert space of the environment per- condition of nonoverlapping partitions guarantees that fectly correlated with the pointer states of the system. all of the corresponding measurements commute, and ͕͉᭪͘ ͉͖͘ On the other hand, with respect to the , basis, that the information can indeed be extracted indepen- the redundancy ratio for ͉␺SE͘ of Eq. (7.2) is only ϳ1 E dently from each environment fragment k . (see also Zurek, 1983, 2000). Redundancy measures the The preferred basis of S can now be defined by maxi- number of errors that can obliterate the difference be- mizing RJ(͕͉s͖͘) with respect to the selection of ͕͉s͖͘: tween two records, and in this basis one phase flip is R ϭ R ͕͉͑ ͖͘ clearly enough. This basis dependence of redundancy J max max J s ). (7.17) ͕͉s͖͘;͕ HE ͖ suggests an alternative strategy to seeking preferred k states. This maximum can be sought either by varying the basis To define R in general we can start with mutual infor- of the system only or (as is indicated above) by varying E mation between the subsystems of the environment k both the basis and the partition of the environment. and the system S. As we have already seen in Sec. IV, It remains to be seen whether and under what circum- the definition of mutual information in quantum me- stances the pointer basis ‘‘stands out’’ through its defini- chanics is not straightforward. The basis-independent tion in terms of RJ. The criterion for a well-defined set formula of pointer states ͕͉p͖͘ would be

I ϭI͑S E ͒ϭ ͑S͒ϩ ͑E ͒Ϫ ͑S E ͒ RJ ϭRJ͕͉͑p͖͘)ӷRJ͕͉͑s͖͘), (7.18) k : k H H k H , k (7.11) max is simple to evaluate [although it does have some strange where ͕͉s͖͘ are typical superpositions of states belonging features; see Eqs. (4.30)–(4.36)]. In the present context to different pointer eigenstates. it involves the joint density matrix This definition of preferred states directly employs the notion of multiplicity of records available in the environ- ␳SE ϭTrE E ␳SE , (7.12) JрI k / k ment. Since , it follows that R рR where the trace is carried out over all of the environ- J max I max . (7.19) E ment except for its singled-out fragment k . In the ex- The important feature of either version of R that makes ample of Eq. (7.1), for any of the environment bits, them useful for our purpose is their independence on 2 2 ␳SE ϭ͉a͉ ͉↑͗͘↑͉͉0͗͘0͉ϩ͉b͉ ͉↓͗͘↓͉͉1͗͘1͉. H(S). The dependence on H(S) is in effect normalized k out of R. R characterizes the fan-out of information Given the partitioning of the environment into sub- about the preferred basis throughout the environment, systems, the redundancy ratio can be defined as without reference to what is known about the system. The usual redundancy (in bits) is then ϳR H(S), al- R ϭ I͑S E ͒ ͑S͒ • I͕͑ HE ͖͒ ͚ : k /H . (7.13) though other implementations of this program (Ollivier, k k Poulin, and Zurek, 2002) employ different measures of When R is maximized over all of the possible partitions, redundancy, which may be even more accurate than the R ϭ R redundancy ratio we have described above. Indeed, I max max ͕ HE ͖ (7.14) k what is important here is the general idea of measuring ͕ HE ͖ k the classicality of quantum states through the number of R is obtained. Roughly speaking, I max is the total num- copies they imprint throughout the universe. This is a ber of copies of the information about (the optimal basis very Darwinian approach. We define classicality related

Rev. Mod. Phys., Vol. 75, No. 3, July 2003 762 Wojciech Hubert Zurek: Decoherence, einselection, and the quantum origins of the classical to einselection in ways reminiscent of ‘‘fitness’’ in natu- conform to this binary convention, and the decoherence ral selection: states that spawn most of the (information- times are short (Tegmark, 2000). Thus, even if a cell of theoretic) progeny are the most classical. the observer entangles through a premeasurement with a pure quantum state, the record will become effectively 4. Redundancy ratio rate classical almost instantly. As a result, it will be impossible to ‘‘read it off’’ in any basis except for the einse- The rate of change of redundancy is of interest as lected one. This censorship of records is the key differ- another measure of ‘‘fitness,’’ perhaps closest to the defi- ence between the existential interpretation and Everett’s nitions of fitness used in modeling natural selection. Re- dundancy can increase either as a result of interactions original many-worlds interpretation. between the system and the environment, or because the Decoherence treats the observer as any other macro- environment already correlated with S is passing on the scopic quantum system. There is, however, one feature information to more distant environments. In this sec- distinguishing observers from the rest of the universe: ond case ‘‘genetic information’’ is passed on by the They are aware of the content of their memory. Here we ‘‘progeny’’ of the original state. Even an observer con- are using ‘‘aware’’ in a down-to-earth sense: Quite sim- sulting the environment becomes a part of such a more- ply, observers know what they know. Their information- distant environment. The redundancy rate is defined as processing machinery (which must underlie higher functions of the mind such as ‘‘consciousness’’) can readily d consult the contents of their memory. R˙ ϭ R. (7.20) dt The information stored in memory comes with strings attached. The physical state of the observer is described Either basis-dependent or basis-independent versions of R˙ in part by the data in his records. There is no informa- may be of interest. tion without representation. The information the ob- In general, it may not be easy to compute either R or R˙ server has could be, in principle, deduced from his physi- exactly. This is nevertheless possible in models [such cal state. The observer is, in part, information. as those leading to Eqs. (7.1) and (7.2)]. The simplest Moreover, this information encoded in states of macro- illustrative example corresponds to the c-NOT model of scopic quantum systems (neurons) is by no means secret. decoherence in Fig. 4. One can imagine that the con- As a result of the lack of isolation, the environment, secutive record bits get correlated with the two branches ͉ ͘ ͉ ͘ having redundant copies of the relevant data, knows in (corresponding to 0 and 1 in the control) at discrete detail everything the observer knows. Configurations of moments of time. R(t) would then be the total number R˙ neurons in our brains, while at present undecipherable, of c-NOT’s that have acted over time t, and is the num- are, in principle, as objective and as widely accessible as ber of new c-NOT’s added per unit time. the information about the states of other macroscopic The redundancy rate measures information flow from objects. the system to the environment. Note that, after the first R The observer is what he knows. In the unlikely case of c-NOT in the example of Eqs. (7.1) and (7.2), I will a flagrantly quantum input the physical state of the ob- jump immediately from 0 to 2 bits, while the basis spe- R server’s memory will decohere, resulting almost instantly cific J will increase from 0 to 1. In our model this in the einselected alternatives, each of them represent- initial discrepancy [which reflects quantum discord, Eq. I J ing simultaneously both the observer and his memory. (4.36), between and ] will disappear after the second The ‘‘advertising’’ of this state throughout the environ- c-NOT. R R˙ ment makes it effectively objective. Finally, we note that and, especially, can be used An observer perceiving the universe from within is in to introduce new predictability criteria: The states (or a very different position than an experimental physicist the observables) that are being recorded most redun- studying a state vector of a quantum system. In a labo- dantly are the obvious candidates for the objective ratory, the Hilbert space of the investigated system is states, and therefore for the classical states. typically tiny. Such systems can be isolated, so that often the information loss to the environment can be pre- B. Observers and the existential interpretation vented. Then the evolution is unitary. The experimental- ist can know everything there is to know about it. von Neumann (1932), London and Bauer (1939), and Common criticisms of the approach advocated in this Wigner (1963) have all appealed to the special role of paper are based on an unjustified extrapolation of the the conscious observer. Consciousness was absolved above laboratory situation to the case of the observer from following unitary evolution, and thus, could col- who is a part of the universe. Critics of decoherence lapse the wave packet. Quantum formalism has led us to often note that the differences between the laboratory a different view that nevertheless allows for a compat- example above and the case of the rest of the universe ible conclusion. In essence, macroscopic systems are are merely quantitative: the system under investigation open, and their evolution is almost never unitary. is bigger, etc. So why cannot one analyze, they ask, in- Records maintained by the observers are subject to ein- teractions of the observer and the rest of the universe as selection. In a binary alphabet decoherence will allow before, for a small isolated quantum system? for only the two logical states and prohibit their super- In the context of the existential interpretation the positions (Zurek, 1991). For human observers, neurons analogy with the laboratory is, in effect, turned upside

Rev. Mod. Phys., Vol. 75, No. 3, July 2003 Wojciech Hubert Zurek: Decoherence, einselection, and the quantum origins of the classical 763 down: For, now the observer (or the apparatus, or any- vello, 1999; Jozsa, 2002) in spite of the no-cloning theo- thing effectively classical) is continuously monitored by rem (Dieks 1982; Wootters and Zurek, 1982). In that the rest of the universe. Its state is repeatedly forced limit, and given the same reasonable constraints on the into the einselected states, and very well (very redun- nature of the interactions and on the structure of the dantly) known to the rest of the universe. environment that underlies that definition of R, it would The higher functions of observers, e.g., consciousness, take infinite resources such as action, Eqs. (7.5)–(7.9), to etc., may be at present poorly understood, but it is safe hide or subvert evidence of such an objective history. to assume that they reflect physical processes in the There are differences and parallels between the rela- information-processing hardware of the brain. Hence tively objective histories introduced here and the consis- mental processes are in effect objective, since they must tent histories proposed by Griffiths (1984, 1996), and in- reflect conditional quantum dynamics of open system— vestigated by Gell-Mann and Hartle (1990, 1993, 1997), observer’s network of neurons—and, hence leave an in- Omne`s (1988, 1992, 1994), Halliwell (1999), and others delible imprint on the environment. The observer has no (Dowker and Kent, 1996; Kiefer, 1996). Such histories chance of perceiving either his memory, or any other are defined as time-ordered sequences of projection op- macroscopic part of the universe in some arbitrary su- 1 2 n erators P␣ (t ),P␣ (t ),...,P␣ (t ) and are abbreviated perposition. Moreover, the memory capacity of observ- 1 1 2 2 n n ers is miniscule compared to the information content of ͓P␣͔. Consistency is achieved when they can be com- the universe. So, while observers may know the exact bined into coarse-grained sets (where the projectors de- state of the laboratory systems, their records of the uni- fining a coarse-grained set are given by the sums of the verse will be very fragmentary. By contrast, the universe projectors in the original set) while obeying probability has enough memory capacity to acquire and maintain sum rules: The probability of a bundle of histories detailed records of the states of macroscopic systems should be a sum of the probabilities of the constituent and their histories. Thus, indeed, it appears that con- histories. The corresponding condition can be expressed sciousness does not follow a unitary quantum evolution, in terms of the decoherence functional (Gell-Mann and as the conditional dynamics that implements such Hartle, 1990): ‘‘higher functions’’ must be subject to decoherence and ͓͑ ͔ ͓ ͔͒ einselection (see also Tegmark, 2000). As promised, we D P␣ , P␤ have in a sense recovered postulates of von Neumann, ϭ ͓ n ͑ ͒ 1 ͑ ͒␳ 1 ͑ ͒ n ͑ ͔͒ Tr P␣ tn ...P␣ t1 P␤ t1 ...P␤ tn . London and Bauer, and Wigner, and we have done that n 1 1 n without involing any ‘‘extraphysical’’ postulates. (7.21) Above, the state of the system of interest is described by C. Events, records, and histories the density matrix ␳. Griffiths’ condition is equivalent to the vanishing of the real part of the expression above, ͓ ͔ ͓ ͔ ϭ ␦ Suppose that instead of a monotonous record se- Re͕D( P␣ , P␤ )͖ p␣ ␣,␤ . As Gell-Mann and Hartle quence in the environment basis corresponding to the (1990) emphasize, it is more convenient, and in the con- pointer states of the system ͕͉↑͘,͉↓͖͘ implied by Eq. (7.1) text of an emergent classicality more realistic, to require ͓ ͔ ͓ ͔ ϭ ␦ the observer looking at the environment detects instead that D( P␣ , P␤ ) p␣ ␣,␤ . Both weaker and stronger conditions for the consistency of histories were 000...0111...1000...0111.... considered (Goldstein and Page, 1995; Gell-Mann and Given the appropriate additional assumptions, such se- Hartle, 1997). The problem with all of them is that the quences consisting of long stretches of record 0’s and 1’s resulting histories are very subjective: Given an initial justify inference of the history of the system. Let us fur- density matrix of the universe it is in general quite easy ther assume that the observer’s records come from inter- to specify many different, mutually incompatible consis- cepting a small fragment of the environment. Other ob- tent sets of histories. This subjectivity leads to serious servers will then be able to consult their independently interpretational problems (d’Espagnat, 1989, 1995; accessible environmental records, and will infer (more Dowker and Kent, 1996). Thus a demand for exact con- or less) the same history. Thus, in view of the prepon- sistency as one of the conditions for classicality is both derance of evidence, history defined as a sensible infer- uncomfortable (overly restrictive) and insufficient (since ence from the available records can be probed by many the resulting histories are very nonclassical). Moreover, observers independently, and can be regarded as classi- coarse grainings that help secure approximate consis- cal and objective. tency have to be, in effect, guessed at. The redundancy ratio of the records R is a measure of The attitudes adopted by the practitioners of the this objectivity. Note that this relatively objective exis- consistent-histories approach in view of its unsuitability tence (Zurek, 1998a) is an operational notion, quantified for the role of the cornerstone of emergent classicality by the number of times the state of the system can be differ. Initially, before difficulties became apparent, it determined independently, and not some absolute objec- was hoped that such an approach would answer all of tivity. However, and in a sense that can be rigorously the interpretational questions, perhaps when supple- defined, relative objectivity tends to absolute objectivity mented by a subsidiary condition, i.e., some assumption in the limit R→ϱ. For example, cloning of unknown about favored coarse grainings. At present, some still states becomes possible (Bruss, Ekert, and Macchia- aspire to the original goals of deriving classicality from

Rev. Mod. Phys., Vol. 75, No. 3, July 2003 764 Wojciech Hubert Zurek: Decoherence, einselection, and the quantum origins of the classical consistency alone. Others may uphold the original aims A measurement reveals to the observer his branch of of the program, but they also generally rely on the universal state vector. The correlations established environment-induced decoherence, using in calculations alter the observer’s state, his records, and ‘‘attach’’ him variants of models we have presented in this paper. This to this branch. He will share it with other observers who strategy has been quite successful—after all, decoher- examined the same set of observables, and who have ence leads to consistency. For instance, the special role recorded compatible results. of the hydrodynamic observables (Gell-Mann and It is also possible to imagine a stubborn observer who Hartle, 1990; Dowker and Halliwell, 1992; Brun and insists on measuring either the relative phase between Hartle, 1999; Halliwell, 1999) can be traced to their pre- the two obvious branches of the environment in Eq. dictability, or to their approximate commutativity with (7.2), or the state of the environment in the Hadamard- ͕͉ϩ͘ ͉Ϫ͖͘ the total Hamiltonian [see Eq. (4.41)]. On the other transformed basis , . In either case the distinction hand, the original goals of Griffiths (1984, 1996) have between the two outcomes could determine the state of ͕͉᭪͘ ͉͖͘ been more modest. Using consistent histories, one can the spin in the , basis. However, in that basis R ϭ discuss the sequences of events in an evolving quantum J 1. Hence, while, in principle, these measurements system without logical contradictions. The ‘‘golden can be carried out and yield the correct result, the infor- ͕͉᭪͘ ͉͖͘ middle’’ is advocated by Griffiths and Omne`s (1999) mation concerning the , basis is not redundant who regard consistent histories as a convenient lan- and therefore not objective: Only one stubborn observer R guage, rather than as an explanation of classicality. can access it directly. As a result J will decrease. R ͕͉᭪͘ ͉ ͖͘ The origin of effective classicality can be traced to Whether J( , ) will become larger than R ͕͉↑͘ ͉↓͖͘ decoherence and einselection. As was noted by Gell- J( , ) was before the measurement of the stub- Mann and Hartle (1990), and elucidated by Omne`s born observer will depend on a detailed comparison of (1992, 1994) decoherence suffices to ensure approximate the initial redundancy with the amplification involved, consistency. But consistency is both not enough and too the decoherence, etc. much; it is too easy to accomplish, and does not neces- There is a further significant difference between the sarily lead to classicality (Dowker and Kent, 1996). two stubborn observers considered above. When the ob- What is needed instead is the objectivity of events and server measures the phase between the two sequences of their time-ordered sequences—their histories. As we 0’s and 1’s in Eq. (7.2), correlations between the bits of have seen above, both can appear as a result of einselec- the environment remain. Thus, even after his measure- tion. ment, one could find relatively objective evidence of the We have already provided an operational definition of past event—the past state of the spin—and, in more the relatively objective existence of quantum states. It is complicated cases, of the history. On the other hand, ͕͉ϩ͘ ͉Ϫ͉͘ easy to apply it to events and histories: When many ob- measurement of all the environment bits in the , servers can independently gather compatible evidence basis will obliterate evidence of such a past. concerning an event, we call it relatively objective. Rela- The relatively objective existence of events is the tively objective history is then a time-ordered sequence strongest condition we have considered here. It is a con- of relatively objective events. sequence of the existence of multiple records of the Monitoring of the system by the environment leads to same set of states of the system. It allows for such mani- decoherence and einselection. It will also typically lead festations of classicality as unimpeded cloning. It implies to redundancy and hence to an effectively objective clas- einselection of states most closely monitored by the en- sical existence in the sense of quantum Darwinism. Ob- vironment. Decoherence is clearly weaker and easier to servers can independently access redundant records of accomplish. events and histories imprinted in the environmental de- ‘‘The past exists only insofar as it is recorded in the grees of freedom. The number of observers who can ex- present’’ (a dictum often repeated by Wheeler) may the amine evidence etched in the environment can be of the best summary of the above discussion. The relatively objective reality of a few selected observables in our famil- order of, and is bounded from above by, RJ. Redun- dancy is a measure of this objectivity and classicality. iar universe is measured by their ‘‘Darwinian’’ fitness—by the redundancy with which they are re- As observers record their data, RJ changes. Consider an observer who measures the ‘‘right observable’’ of E corded in the environment. This multiplicity of available [i.e., the one with the eigenstates ͉0͘,͉1͘ in the example of copies of the same information can be regarded as a Eq. (7.1)]. Then his records and, as his records decohere, consequence of amplification, and as a cause of indelibil- also their environment, become a part of the evidence, ity. Multiple records safeguard the objectivity of our and are correlated with the preferred basis of the sys- past. tem. Consequently RJ computed from Eq. (7.14) increases. Every interaction that increases the number of VIII. DECOHERENCE IN THE LABORATORY records also increases RJ. This is obvious for the ‘‘primary’’ interactions with the system, but it is also true for The biggest obstacle in the experimental study of de- the secondary, tertiary, etc., acts of replication of the in- coherence is, paradoxically, its effectiveness. In the mac- formation obtained from the observers who recorded roscopic domain only the einselected states survive. the primary state of the system, from the environment, Their superpositions are next to impossible to prepare. from the environment of the environment, and so on. In the mesoscopic regime one may hope to adjust the

Rev. Mod. Phys., Vol. 75, No. 3, July 2003 Wojciech Hubert Zurek: Decoherence, einselection, and the quantum origins of the classical 765 size of the system, and thus, interpolate between quan- Weinfurter, and Zeilinger, 1995, for implementations of tum and classical. The strength of the coupling to the this ‘‘quantum erasure’’ trick of Hillery and Scully, environment is the other parameter one may employ to 1983). Similar experiments have also been carried out control the decoherence rate. using neutron interferometry (see, for example, Rauch, One of the key consequences of monitoring by the 1998). In all of these experiments one is dealing with a very environment is the inevitable introduction of the simplified situation involving a single microsystem and a Heisenberg uncertainty into the observable complemen- single ‘‘unit’’ of decoherence (RJϳ1) caused by a single tary to the one that is monitored. One can simulate such quantum of the environment. Experiments on a meso- uncertainty without any monitoring environment by in- scopic system monitored by the environment are obvi- troducing classical noise. In each specific run of the ex- ously much harder to devise. Nevertheless, Haroche, periment, for each realization of time-dependent noise, Raimond, Brune, and their colleagues at the Ecole Nor- the quantum system will evolve deterministically. How- male Supe´rieure (Brune et al., 1996; Haroche, 1998; Rai- ever, after averaging over different noise realizations, as mond, Brune, and Haroche, 2001) have carried out a it is discussed in Sec. IV.C, the evolution of the density spectacular experiment of this type, yielding solid evi- matrix describing an ensemble of systems may approxi- dence in support of the basic tenets of the environment- mate decoherence due to an entangling quantum envi- induced transition from quantum to classical. Their sys- ronment. In particular, the master equation may be es- tem is a microwave cavity. It starts in a coherent state sentially the same as that for true decoherence, although with an amplitude corresponding to a few photons. the interpretational implications are more limited. Yet, A Schro¨ dinger-cat state is created by introducing an using such strategies one can simulate much of the dy- atom in a superposition of two Rydberg states, ͉ϩ͘ϭ͉0͘ namics of open quantum systems. ϩ͉1͘: The atom passing through the cavity puts its re- The strategy of simulating decoherence can be taken fractive index in a superposition of two values. Hence further: Not just the effect of the environment, but also the phase of the coherent state shifts by the amount the dynamics of the quantum system can be simulated correlated with the state of the atom, creating an en- by classical means. This can be accomplished when clas- tangled state: sical wave phenomena follow equations of motion re- ͉→͉͑͘0͘ϩ͉1͘)⇒͉%͉͘0͘ϩ͉&͉͘1͘ϭ͉␽͘. (8.1) lated to the Schro¨ dinger equation. We shall discuss experiments that fall into all of the above categories. Arrows indicate relative phase-space locations of coher- Last but not least, while decoherence—through ent states. States of the atom are ͉0͘ and ͉1͘. The ‘‘Schro¨ - einselection—helps solve the measurement problem, it dinger kitten’’ is prepared from this entangled state by is also a major obstacle to quantum information process- measuring the atom in the ͕͉ϩ͘,͉Ϫ͖͘ basis: ing. We shall thus end this section briefly describing ͉␽͘ϭ͉͑%͘ϩ͉&͘)͉ϩ͘ϩ͉͑%͘Ϫ͉&͘)͉Ϫ͘. (8.2) strategies that may allow one to tame decoherence. Thus the atom in the state ͉ϩ͘ implies preparation of a A. Decoherence due to entangling interactions ‘‘positive cat’’ ͉]͘ϭ͉%͘ϩ͉&͘ in the cavity. Such superpositions of coherent states could survive forever if there Several experiments fit this category, and more have was no decoherence. However, radiation leaks out of the been proposed. Decoherence due to emission or scatter- cavity. Hence the environment acquires information ing of photons has been investigated by the MIT group about the state inside. Consequences are tested by pass- of David Pritchard (Chapman et al., 1995) using atomic ing another atom in the state ͉ϩ͘ϭ͉0͘ϩ͉1͘ through the interferometry. Emission or scattering deposits a record cavity. In the absence of decoherence the state would in the environment. It can store information about the evolve as path of the atom providing the photon wavelength is ͉]͉͘ϩ͘ϭ͉͑%͘ϩ͉&͘)͉͑0͘ϩ͉1͘) shorter than the separation between two of the atoms. In the case of emission this record is not redundant, ⇒͉͑↑͉͘0͘ϩ͉→͉͘1͘) since the atom and photon are simply entangled, RJ ϳ1, in any basis. Scattering may involve more photons, ϩ͉͑→͉͘0͘ϩ͉↓͉͘1͘) and a recent careful experiment (Kokorowski et al., ϭ͉͑↑͉͘0͘ϩ͉↓͉͘1͘)ϩͱ2͉→͉͘ϩ͘. (8.3) 2001) has confirmed the saturation of decoherence rate at distances in excess of the photon wavelength (Gallis Above we have omitted the overall normalization, but and Fleming, 1990; Anglin, Paz, and Zurek, 1997). retained the (essential) relative amplitude. There is an intimate connection between interference For the above state, detection of ͉ϩ͘ in the first (pre- and complementarity in the two-slit experiment on one paratory) atom implies the conditional probability of de- hand, and entanglement on the other (Wootters and tection of ͉ϩ͘, pϩ͉ϩϭ3/4, for the second (test) atom. De- Zurek, 1979). Consequently, appropriate measurements coherence will suppress the off-diagonal terms of the of the photon allow one to restore interference fringes in density matrix so that, some time after the preparation, ␳ ͉]͗͘]͉ the conditional subensembles corresponding to a defi- cavity that starts, say, as becomes nite phase between the two photon trajectories (see es- ␳ ϭ͉͑%͗͘%͉ϩ͉&͗͘&͉͒/2 pecially Chapman et al., 1995, as well as Kwiat, Stein- cavity berg, and Chiao, 1992; Pfau et al., 1994; Herzog, Kwiat, ϩz͉͑%͗͘&͉ϩ͉&͗͘%͉͒/2. (8.4)

Rev. Mod. Phys., Vol. 75, No. 3, July 2003 766 Wojciech Hubert Zurek: Decoherence, einselection, and the quantum origins of the classical

When zϭ0 the conditional probability is p(ϩ͉ϩ)ϭ1/2. tor the rotating-wave approximation blurs the distinc- In the intermediate cases intermediate values of this tion between x and p, leading to einselection of coherent and other relevant conditional probabilities are pre- states.) The phase noise would arise in an environment dicted. The rate of decoherence, and consequently, the monitoring the number operator, thus leading to uncer- time-dependent value of z can be estimated from the tainty in phase. Consequently, number eigenstates are cavity quality factor Q, and from the data about the co- einselected. herent state initially present in the cavity. The decoher- The applied noise is classical, and the environment ence rate is a function of the separation of the two com- does not acquire any information about the ion (RI ͉]͘ ponents of the cat . Experimental results agree with ϭ0). Thus, following a particular realization of the predictions. noise, the state of the system is still pure. Nevertheless, The discussion above depends on the special role of an ensemble average over many noise realizations is coherent states. Coherent states are einselected in har- represented by the density matrix that follows an appro- monic oscillators, and hence, in underdamped bosonic priate master equation. Thus, as Wineland, Monroe, and fields (Anglin and Zurek, 1996). Thus they are the their colleagues note, decoherence simulated by classical pointer states of the cavity. Their special role is recognized implicitly above: If number eigenstates were ein- noise could be in each individual case—for each selected, predictions would obviously be quite different. realization—reversed by simply measuring the corre- Therefore, while the Ecole Normale Supe´rieure experi- sponding time-dependent noise either beforehand or af- ment is focused on the decoherence rate, confirmation terwards, and then applying the appropriate unitary of the predicted special role of coherent states in transformation to the state of the system. By contrast, in bosonic fields is its important (albeit implicit) corollary. the case of entangling interactions, two measurements, one preparing the environment before the interaction with the environment, the other following it, would be B. Simulating decoherence with classical noise the least required for a chance of undoing the effect of decoherence. From the fundamental point of view, the distinction The same two papers study the decay of a superposi- between cases in which decoherence is caused by entan- tion of number eigenstates ͉0͘ and ͉2͘ due to an indirect gling interactions with the quantum state of the environ- coupling with the vacuum. This proceeds through en- ment and in which it is simulated by classical noise in the tanglement with the first-order environment (that, in ef- observable complementary to the pointer is essential. fect, consists of the other states of the harmonic oscilla- However, from the engineering point of view (adopted, tor) and a slower transfer of information to the distant for example, by the practitioners of quantum computa- environment. Dynamics involving the system and its tion, see Nielsen and Chuang, 2000, for a discussion) this first-order environment lead to nonmonotonic behavior may not matter. For instance, quantum error-correction of the off-diagonal terms representing coherence. Fur- techniques (Shor, 1995; Steane, 1996; Preskill, 1999) are ther studies of decoherence in the ion-trap setting are capable of dealing with either. Moreover, experimental likely to follow, since this is an attractive implementa- investigations of this subject often involve both. tion of the quantum computer (Cirac and Zoller, 1995). The classic experiment in this category was carried out recently by Wineland, Monroe, and their collabora- tors (Myatt et al., 2000; Turchette et al., 2000). They used an ion trap to study the behavior of individual ions in a 1. Decoherence, noise, and quantum chaos Schro¨ dinger-cat state (Monroe et al., 1996) under the influence of injected classical noise. They also embarked Following a proposal of Graham, Schlautmann, and on a preliminary study of ‘‘environment engineering.’’ Zoller (1993) Raizen and his group (Moore et al., 1994) Superpositions of two coherent states as well as of used a one-dimensional (1D) optical lattice to imple- number eigenstates were subjected to simulated high- ment a variety of 1D chaotic systems including the temperature amplitude and phase ‘‘reservoirs.’’ This was ‘‘standard map.’’ Various aspects of the behavior ex- done through time-dependent modulation of the self- pected from a quantized version of a classically chaotic Hamiltonian of the system. For the amplitude noise system were subsequently found, including, in particular, these are, in effect, random fluctuations of the location dynamical localization (Reichl, 1992; Casati and Chir- of the minimum of the harmonic trap. Phase noise cor- ikov, 1995a). responds to random fluctuations of the trap frequency. Dynamical localization establishes, in a class of driven In either case, the resulting loss of coherence is well quantum chaotic systems, a saturation of momentum described by the exponential decay with time, with an dispersion, and leads to a characteristic exponential exponent that scales with the square of the separation form of its distribution (Casati and Chirikov, 1995a). Lo- between the two components of the macroscopic quan- calization is obviously a challenge to the quantum- tum superposition [e.g., Eq. (5.34)]. The case of the am- classical correspondence, since in these very same sys- plitude noise approximates decoherence in quantum tems the classical prediction has the momentum Brownian motion in that the coordinate is monitored by dispersion growing unbounded, more or less with the the environment, and hence, the momentum is per- square root of time. However, localization sets in after ϳបϪ␣ ␣ϳ turbed. (Note that in an underdamped harmonic oscilla- tL , where 1 (rather than on the much shorter

Rev. Mod. Phys., Vol. 75, No. 3, July 2003 Wojciech Hubert Zurek: Decoherence, einselection, and the quantum origins of the classical 767

Ϫ1 tបϳln ប that we have discussed in Sec. III) so it can be motion that are based on a dipole approximation. How- ignored for macroscopic systems. On the other hand, its ever, it is in good accord with more sophisticated discus- signature is easy to detect. sions that recognize that, for separations of the order of Demonstration of dynamical Anderson localization in the prevalent wavelength in the environment, the dipole the optical-lattice implementation of the ␦-kicked rotor approximation fails and other more complicated behav- and related studies of quantum chaos have been a sig- iors can set in (Gallis and Fleming, 1990; Anglin, Paz, nificant success (Moore et al., 1994). More recently, the and Zurek, 1997; Paz and Zurek, 1999). A similar result attention of both Raizen and his group in Texas as well in a completely quantum case was obtained by Koko- as of Christensen and his group in New Zealand has rowski et al. (2001) using atomic interferometry. shifted towards the effect of decoherence on quantum chaotic evolution (Ammann et al., 1998; Klappauf et al., C. Taming decoherence 1998). In all of the above studies the state of the chaotic In many of the applications of quantum mechanics the system (␦-kicked rotor) was perturbed by spontaneous quantum nature of the information stored or processed emission from the trapped atoms, which was induced by needs to be protected. Thus decoherence is an enemy. decreasing the detuning of the lasers used to set up the Quantum computation is an example of this situation. A optical lattice. In addition, noise was occasionally intro- quantum computer can be thought of as a sophisticated duced into the potential. Both groups found that, as a interference device that works by performing in parallel result of spontaneous emission, localization disappears, a coherent superposition of a multitude of classical com- although the two studies differ in some of the details. putations. Loss of coherence would disrupt the quantum More experiments, including some that allow gentler parallelism essential for the expected speedup. forms of monitoring by the environment (rather than In the absence of the ideal—a completely isolated ab- spontaneous emission noise) appear to be within reach. solutely perfect quantum computer, something easy for a In all of the above cases one deals, in effect, with a theorist to imagine but impossible to attain in the large ensemble of identical atoms. While each atom suf- laboratory—one must deal with imperfect hardware fers repeated disruptions of its evolution due to sponta- ‘‘leaking’’ some of its information to the environment. neous emission, the ensemble evolves smoothly and in And maintaining isolation while simultaneously achiev- accord with the appropriate master equation. The situa- ing a reasonable ‘‘clock time’’ for the quantum computer tion is reminiscent of decoherence simulated by noise. is likely to be difficult since both are in general con- Indeed, experiments that probed the effect of classical trolled by the same interaction [although there are ex- noise on chaotic systems were carried out earlier (Koch, ceptions; for example, in the ion-trap proposal of Cirac 1995). They were, however, analyzed from a point of and Zoller (1995) interaction is in a sense ‘‘on demand,’’ view that does not readily shed light on decoherence. and is turned on by the laser coupling the internal states A novel experimental approach to decoherence and of ions with the vibrational degree of freedom of the ion to irreversibility in open complex quantum systems has chain]. been pursued by Levstein, Pastawski, and their col- The need for error correction in quantum computa- leagues (Levstein, Usaj, and Pastawski, 1998; Levstein tion was realized early on (Zurek, 1984b) but methods et al., 2000). Using NMR techniques they investigated for accomplishing this goal have evolved dramatically the reversibility of dynamics by implementing a version from the Zeno effect suggested then to the very sophis- of spin echo. This promising ‘‘Loschmidt echo’’ ap- ticated (and much more effective) strategies in recent proach has led to renewed interest in the issues that years. This is fortunate. Without error correction even touch on quantum chaos, decoherence, and related sub- fairly modest quantum computations (such as factoring jects (see, for example, Jacquod, Silvestrov, and Beenak- the number 15 in an ion trap with imperfect control of ker, 2001; Jalabert and Pastawski, 2001; Gorin and Selig- the duration of the laser pulses) go rapidly astray as a man, 2002; Prosen and Seligman, 2002). consequence of relatively small imperfections (Miquel, Paz, and Zurek, 1997). Three different, somewhat overlapping, approaches 2. Analog of decoherence in a classical system that aim to control and tame decoherence, or to correct Both the system and the environment are effectively errors caused by decoherence or by the other imperfec- classical in the last category of experiments, represented tions of the hardware, have been proposed. We summa- by the work of Cheng and Raymer (1999). They have rize them very briefly, spelling out main ideas and point- investigated the behavior of transverse spatial coherence ing out references that discuss them in greater detail. during the propagation of an optical beam through a 1. Pointer states and noiseless subsystems dense, random dielectric medium. This problem can be modeled by a Boltzmann-like transport equation for the The most straightforward strategy to suppress deco- Wigner function of the wave field, and exhibits a char- herence is to isolate the system of interest (e.g., the acteristic increase of decoherence rate with the square quantum computer). Failing that, one may try to isolate of the spatial separation, followed by saturation at suffi- some of its observables with degenerate pointer sub- ciently large distances. This saturation contrasts with the spaces, which then constitute niches in the Hilbert space simple models of decoherence in quantum Brownian of the information-processing system that do not get dis-

Rev. Mod. Phys., Vol. 75, No. 3, July 2003 768 Wojciech Hubert Zurek: Decoherence, einselection, and the quantum origins of the classical

rupted in spite of the coupling to the environment. tum computer proposed by Cirac and Zoller (1995). The Decoherence-free subspaces are thus identical in con- answer given by the theory is, of course, that the choice ception with the pointer subspaces introduced some of the einselected basis is predicated on the details of time ago (Zurek, 1982), and satisfy (exactly or approxi- the situation and, in particular, on the nature of the in- mately) the same Eqs. (4.22) and (4.41) or their equiva- teraction between the system and the environment lents [given, for example, in terms of ‘‘Kraus operators’’ (Zurek, 1981, 1982, 1993a). Yet Poyatos, Cirac, and Zol- (Kraus, 1983)] that represent the nonunitary conse- ler (1996) have suggested a scheme suitable for imple- quences of the interaction with the environment in the mentation in an ion trap, in which interaction with the Lindblad (1976) form of the master equation. environment, and, in accord with Eq. (4.41), the pointer Decoherence-free subspaces were (re)discovered in the basis itself, can be adjusted. The key idea is to recognize context of quantum information processing. They ap- that the effective coupling between the vibrational de- pear as a consequence of an exact or approximate sym- grees of freedom of an ion (the system) and the laser metry of the Hamiltonians that govern the evolution of light (which plays the role of the environment) is given the system and its interaction with the environment (Za- by nardi and Rasetti, 1997; Duan and Guo, 1998; Lidar ⍀ Ϫ ␻ ␻ ϩ et al., 1999; Zanardi, 1998, 2001). ϭ ͑␴ i Ltϩ␴ i Lt͒ ͓␬͑ ϩ ͒ϩ␾͔ Hint ϩe Ϫe sin a a . An active extension of this approach aimed at finding 2 quiet corners of the Hilbert space is known as dynamical (8.5) ⍀ ␻ decoupling. There the effectively decoupled subspaces Above, is the Rabi frequency, L the laser frequency, are induced by time-dependent modifications of the evo- ␾ is related to the relative position of the center of the lution of the system deliberately introduced from the trap with respect to the laser standing wave, and ␬ is the outside by time-dependent Hamiltonians and/or mea- Lamb-Dicke parameter of the transition, while ␴Ϫ(␴ϩ) surements (see, for example, Viola and Lloyd, 1998; Za- and a(aϩ) are the annihilation and creation operators of nardi, 2001). A further generalization and unification of the atomic transition and of the harmonic oscillator rep- various techniques leads to the concept of noiseless resenting ion in the trap, respectively. ␾ ␻ quantum subsystems (Knill, Laflamme, and Viola, 2000; By adjusting and L and adopting the appropriate Zanardi, 2001), which may be regarded as a non- set of approximations (which includes the elimination of Abelian (and quite nontrivial) generalization of pointer the internal degrees of freedom of the atom) one is led subspaces. to the master equation for the system, i.e, for the density A sophisticated and elegant strategy that can be re- matrix of the vibrational degree of freedom, garded as a version of the decoherence-free approach ␳˙ ϭ␥͑ ␳ ϩϪ ϩ ␳Ϫ␳ ϩ ͒ was devised independently by Kitaev (1997a, 1997b). He 2f f f f f f . (8.6) has advocated using states that are topologically stable, Above, f is an operator with a form that depends on the ␾ ␻ ␥ and thus, can successfully resist arbitrary interactions adjustable parameters and L in Hint , while is a with the environment. The focus here (in contrast to constant that also depends on ⍀ and ␩. As Poyatos et al. much of the decoherence-free subspace work) is on de- show, one can alter the effective interaction between the vising a system with a self-Hamiltonian that—as a con- slow degree of freedom (the oscillator) and the environ- sequence of the structure of the gap in its energy spec- ment (laser light) by adjusting the parameters of the ac- trum relate to the ‘‘cost’’ of topologically nontrivial tual Hint . excitations—acquires a subspace isolated de facto from The first steps towards realization of these ‘‘environ- the environment. This approach has been further devel- ment engineering’’ proposals were taken by the NIST oped by Bravyi and Kitaev (1998) and by Freedman and group (Myatt et al., 2000; Turchette et al., 2000). Similar Meyer (2001). techniques can be employed to protect deliberately selected states from decoherence (Carvalho et al., 2001). Other ideas aimed at controling and even at exploit- 2. Environment engineering ing decoherence have also been explored in contexts This strategy involves altering the (effective) interac- that range from quantum information processing (Beige tion Hamiltonian between the system and the environ- et al., 2000) to preservation of Schro¨ dinger cats in Bose- ment or influencing the state of the environment to se- Einstein condensates (Dalvit, Dziarmaga, and Zurek, lectively suppress decoherence. There are many ways to 2000). implement it, and we shall describe under this label a variety of proposed techniques (some of which are not all that different from the strategies we have just dis- 3. Error correction and resilient quantum computing cussed) that aim to protect the quantum information stored in selected subspaces of the Hilbert space of the This strategy is perhaps the most sophisticated and system, or even to exploit the pointer states induced or comprehensive, and capable of dealing with the greatest redefined in this fashion. variety of errors in the most hardware-independent The basic question that started this line of research, manner. It is a direct descendant of the error-correction whether one can influence the choice of the preferred techniques employed in dealing with classical informa- pointer states, arose in the context of the ion-trap quan- tion based on redundancy. Multiple copies of the infor-

Rev. Mod. Phys., Vol. 75, No. 3, July 2003 Wojciech Hubert Zurek: Decoherence, einselection, and the quantum origins of the classical 769 mation are made, and the errors are found and cor- surement of the pointer basis by the environment, and rected by sophisticated ‘‘majority voting’’ techniques. can be regarded as such from the standpoint of applica- One might have thought that implementing error cor- tions (e.g., NMR teleportation in the example above). rection in the quantum setting would be difficult for two reasons. To begin with, quantum states and hence, quan- IX. CONCLUDING REMARKS tum information cannot be cloned (Dieks, 1982; Woot- Decoherence, einselection, pointer states, and even ters and Zurek, 1982). Moreover, quantum information the predictability sieve have become familiar to many in is very private, and the measurement that is involved in the past decade. The first goal of this paper was to re- majority voting would infringe on this privacy and de- view these advances and to survey, and—where possible, stroy quantum coherence, making quantum information to address—the remaining difficulties. The second re- classical. Fortunately, both of these difficulties can be lated aim was to preview future developments. This has simultaneously overcome by encoding quantum infor- led to considerations involving information, as well as to mation in entangled states of several qubits. Cloning the operational, physically motivated discussions of turns out not to be necessary. And measurements can be seemingly esoteric concepts such as objectivity. Some of carried out in a way that identifies errors while keeping the material presented (including the Darwinian view of quantum information untouched. Moreover, error cor- the emergence of objectivity through redundancy, as rection is discrete; measurements that reveal error syn- well as the discussion of envariance and probabilities) is dromes have ‘‘yes-no’’ outcomes. Thus, even though the rather new, and a subject of research, hence the word information stored in a qubit represents a continuum of ‘‘preview’’ applies here. possible quantum states (e.g., corresponding to a surface New paradigms often take a long time to gain ground. of the Bloch sphere) error correction is discrete, allaying The atomic theory of matter (which, until the early 20th one of the earliest worries concerning the feasibility of century, was ‘‘just an interpretation’’) is a case in point. quantum computation—the unchecked ‘‘drift’’ of the Some of the most tangible applications and conse- quantum state representing the information (Landauer, quences of new ideas are difficult to recognize immedi- 1995). ately. In the case of atomic theory, Brownian motion is a This strategy [discovered by Shor (1995) and Steane good example. Even when the evidence is available, it is (1996)] has been since investigated by many (Bennett often difficult to decode its significance. et al., 1996; Ekert and Macchiavello, 1996; Laflamme Decoherence and einselection are no exception. They et al., 1996) and codified into a mathematically appeal- have been investigated for about two decades. They are ing formalism (Gottesman, 1996; Knill and Laflamme, the only explanation of classicality that does not require 1997). Moreover, the first examples of successful imple- modifications of quantum theory, as do the alternatives mentation (see, for example, Cory et al., 1999) are al- (Bohm, 1952; Pearle, 1976, 1993; Leggett, 1980, 1998, ready at hand. 2002; Ghirardi, Rimini, and Weber, 1986, 1987; Penrose, Error correction allows one, at least, in principle, to 1986, 1989; Gisin and Percival, 1992, 1993a, 1993b, compute forever, providing that the errors are suitably 1993c; Holland, 1993; Goldstein, 1998). Ideas based on Ϫ small (ϳ10 4 per computational step seems to be the the immersion of the system in the environment have error-probability threshold sufficient for most error- recently gained enough support to be described (by correction schemes). Strategies that accomplish this en- skeptics) as ‘‘the new orthodoxy’’ (Bub, 1997). This is a code qubits in already encoded qubits (Aharonov and dangerous characterization, since it suggests that the in- Ben-Or, 1996; Knill, Laflamme, and Zurek, 1996, 1998a, terpretation based on the recognition of the role of the 1998b; Kitaev, 1997c; Preskill, 1998). The number of lay- environment is both complete and widely accepted. Cer- ers of such concatenations necessary to achieve fault tainly neither is the case. tolerance—the ability to carry out arbitrarily long Many conceptual and technical issues (such as what computations—depends on the size (and the character) constitutes a system) are still open. As for the breadth of of the errors, and on the duration of the computation, acceptance, ‘‘the new orthodoxy’’ seems to be an opti- but when the error probability is smaller than the mistic (mis)characterization of decoherence and einse- threshold, that number of layers is finite. Overviews of lection, especially since this explanation of the transition fault-tolerant computation are already at hand (Preskill, from quantum to classical has (with very few exceptions) 1999; Nielsen and Chuang, 2000, and references not made it into the textbooks. This is intriguing, and therein). may be as much a comment on the way in which quan- An interesting subject related to the above discussion tum physics has been taught, especially on the under- is quantum process tomography, anticipated by Jones graduate level, as on the status of the theory we have (1994), and described in the context of quantum infor- reviewed and its level of acceptance among physicists. mation processing by Chuang and Nielsen (1997) and by Quantum mechanics has been to date, by and large, Poyatos, Cirac, and Zoller (1997). The aim here is to presented in a manner that reflects its historical devel- completely characterize a process, such as a quantum opment. That is, Bohr’s planetary model of the atom is logical gate, and not just a state. The first deliberate still often the point of departure, Hamilton-Jacobi equa- implementation of this procedure (Nielsen, Knill, and tions are used to ‘‘derive’’ the Schro¨ dinger equation, and Laflamme, 1998) has also demonstrated experimentally an oversimplified version of the quantum-classical rela- that einselection is indeed equivalent to an unread mea- tionship (attributed to Bohr, but generally not doing jus-

Rev. Mod. Phys., Vol. 75, No. 3, July 2003 770 Wojciech Hubert Zurek: Decoherence, einselection, and the quantum origins of the classical tice to his much more sophisticated views) with the cor- well, 1989; Barvinsky and Kamenshchik, 1990, 1995; respondence principle, kinship of commutators and Brandenberger, Laflamme, and Mijic, 1990; Paz and Poisson brackets, the Ehrenfest theorem, some version Sinha, 1991, 1992; Castagnino et al., 1993; Kiefer and of the Copenhagen interpretation, and other evidence Zeh, 1995; Mensky and Novikov, 1996) and even (quan- that quantum theory is really not all that different from tum) robotics (Benioff, 1988). Given the limitations of space we have not done justice to most of these subjects, classical—especially when systems of interest become focusing instead on issues of principle. In some areas macroscopic, and all one cares about are averages—is reviews already exist. Thus Giulini et al. (1996) is a valu- presented. able collection of essays, where, for example, decoher- The message seems to be that there is really no prob- ence in field theories is addressed. The dissertation of lem and that quantum mechanics can be ‘‘tamed’’ and Wallace (2002) offers a good (if somewhat philosophi- confined to the microscopic domain. Indeterminacy and cal) summary of the role of decoherence with a rather the double-slit experiment are of course discussed, but different emphasis on similar field-theoretic issues. Con- to prove peaceful coexistence within the elbow room ference proceedings edited by Blanchard et al. (2000) assured by Heisenberg’s principle and complementarity. and, especially, an extensive historical overview of the Entanglement is rarely explored. This is quite consistent foundation of quantum theory from the modern per- with the aim of introductory quantum-mechanics spective by Auletta (2000) are also recommended. More courses, which has been (only slightly unfairly) summed specific technical issues with implications for decoher- up by the memorable phrase ‘‘shut up and calculate.’’ ence and einselection have also been reviewed. For ex- Discussion of measurement is either dealt with through ample, on the subject of master equations there are sev- models based on the Copenhagen interpretation ‘‘old or- eral reviews with very different emphases including thodoxy’’ or not at all. An implicit (and sometimes ex- Alicki and Lendi (1987); Grabert, Schramm, and Ingold plicit) message is that those who ask questions that do (1988); Namiki and Pascazio (1993); as well as—more not lend themselves to an answer through laborious, recently—Paz and Zurek (2001). In some areas, such as preferably perturbative calculations are ‘‘philosophers’’ atomic Bose-Einstein condensation, the study of deco- and should be avoided. herence has only started (Anglin, 1997; Dalvit, Dziar- The above description is of course a caricature. But maga, and Zurek, 2001). In many situations (e.g., quan- given that the calculational techniques of quantum tum optics) a useful supplement to the decoherence theory needed in atomic, nuclear, particle, or condensed- view of the quantum-classical interface is afforded by matter physics are indeed difficult to master, and given quantum trajectories—a study of the state of the system that, to date, most of the applications had nothing to do inferred from the intercepted state of the environment with the nature of quantum states, entanglement, and (see Carmichael, 1993; Gisin and Percival, 1993a, 1993b, such, the attitude of avoiding the most flagrantly quan- 1993c; Wiseman and Milburn, 1993). This approach ‘‘un- tum aspects of quantum theory is easy to understand. ravels’’ the evolving density matrices of open systems Yet, novel applications force one to consider questions into trajectories conditioned upon the measurement car- about the information content, the nature of the quan- ried out on the environment, and may have—especially tum, and the emergence of the classical much more di- in quantum optics—intriguing connections with the ‘‘en- rectly, with a focus on states and correlations, rather vironment as a witness’’ point of view (see Dalvit, Dziar- than on the spectra, cross sections, and the expectation maga, and Zurek, 2001). In other areas, such as con- values. Hence problems that are usually bypassed will densed matter, decoherence phenomena have so many come to the fore. It is hard to brand Schro¨ dinger cats variations and are so pervasive that a separate ‘‘decoher- and entanglement as exotic and make them the center- ent review’’ may be in order, especially as intriguing ex- piece of a marketable device. I believe that as a result perimental puzzles seem to challenge the theory (Mo- decoherence will become part of textbook lore. Indeed, hanty and Webb, 1997; Kravtsov and Altshuler, 2000). at the graduate level there are already some notable ex- Indeed, perhaps the most encouraging development is ceptions among monographs (Peres, 1993) and special- the increase of interest in experiments that test validity ized texts (Walls and Milburn, 1994; Nielsen and of quantum physics beyond the microscopie domain Chuang, 2000). (see, for example, Folman, Kru¨ ger, Schmiedmayer, et al., Moreover, the range of subjects already influenced by 2002; Marshall, Simon, Penrose, and Bouwmeester, decoherence and einselection—by the ideas originally 2002). motivated by the quantum theory of measurements—is The physics of information and computation is a spe- beginning to extend way beyond its original domain. In cial case. Decoherence is obviously a key obstacle in the addition to atomic physics, quantum optics, and quan- implementation of information-processing hardware tum information processing (which were all mentioned that takes advantage of the superposition principle. throughout this review) it stretches from material sci- While we have not focused on quantum information ences (Karlsson, 1998; Chatzidimitriou-Dreismann et al., processing, the discussion has often been couched in lan- 1997, 2001), surface science, where it seems to be an guage inspired by information theory. This is no acci- essential ingredient explaining the emission of electrons dent. It is the belief of this author that many of the (Brodie, 1995; Durakiewicz et al., 2001), through heavy- remaining gaps in our understanding of quantum physics ion collisions (Krzywicki, 1993) to quantum gravity and and its relation to the classical domain—such as the defi- cosmology (Zeh, 1986, 1988, 1992; Kiefer, 1987; Halli- nition of systems, or the still mysterious aspects of

Rev. Mod. Phys., Vol. 75, No. 3, July 2003 Wojciech Hubert Zurek: Decoherence, einselection, and the quantum origins of the classical 771 collapse—will follow the pattern of the predictability Auletta, G., 2000, Foundations and Interpretation of Quantum sieve and be expanded into new areas of investigation by Theory (World Scientific, Singapore). considerations that simultaneously elucidate the nature Bacciagaluppi, G., and M. Hemmo, 1998, in Quantum Mea- of the quantum and of information. surements Beyond Paradox, edited by R. A. Healey and G. Hellman (University of Minnesota, Minneapolis), p. 95. ACKNOWLEDGMENTS Ballentine, L. E., 1970, Rev. Mod. Phys. 42, 358. Ballentine, L. E., Y. Yang, and J. P. Zibin, 1994, Phys. Rev. A John Archibald Wheeler—a quarter century ago— 50, 2854. taught a course on the subject of quantum measure- Barnett, S. M., and S. J. D. Phoenix, 1989, Phys. Rev. A 40, ments at the University of Texas in Austin. The ques- 2404. tions raised then have since evolved into ideas presented Barnum, H., C. M. Caves, J. Finkelstein, C. A. Fuchs, and R. here, partly in collaboration with Juan Pablo Paz, and Schack, 2000, Proc. R. Soc. London, Ser. A 456, 1175. Barvinsky, A. O., and A. Yu. Kamenshchik, 1990, Class. Quan- through interactions with many colleagues, including tum Grav. 7, 2285. Andreas Albrecht, James Anglin, Charles Bennett, Barvinsky, A. O., and A. Yu. Kamenshchik, 1995, Phys. Rev. D Robin Blume-Kohout, Carlton Caves, Isaac Chuang, Di- 52, 743. ego Dalvit, David Deutsch, David Divincenzo, Jacek Beckman, D., D. Gottesman, M. A. Nielsen, and J. Preskill, Dziarmaga, Richard Feynman, Murray Gell-Mann, 2001, Phys. Rev. A 64, 052309. Daniel Gottesmann, Robert Griffiths, Salman Habib, Beige, A., D. Braun, D. Tregenna, and P. L. Knight, 2000, Jonathan Halliwell, Serge Haroche, James Hartle, Chris- Phys. Rev. Lett. 85, 1762. topher Jarzynski, Erich Joos, Emmanuel Knill, Ray- Bell, J. 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