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2004 Decoherence, the measurement problem, and interpretations of quantum mechanics Maximillian Schlosshauer University of Portland, [email protected]

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Citation: Pilot Scholars Version (Modified MLA Style) Schlosshauer, Maximillian, "Decoherence, the measurement problem, and interpretations of quantum mechanics" (2004). Physics Faculty Publications and Presentations. 13. http://pilotscholars.up.edu/phy_facpubs/13

This Journal Article is brought to you for free and open access by the Physics at Pilot Scholars. It has been accepted for inclusion in Physics Faculty Publications and Presentations by an authorized administrator of Pilot Scholars. For more information, please contact [email protected]. arXiv:quant-ph/0312059v4 28 Jun 2005 ∗ program decoherence The III. Contents interpretation and mechanics problem, measurement the Decoherence, V h oeo eoeec nitrrttosof interpretations in decoherence of role The IV. I h esrmn problem measurement The II. lcrncaddress: Electronic .Introduction I. unu mechanics quantum .Mdlinterpretations Modal D. interference of suppression local and Decoherence D. decoherence and transition quantum-to-classical The D. .Gnrlipiain fdchrnefrinterpretation for decoherence of implications General A. subsystems into Resolution A. scheme measurement Quantum A. .Rltv-tt interpretations Relative-state C. scheme measurement Neumann von modified A C. problem preferred-basis The C. .Tesadr n h oehgninterpretation Copenhagen the and standard The B. matrices density reduced of concept The B. outcomes definite of problem The B. .Evrnetidcdsuperselection Environment-induced E. .Evrac,qatmpoaiiis n h onrule Born the and probabilities, quantum Envariance, F. .Slcino uscasclproperties quasiclassical of basis Selection pointer and 2. criterion Stability decoherence 1. for model two-state solvable exactly An 2. formalism General 1. definiteness subjective vs. Objective attribution outcome 3. and Superpositions ensembles 2. and Superpositions 1. .Poet sinetbsdo instantaneous on based assignment Property 2. environment-induced on based assignment Property 1. interpretation” “existential The interpretations Everett 3. problem in preferred-basis Probabilities the and 2. branches Everett 1. Copenhagen the in classicality of concept The 3. and measurements, outcomes Observables, definite of 2. problem The 1. outlook and Summary rule Born 3. the Deducing 2. invariance Environment-assisted 1. states Schmidt instantaneous vs basis problem Pointer preferred-basis the 4. for Implications 3. hsppri neddt lrf e etrso h decoher co the and of mechanics. application features quantum their of key investigate approaches clarify to interpretive and to results, intended remaine recent have is problem, paper foundat measurement the This quantum for the implications notably their yet most decades, two past the have superselection and decoherence Environment-induced W Seattle, Washington, of University Physics, of Department Schlosshauer Maximilian cmd decompositions Schmidt superselection interpretation superselection environment-induced [email protected] ∗ s 11 27 26 26 25 24 23 22 22 21 20 20 19 18 18 17 17 16 15 14 13 12 12 10 10 10 6 5 4 4 4 3 3 1 9 8 7 6 enasbeto nesv eerhover research intensive of subject a been sqecsi h otx ftemain the of context the in nsequences oa rbeso unu mechanics, quantum of problems ional .INTRODUCTION I. niomn neatos nsot eoeec brings envi- decoherence its short, system- In and these of system interactions. effects the surprising environment often of the and states ronment in- the corre- its quantum between or of lations theory formation mathematical resulting adding the the without terpretation), in elements (i.e., within new formalism any entirely quantum sur- studies, standard the then the in program it. immersed decoherence with are continuously The interact but and environment isolated, sys- rounding quantum promoted never realistic idea are that key insight tems The the 1980s. is decoherence early of by the formulation precise in first program the the since debate subject ongoing the an been of have mechanics quantum of foundations .Cnldn remarks Concluding V. neporm nldn t more its including program, ence sigo 89,USA 98195, ashington h mlctoso h eoeec rga o the for program decoherence the of implications The atro ra controversy. great of matter a d References Acknowledgments interpretations histories Consistent G. .Pyia olpetheories collapse Physical E. .Bhinmechanics Bohmian F. .Smayadoutlook and models Summary collapse and 5. decoherence Connecting problem 4. tails The 3. and decoherence of presence Simultaneous problem 2. preferred-basis The 1. remarks Concluding 4. of decompositions on based assignment Property 3. .Smayaddiscussion and Summary 8. environment-induced and Consistency consistency projectors approximate 7. as vs basis Exact pointer vs 6. states systems Schmidt open of 5. histories classicality Consistent and histories 4. of Selection consistency 3. and Probabilities histories 2. of Definition 1. decoherence and trajectories Bohmian entities fundamental 2. as Particles 1. pnaeu localization spontaneous matrix density decohered the superselection fquantum of s 38 38 37 37 36 36 35 35 34 33 33 33 32 31 31 31 30 30 29 29 28 28 27 2 about a local suppression of interference between pre- existence” [from decoherence] (. . . ) significantly ferred states selected by the interaction with the envi- reduces and perhaps even eliminates the role of ronment. the “collapse” of the state vector. Bub (1997) termed decoherence part of the “new or- D’Espagnat, who considers the explanation of our ex- thodoxy” of understanding quantum mechanics—as the periences (i.e., of “appearances”) as the only “sure” re- working physicist’s way of motivating the postulates of quirement of a physical theory, states (d’Espagnat, 2000, quantum mechanics from physical principles. Proponents p. 136) of decoherence called it an “historical accident” (Joos, 2000, p. 13) that the implications for quantum mechan- For macroscopic systems, the appearances are ics and for the associated foundational problems were those of a classical world (no interferences etc.), overlooked for so long. Zurek (2003b, p. 717) suggests even in circumstances, such as those occurring in quantum measurements, where quantum effects The idea that the “openness” of quantum sys- take place and quantum probabilities intervene tems might have anything to do with the transi- (. . . ). Decoherence explains the just mentioned tion from quantum to classical was ignored for appearances and this is a most important result. a very long time, probably because in classi- (. . . ) As long as we remain within the realm of cal physics problems of fundamental importance mere predictions concerning what we shall ob- were always settled in isolated systems. serve (i.e., what will appear to us)—and refrain When the concept of decoherence was first introduced from stating anything concerning “things as they to the broader scientific community by Zurek’s (1991) must be before we observe them”—no break in article in Physics Today, it elicited a series of contentious the linearity of quantum dynamics is necessary. comments from the readership (see the April 1993 issue of In his monumental book on the foundations of quantum Physics Today). In response to his critics, Zurek (2003b, mechanics (QM), Auletta (2000, p. 791) concludes that p. 718) states the Measurement theory could be part of the in- In a field where controversy has reigned for so terpretation of QM only to the extent that it long this resistance to a new paradigm [namely, would still be an open problem, and we think to decoherence] is no surprise. that this is largely no longer the case. Omn`es (2002, p. 2) had this assessment: This is mainly so because, according to Auletta (2000, p. 289), The discovery of decoherence has already much improved our understanding of quantum mechan- decoherence is able to solve practically all the ics. (. . . ) [B]ut its foundation, the range of its problems of Measurement which have been dis- validity and its full meaning are still rather ob- cussed in the previous chapters. scure. This is due most probably to the fact that On the other hand, even leading adherents of decoherence it deals with deep aspects of physics, not yet fully have expressed caution or even doubt that decoherence investigated. has solved the measurement problem. Joos (2000, p. 14) writes In particular, the question whether decoherence provides, or at least suggests, a solution to the measurement prob- Does decoherence solve the measurement prob- lem of quantum mechanics has been discussed for several lem? Clearly not. What decoherence tells us, is years. For example, Anderson (2001, p. 492) writes in an that certain objects appear classical when they essay review are observed. But what is an observation? At some stage, we still have to apply the usual prob- The last chapter (. . . ) deals with the quantum ability rules of quantum theory. measurement problem (. . . ). My main test, al- lowing me to bypass the extensive discussion, was Along these lines, Kiefer and Joos (1999, p. 5) warn that a quick, unsuccessful search in the index for the One often finds explicit or implicit statements to word “decoherence” which describes the process the effect that the above processes are equivalent that used to be called “collapse of the wave func- to the collapse of the wave function (or even solve tion.” the measurement problem). Such statements are Zurek speaks in various places of the “apparent” or “ef- certainly unfounded. fective” collapse of the wave function induced by the in- In a response to Anderson’s (2001, p. 492) comment, teraction with environment (when embedded into a min- Adler (2003, p. 136) states imal additional interpretive framework) and concludes (Zurek, 1998, p. 1793) I do not believe that either detailed theoretical calculations or recent experimental results show A “collapse” in the traditional sense is no longer that decoherence has resolved the difficulties as- necessary. (. . . ) [The] emergence of “objective sociated with quantum measurement theory. 3

Similarly, Bacciagaluppi (2003b, p. 3) writes in some form addressed by the decoherence program. In Sec. III, we then introduce and discuss the main features Claims that simultaneously the measurement of the theory of decoherence, with a particular emphasis problem is real [and] decoherence solves it are on their foundational implications. Finally, in Sec. IV, confused at best. we investigate the role of decoherence in various inter- Zeh asserts (Joos et al., 2003, Ch. 2) pretive approaches of quantum mechanics, in particular with respect to the ability to motivate and support (or Decoherence by itself does not yet solve the disprove) possible solutions to the measurement problem. measurement problem (. . . ). This argument is nonetheless found wide-spread in the literature. (. . . ) It does seem that the measurement problem can only be resolved if the Schr¨odinger dynamics (. . . ) is supplemented by a nonunitary collapse II. THE MEASUREMENT PROBLEM (...). The key achievements of the decoherence program, apart One of the most revolutionary elements introduced into from their implications for conceptual problems, do not physical theory by quantum mechanics is the superposi- seem to be universally understood either. Zurek (1998, tion principle, mathematically founded in the linearity of p. 1800) remarks the Hilbert state space. If 1 and 2 are two states, then quantum mechanics tells us| i that any| i linear combination [The] eventual diagonality of the density matrix α 1 + β 2 also corresponds to a possible state. Whereas (...) is a byproduct (...) but not the essence such| i superpositions| i of states have been experimentally of decoherence. I emphasize this because diago- extensively verified for microscopic systems (for instance, nality of [the density matrix] in some basis has through the observation of interference effects), the appli- been occasionally (mis-)interpreted as a key ac- cation of the formalism to macroscopic systems appears complishment of decoherence. This is mislead- to lead immediately to severe clashes with our experience ing. Any density matrix is diagonal in some ba- of the everyday world. A book has never been ever ob- sis. This has little bearing on the interpretation. served to be in a state of being both “here” and “there” These remarks show that a balanced discussion of the (i.e., to be in a superposition of macroscopically distin- key features of decoherence and their implications for the guishable positions), nor does a Schr¨odinger cat that is foundations of quantum mechanics is overdue. The deco- a superposition of being alive and dead bear much re- herence program has made great progress over the past semblence to reality as we perceive it. The problem is, decade, and it would be inappropriate to ignore its rel- then, how to reconcile the vastness of the Hilbert space of evance in tackling conceptual problems. However, it is possible states with the observation of a comparatively equally important to realize the limitations of decoher- few “classical” macrosopic states, defined by having a ence in providing consistent and noncircular answers to small number of determinate and robust properties such foundational questions. as position and momentum. Why does the world appear An excellent review of the decoherence program has classical to us, in spite of its supposed underlying quan- recently been given by Zurek (2003b). It deals pri- tum nature, which would, in principle, allow for arbitrary marily with the technicalities of decoherence, although superpositions? it contains some discussion on how decoherence can be employed in the context of a relative-state interpreta- tion to motivate basic postulates of quantum mechanics. A helpful first orientation and overview, the entry by Bacciagaluppi (2003a) in the Stanford Encyclopedia of A. Quantum measurement scheme Philosophy features a relatively short (in comparison to the present paper) introduction to the role of decoher- This question is usually illustrated in the context ence in the foundations of quantum mechanics, including of quantum measurement where microscopic superposi- comments on the relationship between decoherence and tions are, via quantum entanglement, amplified into the several popular interpretations of quantum theory. In macroscopic realm and thus lead to very “nonclassical” spite of these valuable recent contributions to the litera- states that do not seem to correspond to what is actually ture, a detailed and self-contained discussion of the role perceived at the end of the measurement. In the ideal of decoherence in the foundations of quantum mechanics measurement scheme devised by von Neumann (1932), seems still to be lacking. This review article is intended a (typically microscopic) system , represented by ba- S to fill the gap. sis vectors sn in a Hilbert space S , interacts with To set the stage, we shall first, in Sec. II, review the a measurement{| i} apparatus , describedH by basis vectors A measurement problem, which illustrates the key difficul- an spanning a Hilbert space A, where the an are ties that are associated with describing quantum mea- assumed{| i} to correspond to macroscopicallyH distinguish-| i surement within the quantum formalism and that are all able “pointer” positions that correspond to the outcome 4

1 of a measurement if is in the state sn . referred to as an “ignorance-interpretable,” or “proper” Now, if is in aS (microscopically| “unproblematic”)i ensemble). S superposition cn sn , and is in the initial “ready” This can be shown explicitely, especially on micro- n | i A state ar , the linearity of the Schr¨odinger equation en- scopic scales, by performing experiments that lead to the | i P tails that the total system , assumed to be represented direct observation of interference patterns instead of the SA by the Hilbert product space S A, evolves according realization of one of the terms in the superposed pure H ⊗H to state, for example, in a setup where electrons pass in- dividually (one at a time) through a double slit. As is c s a t c s a . (2.1) well known, this experiment clearly shows that, within  n| ni| ri −→ n| ni| ni Xn Xn the standard quantum-mechanical formalism, the elec- tron must not be described by either one of the wave This dynamical evolution is often referred to as a pre- functions describing the passage through a particular slit measurement in order to emphasize that the process de- (ψ or ψ ), but only by the superposition of these wave scribed by Eq. (2.1) does not suffice to directly conclude 1 2 functions (ψ1 +ψ2), since the correct density distribution that a measurement has actually been completed. This is ̺ of the pattern on the screen is not given by the sum of so for two reasons. First, the right-hand side is a super- the squared wave functions describing the addition of in- position of system-apparatus states. Thus, without sup- 2 2 dividual passages through a single slit (̺ = ψ1 + ψ2 ), plying an additional physical process (say, some collapse but only by the square of the sum of the individual| | | wave| mechanism) or giving a suitable interpretation of such a functions (̺ = ψ + ψ 2). superposition, it is not clear how to account, given the fi- | 1 2| Put differently, if an ensemble interpretation could be nal composite state, for the definite pointer positions that attached to a superposition, the latter would simply rep- are perceived as the result of an actual measurement— resent an ensemble of more fundamentally determined i.e., why do we seem to perceive the pointer to be in states, and based on the additional knowledge brought one position a but not in a superposition of posi- n about by the results of measurements, we could simply tions? This is| thei problem of definite outcomes. Sec- choose a subensemble consisting of the definite pointer ond, the expansion of the final composite state is in gen- state obtained in the measurement. But then, since the eral not unique, and therefore the measured observable time evolution has been strictly deterministic according is not uniquely defined either. This is the problem of to the Schr¨odinger equation, we could backtrack this the preferred basis. In the literature, the first difficulty subensemble in time and thus also specify the initial is typically referred to as the measurement problem, but state more completely (“postselection”), and therefore the preferred-basis problem is at least equally important, this state necessarily could not be physically identical since it does not make sense even to inquire about specific to the initially prepared state on the left-hand side of outcomes if the set of possible outcomes is not clearly de- Eq. (2.1). fined. We shall therefore regard the measurement prob- lem as composed of both the problem of definite outcomes and the problem of the preferred basis, and discuss these components in more detail in the following. 2. Superpositions and outcome attribution

In the standard (“orthodox”) interpretation of quan- B. The problem of definite outcomes tum mechanics, an observable corresponding to a phys- ical quantity has a definite value if and only if the sys- 1. Superpositions and ensembles tem is in an eigenstate of the observable; if the system is, however, in a superposition of such eigenstates, as in The right-hand side of Eq. (2.1) implies that after the Eq. (2.1), it is, according to the orthodox interpretation, premeasurement the combined system is left in a pure meaningless to speak of the state of the system as having SA state that represents a linear superposition of system- any definite value of the observable at all. (This is fre- pointer states. It is a well-known and important prop- quently referred to as the so-called eigenvalue-eigenstate erty of quantum mechanics that a superposition of states link, or “e-e link” for short.) The e-e link, however, is is fundamentally different from a classical ensemble of by no means forced upon us by the structure of quan- states, where the system actually is in only one of the tum mechanics or by empirical constraints (Bub, 1997). states but we simply do not know in which (this is often The concept of (classical) “values” that can be ascribed through the e-e link based on observables and the exis- tence of exact eigenstates of these observables has there- fore frequently been either weakened or altogether aban- 1 Note that von Neumann’s scheme is in sharp contrast to the donded. For instance, outcomes of measurements are Copenhagen interpretation, where measurement is not treated as typically registered in position space (pointer positions, a system-apparatus interaction described by the usual quantum- mechanical formalism, but instead as an independent component etc.), but there exist no exact eigenstates of the position of the theory, to be represented entirely in fundamentally classi- operator, and the pointer states are never exactly mutu- cal terms. ally orthogonal. One might then (explicitely or implic- 5 itly) promote a “fuzzy” e-e link, or give up the concept of not justified, and that subjective definiteness should be observables and values entirely and directly interpret the viewed on a par with objective definitess with respect time-evolved wave functions (working in the Schr¨odinger to a satisfactory solution to the measurement problem. picture) and the corresponding density matrices. Also, We demand objective definiteness because we experience if it is regarded as sufficient to explain our perceptions definiteness on the subjective level of observation, and rather than describe the “absolute” state of the entire it should not be viewed as an a priori requirement for universe (see the argument below), one might only re- a physical theory. If we knew independently of our ex- quire that the (exact or fuzzy) e-e link hold in a “rela- perience that definiteness existed in nature, subjective tive” sense, i.e., for the state of the rest of the universe definiteness would presumably follow as soon as we had relative to the state of the observer. employed a simple model that connected the “external” Then, to solve the problem of definite outcomes, some physical phenomena with our “internal” perceptual and interpretations (for example, modal interpretations and cognitive apparatus, where the expected simplicity of relative-state interpretations) interpret the final-state su- such a model can be justified by referring to the pre- perposition in such a way as to explain the existence, or sumed identity of the physical laws governing external at least the subjective perception, of “outcomes” even and internal processes. But since knowledge is based on if the final composite state has the form of a super- experience, that is, on observation, the existence of ob- position. Other interpretations attempt to solve the jective definiteness could only be derived from the obser- measurement problem by modifying the strictly unitary vation of definiteness. And, moreover, observation tells Schr¨odinger dynamics. Most prominently, the ortho- us that definiteness is in fact not a universal property dox interpretation postulates a collapse mechanism that of nature, but rather a property of macroscopic objects, transforms a pure-state density matrix into an ignorance- where the borderline to the macroscopic realm is diffi- interpretable ensemble of individual states (a “proper cult to draw precisely; mesoscopic interference experi- mixture”). Wave-function collapse theories add stochas- ments have demonstrated clearly the blurriness of the tic terms to the Schr¨odinger equation that induce an ef- boundary. Given the lack of a precise definition of the fective (albeit only approximate) collapse for states of boundary, any demand for fundamental definiteness on macroscopic systems (Ghirardi et al., 1986; Gisin, 1984; the objective level should be based on a much deeper and Pearle, 1979, 1999), while other authors suggested that more general commitment to a definiteness that applies collapse occurs at the level of the mind of a conscious ob- to every physical entity (or system) across the board, re- server (Stapp, 1993; Wigner, 1963). Bohmian mechanics, gardless of spatial size, physical property, and the like. on the other hand, upholds a unitary time evolution of Therefore, if we realize that the often deeply felt com- the wavefunction, but introduces an additional dynam- mitment to a general objective definiteness is only based ical law that explicitely governs the always-determinate on our experience of macroscopic systems, and that this positions of all particles in the system. definiteness in fact fails in an observable manner for mi- croscopic and even certain mesoscopic systems, the au- thor sees no compelling grounds on which objective def- 3. Objective vs. subjective definiteness initeness must be demanded as part of a satisfactory physical theory, provided that the theory can account for subjective, observational definiteness in agreement with In general, (macroscopic) definiteness—and thus a so- our experience. Thus the author suggests that the same lution to the problem of outcomes in the theory of quan- legitimacy be attributed to proposals for a solution of tum measurement—can be achieved either on an onto- the measurement problem that achieve “only” subjective logical (objective) or an observational (subjective) level. but not objective definiteness—after all, the measure- Objective definiteness aims at ensuring “actual” definite- ment problem arises solely from a clash of our experi- ness in the macroscopic realm, whereas subjective defi- ence with certain implications of the quantum formalism. niteness only attempts to explain why the macroscopic D’Espagnat (2000, pp. 134–135) has advocated a similar world appears to be definite—and thus does not make viewpoint: any claims about definiteness of the underlying physi- cal reality (whatever this reality might be). This raises The fact that we perceive such “things” as macro- the question of the significance of this distinction with scopic objects lying at distinct places is due, respect to the formation of a satisfactory theory of the partly at least, to the structure of our sensory and physical world. It might appear that a solution to the intellectual equipment. We should not, there- measurement problem based on ensuring subjective, but fore, take it as being part of the body of sure not objective, definiteness is merely good “for all prac- knowledge that we have to take into account for tical purposes”—abbreviated, rather disparagingly, as defining a quantum state. (. . . ) In fact, scien- “FAPP” by Bell (1990)—and thus not capable of solv- tists most righly claim that the purpose of science ing the “fundamental” problem that would seem relevant is to describe human experience, not to describe to the construction of the “precise theory” that Bell de- “what really is”; and as long as we only want to manded so vehemently. describe human experience, that is, as long as we It seems to the author, however, that this critism is are content with being able to predict what will 6

be observed in all possible circumstances (. . . ) using the eigenstates x 1,2 of the observable σx (which we need not postulate the existence—in some represents a measurement| ±i of the spin orientation along absolute sense—of unobserved (i.e., not yet ob- the x axis) as basis vectors, we get served) objects lying at definite places in ordinary 1 3-dimensional space. ψ = ( x+ x x x+ ). (2.5) | i √2 | i1| −i2 − | −i1| i2

C. The preferred-basis problem Now suppose that system 2 acts as a measuring device for the spin of system 1. Then Eqs. (2.4) and (2.5) imply that The second difficulty associated with quantum mea- the measuring device has established a correlation with surement is known as the preferred-basis problem, which both the z and the x spin of system 1. This means that, if demonstrates that the measured observable is in general we interpret the formation of such a correlation as a mea- not uniquely defined by Eq. (2.1). For any choice of sys- surement in the spirit of the von Neumann scheme (with- out assuming a collapse), our apparatus (system 2) could tem states sn , we can find corresponding apparatus {| i} be considered as having measured also the x spin once it states an , and vice versa, to equivalently rewrite the final state{| i} emerging from the premeasurement interac- has measured the z spin, and vice versa—in spite of the tion, i.e., the right-hand side of Eq. (2.1). In general, noncommutativity of the corresponding spin observables however, for some choice of apparatus states the corre- σz and σx. Moreover, since we can rewrite Eq. (2.4) in sponding new system states will not be mutually orthog- infinitely many ways, it appears that once the apparatus onal, so that the observable associated with these states has measured the spin of system 1 along one direction, it will not be Hermitian, which is usually not desired (how- can also be regarded as having measured the spin along ever, not forbidden—see the discussion by Zurek, 2003b). any other direction, again in apparent contradiction with Conversely, to ensure distinguishable outcomes, we must, quantum mechanics due to the noncommutativity of the in general, require the (at least approximate) orthogonal- spin observables corresponding to different spatial orien- ity of the apparatus (pointer) states, and it then follows tations. from the biorthogonal decomposition theorem that the It thus seems that quantum mechanics has nothing to expansion of the final premeasurement system-apparatus say about which observable(s) of the system is (are) being state of Eq. (2.1), recorded, via the formation of quantum correlations, by the apparatus. This can be stated in a general theorem ψ = c s a , (2.2) (Auletta, 2000; Zurek, 1981): When quantum mechanics | i n| ni| ni is applied to an isolated composite object consisting of a Xn system and an apparatus , it cannot determine which S A is unique, but only if all coefficients cn are distinct. Oth- observable of the system has been measured—in obvious erwise, we can in general rewrite the state in terms of contrast to our experience of the workings of measuring different state vectors, devices that seem to be “designed” to measure certain quantities. ψ = c′ s′ a′ , (2.3) | i n| ni| ni Xn D. The quantum-to-classical transition and decoherence such that the same postmeasurement state seems to cor- respond to two different measurements, that is, of the In essence, as we have seen above, the measurement observables A = λ s s and B = λ′ s′ s′ problem deals with the transition from a quantum world, n n| nih n| n n| nih n| P P described by essentially arbitrary linear superpositions of of the system,b respectively, although inb general A and B do not commute. state vectors, to our perception of “classical” states in the As an example, consider a Hilbert space = b b macroscopic world, that is, a comparatively small subset H H1 ⊗ H2 where 1 and 2 are two-dimensional spin spaces with of the states allowed by the quantum-mechanical super- statesH correspondingH to spin up or spin down along a position principle, having only a few, but determinate given axis. Suppose we are given an entangled spin state and robust, properties, such as position, momentum, etc. of the Einstein-Podolsky-Rosen form (Einstein et al., The question of why and how our experience of a “clas- 1935) sical” world emerges from quantum mechanics thus lies at the heart of the foundational problems of quantum 1 theory. ψ = ( z+ z z z+ ), (2.4) | i √2 | i1| −i2 − | −i1| i2 Decoherence has been claimed to provide an explana- tion for this quantum-to-classical transition by appeal- where z represents the eigenstates of the observable ing to the ubiquitous immersion of virtually all physical | ±i1,2 σz corresponding to spin up or spin down along the z axis systems in their environment (“environmental monitor- of the two systems 1 and 2. The state ψ can however ing”). This trend can also be read off nicely from the equivalently be expressed in the spin basis| i corresponding titles of some papers and books on decoherence, for ex- to any other orientation in space. For example, when ample, “The emergence of classical properties through 7 interaction with the environment” (Joos and Zeh, 1985), sical” properties of our experience. Interference between “Decoherence and the transition from quantum to clas- such states gets locally suppressed and is thus claimed to sical” (Zurek, 1991), and “Decoherence and the appear- become inaccessible to the observer. ance of a classical world in quantum theory” (Joos et al., Probably the most surprising aspect of decoherence 2003). We shall critically investigate in this paper to is the effectiveness of the system-environment interac- what extent the appeal to decoherence for an explana- tions. Decoherence typically takes place on extremely tion of the quantum-to-classical transition is justified. short time scales and requires the presence of only a minimal environment (Joos and Zeh, 1985). Due to the large number of degrees of freedom of the envi- III. THE DECOHERENCE PROGRAM ronment, it is usually very difficult to undo system- environment entanglement, which has been claimed as a As remarked earlier, the theory of decoherence is based source of our impression of irreversibility in nature (see, on a study of the effects brought about by the inter- for example, Kiefer and Joos, 1999; Zeh, 2001; Zurek, action of physical systems with their environment. In 1982, 2003b; Zurek and Paz, 1994). In general, the ef- classical physics, the environment is usually viewed as fect of decoherence increases with the size of the sys- a kind of disturbance, or noise, that perturbs the sys- tem (from microscopic to macroscopic scales), but it is tem under consideration in such a way as to negatively important to note that there exist, admittedly some- influence the study of its “objective” properties. There- what exotic, examples for which the decohering influ- fore science has established the idealization of isolated ence of the environment can be sufficiently shielded to systems, with experimental physics aiming at eliminat- lead to mesoscopic and even macroscopic superpositions. ing any outer sources of disturbance as much as possible One such example would be the case of superconduct- in order to discover the “true” underlying nature of the ing quantum interference devices (SQUIDs), in which su- system under study. perpositions of macroscopic currents become observable The distinctly nonclassical phenomenon of quantum (Friedman et al., 2000; van der Wal et al., 2000). Con- entanglement, however, has demonstrated that the cor- versely, some microscopic systems (for instance, certain relations between two systems can be of fundamental im- chiral molecules that exist in different distinct spatial portance and can lead to properties that are not present configurations) can be subject to remarkably strong de- in the individual systems.2 The earlier view of phenom- coherence. ena arising from quantum entanglement as “paradoxa” The decoherence program has dealt with the following has generally been replaced by the recognition of entan- two main consequences of environmental interaction: glement as a fundamental property of nature. The decoherence program3 is based on the idea that (1) Environment-induced decoherence. The fast local such quantum correlations are ubiquitous; that nearly suppression of interference between different states every physical system must interact in some way with of the system. However, since only unitary time its environment (for example, with the surrounding pho- evolution is employed, global phase coherence is tons that then create the visual experience within the not actually destroyed—it becomes absent from observer), which typically consists of a large number the local density matrix that describes the sys- of degrees of freedom that are hardly ever fully con- tem alone, but remains fully present in the to- 4 trolled. Only in very special cases of typically micro- tal system-environment composition. We shall scopic (atomic) phenomena, so goes the claim of the de- discuss environment-induced local decoherence in coherence program, is the idealization of isolated systems more detail in Sec. III.D. applicable so that the predictions of linear quantum me- chanics (i.e., a large class of superpositions of states) (2) Environment-induced superselection. The selection can actually be observationally confirmed. In the ma- of preferred sets of states, often referred to as jority of the cases accessible to our experience, however, “pointer states,” that are robust (in the sense of interaction with the environment is so dominant as to retaining correlations over time) in spite of their preclude the observation of the “pure” quantum world, immersion in the environment. These states are imposing effective superselection rules (Cisnerosy et al., determined by the form of the interaction between 1998; Galindo et al., 1962; Giulini, 2000; Wick et al., the system and its environment and are suggested 1952, 1970; Wightman, 1995) onto the space of observ- to correspond to the “classical” states of our ex- able states that lead to states corresponding to the “clas- perience. We shall consider this mechanism in Sec. III.E.

2 Broadly speaking, this means that the (quantum-mechanical) whole is different from the sum of its parts. 4 Note that the persistence of coherence in the total state is impor- 3 For key ideas and concepts, see Joos and Zeh (1985); Joos et al. tant to ensure the possibility of describing special cases in which (2003); K¨ubler and Zeh (1973); Zeh (1970, 1973, 1995, 1997, mesoscopic or macrosopic superpositions have been experimen- 2000); Zurek (1981, 1982, 1991, 1993, 2003b). tally realized. 8

Another, more recent aspect of the decoherence program, In the absence of systems, the problem of inter- termed enviroment-assisted invariance or “envariance,” pretation seems to disappear. There is simply was introduced by Zurek (2003b,c, 2004b) and further no need for “collapse” in a universe with no sys- developed in Zurek (2004a). In particular, Zurek used tems. Our experience of the classical reality does envariance to explain the emergence of probabilities in not apply to the universe as a whole, seen from quantum mechanics and to derive Born’s rule based on the outside, but to the systems within it. certain assumptions. We shall review envariance and Zurek’s derivation of the Born rule in Sec. III.F. Moreover, terms like “observation,” “correlation,” and “interaction” will naturally make little sense without a Finally, let us emphasize that decoherence arises from a division into systems. Zeh has suggested that the locality direct application of the quantum mechanical formalism of the observer defines an observation in the sense that to a description of the interaction of a physical system any observation arises from the ignorance of a part of the with its environment. By itself, decoherence is therefore universe; and that this also defines the “facts” that can neither an interpretation nor a modification of quantum occur in a quantum system. Landsman (1995, pp. 45–46) mechanics. Yet the implications of decoherence need to argues similarly: be interpreted in the context of the different interpre- tations of quantum mechanics. Also, since decoherence The essence of a “measurement,” “fact” or effects have been studied extensively in both theoretical “event” in quantum mechanics lies in the non- models and experiments (for a survey, see, for example, observation, or irrelevance, of a certain part of Joos et al., 2003; Zurek, 2003b), their existence can be the system in question. (. . . ) A world without taken as a well-confirmed fact. parts declared or forced to be irrelevant is a world without facts.

However, the assumption of a decomposition of the uni- A. Resolution into subsystems verse into subsystems—as necessary as it appears to be for the emergence of the measurement problem and for Note that decoherence derives from the presupposition the definition of the decoherence program—is definitely of the existence and the possibility of a division of the nontrivial. By definition, the universe as a whole is a world into “system(s)” and “environment.” In the deco- closed system, and therefore there are no “unobserved herence program, the term “environment” is usually un- degrees of freedom” of an external environment which derstood as the “remainder” of the system, in the sense would allow for the application of the theory of decoher- that its degrees of freedom are typically not (cannot be, ence to determine the space of quasiclassical observables do not need to be) controlled and are not directly rele- of the universe in its entirety. Also, there exists no gen- vant to the observation under consideration (for example, eral criterion for how the total Hilbert space is to be the many microsopic degrees of freedom of the system), divided into subsystems, while at the same time much but that nonetheless the environment includes “all those of what is called a property of the system will depend degrees of freedom which contribute significantly to the on its correlation with other systems. This problem be- evolution of the state of the apparatus” (Zurek, 1981, comes particularly acute if one would like decoherence p. 1520). not only to motivate explanations for the subjective per- ception of classicality (as in Zurek’s “existential inter- This system–environment dualism is generally associ- pretation,” see Zurek, 1993, 1998, 2003b, and Sec. IV.C ated with quantum entanglement, which always describes below), but moreover to allow for the definition of quasi- a correlation between parts of the universe. As long as classical “macrofacts.” Zurek (1998, p. 1820) admits this the universe is not resolved into individual subsystems, severe conceptual difficulty: there is no measurement problem: the state vector Ψ 5 | i of the entire universe evolves deterministically accord- In particular, one issue which has been often ing to the Schr¨odinger equation i~ ∂ Ψ = H Ψ , which taken for granted is looming big, as a founda- ∂t | i | i poses no interpretive difficulty. Only when we decom- tion of the whole decoherence program. It is the b pose the total Hilbert-state space of the universe into question of what are the “systems” which play H a product of two spaces 1 2, and accordingly form such a crucial role in all the discussions of the H ⊗ H the joint-state vector Ψ = Ψ1 Ψ2 , and want to ascribe emergent classicality. (. . . ) [A] compelling ex- | i | i| i an individual state (besides the joint state that describes planation of what are the systems—how to define a correlation) to one of the two systems (say, the appara- them given, say, the overall Hamiltonian in some tus), does the measurement problem arise. Zurek (2003b, suitably large Hilbert space—would be undoubt- p. 718) puts it like this: edly most useful.

A frequently proposed idea is to abandon the notion of an “absolute” resolution and instead postulate the intrinsic 5 If we dare to postulate this total state—see counterarguments by relativity of the distinct state spaces and properties that Auletta (2000). emerge through the correlation between these relatively 9 defined spaces (see, for example, the proposals, unrelated in some (unknown) pure state. After the interaction, i.e., to decoherence, of Everett, 1957; Mermin, 1998a,b; and after the correlation between the systems is established, Rovelli, 1996). This relative view of systems and corre- the observer has access to only one of the systems, say, lations has counterintuitive, in the sense of nonclassical, system 1; everything that can be known about the state implications. However, as in the case of quantum entan- of the composite system must therefore be derived from glement, these implications need not be taken as para- measurements on system 1, which will yield the possible doxa that demand further resolution. Accepting some outcomes of system 1 and their probability distribution. properties of nature as counterintuitive is indeed a satis- All information that can be extracted by the observer factory path to take in order to arrive at a description of is then, exhaustively and correctly, contained in the re- nature that is as complete and objective as is allowed by duced density matrix of system 1, assuming that the Born the range of our experience (which is based on inherently rule for quantum probabilities holds. local observations). Let us return to the Einstein-Podolsky-Rosen-type ex- ample, Eqs. (3.1) and (3.2). If we assume that the states of system 2 are orthogonal, 2 + 2 = 0, ρ1 becomes B. The concept of reduced density matrices diagonal, h |−i

Since reduced density matrices are a key tool of deco- 1 1 ρ = Tr ψ ψ = ( + + ) + ( ) . (3.4) herence, it will be worthwile to briefly review their ba- 1 2| ih | 2 | ih | 1 2 |−ih−| 1 sic properties and interpretation in the following. The concept of reduced density matrices emerged in the ear- But this density matrix is formally identical to the den- liest days of quantum mechanics (Furry, 1936; Landau, sity matrix that would be obtained if system 1 were in a mixed state, i.e., in either one of the two states + and 1927; von Neumann, 1932; for some historical remarks, | i1 see Pessoa, 1998). In the context of a system of two en- 1 with equal probabilties—as opposed to the super- |−iposition ψ , in which both terms are considered present, tangled systems in a pure state of the Einstein-Podolsky- | i Rosen-type, which could in principle be confirmed by suitable inter- ference experiments. This implies that a measurement 1 of an observable that only pertains to system 1 cannot ψ = ( + + ), (3.1) 6 | i √2 | i1|−i2 − |−i1| i2 discriminate between the two cases, pure vs mixed state. However, note that the formal identification of the re- it had been realized early that for an observable O that duced density matrix with a mixed-state density matrix pertains only to system 1, O = O I , the pure-state is easily misinterpreted as implying that the state of the 1 2 b density matrix ρ = ψ ψ yields, according⊗ to the trace system can be viewed as mixed too (see also the discus- | ih | b b b rule O = Tr(ρO) and given the usual Born rule for sion by d’Espagnat, 1988). Density matrices are only a calculatingh i probabilities, exactly the same statistics as calculational tool for computing the probability distri- the reducedb densityb matrix ρ obtained by tracing over bution of a set of possible outcomes of measurements; 1 they do not specify the state of the system.7 Since the the degrees of freedom of system 2 (i.e., the states + 2 and ), | i two systems are entangled and the total composite sys- |−i2 tem is still described by a superposition, it follows from the standard rules of quantum mechanics that no indi- ρ1 = Tr2 ψ ψ = 2 + ψ ψ + 2 + 2 ψ ψ 2, (3.2) | ih | h | ih | i h−| ih |−i vidual definite state can be attributed to one of the sys- since it is easy to show that, for this observable O, tems. The reduced density matrix looks like a mixed- state density matrix because, if one actually measured b an observable of the system, one would expect to get a O ψ = Tr(ρO) = Tr1(ρ1O1). (3.3) h i definite outcome with a certain probability; in terms of This result holdsb in generalb for any pureb state ψ = measurement statistics, this is equivalent to the situation | i in which the system is in one of the states from the set of i αi φi 1 φi 2 φi N of a resolution of a system into | i | i ···| i possible outcomes from the beginning, that is, before the PN subsystems, where the φi j are assumed to form orthonormal basis sets in their{| i respective} Hilbert spaces measurement. As Pessoa (1998, p. 432) puts it, “taking , j =1 N. For any observable O that pertains only a partial trace amounts to the statistical version of the Hj · · · to system j, O = I I I O I I , projection postulate.” 1 ⊗ 2 ⊗···⊗ j−1 ⊗b j ⊗ j+1 ⊗···⊗ N the statisticsb of Ob generatedb byb applyingb b the trace ruleb will be identical regardless of whether we use the pure- b state density matrix ρ = ψ ψ or the reduced density 6 | ih | As discussed by Bub (1997, pp. 208–210), this result also holds matrix ρj = Tr1,...,j−1,j+1,...,N ψ ψ , since again O = for any observable of the composite system that factorizes into | ih | h i the form O = O1 ⊗ O2, where O1 and O2 do not commute with Tr(ρO) = Trj (ρj Oj ). b the projection operators (|±ih±|)1 and (|±ih±|)2, respectively. The typical situation in which the reduced density ma- 7 b b b b b b b In this context we note that any nonpure density matrix can be trix arises is this: Before a premeasurement-type inter- written in many different ways, demonstrating that any partition action, the observer knows that each individual system is in a particular ensemble of quantum states is arbitrary. 10

C. A modified von Neumann measurement scheme become known as environment-induced decoherence, and it has also frequently been claimed to imply at least a Let us now reconsider the von Neumann model for partial solution to the measurement problem. ideal quantum-mechanical measurement, Eq. (3.5), but now with the environment included. We shall denote the environment by and represent its state before the mea- 1. General formalism surement interactionE by the initial state vector e in | 0i a Hilbert space E . As usual, let us assume that the In Sec. III.B, we have already introduced the concept state space of theH composite object system-apparatus- of local (or reduced) density matrices and pointed out environment is given by the tensor product of the indi- some caveats on their interpretation. In the context of vidual Hilbert spaces, S A E . The linearity of the decoherence program, reduced density matrices arise the Schr¨odinger equationH then⊗ H yields⊗ H the following time as follows. Any observation will typically be restricted to evolution of the entire system , the system-apparatus component, , while the many SAE degrees of freedom of the environmentSA remain unob- (1) E c s a e c s a e served. Of course, typically some degrees of freedom of  n| ni| ri| 0i −→  n| ni| ni| 0i the environment will always be included in our obser- Xn Xn vation (e.g., some of the photons scattered off the ap- (2) c s a e , (3.5) paratus) and we shall accordingly include them in the −→ n| ni| ni| ni Xn “observed part of the universe.” The crucial point is that there stillSA remains a comparatively large number where the e are the states of the environment associ- n of environmental degrees of freedom that will not be ob- ated with the| i different pointer states a of the measur- n served directly. ing apparatus. Note that while for two| subsystems,i say, Suppose then that the operator O represents an ob- and , there always exists a diagonal (“Schmidt”) de- SA S A servable of only. Its expectation value OSA is given composition of the final state of the form cn sn an , b n | i| i by SA h i for three subsystems (for example, , ,P and ), a de- b composition of the form c s aS Ae is notE always n n| ni| ni| ni O = Tr(ρ [O I ]) = Tr (ρ O ), (3.6) possible. This implies thatP the total Hamiltonian that h SAi SAE SA ⊗ E SA SA SA induces a time evolution of the above kind, Eq. (3.5), whereb the densityb matrixb ρb of the totalb b combi- must be of a special form.8 SAE nation, SAE Typically, the e will be product states of many mi- | ni b crosopic subsystem states εn i corresponding to the in- ∗ | i ρSAE = cmcn sm am em sn an en , (3.7) dividual parts that form the environment, i.e., en = | i| i| ih |h |h | Xmn ε ε ε . We see that a nonseparable| andi in | ni1| ni2| ni3 · · · b most cases, for all practical purposes, irreversible (due to has, for all practical purposes of statistical prediction, the enormous number of degrees of freedom of the en- been replaced by the local (or reduced) density matrix vironment) correlation has been established between the ρSA, obtained by “tracing out the unobserved degrees of states of the system–apparatus combination and the the environment,” that is, different states of the environment . Note thatSA Eq. (3.5) b E also implies that the environment has recorded the state ρ = Tr (ρ )= c c∗ s a s a e e . SA E SAE m n| mi| mih n|h n|h n| mi of the system—and, equivalently, the state of the system- Xmn apparatus composition. The environment, composed of b b (3.8) many subsystems, thus acts as an amplifying, higher- So far, ρSA contains characteristic interference terms order measuring device. sm am sn an , m = n, since we cannot assume from the| i| outsetihb that|h the| basis6 vectors e of the environment | mi are necessarily mutually orthogonal, i.e., that en em = D. Decoherence and local suppression of interference 0 if m = n. Many explicit physical models forh the| inter-i action6 of a system with the environment (see Sec. III.D.2 Interaction with the environment typically leads to a below for a simple example), however, have shown that rapid vanishing of the diagonal terms in the local density due to the large number of subsystems that compose the matrix describing the probability distribution for the out- environment, the pointer states en of the environment comes of measurements on the system. This effect has rapidly approach orthogonality, |e ie (t) δ , such h n| mi → n,m that the reduced density matrix ρSA becomes approxi- mately orthogonal in the “pointer basis” an ; that is, b {| i} 8 For an example of such a Hamiltonian, see the model of Zurek t d 2 ρSA ρSA cn sn an sn an (1981, 1982) and its outline in Sec. III.D.2 below. For a criti- −→ ≈ | | | i| ih |h | Xn cal comment regarding limitations on the form of the evolution b b operator and the possibility of a resulting disagreement with ex- = c 2P (S) P (A). (3.9) | n| n ⊗ n perimental evidence, see Pessoa (1998). Xn b b 11

(S) (A) Here, Pn and Pn are the projection operators onto where the interference coefficient z(t) which determines the eigenstates of and , respectively. Therefore the the weight of the off-diagonal elements in the reduced interferenceb termsb S have vanishedA in this local represen- density matrix is given by tation, i.e., phase coherence has been locally lost. This is precisely the effect referred to as environment-induced N z(t)= E (t) E (t) = ( α 2eigk t + β 2e−igk t), decoherence. The decohered local density matrices de- h ⇑ | ⇓ i | k| | k| scribing the probability distribution of the outcomes of kY=1 (3.15) a measurement on the system-apparatus combination is and thus formally (approximately) identical to the corresponding mixed-state density matrix. But as we pointed out in N Sec. III.B, we must be careful in interpreting this state z(t) 2 = 1 + [( α 2 β 2)2 1] sin2 2g t . (3.16) | | { | k| − | k| − k } of affairs, since full coherence is retained in the total den- kY=1 sity matrix ρSAE . At t = 0, z(t) = 1, i.e., the interference terms are fully present, as expected. If α 2 = 0 or 1 for each k, i.e., | k| 2. An exactly solvable two-state model for decoherence if the environment is in an eigenstate of the interaction Hamiltonian HSE of the type 1 2 3 N , and/or if 2g t = mπ (m =0, 1,...), then|↑ i|↑z(i|↓t)2 i···|↑1 so coherencei To see how the approximate mutual orthogonality of k b ≡ the environmental state vectors arises, let us discuss a is retained over time. However, under realistic circum- simple model first introduced by Zurek (1982). Consider stances, we can typically assume a random distribution a system with two spin states , that inter- of the initial states of the environment (i.e., of coefficients acts withS an environment described{|⇑i |⇓i} by a collection αk, βk) and of the coupling coefficients gk. Then, in the of N other two-state spinsE represented by , , long-time average, {|↑ki |↓ki} k = 1 N. The self-Hamiltonians HS and HE and the N · · · 2 −N 2 2 2 N→∞ self-interaction Hamiltonian HEE of the environment are z(t) t→∞ 2 [1 + ( αk βk ) ] 0, taken to be equal to zero. Only the interactionb b Hamilto- h| | i ≃ | | − | | −→ b kY=1 nian HSE that describes the coupling of the spin of the (3.17) system to the spins of the environment is assumed to be so the off-diagonal terms in the reduced density matrix nonzerob and of the form become strongly damped for large N. It can also be shown directly that, given very general H = ( ) g ( ) I ′ , assumptions about the distribution of the couplings g SE |⇑ih⇑|−|⇓ih⇓| ⊗ k |↑kih↑k|−|↓kih↓k| k k Xk kO′ 6=k (namely, requiring their initial distribution to have finite b (3.10)b variance), z(t) exhibits a Gaussian time dependence of 2 2 iAt −B t /2 where the gk are coupling constants and Ik = ( k k + the form z(t) e e , where A and B are real ) is the identity operator for the kth environmen-|↑ ih↑ | constants (Zurek∝ et al., 2003). For the special case in |↓kih↓k| b tal spin. Applied to the initial state before the interaction which αk = α and gk = g for all k, this behavior of z(t) is turned on, can be immediately seen by first rewriting z(t) as the binomial expansion N 2 igt 2 −igt N ψ(0) = (a + b ) (αk k + βk k ), (3.11) z(t) = ( α e + β e ) | i |⇑i |⇓i |↑ i |↓ i | | | | Ok=1 N N = α 2l β 2(N−l)eig(2l−N)t. (3.18) this Hamiltonian yields a time evolution of the state given  l | | | | Xl=0 by For large N, the binomial distribution can then be ap- ψ(t) = a E⇑(t) + b E⇓(t) , (3.12) proximated by a Gaussian, | i |⇑i| i |⇓i| i 2 2 2 2 where the two environmental states E (t) and E (t) −(l−N|α| ) /(2N|α| |β| ) ⇑ ⇓ N 2l 2(N−l) e are | i | i α β , (3.19)  l | | | | ≈ 2πN α 2 β 2 | | | | N p E (t) = E ( t) = (α eigkt +β e−igkt ). in which case z(t) becomes | ⇑ i | ⇓ − i k |↑ki k |↓ki Ok=1 N 2 2 2 2 (3.13) e−(l−N|α| ) /(2N|α| |β| ) z(t)= eig(2l−N)t, (3.20) The reduced density matrix ρS (t) = TrE ( ψ(t) ψ(t) ) is 2πN α 2 β 2 | ih | Xl=0 | | | | then p that is, z(t) is the Fourier transform of an (approxi- ρ (t) = a 2 + b 2 S | | |⇑ih⇑| | | |⇓ih⇓| mately) Gaussian distribution and is therefore itself (ap- +z(t)ab∗ + z∗(t)a∗b , (3.14) proximately) Gaussian. |⇑ih⇓| |⇓ih⇑| 12

Detailed model calculations, in which the environ- sion of the postmeasurement state and thereby leaves ment is typically represented by a more sophisticated open the question of which observable can be considered model consisting of a collection of harmonic oscillators as having been measured by the apparatus. This situa- (Caldeira and Leggett, 1983; Hu et al., 1992; Joos et al., tion is changed by the inclusion of the environment states 2003; Unruh and Zurek, 1989; Zurek, 2003b; Zurek et al., in Eq. (3.5), for the following two reasons: 1993), have shown that the damping occurs on extremely short decoherence time scales τD, which are typically (1) Environment-induced superselection of a preferred many orders of magnitude shorter than the thermal relax- basis. The interaction between the apparatus and ation. Even microscopic systems such as large molecules the environment singles out a set of mutually com- are rapidly decohered by the interaction with thermal muting observables. radiation on a time scale that is much shorter than any practical observation could resolve; for mesoscopic sys- (2) The existence of a tridecompositional uniqueness tems such as dust particles, the 3K cosmic microwave theorem (Bub, 1997; Clifton, 1994; Elby and Bub, 1994). If a state ψ in a Hilbert space background radiation is sufficient to yield strong and im- | i H1 ⊗H2 ⊗H3 mediate decoherence (Joos and Zeh, 1985; Zurek, 1991). can be decomposed into the diagonal (“Schmidt”) Within τ , z(t) approaches zero and remains close to form ψ = i αi φi 1 φi 2 φi 3, the expansion is D | | unique| providedi that| i the| i | φ i and φ are zero, fluctuating with an average standard deviation of P {| ii1} {| ii2} the random-walk type σ √N (Zurek, 1982). However, sets of linearly independent, normalized vectors in and , respectively, and that φ is a set the multiple periodicity∼ of z(t) implies that coherence, 1 2 i 3 Hof mutuallyH noncollinear normalized{| vectorsi } in . and thus the purity of the reduced density matrix, will H3 reappear after a certain time τ , which can be shown to This can be generalized to an N-decompositional r uniqueness theorem, in which N 3. Note that be very long and of the Poincar´etype with τr N!. For ≥ macroscopic environments of realistic but finite∼ sizes, τ it is not always possible to decompose an arbitrary r pure state of more than two systems (N 3) into can exceed the lifetime of the universe (Zurek, 1982), but ≥ nevertheless always remains finite. the Schmidt form ψ = i αi φi 1 φi 2 φi N , but if the decomposition| i exists,| itsi | uniquenessi ···| i is From a conceptual point of view, recurrence of coher- P guaranteed. ence is of little relevance. The recurrence time could only be infinitely long in the hypothetical case of an in- The tridecompositional uniqueness theorem ensures finitely large environment. In this situation, off-diagonal that the expansion of the final state in Eq. (3.5) is unique, terms in the reduced density matrix would be irreversibly which fixes the ambiguity in the choice of the set of pos- damped and lost in the limit t , which has some- → ∞ sible outcomes. It demonstrates that the inclusion of (at times been regarded as describing a physical collapse of least) a third “system” (here referred to as the environ- the state vector (Hepp, 1972). But the assumption of ment) is necessary to remove the basis ambiguity. infinite sizes and times is never realized in nature (Bell, Of course, given any pure state in the composite 1975), nor can information ever be truly lost (as achieved Hilbert space , the tridecompositional by a “true” state vector collapse) through unitary time 1 2 3 uniqueness theoremH ⊗ neither H ⊗ H tells us whether a Schmidt evolution—full coherence is always retained at all times decomposition exists nor specifies the unique expansion in the total density matrix ρ (t)= ψ(t) ψ(t) . SAE | ih | itself (provided the decomposition is possible), and since We can therefore state the general conclusion that, the precise states of the environment are generally not except for exceptionally well-isolated and carefully pre- known, an additional criterion is needed that determines pared microsopic and mesoscopic systems, the interac- what the preferred states will be. tion of the system with the environment causes the off- diagonal terms of the local density matrix, expressed in the pointer basis and describing the probability distribu- 1. Stability criterion and pointer basis tion of the possible outcomes of a measurement on the system, to become extremely small in a very short pe- The decoherence program has attempted to define such riod of time, and that this process is irreversible for all a criterion based on the interaction with the environ- practical purposes. ment and the idea of robustness and preservation of cor- relations. The environment thus plays a double role in suggesting a solution to the preferred-basis problem: it E. Environment-induced superselection selects a preferred pointer basis, and it guarantees its uniqueness via the tridecompositional uniqueness theo- Let us now turn to the second main consequence rem. of the interaction with the environment, namely, the In order to motivate the basis superselection approach environment-induced selection of stable preferred-basis proposed by the decoherence program, we note that in states. We discussed in Sec. II.C the fact that step (2) of Eq. (3.5) we tacitly assumed that interaction the quantum-mechanical measurement scheme as repre- with the environment does not disturb the established sented by Eq. (2.1) does not uniquely define the expan- correlation between the state of the system, s , and | ni 13

the corresponding pointer state an . This assumption are measured by the apparatus, that is, reliably recorded can be viewed as a generalization of| thei concept of “faith- through the formation of dynamically stable quantum ful measurement” to the realistic case in which the envi- correlations. The tridecompositional uniqueness theorem ronment is included. Faithful measurement in the usual then guarantees the uniqueness of the expansion of the sense concerns step (1), namely, the requirement that the final state ψ = c s a e (where no constraints | i n n| ni| ni| ni measuring apparatus act as a reliable “mirror” of the on the cn have toP be imposed) and thereby the uniqueness states of the system A by forming only correlations of of the preferred pointer basis. S the form sn an but not sm an with m = n. But Other criteria similar to the commutativity require- | i| i | i| i 6 since any realistic measurement process must include the ment, Eq. (3.21), have been suggested for the selection inevitable coupling of the apparatus to its environment, of the preferred pointer basis because it turns out that in the measurement could hardly be considered faithful as a realistic cases the simple relation of Eq. (3.21) can usually whole if the interaction with the environment disturbed only be fulfilled approximately (Zurek, 1993; Zurek et al., the correlations between the system and the apparatus.9 1993). More general criteria, for example, have been 2 2 It was therefore first suggested by Zurek (1981) that based on the von Neumann entropy TrρΨ(t) ln ρΨ(t), 2 − the preferred pointer basis be taken as the basis which or the purity TrρΨ(t), with the goal of finding the most “contains a reliable record of the state of the system ” robust states or the states which become least entan- S (Zurek, 1981, p. 1519), i.e., the basis in which the system- gled with the environment in the course of the evolution apparatus correlations sn an are left undisturbed by (Zurek, 1993, 1998, 2003b; Zurek et al., 1993). Pointer | i| i the subsequent formation of correlations with the envi- states are obtained by extremizing the measure (i.e., min- ronment (the stability criterion). One can then find a imizing entropy, or maximizing purity, etc.) over the ini- sufficient criterion for dynamically stable pointer states tial state Ψ and requiring the resulting states to be ro- that preserve the system–apparatus correlations in spite bust when| varyingi the time t. Application of this method of the interaction of the apparatus with the environ- leads to a ranking of the possible pointer states with re- ment by requiring all pointer state projection opera- spect to their “classicality,” i.e., their robustness with (A) tors Pn = an an to commute with the apparatus- respect to interaction with the environment, and thus al- | ih | 10 environment interaction Hamiltonian HAE , lows for the selection of preferred pointer basis in terms b of the “most classical” pointer states (the “predictability (A) b sieve”; see Zurek, 1993; Zurek et al., 1993). Although [Pn , HAE ] = 0 for all n. (3.21) the proposed criteria differ somewhat and other mean- This implies thatb anyb correlation of the measured system ingful criteria are likely to be suggested in the future, (or any other system, for instance an observer) with the it is hoped that in the macrosopic limit the resulting eigenstates of a preferred apparatus observable, stable pointer states obtained from different criteria will turn out to be very similar (Zurek, 2003b). For some (A) toy models (in particular, for harmonic-oscillator mod- OA = λnPn , (3.22) Xn els that lead to coherent states as pointer states), this b b has already been verified explicitly (see, for example, is preserved, and that the states of the environment re- Di´osi and Kiefer, 2000; Eisert, 2004; Joos et al., 2003; (A) liably mirror the pointer states Pn . In this case, the K¨ubler and Zeh, 1973; Zurek, 1993). environment can be regarded as carrying out a nonde- molition measurement on the apparatus.b The commu- tativity requirement, Eq. (3.21), is obviously fulfilled if 2. Selection of quasiclassical properties HAE is a function of OA, HAE = HAE (OA). Conversely, system-apparatus correlations in which the states of the b b b b b System-environment interaction Hamiltonians fre- apparatus are not eigenstates of an observable that com- quently describe a scattering process of surrounding par- mutes with HAE will in general be rapidly destroyed by ticles (photons, air molecules, etc.) interacting with the the interaction. system under study. Since the force laws describing such b Put the other way around, this implies that the envi- processes typically depend on some power of distance ronment determines through the form of the interaction (such as r−2 in Newton’s or Coulomb’s force law), ∝ Hamiltonian HAE , a preferred apparatus observable OA, the interaction Hamiltonian will usually commute with Eq. (3.22), and thereby also the states of the system that the position basis, such that, according the commutativ- b b ity requirement of Eq. (3.21), the preferred basis will be in position space. The fact that position is frequently the determinate property of our experience can then be 9 For fundamental limitations on the precision of von Neumann explained by referring to the dependence of most inter- measurements of operators that do not commute with a glob- actions on distance (Zurek, 1981, 1982, 1991). ally conserved quantity, see the Wigner-Araki-Yanase theorem (Araki and Yanase, 1960; Wigner, 1952). This holds, in particular, for mesoscopic and macro- 10 For simplicity, we assume here that the environment E interacts scopic systems, as demonstrated, for instance, by the directly only with the apparatus A, but not with the system S. pioneering study of Joos and Zeh (1985), in which sur- 14 rounding photons and air molecules are shown to contin- by HSE , i.e., the interaction with the environment, uously “measure” the spatial structure of dust particles, the pointer states will be eigenstates of H (and b SE leading to rapid decoherence into an apparent (improper) thus typically eigenstates of position). This case mixture of wave packets that are sharply peaked in po- corresponds to the typical quantum measurementb sition space. Similar results sometimes even hold for mi- setting; see, for example, the model of Zurek (1981, croscopic systems (usually found in energy eigenstates; 1982), which is outlined in Sec. III.D.2 above. see below) when they occur in distinct spatial struc- tures that couple strongly to the surrounding medium. (2) When the interaction with the environment is weak For instance, chiral molecules such as sugar are always and HS dominates the evolution of the system (that observed to be in chirality eigenstates (left-handed and is, when the environment is “slow” in the above b right-handed) which are superpositions of different en- sense), a case that frequently occurs in the micro- ergy eigenstates (Harris and Stodolsky, 1981; Zeh, 2000). scopic domain, pointer states will arise that are en- This is explained by the fact that the spatial structure ergy eigenstates of HS (Paz and Zurek, 1999). of these molecules is continuously “monitored” by the environment, for example, through the scattering of air (3) In the intermediateb case, when the evolution of molecules, which gives rise to a much stronger coupling the system is governed by HSE and HS in roughly equal strength, the resulting preferred states will than could typically be achieved by a measuring device b b that was intended to measure, say, parity or energy; fur- represent a “compromise” between the first two thermore, any attempt to prepare such molecules in en- cases; for instance, the frequently studied model ergy eigenstates would lead to immediate decoherence of quantum Brownian motion has shown the emer- into environmentally stable (“dynamically robust”) chi- gence of pointer states localized in phase space, rality eigenstates, thus selecting position as the preferred i.e., in both position and momentum (Eisert, 2004; basis. Joos et al., 2003; Unruh and Zurek, 1989; Zurek, On the other hand, it is well known that many systems, 2003b; Zurek et al., 1993). especially in the microsopic domain, are typically found in energy eigenstates, even if the interaction Hamiltonian 3. Implications for the preferred-basis problem depends on a different observable than energy, e.g., posi- tion. Paz and Zurek (1999) have shown that this situa- The decoherence program proposes that the preferred tion arises when the predominant frequencies present in basis be selected by the requirement that correlations the environment are significantly lower than the intrinsic be preserved in spite of the interaction with the en- frequencies of the system, that is, when the separation vironment, and thus be chosen through the form of between the energy states of the system is greater than the system-environment interaction Hamiltonian. This the largest energies available in the environment. Then, seems certainly reasonable, since only such “robust” the environment will only be able to monitor quantities states will in general be observable—and, after all, we that are constants of motion. In the case of nondegener- solely seek an explanation for our experience (see the acy, this will be energy, thus leading to the environment- discussion in Sec. II.B.3). Although only particular ex- induced superselection of energy eigenstates for the sys- amples have been studied (for a survey and references, tem. see, for example, Blanchard et al., 2000; Joos et al., 2003; Another example of environment-induced superselec- Zurek, 2003b), the results thus far suggest that the se- tion that has been studied is related to the fact that only lected properties are in agreement with our observation: eigenstates of the charge operator are observed, but never for mesoscopic and macroscopic objects the distance- superpositions of different charges. The existence of the dependent scattering interaction with surrounding air corresponding superselection rules was first only postu- molecules, photons, etc., will in general give rise to im- lated (Wick et al., 1952, 1970), but could subsequently mediate decoherence into spatially localized wave pack- be explained in the framework of decoherence by refer- ets and thus select position as the preferred basis. On ring to the interaction of the charge with its own Coulomb the other hand, when the environment is comparably (far) field, which takes the role of an “environment,” lead- “slow,” as is frequently the case for microsopic systems, ing to immediate decoherence of charge superpositions environment-induced superselection will typically yield into an apparent mixture of charge eigenstates (Giulini, energy eigenstates as the preferred states. 2000; Giulini et al., 1995). The clear merit of the approach of environment- In general, three different cases have typically been induced superselection lies in the fact that the preferred distinguished (for example, in Paz and Zurek, 1999) for basis is not chosen in an ad hoc manner simply to the kind of pointer observable emerging from an inter- make our measurement records determinate or to match action with the environment, depending on the relative our experience of which physical quantities are usually strengths of the system’s self-Hamiltonian HS and of the perceived as determinate (for example, position). In- system-environment interaction Hamiltonian H : stead the selection is motivated on physical, observer-free b SE grounds, that is, through the system-environment inter- (1) When the dynamics of the system are dominatedb action Hamiltonian. The vast space of possible quantum- 15 mechanical superpositions is reduced so much because has not yet been fully explored. the laws governing physical interactions depend only on Finally, a fundamental conceptual difficulty of the a few physical quantities (position, momentum, charge, decoherence-based approach to the preferred-basis prob- and the like), and the fact that precisely these are the lem is the lack of a general criterion for what defines the properties that appear determinate to us is explained by systems and the “unobserved” degrees of freedom of the the dependence of the preferred basis on the form of the environment (see the discussion in Sec. III.A). While in interaction. The appearance of “classicality” is therefore many laboratory-type situations, the division into sys- grounded in the structure of the physical laws—certainly tem and environment might seem straightforward, it is a highly satisfying and reasonable approach. not clear a priori how quasiclassical observables can be The above argument in favor of the approach of defined through environment-induced superselection on environment-induced superselection could, of course, be a larger and more general scale, when larger parts of considered as inadequate on a fundamental level: All the universe are considered where the split into subsys- physical laws are discovered and formulated by us, so tems is not suggested by some specific system-apparatus- they can contain only the determinate quantities of our surroundings setup. experience. These are the only quantities we can perceive To summarize, environment-induced superselection of and thus include in a physical law. Thus the derivation a preferred basis (i) proposes an explanation for why a of determinacy from the structure of our physical laws particular pointer basis gets chosen at all—by arguing might seem circular. However, we argue again that it that it is only the pointer basis that leads to stable, and suffices to demand a subjective solution to the preferred- thus perceivable, records when the interaction of the ap- basis problem—that is, to provide an answer to the ques- paratus with the environment is taken into account; and tion of why we perceive only such a small subset of prop- (ii) argues that the preferred basis will correspond to a erties as determinate, not whether there really are de- subset of the set of the determinate properties of our terminate properties (on an ontological level) and what experience, since the governing interaction Hamiltonian they are (cf. the remarks in Sec. II.B.3). will depend solely on these quantities. But it does not tell We might also worry about the generality of this us precisely what the pointer basis will be in any given approach. One would need to show that any such physical situation, since it will usually be hardly possible environment-induced superselection leads,in fact, to pre- to write down explicitly the relevant interaction Hamil- cisely those properties that appear determinate to us. tonian in realistic cases. This also means that it will But this would require precise knowledge of the system be difficult to argue that any proposed criterion based and the interaction Hamiltonian. For simple toy models, on the interaction with the environment will always and the relevant Hamiltonians can be written down explicitly. in all generality lead to exactly those properties that we In more complicated and realistic cases, this will in gen- perceive as determinate. eral be very difficult, if not impossible, since the form of More work remains to be done, therefore, to fully ex- the Hamiltonian will depend on the particular system or plore the general validity and applicability of the ap- apparatus and the monitoring environment under consid- proach of environment-induced superselection. But since eration, where, in addition, the environment is not only the results obtained thus far from toy models have been difficult to define precisely, but also constantly changing, in promising agreement with empirical data, there is little uncontrollable, and, in essence, infinitely large. reason to doubt that the decoherence program has pro- But the situation is not as hopeless as it might sound, posed a very valuable criterion for explaining the emer- since we know that the interaction Hamiltonian will, in gence of preferred states and their robustness. The fact general, be based on the set of known physical laws that the approach is derived from physical principles which, in turn, employ only a relatively small number should be counted additionally in its favor. of physical quantities. So as long as we assume the sta- bility criterion and consider the set of known physical quantities as complete, we can automatically anticipate 4. Pointer basis vs instantaneous Schmidt states that the preferred basis will be a member of this set. The remaining, yet very relevant, question is then which sub- The so-called Schmidt basis, obtained by diagonaliz- set of these properties will be chosen in a specific physi- ing the (reduced) density matrix of the system at each cal situation (for example, will the system preferably be instant of time, has been frequently studied with respect found in an eigenstate of energy or position?), and to to its ability to yield a preferred basis (see, for exam- what extent this will match the experimental evidence. ple, Albrecht, 1992, 1993; Zeh, 1973), having led some to To give an answer, one will usually need a more detailed consider the Schmidt basis states as describing “instan- knowledge of the interaction Hamiltonian and of its rela- taneous pointer states” (Albrecht, 1992). However, as it tive strength with respect to the self-Hamiltonian of the has been emphasized (for example, by Zurek, 1993), any system in order to verify the approach. Besides, as men- density matrix is diagonal in some basis, and this ba- tioned in Sec. III.E, there exist other criteria than the sis will in general not play any special interpretive role. commutativity requirement, and whether they all lead Pointer states that are supposed to correspond to qua- to the same determinate properties is a question that siclassical stable observables must be derived from an 16 explicit criterion for classicality (typically, the stability by other authors, who assessed the approach, pointed out criterion); the simple mathematical diagonalization pro- more clearly the assumptions entering into the deriva- cedure of the instantaneous density matrix will gener- tion, and presented variants of the proof (Barnum, 2003; ally not suffice to determine a quasiclassical pointer basis Mohrhoff, 2004; Schlosshauer and Fine, 2005). An ex- (see the studies by Barvinsky and Kamenshchik, 1995; panded treatment of envariance and quantum probabili- Kent and McElwaine, 1997). ties that addresses some of the issues discussed in these In a more refined method, one refrains from comput- papers and that offers an interesting outlook on fur- ing instantaneous Schmidt states and instead allows for a ther implications of envariance can be found in Zurek characteristic decoherence time τD to pass during which (2004a). In our outline of the theory of envariance, we the reduced density matrix decoheres (a process that can shall follow this most recent treatment, as it spells out be described by an appropriate master equation) and the derivation and the required assumptions more ex- becomes approximately diagonal in the stable pointer plicitly and in greater detail and clarity than in Zurek’s basis, the basis that is selected by the stability crite- earlier (2003b; 2003c; 2004b) papers (cf. also the remarks rion. Schmidt states are then calculated by diagonalizing of Schlosshauer and Fine, 2005). the decohered density matrix. Since decoherence usually We include a discussion of Zurek’s proposal here for leads to rapid diagonality of the reduced density matrix two reasons. First, the derivation is based on the in- in the stability-selected pointer basis to a very good ap- clusion of an environment , entangled with the system proximation, the resulting Schmidt states are typically of interest to which probabilitiesE of measurement out- very similar to the pointer basis except when the pointer comesS are to be assigned, and thus it matches well the states are very nearly degenerate. The latter situation spirit of the decoherence program. Second, and more is readily illustrated by considering the approximately importantly, despite the contributions of decoherence to diagonalized decohered density matrix explaining the emergence of subjective classicality from quantum mechanics, a consistent derivation of classical- 1/2+ δ ω∗ ρ = , (3.23) ity (including a motivation for some of the axioms of  ω 1/2 δ  − quantum mechanics, as suggested by Zurek, 2003b) re- quires the separate derivation of the Born rule. The deco- where ω 1 (strong decoherence) and δ 1 (near- herence program relies heavily on the concept of reduced degeneracy;| | ≪Albrecht, 1993). If decoherence led≪ to exact density matrices and the related formalism and interpre- diagonality, ω = 0, the eigenstates would be, for any fixed tation of the trace operation, see Eq. (3.6), which presup- value of δ, proportional to (0, 1) and (1, 0) (correspond- pose Born’s rule. Therefore decoherence itself cannot be ing to the “ideal” pointer states). However, for fixed used to derive the Born rule (as was tried, for example, ω > 0 (approximate diagonality) and δ 0 (degener- by Deutsch, 1999 and Zurek, 1998) since otherwise the acy), the eigenstates become proportional→ to ( ω /ω, 1), argument would be rendered circular (Zeh, 1997; Zurek, which implies that, in the case of degeneracy, the±| Schmidt| 2003b). decomposition of the reduced density matrix can yield preferred states that are very different from the stable There have been various attempts in the past to re- pointer states, even if the decohered, rather than the in- place the postulate of the Born rule by a derivation. stantaneous, reduced density matrix is diagonalized. Gleason’s (1957) theorem has shown that if one im- In summary, it is important to emphasize that stability poses the condition that for any orthonormal basis of (or a similar criterion) is the relevant requirement for a given Hilbert space the sum of the probabilities asso- the emergence of a preferred quasiclassical basis, which ciated with each basis vector must add up to one, the cannot, in general, be achieved by simply diagonalizing Born rule is the only possibility for the calculation of the instantaneous reduced density matrix. However, the probabilities. However, Gleason’s proof provides little in- eigenstates of the decohered reduced density matrix will, sight into the physical meaning of the Born probabilities, in many situations, approximate the quasiclassical stable and therefore various other attempts, typically based on pointer states well, especially when these pointer states a relative frequencies approach (i.e., on a counting ar- are sufficiently nondegenerate. gument), have been made towards a derivation of the Born rule in a no-collapse (and usually relative-state) setting (see, for example, Deutsch, 1999; DeWitt, 1971; F. Envariance, quantum probabilities, and the Born rule DeWitt and Graham, 1973; Everett, 1957; Farhi et al., 1989; Geroch, 1984; Graham, 1973; Hartle, 1968). How- In the following, we shall review an interesting ever, it was pointed out that these approaches fail due and promising approach introduced recently by Zurek to the use of circular arguments (Barnum et al., 2000; (2003b,c, 2004a,b) that aims to explain the emergence of Kent, 1990; Squires, 1990; Stein, 1984); cf. also Wallace quantum probabilities and to deduce the Born rule based (2003b) and Saunders (2002). on a mechanism termed “environment-assisted invari- Zurek’s recently developed theory of envariance pro- ance,” or “envariance” for short, a particular symmetry vides a promising new strategy for deriving, given certain property of entangled quantum states. The original expo- assumptions, the Born rule in a manner that avoids the sition of Zurek (2003b) was followed up by several articles circularities of the earlier approaches. We shall outline 17 the concept of envariance in the following and show how (A3) implies that the final state of (after the trans- it can lead to Born’s rule. formation and countertransformation)S is identical to the initial state of , the first transformation uS IE cannot have altered theS state of either. Thus, using⊗ assump- 1. Environment-assisted invariance tion (A2), it follows thatS an envariant transformationb b uS IE acting on ψSE leaves any measurable properties Zurek introduces his definition of envariance as follows. of ⊗unchanged, in| particulari the probabilities associated Consider a composite state ψ (where, as usual, S b | SE i S withb outcomes of measurements performed on . refers to the “system” and to some “environment”) in Let us now consider two different envariantS transfor- a Hilbert space given by theE tensor product , HS ⊗ HE mations: A phase transformation of the form and a pair of unitary transformations US = uS IE and ⊗ iξ1 iξ2 UE = IS uE acting on and , respectively. If ψSE is uS (ξ1, ξ2)= e s1 s1 + e s2 s2 (3.26) ⊗ S E b b | b i | ih | | ih | invariant under the combined application of US and UE , that changes the phases associated with the Schmidt b b b b product states sk ek in Eq. (3.25), and a swap trans- U (U ψ )= ψ , b (3.24)b | i| i E S | SE i | SE i formation ψ is called envariantb b under u . In other words, the u (1 2) = eiξ12 s s + eiξ21 s s (3.27) | SE i S S ↔ | 1ih 2| | 2ih 1| change in ψSE induced by acting on via US can be | i b S that exchangesb the pairing of the sk with the el . Based undone by acting on via U . Note that envariance is | i | i E b on the assumptions (A1)–(A3) mentioned above, envari- a distinctly quantumE feature, absent from pure classical b ance of ψSE under these transformations means that states, and a consequence of quantum entanglement. measurable| propertiesi of cannot depend on the phases The main argument of Zurek’s derivation is based on a S ϕk in the Schmidt expansion of ψSE , Eq. (3.25). Simi- study of a composite pure state in the diagonal Schmidt | i larily, it follows that a swap uS(1 2) leaves the state decomposition of unchanged, and that the consequences↔ of the swap cannotS be detected by any measurementb that pertains to 1 iϕ1 iϕ2 ψSE = e s1 e1 + e s1 e1 , (3.25) alone. | i √2 | i| i | i| i S
where the s and e are sets of orthonormal basis {| ki} {| ki} 2. Deducing the Born rule vectors that span the Hilbert spaces S and E , respec- tively. The case of higher-dimensionalH stateH spaces can Together with an additional assumption, this result be treated similarly, and a generalization to expansion can then be used to show that the probabilities of the coefficients of different magnitudes can be made by ap- “outcomes” s appearing in the Schmidt decomposi- plication of a standard counting argument (Zurek, 2003c, k tion of ψ |musti be equal, thus arriving at Born’s rule 2004a). The Schmidt states s are identified with the SE k for the| speciali case of a state-vector expansion with co- outcomes, or “events” (Zurek| , i2004b, p. 12), to which efficients of equal magnitude. Zurek (2004a) offers three probabilities are to be assigned. possibilities for such an assumption. Here we shall limit Zurek now states three simple assumptions, called our discussion to one of these possible assumptions (see “facts” (Zurek, 2004a, p. 4; see also the discussion in also the comments in Schlosshauer and Fine, 2005): Schlosshauer and Fine, 2005): (A4) The Schmidt product states s e appearing in | ki| ki (A1) A unitary transformation of the form IS the state-vector expansion of ψSE imply a direct does not alter the state of . ···⊗ ⊗··· and perfect correlation of the| measurementi out- S b comes associated with the sk and ek . That is, (A2) All measurable properties of , including probabil- | i | i S if an observable OS = skl sk sl is measured on ities of outcomes of measurements on , are fully | ih | S and sk is obtained,P a subsequent measurement determined by the state of . S | i b S of O = e e e on will yield e with cer- E kl| kih l| E | ki (A3) The state of is completely specified by the global tainty (i.e., with probability equal to one). b P composite stateS vector ψ . | SE i This assumption explicitly introduces a probability Given these assumptions, one can show that the state concept into the derivation. (Similarly, the two other pos- of and any measurable properties of cannot be af- sible assumptions suggested by Zurek establish a connec- fectedS by envariant transformations. TheS proof goes as tion between the state of and probabilities of outcomes of measurements on .) S follows. The effect of an envariant transformation uS IE S ⊗ Then, denoting the probability for the outcome sk by acting on ψSE can be undone by a corresponding “coun- | i | i b b p( sk , ψSE ) when the composite system is described tertransformation” IS uE that restores the original state | i | i SE ⊗ by the state vector ψSE , this assumption implies that vector ψSE . Since it follows from (A1) that the latter | i transformation| i hasb leftb the state of unchanged, but p( s ; ψ )= p( e ; ψ ). (3.28) S | ki | SE i | ki | SE i 18

After acting on ψ with the envariant swap transfor- is, for environment-induced superselection) without an | SE i mation US = uS(1 2) IE [see Eq. (3.27)] and using appeal to the trace operation or to reduced density ma- assumption (A4) again,↔ we⊗ get trices. This could open up the possibility of a redevelop- b b b ment of the decoherence program based on fundamental p( s1 ; US ψSE )= p( e2 ; US ψSE ), quantum-mechanical principles that do not require one to | i | i | i | i (3.29) presuppose the Born rule; this also might shed new light, p( s2 ; UbS ψSE )= p( e1 ; UbS ψSE ). | i | i | i | i for example, on the interpretation of reduced density ma- b b trices that has led to much controversy in discussions of Now, when a “counterswap” UE = IS uE (1 2) is decoherence (see Sec. III.B). As of now, the development applied to ψ , the original state vector⊗ ψ ↔ is re- | SE i b b | SE i of such ideas is at a very early stage, but we can expect stored, i.e., UE (US ψSE ) = ψSE . It then follows from further interesting results derived from envariance in the | i | i assumptions (A2) and (A3) listed above that near future. b b p( s ; U U ψ )= p( s ; ψ ). (3.30) | ki E S | SE i | ki | SE i Furthermore, assumptionsb b (A1) and (A2) imply that the IV. THE ROLE OF DECOHERENCE IN INTERPRETATIONS OF QUANTUM MECHANICS first and second swap cannot have affected the measur- able properties of and , respectively, particularly not the probabilities forE outcomesS of measurements on ( ), It was not until the early 1970s that the importance E S of the interaction of physical systems with their envi- ronments for a realistic quantum-mechanical description p( sk ; UE US ψSE )= p( sk ; US ψSE ), of these systems was realized and a proper viewpoint | i | i | i | i (3.31) on such interactions was established (Zeh, 1970, 1973). p( ekb; UbS ψSE )= p( ek ; bψSE ). | i | i | i | i It took another decade for the first concise formulation of the theory of decoherence (Zurek, 1981, 1982) to be Combining Eqs.b (3.28)–(3.31) yields worked out and for numerical studies to be made that (3.30) showed the ubiquity and effectiveness of decoherence ef- p( s ; ψ ) = p( s ; U U ψ ) | 1i | SE i | 1i E S | SE i fects (Joos and Zeh, 1985). Of course, by that time, (3.31) several interpretive approaches to quantum mechanics = p( s ; Ub bψ ) | 1i S | SE i had already been established, for example, Everett-style (3.29) = p( e ; Ub ψ ) relative-state interpretations (Everett, 1957), the con- | 2i S| SE i (3.31) cept of modal interpretations introduced by van Fraassen = p( e ; bψ ) | 2i | SE i (1973, 1991), and the pilot-wave theory of de Broglie and (3.28) Bohm (Bohm, 1952). = p( s ; ψ ), (3.32) | 2i | SE i When the relevance of decoherence effects was recog- which establishes the desired result p( s1 ; ψSE ) = nized by (parts of) the scientific community, decoherence p( s ; ψ ). The general case of unequal| coefficientsi | i in provided a motivation for a fresh look at the existing in- | 2i | SE i the Schmidt decomposition of ψSE can then be treated terpretations and for the introduction of changes and ex- by means of a simple counting| methodi (Zurek, 2003c, tensions to these interpretations, as well as for new inter- 2004a), leading to Born’s rule for probabilities that are pretations. Some of the central questions in this context rational numbers. Using a continuity argument, this re- were, and still are, the following: sult can be further generalized to include probabilities that cannot be expressed as rational numbers (Zurek, 1. Can decoherence by itself solve certain foundational 2004a). issues at least for all practical purposes, such as to make certain interpretive additives superfluous? What, then, are the crucial remaining foundational 3. Summary and outlook problems?

If one grants the stated assumptions, Zurek’s devel- 2. Can decoherence protect an interpretation from opment of the theory of envariance offers a novel and empirical disproof? promising way of deducing Born’s rule in a noncircular manner. Compared to the relatively well-studied field of 3. Conversely, can decoherence provide a mechanism decoherence, envariance and its consequences have only to exclude an interpretive strategy as incompati- begun to be explored. In this review, we have focused ble with quantum mechanics and/or as empirically on envariance in the context of a derivation of the Born inadequate? rule, but other far-reaching implications of envariance have recently been suggested by Zurek (2004a). For 4. Can decoherence physically motivate some of the example, envariance could also account for the emer- assumptions on which an interpretation is based gence of an environment-selected preferred basis (that and give them a more precise meaning? 19

5. Can decoherence serve as an amalgam that would The hope is then that environment-induced super- unify and simplify a spectrum of different interpre- selection of a preferred basis can provide a universal tations? mechanism that fulfills the above criteria and solves the preferred-basis problem on strictly physical grounds. These and other questions have been widely discussed, both in the context of particular interpretations and A popular reading of the decoherence program typi- with respect to the general implications of decoherence cally goes as follows. First, the interaction of the system for any interpretation of quantum mechanics. In par- with the environment selects a preferred basis, i.e., a par- ticular, interpretations that uphold the universal va- ticular set of quasiclassical robust states that commute, lidity of the unitary Schr¨odinger time evolution, most at least approximately, with the Hamiltonian governing notably relative-state and modal interpretations, have the system–environment interaction. Since the form of frequently incorporated environment-induced superselec- the interaction Hamiltonians usually depends on familiar tion of a preferred basis and decoherence into their frame- “classical” quantities, the preferred states will typically work. It is the purpose of this section to critically inves- also correspond to the small set of “classical” properties. tigate the implications of decoherence for the existing in- Decoherence then quickly damps superpositions between terpretations of quantum mechanics, with particular at- the localized preferred states when only the system is con- tention to the questions outlined above. sidered. This is taken as an explanation of the appear- ance to a local observer of a “classical” world of determi- nate, “objective” (in the sense of being robust) proper- A. General implications of decoherence for interpretations ties. The tempting interpretation of these achievements is then to conclude that this accounts for the observation When measurements are understood as ubiquitous in- of unique (via environment-induced superselection) and teractions that lead to the formation of quantum correla- definite (via decoherence) pointer states at the end of the tions, the selection of a preferred basis becomes in most measurement, and the measurement problem appears to cases a fundamental requirement. This also corresponds, be solved, at least for all practical purposes. in general, to the question of what properties are being ascribed to systems (or worlds, minds, etc.). Thus the However, the crucial difficulty in the above reasoning preferred-basis problem is at the heart of any interpreta- is justifying the second step: How is one to interpret the tion of quantum mechanics. Some of the difficulties that local suppression of interference in spite of the fact that must be faced in solving the preferred-basis problem are full coherence is retained in the total state that describes the system-environment combination? While the local (i) to decide whether the selection of any preferred ba- destruction of interference allows one to infer the emer- sis (or quantity or property) is justified at all or gence of an (improper) ensemble of individually localized only an artefact of our subjective experience; components of the wave function, one still needs to im- pose an interpretive framework that explains why only (ii) if we decide on (i) in the positive, to select those one of the localized states is realized and/or perceived. determinate quantity or quantities (what appears This has been done in various interpretations of quan- determinate to us does not need to be appear de- tum mechanics, typically on the basis of the decohered terminate to other kinds of observers, nor does it reduced density matrix to ensure consistency with the need to be the “true” determinate property); predictions of the Schr¨odinger dynamics and thus empir- (iii) to avoid any ad hoc character of the choice and any ical adequacy. possible empirical inadequacy or inconsistency with In this context, one might raise the question whether the confirmed predictions of quantum mechanics; retention of full coherence in the composite state of (iv) if a multitude of quantities is selected that apply the system-environment combination could ever lead to differently among different systems, to be able to empirical conflicts with the ascription of definite val- formulate specific rules that specify the determi- ues to (mesoscopic and macroscopic) systems in some nate quantity or quantities under every circum- decoherence-based interpretive approach. After all, one stance; could think of enlarging the system so as to include the environment in such a way that measurements could now (v) to ensure that the basis is chosen such that if the actually reveal the persisting quantum coherence even on system is embedded in a larger (composite) system, a macroscopic level. However, Zurek (1982) asserted that the principle of property composition holds, i.e., the such measurements would be impossible to carry out in property selected by the basis of the original system practice, a statement that was supported by a simple should also persist when the system is considered model calculation by Omn`es (1992, p. 356) for a body as part of a larger composite system.11 with a macrosopic number (1024) of degrees of freedom.

11 This is a problem encountered in some modal interpretations (see Clifton, 1996). 20

B. The standard and the Copenhagen interpretation Note that, with respect to the global postmeasurement state vector, given by the final step in Eq. (3.5), the in- As is well known, the standard interpretation (“ortho- teraction with the environment has only led to additional dox” quantum mechanics) postulates that every measure- entanglement. it has not transformed the state vector in ment induces a discontinuous break in the unitary time any way, since the rapidly increasing orthogonality of the evolution of the state through the collapse of the total states of the environment associated with the different wave function onto one of its terms in the state-vector pointer positions has not influenced the state description expansion (uniquely determined by the eigenbasis of the at all. In brief, the entanglement brought about by inter- measured observable), which selects a single term in the action with the environment could even be considered as superposition as representing the outcome. The nature making the measurement problem worse. Bacciagaluppi of the collapse is not at all explained, and thus the defi- (2003a, Sec. 3.2) puts it like this: nition of measurement remains unclear. Macroscopic su- Intuitively, if the environment is carrying out, perpositions are not a priori forbidden, but are never without our intervention, lots of approximate observed since any observation would entail a measure- position measurements, then the measurement mentlike interaction. In the following, we shall also con- problem ought to apply more widely, also to these sider a “Copenhagen” variant of the standard interpreta- spontaneously occurring measurements. (. . . ) tion, which adds an additional key element, postulating The state of the object and the environment the necessity of classical concepts in order to describe could be a superposition of zillions of very well quantum phenomena, including measurements. localised terms, each with slightly different po- sitions, and which are collectively spread over a macroscopic distance, even in the case of every- 1. The problem of definite outcomes day objects. (. . . ) If everything is in interaction with everything else, everything is entangled with The interpretive rule of orthodox quantum mechanics everything else, and that is a worse problem than that tells us when we can speak of outcomes is given by the entanglement of measuring apparatuses with 12 the e-e link. This is an “objective” criterion since it the measured probes. allows us to infer the existence of a definite state of the system to which a value of a physical quantity can be as- Only once we have formed the reduced pure-state den- cribed. Within this interpretive framework (and without sity matrix ρSA, Eq. (3.8), can the orthogonality of the presuming the collapse postulate) decoherence cannot environmental states have an effect; then, ρSA dynami- b d solve the problem of outcomes: Phase coherence between cally evolves into the improper ensemble ρSA [Eq. (3.9)]. b macroscopically different pointer states is preserved in However, as pointed out in our general discussion of re- b the state that includes the environment, and we can al- duced density matrices in Sec. III.B, the orthodox rule of ways enlarge the system so as to include (at least parts of) interpreting superpositions prohibits regarding the com- the environment. In other words, the superposition of dif- ponents in the sum of Eq. (3.9) as corresponding to in- ferent pointer positions still exists, coherence is only “de- dividual well-defined quantum states. localized into the larger system” (Kiefer and Joos, 1999, Rather than considering the postdecoherence state of p. 5), that is, into the environment—or, as Joos and Zeh the system (or, more precisely, of the system-apparatus combination ), we can instead analyze the influence (1985, p. 224) put it, “the interference terms still exist, SA but they are not there”—and the process of decoherence of decoherence on the expectation values of observables pertaining to ; after all, such expectation values are could in principle always be reversed. Therefore, if we SA assume the orthodox e-e link to establish the existence what local observers would measure in order to arrive at conclusions about . The diagonalized reduced of determinate values of physical quantities, decoherence SA cannot ensure that the measuring device actually ever is density matrix, Eq. (3.9), together with the trace rela- in a definite pointer state (unless, of course, the system tion, Eq. (3.6), implies that, for all practical purposes, the statistics of the system will be indistinguish- is initially in an eigenstate of the observable), or that SA measurements have outcomes at all. Much of the general able from that of a proper mixture (ensemble) by any local observation on . That is, given (i) the trace criticism directed against decoherence with respect to its SA ability to solve the measurement problem (at least in the rule O = Tr(ρO) and (ii) the interpretation of O as h i h i context of the standard interpretation) has been centered the expectation value of an observable O, the expecta- b b b on this argument. tion value of anyb observable O restricted to the local SA b system will be for all practical purposes identical to the expectationSA value of thisb observable if had been SA in one of the states sn an (as if were described 12 It is not particularly relevant for the subsequent discussion by an ensemble of states).| i| Ini otherSA words, decoherence whether the e-e link is assumed in its “exact” form, i.e., requir- has effectively removed any interference terms (such as ing the exact eigenstates of an observable, or a “fuzzy” form that s a a s where m = n) from the calculation of allows the ascription of definiteness based on only approximate | mi| mih n|h n| 6 eigenstates or on wave functions with (tiny) “tails.” the trace Tr(ρSAOSA) and thereby from the calculation

b b 21 of the expectation value OSA . It has therefore been ferred basis). What then determines the observable that claimed that formal equivalenceh i—i.e., the fact that de- is being measured? As our discussion in Sec. II.C has coherence transforms the reducedb density matrix into a demonstrated, the derivation of the measured observable form identical to that of a density matrix representing an from the particular form of a given state-vector expan- ensemble of pure states—yields observational equivalence sion can lead to paradoxial results since this expansion in the sense above, namely, the (local) indistiguishability is in general nonunique, so the observable must be cho- of the expectation values derived from these two types of sen by other means. In the standard and Copenhagen density matrices via the trace rule. interpretations, it is essentially the “user” who simply But we must be careful in interpreting the correspon- “chooses” the particular observable to be measured and dence between the mathematical formalism (such as the thus determines which properties the system possesses. trace rule) and the common terms employed in describ- This positivist point of view has, of course, led to a ing “the world.” In quantum mechanics, the identifica- lot of controversy, since it runs counter to the attempt to tion of the expression “Tr(ρA)” as the expectation value establish an observer-independent reality that has been of a quantity relies on the mathematical fact that, when the central pursuit of natural science since its beginning. writing out this trace, it is found to be equal to a sum Moreover, in practice, one certainly does not have the over the possible outcomes of the measurement, weighted freedom to choose any arbitrary observable and mea- by the Born probabilities for the system to be “thrown” sure it; instead, one has “instruments” (including one’s into a particular state corresponding to each of these out- senses) that are designed to measure a particular observ- comes in the course of a measurement. This certainly able. For most (and maybe all) practical purposes, this represents our common-sense intuition about the mean- will ultimately boil down to a single relevant observable, ing of expectation values as the sum over possible values namely, position. But what, then, makes the instruments that can appear in a given measurement, multiplied by designed for such a particular observable? the relative frequency of actual occurrence of these val- Answering this crucial question essentially means ues in a series of such measurements. This interpreta- abandoning the orthodox view of treating measurements tion, however, presumes (i) that measurements have out- as a “black box” process that has little, if any, relation comes, (ii) that measurements lead to definite “values,” to the workings of actual physical measurements (where (iii) that measurable physical quantities are identified as measurements can here be understood in the broadest operators (observables) in a Hilbert space, and (iv) that sense of a “monitoring” of the state of the system). The the modulus square of the expansion coefficients of the first key point, the formalization of measurements as a state in terms of the eigenbasis of the observable can be formation of quantum correlations between system and interpreted as representing probabilities of actual mea- apparatus, goes back to the early years of quantum me- surement outcomes (Born rule). chanics and is reflected in the measurement scheme of Thus decoherence brings about an apparent (and ap- von von Neumann (1932), but it does not resolve the proximate) mixture of states that seem, based on the issue of how the choice of observables is made. The models studied, to correspond well to those states that we second key point, the explicit inclusion of the environ- perceive as determinate. Moreover, our observation tells ment in a description of the measurement process, was us that this apparent mixture indeed looks like a proper brought into quantum theory by the studies of deco- ensemble in a measurement situation, as we observe that herence. Zurek’s (1981) stability criterion discussed in measurements lead to the “realization” of precisely one Sec. III.E has shown that measurements must be of such state in the “ensemble.” But within the framework of the a nature as to establish stable records, where stability orthodox interpretation, decoherence cannot explain this is to be understood as preserving the system-apparatus crucial step from an apparent mixture to the existence correlations in spite of the inevitable interaction with the and/or perception of single outcomes. surrounding environment. The “user” cannot choose the observables arbitrarily, but must design a measuring de- vice whose interaction with the environment is such as to 2. Observables, measurements, and environment-induced ensure stable records in the sense above (which, in turn, superselection defines a measuring device for this observable). In the reading of orthodox quantum mechanics, this can be in- In the standard and Copenhagen interpretationS, terpreted as the environment determining the properties property ascription is determined by an observable that of the system. represents the measurement of a physical quantity and In this sense, the decoherence program has embedded that in turn defines the preferred basis. However, any the rather formal concept of measurement as proposed Hermitian operator can play the role of an observable, by the standard and Copenhagen interpretations—with and thus any given state has the potential for an in- its vague notion of observables that are seemingly freely finite number of different properties whose attribution chosen by the observer—in a more realistic and physical is usually mutually exclusive unless the corresponding framework. This is accomplished via the specification of observables commute (in which case they share a com- observer-free criteria for the selection of the measured ob- mon eigenbasis which preserves the uniqueness of the pre- servable through the physical structure of the measuring 22 device and its interaction with the environment, which ence program, it is reasonable to anticipate that decoher- is, in most cases, needed to amplify the measurement ence embedded in some additional interpretive structure record and thereby to make it accessible to the external could lead to a complete and consistent derivation of the observer. classical world from quantum mechanical principles. This would make the assumption of an intrinsically classical apparatus (which has to be treated outside of the realm 3. The concept of classicality in the Copenhagen interpretation of quantum mechanics) appear as neither a necessary nor a viable postulate. Bacciagaluppi (2003b, p. 22) refers to The Copenhagen interpretation additionally postu- this strategy as “having Bohr’s cake and eating it”: ac- lates that classicality is not to be derived from quan- knowledging the correctness of Bohr’s notion of the ne- tum mechanics, for example, as the macroscopic limit cessity of a classical world (“having Bohr’s cake”), but of an underlying quantum structure (as is in some sense being able to view the classical world as part of and as assumed, but not explicitely derived, in the standard in- emerging from a purely quantum-mechanical world. terpretation), but instead that it be viewed as an indis- pensable and irreducible element of a complete quantum theory—and, in fact, be considered as a concept prior to C. Relative-state interpretations quantum theory. In particular, the Copenhagen inter- pretation assumes the existence of macroscopic measure- Everett’s original (1957) proposal of a relative-state ment apparatuses that obey classical physics and that interpretation of quantum mechanics has motivated sev- are not supposed to be described in quantum mechani- eral strands of interpretation, presumably owing to the cal terms (in sharp contrast to the von Neumann mea- fact that Everett himself never clearly spelled out how surement scheme, which rather belongs to the standard his theory was supposed to work. The system-observer interpretation); such a classical apparatus is considered duality of orthodox quantum mechanics introduces into necessary in order to make quantum-mechanical phenom- the theory external “observers” who are not described by ena accessible to us in terms of the “classical” world of the deterministic laws of quantum systems but instead our experience. This strict dualism between the system follow a stochastic indeterminism. This approach obvi- , to be described by quantum mechanics, and the appa- ously runs into problems when the universe as a whole S ratus , obeying classical physics, also entails the exis- is considered: by definition, there cannot be any exter- A tence of an essentially fixed boundary between and , nal observers. The central idea of Everett’s proposal is S A which separates the microworld from the macroworld (the then to abandon duality and instead (i) to assume the “Heisenberg cut”). This boundary cannot be moved sig- existence of a total state Ψ representing the state of nificantly without destroying the observed phenomenon the entire universe and (ii)| toi uphold the universal valid- (i.e., the full interacting compound ). ity of the Schr¨odinger evolution, while (iii) postulating SA Especially in the light of the insights gained from de- that all terms in the superposition of the total state at coherence it seems impossible to uphold the notion of a the completion of the measurement actually correspond fixed quantum–classical boundary on a fundamental level to physical states. Each such physical state can be un- of the theory. Environment-induced superselection and derstood as relative (a) to the state of the other part in suppression of interference have demonstrated how qua- the composite system (as in Everett’s original proposal; siclassical robust states can emerge, or remain absent, also see Mermin, 1998a; Rovelli, 1996), (b) to a particular using the quantum formalism alone and over a broad “branch” of a constantly “splitting” universe (the many- range of microscopic to macroscopic scales, and have es- worlds interpretations, popularized by DeWitt, 1970 and tablished the notion that the boundary between and Deutsch, 1985), or (c) to a particular “mind” in the set is to a large extent movable towards . SimilarS resultsA of minds of the conscious observer (the many-minds in- have been obtained from the generalA study of quantum terpretation; see, for example, Lockwood, 1996). In other nondemolition measurements (see, for example, Chap. 19 words, every term in the final-state superposition can be of Auletta, 2000) which include the monitoring of a sys- viewed as representing an equally “real” physical state of tem by its environment. Also note that since the appa- affairs that is realized in a different “branch of reality.” ratus is described in classical terms, it is macroscopic by Decoherence adherents have typically been inclined definition; but not every apparatus must be macrosopic: towards relative-state interpretations (for instance Zeh, the actual “instrument” could well be microscopic. Only 1970, 1973, 1993; Zurek, 1998), presumably because the the “amplifier” must be macrosopic. As an example, con- Everett approach takes unitary quantum mechanics es- sider Zurek’s (1981) toy model of decoherence, outlined sentially “as is,” with a minimum of added interpretive in Sec. III.D.2, in which the instrument can be repre- elements. This matches well the spirit of the decoher- sented by a bistable atom while the environment plays ence program, which attempts to explain the emergence the role of the amplifier; a more realistic example is a of classicality purely from the formalism of basic quan- macrosopic detector of gravitational waves that is treated tum mechanics. It may also seem natural to identify the as a quantum-mechanical harmonic oscillator. decohering components of the wave function with differ- Based on the progress already achieved by the decoher- ent Everett branches. Conversely, proponents of relative- 23 state interpretations have frequently employed the mech- ployed to solve the preferred-basis problem of relative- anism of decoherence in solving the difficulties associ- state interpretations (see, for example, Butterfield, 2001; ated with this class of interpretations (see, for example, Wallace, 2002, 2003a; Zurek, 1998). There are several Deutsch, 1985, 1996, 2002; Saunders, 1995, 1997, 1998; advantages in a decoherence-related approach to select- Vaidman, 1998; Wallace, 2002, 2003a). ing the preferred Everett bases: First, no a priori ex- There are many different readings and versions of istence of a preferred basis needs to be postulated, but relative-state interpretations, especially with respect to instead the preferred basis arises naturally from the phys- what defines the “branches,” “worlds,” and “minds”; ical criterion of robustness. Second, the selection will whether we deal with one, a multitude, or an infinity be likely to yield empirical adequacy, since the decoher- of such worlds and minds; and whether there is an ac- ence program is derived solely from the well-confirmed tual (physical) or only perspectival splitting of the worlds Schr¨odinger dynamics (modulo the possibility that ro- and minds into different branches corresponding to the bustness may not be the universally valid criterion). terms in the superposition. Does the world or mind split Lastly, the decohered components of the wave function into two separate copies (thus somehow doubling all the evolve in such a way that they can be reidentified over matter contained in the orginal system), or is there just time (forming “trajectories” in the preferred state spaces) a “reassignment” of states to a multitude of worlds or and thus can be used to define stable, temporally ex- minds of constant (typically infinite) number, or is there tended Everett branches. Similarly, such trajectories can only one physically existing world or mind in which each be used to ensure robust observer record states and/or branch corresponds to different “aspects” (whatever they environmental states that make information about the are). Regardless, in the following discussion of the key state of the system of interest widely accessible to ob- implications of decoherence, the precise details and dif- servers (see, for example, Zurek’s “existential interpreta- ferences of these various strands of interpretation will, tion,” outlined in Sec. IV.C.3 below). for the most part, be largely irrelevant. Relative-state interpretations face two core difficulties. While the idea of directly associating the environment- First, the preferred-basis problem: If states are only rel- selected basis states with Everett worlds seems natural ative, the question arises, relative to what? What deter- and straightforward, it has also been subject to criti- mines the particular basis terms that are used to define cism. Stapp (2002) has argued that an Everett-type the branches, which in turn define the worlds or minds interpretation must aim at determining a denumerable in the next instant of time? When precisely does the set of distinct branches that correspond to the appar- “splitting” occur? Which properties are made determi- ently discrete events of our experience. Among these nate in each branch, and how are they connected to the branches one must be able to assign determinate values determinate properties of our experience? Second, what and finite probabilities according to the usual rules and is the meaning of probabilities, since every outcome ac- therefore one would need to be able to specify a denu- tually occurs in some world or mind, and how can Born’s merable set of mutually orthogonal projection operators. rule be motivated in such an interpretive framework? It is well known, however (Zurek, 1998), that the pre- ferred states chosen through the interaction with the en- vironment via the stability criterion frequently form an overcomplete set of states—often a continuum of narrow 1. Everett branches and the preferred-basis problem Gaussian-type wave packets, for example, the coherent states of harmonic-oscillator models that are not neces- Stapp (2002, p. 1043) stated the requirement that sarily orthogonal (i.e., the Gaussians may overlap; see “a many-worlds interpretation of quantum theory exists K¨ubler and Zeh, 1973; Zurek et al., 1993). Stapp there- only to the extent that the associated basis problem is fore considers this approach to the preferred-basis prob- solved.” In the context of relative-state interpretations, lem in relative-state interpretations to be unsatisfactory. the preferred-basis problem is not only much more severe Zurek (2003a) has rebutted this criticism by pointing out than in the orthodox interpretation, but also more fun- that a collection of harmonic oscillators that would lead damental for several reasons: (i) The branching occurs to such overcomplete sets of Gaussians cannot serve as an continuously and essentially everywhere; in the general adequate model of the human brain, and it is ultimately sense of measurements understood as the formation of only in the brain where the perception of denumerability quantum correlations, every newly formed correlation, and mutual exclusiveness of events must be accounted whether it pertains to microscopic or macroscopic sys- for (see Sec. II.B.3); when neurons are more appropri- tems, corresponds to a branching. (ii) The ontological ately modeled as two-state systems, the issue raised by implications are much more drastic, at least in those Stapp disappears (for a discussion of decoherence in a relative-state interpretations, which assume an actual simple two-state model, see Sec. III.D.2).13 “splitting” of worlds or minds, since the choice of the basis determines the resulting “world” or “mind” as a whole. The environment-based basis superselection criteria 13 For interesting quantitative results on the role of decoherence in of the decoherence program have frequently been em- neuronal processes, see Tegmark (2000). 24

The approach of using environment-induced superse- the precise structure that is needed to give meaning to lection and decoherence to define the Everett branches Everett’s basic concept. has also been critized on grounds of being “conceptually However, as pointed out in Sec. III.E.4, emerging ba- approximate,” since the stability criterion generally leads sis states based on the instantaneous Schmidt states only to an approximate specification of a preferred basis will frequently have properties that are very different and therefore cannot give an “exact” definition of the Ev- from those selected by the stability criterion and that erett branches (see, for example, the comments of Kent, are undesirably nonclassical. For example, they may 1990; Zeh, 1973, and also the well-known “anti-FAPP” lack the spatial localization of the robustness-selected position of Bell, 1982). Wallace (2003a, pp. 90–91) has Gaussians (Stapp, 2002). The question to what ex- argued against such an objection as tent the Schmidt basis states correspond to classical properties in Everett-style interpretations was investi- (. . . ) arising from a view implicit in much dis- gated in detail by Barvinsky and Kamenshchik (1995). cussion of Everett-style interpretations: that cer- The authors study the similarity of the states selected tain concepts and objects in quantum mechan- by the Schmidt decomposition to coherent states (i.e., ics must either enter the theory formally in its minimum-uncertainty Gaussians) that are chosen as axiomatic structure, or be regarded as illusion. the “yardstick states” representing classicality (see also (. . . ) [Instead] the emergence of a classical world Eisert, 2004). For the investigated toy models it is found from quantum mechanics is to be understood in that only subsets of the Everett worlds corresponding terms of the emergence from the theory of cer- to the Schmidt decomposition exhibit classicality in this tain sorts of structures and patterns, and . . . this sense; furthermore, the degree of robustness of classical- means that we have no need (as well as no hope!) ity in these branches is very sensitive to the choice of of the precision which Kent [in his (1990) critique] the initial state and the interaction Hamiltonian, such and others (. . . ) demand. that classicality emerges typically only temporarily, and Accordingly, in view of our argument in Sec. II.B.3 the Schmidt basis generally lacks robustness under time for considering subjective solutions to the measurement evolution. Similar difficulties with the Schmidt basis problem as sufficient, there is no a priori reason to doubt approach have been described by Kent and McElwaine that an “approximate” criterion for the selection of the (1997). preferred basis can give a meaningful definition of the These findings indicate that a selection criterion based Everett branches—one that is empirically adequate and on robustness provides a much more meaningful, phys- that accounts for our experiences. The environment- ically transparent, and general rule for the selection of superselected basis emerges naturally from the physically quasiclassical branches in relative-state interpretations, very reasonable criterion of robustness together with the especially with respect to its ability to lead to wave- purely quantum mechanical effect of decoherence. It function components representing quasiclassical proper- would be rather difficult to imagine rgar ab axiomatically ties that can be reidentified over time (which a simple introduced “exact” rule could be able to select preferred diagonalization of the reduced density matrix at each in- bases in a manner that is similarly physically motivated stant of time does not, in general, allow for). and capable of ensuring empirical adequacy. Besides using the environment-superselected pointer states to describe the Everett branches, various authors 2. Probabilities in Everett interpretations have directly used the instantaneous Schmidt decompo- sition of the composite state (or, equivalently, the set of Various attempts unrelated to decoherence have been orthogonal eigenstates of the reduced density matrix) to made to find a consistent derivation of the Born probabil- define the preferred basis (see also Sec. III.E.4). This ities (for instance, Deutsch, 1999; DeWitt, 1971; Everett, approach is easier to implement than the explicit search 1957; Geroch, 1984; Graham, 1973; Hartle, 1968) in the for dynamically stable pointer states since the preferred explicit or implicit context of a relative-state interpre- basis follows directly from a simple mathematical diag- tation, but several arguments have been presented that onalization procedure at each instant of time. Further- show that these approaches fail.14 When the effects of more, it has been favored by some (e.g., Zeh, 1973) since decoherence and environment-induced superselection are it gives an “exact” rule for basis selection in relative- included, it seems natural to identify the diagonal ele- state interpretations; the consistently quantum origin ments of the decohered reduced density matrix (in the of the Schmidt decomposition, which matches well the environment-superselected basis) with the set of possible “pure quantum-mechanics” spirit of Everett’s proposal (where the formalism of quantum mechanics supplies its own interpretation), has also been counted as an advan- tage (Barvinsky and Kamenshchik, 1995). In an earlier 14 See, for example, the critiques of Barnum et al. (2000); Kent work, Deutsch (1985) attributed a fundamental role to (1990); Squires (1990); Stein (1984); however, also note the ar- the Schmidt decomposition in relative-state interpreta- guments of Wallace (2003b) and Gill (2003), defending the ap- tions as defining an “interpretation basis” that imposes proach of Deutsch (1999); see also Saunders (2002). 25

elementary events and to interpret the corresponding co- “objective” state. Alternatively, the observer can take efficients as relative frequencies of worlds (or minds, etc.) advantage of the redundant records of the state of the in the Everett theory, assuming a typically infinite mul- system as monitored by the environment. By intercept- titude of worlds, minds, etc. Since decoherence enables ing parts of this environment, for example, a fraction one to reidentify the individual localized components of of the surrounding photons, he can determine the state the wave function over time (describing, for example, ob- of the system essentially without perturbing it (cf. also servers and their measurement outcomes attached to in- the related recent ideas of “quantum Darwinism” and the dividual well-defined “worlds”), this leads to a natural role of the environment as a “witness,” see Ollivier et al., interpretation of the Born probabilities as empirical fre- 2003; Zurek, 2000, 2003b, 2004b).16 quencies. Zurek emphasizes the importance of stable records However, decoherence cannot yield an actual deriva- for observers, i.e., robust correlations between the tion of the Born rule (for attempts in this direction, see environment-selected states and the memory states of the Deutsch, 1999; Zurek, 1998). As mentioned before, this observer. Information must be represented physically, is so because the key elements of the decoherence pro- and thus the “objective” state of the observer who has gram, the formalism and the interpretation of reduced detected one of the potential outcomes of a measurement density matrices and the trace rule, presume the Born must be physically distinct and objectively different from rule. Attempts to consistently derive probabilities from the state of an observer who has recorded an alternative reduced density matrices and from the trace rule are outcome (since the record states can be determined from therefore subject to the charge of circularity (Zeh, 1997; the outside without perturbing them—see the previous Zurek, 2003b). In Sec. III.F, we outlined a recent pro- paragraph). The different objective states of the ob- posal by Zurek (2003c) that evades this circularity and server are, via quantum correlations, attached to differ- deduces the Born rule from envariance, a symmetry prop- ent branches defined by the environment-selected robust erty of entangled systems, and from certain assumptions states; they thus ultimately label the different branches of about the connection between the state of the system the universal state vector. This is claimed to lead to the S of interest, the state vector of the composite system perception of classicality; the impossibility of perceiving SE that includes an environment entangled with , and arbitrary superpositions is explained via the quick sup- E S probabilities of outcomes of measurements performed on pression of interference between different memory states . Decoherence combined with this approach provides a induced by decoherence, where each (physically distinct) S framework in which quantum probabilities and the Born memory state represents an individual observer identity. rule can be given a rather natural motivation, definition, A similar argument has been given by Zeh (1993) who and interpretation in the context of relative-state inter- employs decoherence together with an (implicit) branch- pretations. ing process to explain the perception of definite out- comes:

3. The “existential interpretation” [A]fter an observation one need not necessarily conclude that only one component now exists A well-known Everett-type interpretation that relies but only that only one component is observed. heavily on decoherence has been proposed by Zurek (. . . ) Superposed world components describing (1993, 1998; see also the recent reevaluation in Zurek, the registration of different macroscopic proper- 2004a). This approach, termed the “existential interpre- ties by the “same” observer are dynamically en- tation,” defines the reality, or objective existence, of a tirely independent of one another: they describe state as the possibility of finding out what the state is different observers. (. . . ) He who considers this and simultaneously leaving it unperturbed, similar to a conclusion of an indeterminism or splitting of the classical state.15 Zurek assigns a “relative objective ex- observer’s identity, derived from the Schr¨odinger istence” to the robust states (identified with elementary equation in the form of dynamically decoupling “events”) selected by the environmental stability crite- (“branching”) wave packets on a fundamental rion. By measuring properties of the system-environment global configuration space, as unacceptable or interaction Hamiltonian and employing the robustness “extravagant” may instead dynamically formal- criterion, the observer can, at least in principle, deter- ize the superfluous hypothesis of a disappearance mine the set of observables that can be measured on of the “other” components by whatever method the system without perturbing it and thus find out its he prefers, but he should be aware that he may thereby also create his own problems: Any devia- tion from the global Schr¨odinger equation must in

15 This intrinsically requires the notion of open systems, since in isolated systems, the observer would need to know in advance what observables commute with the state of the system, in order to perform a nondemolition measurement that avoids repreparing 16 The partial ignorance is necessary to avoid redefinition of the the state of the system. state of the system. 26

principle lead to observable effects, and it should Such an attribution of possible properties must ful- be recalled that none have ever been discovered. fill certain requirements. For instance, probabilities for outcomes of measurements should be consistent with the The existential interpretation has recently been con- usual Born probabilities of standard quantum mechan- nected to the theory of envariance (see Zurek, 2004a, ics; it should be possible to recover our experience of and Sec. III.F). In particular, the derivation of Born’s classicality in the perception of macroscopic objects; and rule based on envariance as outlined in Sec. III.F can an explicit time evolution of properties and their prob- be recast in the framework of the existential interpreta- abilities should be definable that is consistent with the tion such that probabilities refer explicitly to the future results of the Schr¨odinger equation. As we shall see in record state of an observer. Such a concept of proba- the following, decoherence has frequently been employed bility bears similarities with classical probability theory in modal interpretations to motivate and define rules for (for more details on these ideas, see Zurek, 2004a). property ascription. Dieks (1994a,b) has argued that one The existential interpretation continues Everett’s goal of the central goals of modal approaches is to provide an of interpreting quantum mechanics using the quantum- interpretation for decoherence. mechanical formalism itself. Zurek takes the standard no-collapse quantum theory “as is” and explores to what extent the incorporation of environment-induced supers- election and decoherence (and recently also envariance) 1. Property assignment based on environment-induced could form a viable interpretation that would, with a superselection minimal additional interpretive framework, be capable of accounting for the perception of definite outcomes and The intrinsic difficulty of modal interpretations is to of explaining the origin and nature of probabilities. avoid any ad hoc character of the property assignment, yet to find generally applicable rules that lead to a selec- tion of possible properties that include the determinate D. Modal interpretations properties of our experience. To solve this problem, var- ious modal interpretations have embraced the results of The first type of modal interpretation was suggested the decoherence program. A natural approach would be by van Fraassen (1973, 1991), based on his program of to employ the environment-induced superselection of a “constructive empiricism,” which proposes to take only preferred basis—since it is based on an entirely physical empirical adequacy, but not necessarily “truth,” as the and very general criterion (namely, the stability require- goal of science. Since then, a large number of interpre- ment) and has, for the cases studied, been shown to give tations of quantum mechanics have been suggested that results that agree well with our experience, thus match- can be considered as modal (for a review and discussion ing van Fraassen’s goal of empirical adequacy—to yield of some of the basic properties and problems of such in- sets of possible quasiclassical properties associated with terpretations, see Clifton, 1996). the correct probabilities. In general, the approach of modal interpretations con- Furthermore, since the decoherence program is based sists in weakening the orthodox e-e link by allowing for solely on Schr¨odinger dynamics, the task of defining a the assignment of definite measurement outcomes even time evolution of the “property states” and their asso- if the system is not in an eigenstate of the observable ciated probabilities that is in agreement with the re- representing the measurement. In this way, one can pre- sults of unitary quantum mechanics would presumably serve a purely unitary time evolution without the need be easier than in a model of property assignment in for an additional collapse postulate to account for definite which the set of possibilities does not arise dynami- measurement results. Of course, this immediately raises cally via the Schr¨odinger equation alone (for a detailed the question of how physical properties that are per- proposal for modal dynamics of the latter type, see ceived through measurements and measurement results Bacciagaluppi and Dickson, 1999). The need for explicit are connected to the state, since the bidirectional link is dynamics of property states in modal interpretations is broken between the eigenstate of the observable (which controversial. One can argue that it suffices to show corresponds to the physical property) and the eigenvalue that at each instant of time, the set of possibly possessed (which represents the manifestation of the value of this properties that can be ascribed to the system is empiri- physical property in a measurement). The general goal cally adequate, in the sense of containing the properties of modal interpretations is then to specify rules that de- of our experience, especially with respect to the proper- termine a catalog of possible properties of a system de- ties of macroscopic objects (this is essentially the view scribed by the density matrix ρ at time t. Two different of, for example, van Fraassen, 1973, 1991). On the other views are typically distinguished: a semantic approach hand, this cannot ensure that these properties behave that only changes the way of talking about the connec- over time in agreement with our experience (for instance, tion between properties and state; and a realistic view that macroscopic objects that are left undisturbed do that provides a different specification of what the pos- not change their position in space spontaneously in an sible properties of a system really are, given the state observable manner). In other words, the emergence of vector (or the density matrix). classicality is to be tied not only to determinate prop- 27 erties at each instant of time, but also to the existence nalization represent the determinate properties of our of quasiclassical “trajectories” in property space. Since experience. As outlined in Sec. III.E.4, the states se- decoherence allows one to reidentify components of the lected by the (instantaneous) orthogonal decomposition decohered density matrix over time, this could be used of the reduced density matrix will in general differ from to derive property states with continuous, quasiclassical the robust “pointer states” chosen by the stability cri- trajectorylike time evolution based on Schr¨odinger dy- terion of the decoherence program and may have dis- namics alone. For some discussions of this approach, see tinctly nonclassical properties. That this will be the Hemmo (1996) and Bacciagaluppi and Dickson (1999). case especially when the states selected by the orthog- The fact that the states emerging from decoherence onal decomposition are close to degeneracy has already and the stability criterion are sometimes nonorthogonal been indicated in Sec. III.E.4. It has also been explored or form a continuum will presumably be of even less rel- in more detail in the context of modal interpretations evance in modal interpretations than in Everett-style in- by Bacciagaluppi et al. (1995) and Donald (1998), who terpretations (see Sec. IV.C) since the goal here is solely showed that in the case of near degeneracy (as it typically to specify sets of possible properties, of which only one occurs for macroscopic systems with many degrees of set actually gets assigned to the system. Hence an “over- freedom), the resulting projectors will be extremely sensi- lap” of the sets is not necessarily a problem (modulo tive to the precise form of the state (Bacciagaluppi et al., the potential difficulty of a straightforward assignment 1995). Clearly such sensitivity is undesired since the pro- of probabilities in such a situation). jectors, and thus the properties of the system, will not be well behaved under the inevitable approximations em- ployed in physics (Donald, 1998). 2. Property assignment based on instantaneous Schmidt decompositions 3. Property assignment based on decompositions of the Since it is usually rather difficult to determine explic- decohered density matrix itly the robust “pointer states” through the stability (or a similar) criterion, it would not be easy to specify a gen- Other authors therefore have appealed to the or- eral rule for property assignment based on environment- thogonal decomposition of the decohered reduced den- induced superselection. To simplify this situation, sev- sity matrix (instead of the decomposition of the in- eral modal interpretations have restricted themselves to stantaneous density matrix) which has led to notewor- the orthogonal decomposition of the density matrix to thy results. When the system is represented by only define the set of properties that can be assigned (see, for a finite-dimensional Hilbert space, a discrete model of instance, Bub, 1997; Dieks, 1989; Healey, 1989; Kochen, decoherence, the resulting states were indeed found to 1985; Vermaas and Dieks, 1995). For example, the ap- be typically close to the robust states selected by the proach of Dieks (1989) recognizes, by referring to the stability criterion (for macroscopic systems, this typi- decoherence program, the relevance of the environment cally meant localization in position space), unless again by considering a composite system-environment state the final composite state was very nearly degenerate vector and its diagonal Schmidt decomposition, ψ = (Bacciagaluppi and Hemmo, 1996; Bene, 2001; see also S E | i √pk φ φ , which always exists. Possible proper- Sec. III.E.4). Thus, in sufficiently nondegenerate cases, k | k i| k i tiesP that can be assigned to the system are then repre- decoherence can ensure that the definite properties se- S S sented by the Schmidt projectors Pk = λk φk φk . Al- lected by modal interpretations of the Dieks type will be though all terms are present in the Schmidt| ih expansion| appropriately close to the properties corresponding to the (that Dieks calls the “mathematicalb state”), the “phys- ideal pointer states if the modal properties are based on ical state” is postulated to be given by only one of the the orthogonal decomposition of the reduced decohered terms, with probability pk. A generalization of this ap- density matrix. proach to a decomposition into any number of subsystems On the other hand, Bacciagaluppi (2000) showed that has been described by Vermaas and Dieks (1995). In this in the more general and realistic case of an infinite- sense, the Schmidt decomposition itself is taken to define dimensional state space of the system, when one em- an interpretation of quantum mechanics. Dieks (1995) ploys a continuous model of decoherence (namely, that suggested a physical motivation for the Schmidt decom- of Joos and Zeh, 1985), the predictions of the modal ap- position in modal interpretations based on the assumed proach (Dieks, 1989; Vermaas and Dieks, 1995) and those requirement of a one-to-one correspondence between the of decoherence can differ significantly. It was demon- properties of the system and its environment. For a com- strated that the definite properties obtained from the ment on the violation of the property composition prin- orthogonal decomposition of the decohered density ma- ciple in such interpretations, see the analysis of Clifton trix were highly delocalized (that is, smeared out over (1996). the entire spread of the state), although the coherence A central problem associated with the approach of length of the density matrix itself was shown to be very orthogonal decomposition is that it is not at all clear small, so that decoherence indicated very localized prop- that the properties determined by the Schmidt diago- erties. Thus, based on these results (and similar ones 28 of Donald, 1998), decoherence can be used to argue for tion condition c (t) 2 = 1 for all t. This process n | n | the physical inadequacy of the rule for the assignment is known as stochasticP dynamical reduction. Eventually 2 of definite properties as proposed by Dieks (1989) and one amplitude cn(t) 1, while all other squared co- Vermaas and Dieks (1995). efficients 0 (the| “gambler’s| → ruin game” mechanism), → More generally, if the definite properties selected by the where c (t) 2 1 with probability c (t = 0) 2 (the | n | → | n | modal interpretation fail to mesh with the results of de- squared coefficients in the initial precollapse state-vector coherence (in particular, when they also lack the desired expansion) in agreement with the Born probability inter- classicality and correspondence to the determinate prop- pretation of the expansion coefficients. erties of our experience), we are given reason to doubt These early models exhibit two main difficulties. First, whether the proposed rules for property assignment have the preferred-basis problem: What determines the terms sufficient physical motivation, legitimacy, or generality. in the state-vector expansion into which the state vector gets reduced? Why does reduction lead to precisely the distinct macroscopic states of our experience and not su- 4. Concluding remarks perpositions thereof? Second, how can one account for the fact that the effectiveness of collapsing superpositions There are many different proposals that can be increases when going from microscopic to macroscopic grouped under the heading of modal interpretations. scales? They all share the problem of motivating and verifying These problems motivated spontaneous localization a consistent system of property assignment. Using the models, initially proposed by Ghirardi, Rimini, and We- robust pointer states selected by interaction with the en- ber (GRW; Ghirardi et al., 1986). Here state-vector re- vironment and by the stability criterion is a step in the duction is not implemented as a dynamical process (i.e., right direction, but the difficulty remains to derive a gen- as a continuous evolution over time), but instead oc- eral rule for property assignment from this method that curs instantaneously and spontaneously, leading to a would yield explicitly the sets of possibilities in every spatial localization of the wave function. To be pre- situation. In certain cases, for example, close to degen- cise, the N-particle wave function ψ(x1,..., xN ) is at eracy and in Hilbert-state spaces of infinite dimension, random intervals multiplied by a Gaussian of the form the simpler approach of deriving the possible properties exp (X x )2/2∆2 (this process is often called a “hit” − − k from the orthogonal decomposition of the decohered re- or a “jump”), and the resulting product is subsequently duced density matrix fails to yield the sharply localized, normalized. The occurrence of these hits is not explained, quasiclassical pointer states as selected by environmental but rather postulated as a new fundamental physical robustness criteria. These are the cases in which decoher- mechanism. Both the coordinate xk and the “center of ence can play a vital role in helping to identify inadequate the hit” X are chosen at random, but the probability for rules for property assignment in modal interpretations. a specific X is postulated to be given by the squared inner product of ψ(x1,..., xN ) with the Gaussian (and there- fore hits are more likely to occur where ψ 2, viewed as a | | E. Physical collapse theories function of xk only, is large). The mean frequency ν of hits for a single microscopic particle is chosen so as to ef- The basic idea of physical collapse theories is to in- fectively preserve unitary time evolution for microscopic troduce an explicit modification of the Schr¨odinger time systems, while ensuring that for macroscopic objects con- evolution to achieve a physical mechanism for state- taining a very large number N of particles the localiza- vector reduction (for an extensive recent review, see tion occurs rapidly (on the order of Nν), in such a way as Bassi and Ghirardi, 2003). This is in general motivated to preclude the persistence of spatially separated macro- by a “realist” interpretation of the state vector, that is, scopic superpositons (such as the pointer’s being in a the state vector is directly identified with a physical state, superpositon of “up” and “down”) on time scales shorter which then requires reduction to one of the terms in the than realistic observations could resolve. Ghirardi et al. superposition to establish equivalence to the observed de- (1986) chose ν 10−16 s−1, so a macrosopic system with ≈ terminate properties of physical states, at least as far as N 1023 particles undergoes localization on average ev- ≈ −7 the macroscopic realm is concerned. ery 10 s. Inevitable coupling to the environment can in The first proposals for theories of this type were made general be expected to lead to a further drastic increase by Pearle (1976, 1982, 1979) and Gisin (1984), who de- of N and therefore to an even higher localization rate. veloped dynamical reduction models that modify unitary Note, however, that the localization process itself is in- dynamics such that a superposition of quantum states dependent of any interaction with environment, in sharp evolves continuously into one of its terms (see also the contrast to the decoherence approach. review by Pearle, 1999). Typically, terms represent- Subsequently, the ideas of the stochastic dynamical ing external white noise are added to the Schr¨odinger reduction and GRW theory were combined into con- 2 equation, causing the squared amplitudes cn(t) in the tinuous spontaneous localization models (Ghirardi et al., state-vector expansion Ψ(t) = c (t) ψ| to| fluctu- 1990; Pearle, 1989) in which localization of the GRW type | i n n | ni ate randomly in time, while maintainingP the normaliza- can be shown to emerge from a nonunitary, nonlinear Itˆo 29

stochastic differential equation, namely, the Schr¨odinger On the one hand, the selection of position as the pre- equation augmented by spatially correlated Brownian ferred basis is supported by the decoherence program, motion terms (see also Di´osi, 1988, 1989). The partic- since physical interactions frequently are governed by ular choice of stochastic term determines the preferred distance-dependent laws. Given the stability criterion basis. Frequently, the stochastic term has been based on or a similar requirement, this leads to position as the the mass density which yields a GRW-type spatial local- preferred observable. In this sense, decoherence provides ization (Di´osi, 1989; Ghirardi et al., 1990; Pearle, 1989), a physical motivation for the assumption of the GRW but stochastic terms driven by the Hamiltonian, leading model. On the other hand, it makes this assumption ap- to a reduction on an energy basis, have also been stud- pear as too restrictive as it cannot account for cases in ied (Adler, 2002; Adler et al., 2001; Adler and Horwitz, which position is not the preferred basis—for instance, 2000; Bedford and Wang, 1975, 1977; Fivel, 1997; in microscopic systems where typically energy is the ro- Hughston, 1996; Milburn, 1991; Percival, 1995, 1998). If bust observable, or in the superposition of (macroscopic) we focus on the first type of term, the Ghirardi-Rimini- currents in SQUIDs. The GRW model simply excludes Weber theory and continuous spontaneous localization such cases by choosing the parameters of the spontaneous become phenomenologically similar, and we shall refer to localization process such that microscopic systems re- them jointly as “spontaneous localization” models in the main generally unaffected by any state vector reduction. following discussion whenever it is unnecessary to distin- The basis selection approach proposed by the decoher- guish them explicitly. ence program is therefore much more general and also avoids the ad hoc character of the GRW theory by allow- ing for a range of preferred observables and motivating 1. The preferred-basis problem their choice on physical grounds. A similar argument can be made with respect to the Physical reduction theories typically remove wave- continuous spontaneous localization approach. Here, function collapse from the restrictive context of the or- one essentially preselects a preferred basis through the thodox interpretation (where the external observer arbi- particular choice of the stochastic terms added to the trarily selects the measured observable and thus deter- Schr¨odinger equation. This allows for a greater range of mines the preferred basis), and rather understand reduc- possible preferred bases, for instance by combining terms tion as a universal mechanism that acts constantly on driven by the Hamiltonian and by the mass density, lead- every state vector regardless of an explicit measurement ing to a competition between localization in energy and situation. In view of this it is particularly important to position space (corresponding to the two most frequently provide a definition for the states into which the wave observed eigenstates). Nonetheless, any particular choice function collapses. of terms will again be subject to the charge of possessing As mentioned before, the original stochastic dynamical an ad hoc flavor, in contrast to the physical definition reduction models suffer from this preferred-basis prob- of the preferred basis derived from the structure of the lem. Taking into account environment-induced supers- unmodified Hamiltonian as suggested by environment- election of a preferred basis could help resolve this is- induced selection. sue. Decoherence has been shown to occur, especially for mesoscopic and macroscopic objects, on extremely short time scales, and thus would presumably be able to bring 2. Simultaneous presence of decoherence and spontaneous about basis selection much faster than the time required localization for dynamical fluctuations to establish a “winning” ex- pansion coefficient. Since decoherence can be considered as an omnipresent In contrast, the GRW theory solves the preferred-basis phenomenon that has been extensively verified both theo- problem by postulating a mechanism that leads to re- retically and experimentally, the assumption that a phys- duction to a particular state vector in an expansion on a ical collapse theory holds means that the evolution of a position basis, i.e., position is assumed to be the univer- system must be guided by both decoherence effects and sal preferred basis. State-vector reduction then amounts the reduction mechanism. to simply modifying the functional shape of the pro- Let us first consider the situation in which decoherence jection of the state vector ψ onto the position basis and the localization mechanism act constructively in the x ,..., x . This choice can| i be motivated by the in- 1 N same direction, i.e., towards a common preferred basis. hsight that essentially| all (human) observations must be This raises the question in which order these two effects grounded in a position measurement.17 influence the evolution of the system (Bacciagaluppi, 2003a). If localization occurs on a shorter time scale than environment-induced superselection of a preferred basis

17 This measurement may ultimately occur only in the brain of the observer; see the objection to the GRW model by Albert and Vaidman (1989). With respect to the general pref- erence for position as the basis of measurements, see also the comment by Bell (1982). 30 and suppression of local interference, decoherence will in nite state, and thereby to ensure the objective attribu- most cases have very little influence on the evolution of tion of determinate properties to the system.18 In this the system, since typically the system will already have sense, collapse models are as much “fine for all practi- evolved into a reduced state. Conversely, if decoherence cal purposes” (to paraphrase Bell, 1990) as decoherence effects act more quickly on the system than the local- is, where perfect orthogonality of the environment states ization mechanism, the interaction with the environment is only attained as t . The severity of the conse- will presumably lead to the preparation of quasiclassical quences, however, is not→ ∞ equivalent for the two strate- robust states that are subsequently chosen by the local- gies. Since collapse models directly change the state vec- ization mechanism. As pointed out in Sec. III.D, decoher- tor, a single outcome is at least approximately selected, ence usually occurs on extremely short time scales, which and it only requires a “slight” weakening of the e-e link can be shown to be significantly smaller than the action to make this state of affairs correspond to the (objec- of the spontaneous localization process for most cases tive) existence of a determinate physical property. In the (for studies related to the GRW model, see Tegmark, case of decoherence, the lack of a precise destruction of 1993 and Benatti et al., 1995). This indicates that deco- interference terms is not the main problem; even if ex- herence will typically play an important role even in the act orthogonality of the environment states were ensured presence of physical wave-function reduction. at all times, the resulting reduced density matrix would The second case occurs when decoherence leads to the represent an improper mixture, with no outcome having selection of a different preferred basis than the reduc- been singled out according to the e-e link. This would be tion basis specified by the localization mechanism. As the case regardless of whether the e-e link is expressed in remarked by Bacciagaluppi (2003a,b) in the context of the strong or weakened form, and we would still have to the GRW theory, one might then imagine the collapse ei- supply some additional interpretative framework to ex- ther to occur only at the level of the environment (which plain our perception of outcomes (see also the comment would then serve as an amplifying and recording device by Ghirardi et al., 1987). with different localization properties than the system un- der study), or to lead to an explicit competition between decoherence and localization effects. 4. Connecting decoherence and collapse models

It was realized early that there exists a striking formal 3. The tails problem similarity of the equations that govern the time evolu- tion of density matrices in the GRW approach and in The clear advantage of physical collapse models over models of decoherence. For example, the GRW equation the consideration of decoherence-induced effects alone for for a single free mass point reads [Ghirardi et al., 1986, a solution to the measurement problem lies in the fact Eq. (3.5)] that an actual state reduction is achieved such that one may be tempted to conclude that at the conclusion of ′ 2 2 ∂ρ(x, x ,t) 1 ∂ ∂ ′ 2 the reduction process the system actually is in a deter- i = ρ iΛ(x x ) ρ, (4.1) ∂t 2m∂x2 − ∂x′2  − − minate state. However, all collapse models achieve only an approximate (“for all practical purposes”) reduction where the second term on the right-hand side accounts of the wave function. In the case of dynamical reduc- for the destruction of spatially separated interference tion models, the state will always retain small interfer- terms. A simple model for environment-induced decoher- ence terms for finite times. Similarly, in the GRW the- ence yields a very similar equation [Joos and Zeh, 1985, ory the width ∆ of the multiplying Gaussian cannot be Eq. (3.75); see also the comment by Joos, 1987]. Thus the made arbitrarily small, and therefore the reduced wave physical justification for an ad hoc postulate of an explicit packet cannot be made infinitely sharply localized in po- reduction-inducing mechanism could be questioned (of sition space, since this would entail an infinitely large en- course modulo the important interpretive difference be- ergy gain by the system via the time-energy uncertainty tween the approximately proper ensembles arising from relation, which would certainly show up experimentally collapse models and the improper ensembles resulting (Ghirardi et al., 1986, chose ∆ 10−5 cm). This need from decoherence; see also Ghirardi et al., 1987). More for only an approximate reduction≈ leads to wave function constructively, the similarity of the governing equations “tails” (Albert and Loewer, 1996), that is, in any region might enable one to choose the free parameters in collapse in space and at any time t > 0, the wave function will models on physical grounds rather than on the basis of remain nonzero if it has been nonzero at t = 0 (before empirical adequacy. Conversely, this similiarity can also the collapse), and thus there will be always a part of the system that is not “here.” Physical collapse models that achieve reduction only “for all practical purposes” require a modification, 18 It should be noted, however, that such “fuzzy” e-e links may in namely, a weakening, of the orthodox e-e link to allow turn lead to difficulties, as the discussion of Lewis’s “counting one to speak of the system’s actually being in a defi- anomaly” has shown (Lewis, 1997). 31

be viewed as leading to a “protection” of physical col- 5. Summary and outlook lapse theories from empirical disproof. This is so because the inevitable and ubiquitous interaction with the envi- Decoherence has the distinct advantage of being de- ronment will always, for all practical purposes of obser- rived directly from the laws of standard quantum me- vation (that is, of statistical prediction), result in (local) chanics, whereas current collapse models are required to density matrices that are formally very similar to those of postulate their reduction mechanism as a new fundamen- collapse models. What is measured is not the state vector tal law of nature. On the other hand, collapse models itself, but the probability distribution of outcomes, i.e., yield, at least for all practical purposes, proper mixtures, values of a physical quantity and their frequency, and so they are capable of providing an “objective” solution this information is equivalently contained in the state to the measurement problem. The formal similarity be- vector and the density matrix. Measurements with their tween the time evolution equations of the collapse and intrinsically local character will presumably be unable to decoherence models nourishes hopes that the postulated distinguish between the probability distribution given by reduction mechanisms of collapse models could possibly the decohered reduced density matrix and the probability be derived from the ubiquituous and inevitable interac- distribution defined by an (approximately) proper mix- tion of every physical system with its environment and ture obtained from a physical collapse. In other words, as the resulting decoherence effects. We may therefore re- long as the free parameters of collapse theories are cho- gard collapse models and decoherence not as mutually sen in agreement with those determined from decoher- exclusive alternatives for a solution to the measurement ence, models for state-vector reduction can be expected problem, but rather as potential candidates for a fruitful to be empirically adequate since decoherence is an effect unification. For a vague proposal along these lines, see that will be present with near certainty in every realistic Pessoa (1998); cf. also Di´osi (1989) and Pearle (1999) (especially macroscopic) physical system. for speculations that quantum gravity might act as a collapse-inducing universal “environment.” One might of course speculate that the simultaneous presence of both decoherence and reduction effects might actually allow for an experimental disproof of collapse F. Bohmian mechanics theories by preparing states that differ in an observ- able manner from the predictions of the reduction mod- Bohm’s approach (Bohm, 1952; Bohm and Bub, els.19 If we acknowledge the existence of interpretations 1966; Bohm and Hiley, 1993) is a modification of of quantum mechanics that employ only decoherence- de Broglie’s (1930) original “pilot-wave” proposal. In induced suppression of interference to explain the per- Bohmian mechanics, a system containing N (nonrela- ception of apparent collapses (as is, for example, claimed tivistic) particles is described by a wave function ψ(t) by the “existential interpretation” of Zurek, 1993, 1998; and the configuration (t) = q (t),..., q (t) R3N see Sec. IV.C.3), we will not be able to distinguish exper- 1 N of particle positions q (Qt), i.e., the state of the system∈ is imentally between a “true” collapse and a mere suppres- i
represented by (ψ, ) for each instant t. The evolution of sion of interference as explained by decoherence. Instead, the system is guidedQ by two equations. The wave function an experimental situation is required in which the col- ψ(t) is transformed as usual via the standard Schr¨odinger lapse model predicts a collapse, but in which no suppres- ~ sion of interference through decoherence arises. Again, equation, i (∂/∂t)ψ = Hψ, while the particle positions q (t) of the configuration (t) evolve according to the the problem in the realization of such an experiment is i b Q that it is very difficult to shield a system from decoher- “guiding equation” ence effects, especially since we will typically require a ∗ dq ~ ψ i ψ mesoscopic or macroscopic system in which the reduction i vψ q q q q q = i ( 1,..., N ) Im ∇∗ ( 1,..., N ), is efficient enough to be observed. For example, based on dt ≡ mi ψ ψ explicit numerical estimates, Tegmark (1993) has shown (4.2) that decoherence due to scattering of environmental par- where mi is the mass of the ith particle. Thus the par- ticles follow determinate trajectories described by (t), ticles such as air molecules or photons will have a much Q stronger influence than the proposed GRW effect of spon- with the distribution of (t) being given by the quantum equilibrium distributionQρ = ψ 2. taneous localization (see also Bassi and Ghirardi, 2003; | | Benatti et al., 1995; for different results for energy-driven reduction models, cf. Adler, 2002). 1. Particles as fundamental entities

19 Bohm’s theory has been critized for ascribing funda- For proposed experiments to detect the GRW collapse, see for mental ontological status to particles. General arguments example Squires (1991) and Rae (1990). For experiments that could potentially demonstrate deviations from the predictions against particles on a fundamental level of any relativis- of quantum theory when dynamical state-vector reduction is tic quantum theory have been frequently given (see, for present, see Pearle (1984, 1986). instance, Malament, 1996, and Halvorson and Clifton, 32

2002).20 Moreover, and this is the point we would like a relativistic quantum field theory, Bohm’s a priori as- to discuss in this section, it has been argued that the ap- sumption of particles at a fundamental level of the theory pearance of particles (“discontinuities in space”) could be appears seriously challenged. derived from the continuous process of decoherence, lead- ing to claims that no fundamental role need be attributed to particles (Zeh, 1993, 1999, 2003). Based on decohered 2. Bohmian trajectories and decoherence density matrices of mesoscopic and macroscopic systems that essentially always represent quasi-ensembles of nar- A well-known property of Bohmian mechanics is the row wave packets in position space, Zeh (1993, p. 190) fact that its trajectories are often highly nonclassical (see, holds that such wave packets can be viewed as represent- 21 for example, Appleby, 1999b; Bohm and Hiley, 1993; ing individual “particle” positions: Holland, 1993). This poses the serious problem of how Bohm’s theory can explain the existence of quasiclassical All particle aspects observed in measurements of trajectories on a macroscopic level. quantum fields (like spots on a plate, tracks in a bubble chamber, or clicks of a counter) can be Bohm and Hiley (1993) considered the scattering of a understood by taking into account this decoher- beam of environmental particles on a macroscopic sys- ence of the relevant local (i.e., subsystem) density tem, a process that is known to give rise to decoherence et al. matrix. (Joos and Zeh, 1985; Joos , 2003). The authors demonstrate that this scattering yields quasiclassical tra- The first question is then whether a narrow wave packet jectories for the system. It has further been shown that in position space can be identified with the subjective ex- for isolated systems, the Bohm theory will typically not perience of a “particle.” The answer appears to be yes: give the correct classical limit (Appleby, 1999b). It was our notion of “particles” hinges on the property of local- thus suggested that the inclusion of the environment and izability, i.e., the definition of a region of space Ω R3 of the resulting decoherence effects might be helpful in re- in which the system (that is, the support of the∈ wave covering quasiclassical trajectories in Bohmian mechanics function) is entirely contained. Although the nature of (Allori, 2001; Allori et al., 2002; Allori and Zangh`ı, 2001; the Schr¨odinger dynamics implies that any wave func- Appleby, 1999a; Sanz and Borondo, 2003; Zeh, 1999). tion will have nonvanishing support (“tails”) outside of We mentioned before that the interaction between a any finite spatial region Ω and therefore exact localizat- macroscopic system and its environment will typically ibility will never be achieved, we only need to demand lead to a rapid approximate diagonalization of the re- approximate localizability to account for our experience duced density matrix in position space, and thus to spa- of particle aspects. tially localized wave packets that follow (approximately) However, to interpret the ensembles of narrow wave Hamiltonian trajectories. [This observation also provides packets resulting from decoherence as leading to the per- a physical motivation for the choice of position as the fun- ception of individual particles, we must embed standard damental preferred basis in Bohm’s theory, in agreement quantum mechanics (with decoherence) into an addi- with Bell’s (1982) well-known comment that “in physics tional interpretive framework that explains why only one the only observations we must consider are position ob- of the wavepackets is perceived;22 that is, we do need servations, if only the positions of instrument pointers.”] to add some interpretive rule to get from the improper The intuitive step is then to associate these trajectories ensemble emerging from decoherence to the perception with the particle trajectories (t) of the Bohm theory. Q of individual terms, so decoherence alone does not nec- As pointed out by Bacciagaluppi (2003b), a great ad- essarily make Bohm’s particle concept superfluous. But vantage of this strategy lies in the fact that the same it suggests that the postulate of particles as fundamental approach would allow for a recovery of both quantum entities could be unnecessary, and taken together with and classical phenomena. the difficulties in reconciling such a particle theory with However, a careful analysis by Appleby (1999a) showed that this decoherence-induced diagonalization in the po- sition basis alone will in general not suffice to yield quasi- classical trajectories in Bohm’s theory; only under certain 20 On the other hand, there are proposals for a “Bohmian mechan- additional assumptions will processes that lead to deco- ics of quantum fields,” i.e., a theory that embeds quantum field herence also give correct quasiclassical Bohmian trajecto- theory into a Bohmian-style framework (D¨urr et al., 2003, 2004). ries for macroscopic systems (Appleby described the ex- 21 Schr¨odinger (1926) had made an attempt into a similar direction but had failed since the Schr¨odinger equation tends to continu- ample of the long-time limit of a system that has initially ously spread out any localized wavepacket when it is considered been prepared in an energy eigenstate). Interesting re- as describing an isolated system. The inclusion of an interact- sults were also reported by Allori and co-workers (Allori, ing environment and thus decoherence counteracts the spread 2001; Allori et al., 2002; Allori and Zangh`ı, 2001). They and opens up the possibility of maintaining narrow wave packets demonstrated that decoherence effects can play the role over time (Joos and Zeh, 1985). 22 Zeh himself, like Zurek (1998), adheres to an Everett-style of preserving classical properties of Bohmian trajecto- branching to which distinct observers are attached (Zeh, 1993); ries. Furthermore, they showed that while in standard see also the quote in Sec. IV.C. quantum mechanics it is important to maintain narrow 33 wave packets to account for the emergence of classicality, where U(t0,t) is the operator that dynamically propa- the Bohmian description of a system by both its wave gates the state vector from t0 to t. function and its configuration allows for the derivation A possible, “maximally fine-grained” history is defined of quasiclassical behavior from highly delocalized wave by the sequence of times t1 < t2 < < tn and by the functions. Sanz and Borondo (2003) studied the double- choice of one projection operator in the· · · set (i) for each P slit experiment in the framework of Bohmian mechan- time point ti in the sequence, i.e., by the set ics and in the presence of decoherence and showed that even when coherence is fully lost, and thus interference is (1) (2) (n) {α} = P 1 (t1), P 2 (t2),..., P n (tn) . (4.5) absent, nonlocal quantum correlations remain that influ- H { α α α } ence the dynamics of the particles in the Bohm theory, b b b We also define the set H = of all possible histories demonstrating that in this example decoherence does not {α} for a given time sequence t{H

G. Consistent histories interpretations Q(i)(t )= π(i)P (i)(t ), π(i) 0, 1 (4.6) βi i αi αi i αi ∈{ } Xαi The consistent- (or decoherent-) histories approach b b was introduced by Griffiths (1984, 1993, 1996) and fur- (i) of the original projection operators P i (t ). ther developed by Omn`es (1988a,b,c, 1990, 1992, 1994, α i (i) 2002), Gell-Mann and Hartle (1990, 1991a, 1993, 1991b), So far, the projection operators Pbαi (ti) chosen at a Dowker and Halliwell (1992), and others. Reviews of certain instant ti in time in order to form a history {α} b H the program can be found in the papers by Omn`es were independent of the choice of the projection opera- (1992) and Halliwell (1993, 1996), as well as in the tors at earlier times t0 H D t . For each time point t in this sequence,· · · we consider an 0 i decoherence functional (i) (i) where the so-called (α, β) is de- exhaustive set = Pαi (ti) αi =1 mi ,1 i n, fined by (Gell-Mann and Hartle, 1990) D of mutually orthogonalP { Hermitian| projection· · · } operators≤ ≤ (i) b Pαi (ti), obeying (n) (1) (1) (n) (α, β) = Tr Pαn (tn) Pα1 (t1)ρ0Pβ1 (t1) Pβn (tn) . (i) (i) (i) (i) D · · · · · · b P (t )=1, P (t )P (t )= δ i i P (t ),  (4.10) αi i αi i βi i α ,β αi i b b b b Xαi If we instead work in the Schr¨odinger picture, the deco- b b b b (4.3) herence functional is and evolving, using the Heisenberg picture, according to

(i) † (i) Pαi (t)= U (t0,t)Pαi (t0)U(t0,t), (4.4) b b 34

(α, β) =Tr P (n)U(t ,t ) P (1)ρ(t )P (1) U †(t ,t )P (n)(t ) . (4.11) D αn n−1 n · · · α1 1 β1 · · · n−1 n βn n   Consider now the coarse-grained historyb that arises fromb a combinationb of the two maximallyb fine-grained histories and , H{α} H{β} = P (1)(t )+ P (1)(t ), P (2)(t )+ P (2)(t ), ..., P (n)(t )+ P (n)(t ) . (4.12) H{α∨β} { α1 1 β1 1 α2 2 β2 2 αn n βn n } b b (ib) (i)b b b We interpret each combined projection operator Pαi (ti)+ Pβi (ti) as stating that, at time ti, the system was in the (i) (i) range described by the union of Pαi (ti) and Pβi (bti). Accordingly,b we would like to require that the probability for a history containing such a combined projection operator be equivalently calculable from the sum of the probabilities b b (i) (i) of the two histories containing the individual projectors Pαi (ti) and Pβi (ti), that is, b b Tr P (n)(t ) P (i)(t )+ P (i)(t ) P (1)(t )ρ P (1)(t ) P (i)(t )+ P (i)(t ) P (n)(t ) αn n · · · αi i βi i · · · α1 1 0 α1 1 · · · αi i βi i · · · αn n 

 b=! Tr P (n)(t b) P (i)(tb) P (1)(t b)ρ P (1)(t b) P (i)(t )b P (n)(t b) b αn n · · · αi i · · · α1 1 0 α1 1 · · · αi i · · · αn n  (i) (i)  + Trb P (n)(t ) b P (t ) b P (1)(t )ρb P (1)(t ) b P (t ) b P (n)(t ) . αn n · · · βi i · · · α1 1 0 α1 1 · · · βi i · · · αn n   It can be easily shown thatb this relationb holds ifb and onlyb if b b

Re Tr P (n)(t ) P (i)(t ) P (1)(t )ρ P (1)(t ) P (i)(t ) P (n)(t ) = 0 if α = β . (4.13) αn n · · · αi i · · · α1 1 0 α1 1 · · · βi i · · · αn n i 6 i    b b b b b b

Generalizing this two-projector case to the coarse-grained 3. Selection of histories and classicality history of Eq. (4.12), we arrive at the (sufficient H{α∨β} and necessary) consistency condition for two histories Even when the stronger consistency criterion (4.15) is and (Griffiths, 1984; Omn`es, 1990, 1992), imposed on the set H of possible histories, the number of H{α} H{β} mutually incompatible consistent histories remains rela- tively large (d’Espagnat, 1989; Dowker and Kent, 1996). Re[ (α, β)] = δ (α, α). (4.14) D α,βD It is not at all clear a priori that at least some of these histories should represent any meaningful set of proposi- If this relation is violated, the usual sum rule for calcu- tions about the world of our observation. Even if a col- lating probabilities does not apply. This situation arises lection of such “meaningful” histories is found, it leaves when quantum interference between the two combined open the question how to select such histories and which additional criteria would need to be invoked. histories {α} and {β} is present. Therefore, to ensure that the standardH lawsH of probability theory also hold for Griffith’s (1984) original aim in formulating the con- coarse-grained histories, the set H of possible histories sistency criterion was only to allow for a consistent must be consistent in the above sense. description of sequences of events in closed quan- tum systems without running into logical contradic- However, Gell-Mann and Hartle (1990) have pointed tions.23 Commonly, however, consistency has also out that when decoherence effects are included to model been tied to the emergence of classicality. For ex- the emergence of classicality, it is more natural to require ample, the consistency criterion corresponds to the demand for the absence of quantum interference—a property of classicality—between two combined histo- ries. It has become clear that most consistent histo- (α, β)= δ (α, α). (4.15) D α,βD ries are in fact flagrantly nonclassical (Albrecht, 1993; Dowker and Kent, 1995, 1996; Gell-Mann and Hartle, Condition (4.14) has often been referred to as weak de- 1990, 1991b; Paz and Zurek, 1993; Zurek, 1993). For in- coherence, and Eq. (4.15) as medium decoherence (for a proposal of a criterion for strong decoherence, see H Gell-Mann and Hartle, 1998). The set of histories 23 is called consistent (or decoherent) when all its mem- However, Goldstein (1998) used a simple example to argue that the consistent-histories approach can lead to contradictions with bers fulfill the consistency condition, Eqs. (4.14) or H{α} respect to a combinination of joint probabilities, even if the con- (4.15), i.e., when they can be regarded as independent sistency criterion is imposed; see also the subsequent exchange (noninterfering). of letters in the February 1999 issue of Physics Today. 35

(i) (i) stance, when the projection operators Pαi (ti) are chosen ators of the form Pαi (ti) IE , will be directly related to to be the time-evolved eigenstates of the initial density that of open-system histories,⊗ represented by a nonuni- b b b matrix ρ(t0), the consistency condition will automatically tarily evolving reduced density matrix ρS (ti) and “re- be fulfilled, yet the histories composed of these projection (i) duced” projection operators Pαi (ti) (Zurek, 1993). operators have been shown to result in highly nonclassical macroscopic superpositions when applied to standard ex- b amples such as quantum measurement or Schr¨odinger’s 5. Schmidt states vs pointer basis as projectors cat. This demonstrates that the consistency condition cannot serve as a sufficient criterion for classicality. The ability of the instantaneous eigenstates of the reduced density matrix (Schmidt states; see also Sec. III.E.4) to serve as projectors for consistent histo- 4. Consistent histories of open systems ries and possibly to lead to the emergence of quasiclassi- cal histories has been studied in much detail (Albrecht, Various authors have appealed to interaction with the 1992, 1993; Kent and McElwaine, 1997; Paz and Zurek, environment and the resulting decoherence effects in 1993; Zurek, 1993). Paz and Zurek (1993) have shown defining additional criteria that would select quasiclassi- (i) that Schmidt projectors Pαi , defined by their commuta- cal histories and would also lead to a physical motivation tivity with the evolved, path-projected reduced density for the consistency criterion (see, for example, Albrecht, matrix, b 1992, 1993; Anastopoulos, 1996; Dowker and Halliwell, 1992; Finkelstein, 1993; Gell-Mann and Hartle, 1990, P (i),U(t ,t ) U(t ,t )P (1)ρ (t ) 1998; Halliwell, 2001; Paz and Zurek, 1993; Twamley, αi i−1 i {··· 1 2 α1 S 1 1993b; Zurek, 1993). This approach intrinsically requires  (1) † † b P 1 U (t1,t2) Ub (ti−1,ti) =0, (4.16) × α ···} the notion of local, open systems and the split of the uni-  verse into subsystems, in contrast to the original aim of will always giveb rise to an infinite number of sets of con- the consistent-histories approach to describe the evolu- sistent histories (“Schmidt histories”). However, these tion of a single closed, undivided system (typically the histories are branchdependent [see Eq. (4.7)] and usually entire universe). The decoherence-based studies then as- extremely unstable, since small modifications of the time sume the usual decomposition of the total Hilbert space sequence used for the projections (for instance by delet- into a space S , corresponding to the system , and ing a time point) will typically lead to drastic changes H H S E of an environment . One then describes the histo- in the resulting history, indicating that Schmidt histo- Hries of the system byE employing projection operators S ries are usually very nonclassical (Paz and Zurek, 1993; that act only on the system, i.e., that are of the form Zurek, 1993). (i) (i) Pαi (ti) IE , where Pαi (ti) acts only on S and IE is This situation is changed when the time points t are ⊗ H i the identity operator in E . chosen such that the intervals (t t ) are larger than b b b H b i+1 − i This raises the question of when, i.e., under which cir- the typical decoherence time τD of the system over which cumstances, the reduced density matrix ρS = TrE ρSE of the reduced density matrix becomes approximately di- the system suffices to calculate the decoherence func- agonal in the preferred pointer basis chosen through tional. TheS reduced density matrix arises from a nonuni- environment-induced superselection (see also the discus- tary trace over at every time point ti, whereas the de- sion in Sec. III.E.4). When the resulting pointer states, coherence functionalE of Eq. (4.11) employs the full, uni- rather than the instantaneous Schmidt states, are used to tarily evolving density matrix ρSE for all times ti < tf define the projection operators, stable quasiclassical his- and only applies a nonunitary trace operation (over both tories will typically emerge (Paz and Zurek, 1993; Zurek, and ) at the final time tf . Paz and Zurek (1993) have 1993). In this sense, it has been suggested that inter- answeredS E this (rather technical) question by showing that action with the environment can provide the missing se- the decoherence functional can be expressed entirely in lection criterion that ensures the quasiclassicality of his- terms of the reduced density matrix if the time evolu- tories, i.e., their stability (predictability), and the corre- tion of the reduced density matrix is independent of the spondence of the projection operators (the pointer basis) correlations dynamically formed between the system and to the preferred determinate quantities of our experience. the environment. A necessary (but not always sufficient) The approximate noninterference, and thus consis- condition for this requirement to be satisfied is given by tency, of histories based on local density operators (en- demanding that the reduced dynamics be governed by a ergy, number, momentum, charge etc.) as quasiclassi- master equation that is local in time. cal projectors (the so-called hydrodynamic observables, When a “reduced” decoherence functional exists, at see Dowker and Halliwell, 1992; Gell-Mann and Hartle, least to a good approximation, i.e., when the reduced 1991b; Halliwell, 1998) has been attributed to the dy- dynamics are sufficiently insensitive to the formation namical stability exhibited by the eigenstates of the local of system-environment correlations, the consistency of density operators. This stability leads to decoherence whole-universe histories, described by a unitarily evolv- in the corresponding basis (Halliwell, 1998, 1999). It has ing density matrix ρSE and sequences of projection oper- been argued by Zurek (2003b) that this behavior and thus 36 the special quasiclassical properties of hydrodynamic ob- automatically be (approximately) consistent. servables can be explained by the fact that these observ- This approach has been explored by Twamley (1993b), ables obey the commutativity criterion, Eq. (3.21), of the who carried out detailed calculations in the context of environment-induced superselection approach. a quantum optical model of phase-space decoherence and compared the results with two-time projected phase- space histories of the same model system. It was found 6. Exact vs approximate consistency that when the parameters of the interacting environment were changed such that the degree of diagonality of the In the idealized case where the pointer states lead to an reduced density matrix in the emerging preferred pointer exact diagonalization of the reduced density matrix, his- basis was increased, histories in that basis also became tories composed of the corresponding pointer projectors more consistent. For a similar model, however, Twamley will automatically be consistent. However, under real- (1993a) also showed that consistency and diagonality in istic circumstances decoherence will typically lead only a pointer basis as possible criteria for the emergence of to approximate diagonality in the pointer basis. This quasiclassicality may exhibit a very different dependence implies that the consistency criterion will not be ful- on the initial conditions. filled exactly and that hence the probability sum rules will only hold approximately—although usually, due to Extensive studies on the connection between Schmidt the efficiency of decoherence, to a very good approxima- states, pointer states and consistent quasiclassical his- tion (Albrecht, 1992, 1993; Gell-Mann and Hartle, 1991b; tories have also been made by Albrecht (1992, 1993), Griffiths, 1984; Omn`es, 1992, 1994; Paz and Zurek, 1993; based on analytical calculations and numerical simula- Twamley, 1993b; Zurek, 1993). Hence, the consistency tions of toy models for decoherence, including detailed criterion has been viewed both as overly restrictive, since numerical results on the violation of the sum rule for the quasiclassical pointer projectors rarely obey the con- histories composed of different (Schmidt and pointer) sistency equations exactly, and as insufficient, because it projector bases. It was demonstrated that the presence does not give rise to constraints that would single out of stable system-environment correlations (“records”), quasiclassical histories. as demanded by the criterion for the selection of the A relaxation of the consistency criterion has therefore pointer basis, was of great importance in making cer- been suggested, leading to “approximately consistent his- tain histories consistent. The relevance of “records” for tories” whose decoherence functional would be allowed to the consistent-histories approach in ensuring the “perma- contain nonvanishing off-diagonal terms (corresponding nence of the past” has also been emphasized by other au- to a violation of the probability sum rules) as long as the thors, for example, by Paz and Zurek (1993) and Zurek net effect of all the off-diagonal terms was “small” in the (1993, 2003b), and in the “strong decoherence” criterion sense of remaining below the experimentally detectable by Gell-Mann and Hartle (1998). The redundancy with level (see, for example, Dowker and Halliwell, 1992; which information about the system is recorded in the Gell-Mann and Hartle, 1991b). Gell-Mann and Hartle environment and can thus be found out by different ob- (1991b) have even ascribed a fundamental role to such servers without measurably disturbing the system itself approximately consistent histories, a move that has has been suggested to allow for the notion of “objec- sparked much controversy and has been considered as tively existing histories,” based on environment-selected unnecessary and irrelevant by some (Dowker and Kent, projectors that represent sequences of “objectively exist- 1995, 1996). Indeed, if only approximate consistency is ing” quasiclassical events (Paz and Zurek, 1993; Zurek, demanded, it is difficult to regard this condition as a fun- 1993, 2003b, 2004b). damental concept of a physical theory, and the question In general, damping of quantum coherence caused by of how much consistency is required will inevitably arise. decoherence will necessarily lead to a loss of quantum interference between individual histories (but not vice versa—see the discussion by Twamley, 1993b), since 7. Consistency and environment-induced superselection the final trace operation over the environment in the decoherence functional will make the off-diagonal ele- The relationship between consistency and ments very small due to the decoherence-induced approx- environment-induced superselection, and therefore imate mutual orthogonality of the environmental states. the connection between the decoherence functional Finkelstein (1993) has used this observation to propose a and the diagonalization of the reduced density matrix new decoherence condition that coincides with the origi- through environmental decoherence, has been investi- nal definition, Eqs. (4.10) and (4.11), except for restrict- gated by various authors. The basic idea, promoted, for ing the trace to , rather than tracing over both and example, by Zurek (1993) and Paz and Zurek (1993), . It was shownE that this condition not only impliesS the is to suggest that if the interaction with the environ- consistencyE condition of Eq. (4.15), but also characterizes ment leads to rapid superselection of a preferred basis, those histories that decohere due to interaction with the which approximately diagonalizes the local density environment and that lead to the formation of “records” matrix, coarse-grained histories defined in this basis will of the state of the system in the environment. 37

8. Summary and discussion consistency, and of the aims of the consistent-histories approach, like the focus on description of closed systems. The core difficulty associated with the consistent- histories approach has been to explain the emergence of the classical world of our experience from the underlying V. CONCLUDING REMARKS quantum nature. Initially, it was hoped that classical- ity could be derived from the consistency criterion alone. We have presented an extensive discussion of the role Soon, however, the status and the role of this criterion of decoherence in the foundations of quantum mechanics, in the formalism and its proper interpretation became with a particular focus on the implications of decoherence rather unclear and diffuse, since exact consistency was for the measurement problem in the context of various shown to provide neither a necessary nor a sufficient cri- interpretations of quantum mechanics. terion for the selection of quasiclassical histories. A key achievement of the decoherence program is the The inclusion of decoherence effects into the consis- recognition that openness in quantum systems is impor- tent histories approach, leading to the emergence of sta- tant for their realistic description. The well-known phe- ble quasiclassical pointer states, has been found to yield nomenon of quantum entanglement had already, early a highly efficient mechanism and a sensitive criterion in the history of quantum mechanics, demonstrated that for singling out quasiclassical observables that simulta- correlations between systems can lead to “paradoxical” neously fulfill the consistency criterion to a very good properties of the composite system that cannot be com- approximation due to the suppression of quantum coher- posed from the properties of the individual systems. ence in the state of the system. The central question Nonetheless, the viewpoint of classical physics that the is then: What is the meaning and the remaining role idealization of isolated systems is necessary to arrive at of an explicit consistency criterion in the light of such an “exact description” of physical systems has influenced “natural” mechanisms for the decoherence of histories? quantum theory for a long time. It is the great merit of Can one dispose of this criterion as a key element of the the decoherence program to have emphasized the ubiq- fundamental theory by noting that for all “realistic” his- uity and essential inescapability of system-environment tories consistency will be likely to arise naturally from correlations and to have established the important role environment-induced decoherence alone? of such correlations as factors in the emergence of “clas- The answer to this question may actually depend on sicality” from quantum systems. Decoherence also pro- the viewpoint one takes with respect to the aim of the vides a realistic physical modeling and a generalization of consistent-histories approach. As we have noted be- the quantum measurement process, thus enhancing the fore, the original goal was simply to provide a formalism “black-box” view of measurements in the standard (“or- in which one could, in a measurement-free context, as- thodox”) interpretation and challenging the postulate of sign probabilities to certain sequences of quantum events fundamentally classical measuring devices in the Copen- without logical inconsistencies. The more recent and hagen interpretation. rather opposite aim would be to provide a formalism With respect to the preferred-basis problem of quan- that selects only a very small subset of “meaningful” qua- tum measurement, decoherence provides a very promis- siclassical histories, all of which are consistent with our ing definition of preferred pointer states via a physically world of experience, and whose projectors can be directly meaningful requirement, namely, the robustness crite- interpreted as objective physical events. rion, and it describes methods for selecting operationally The consideration of decoherence effects that can give such states, for example, via the commutativity crite- rise to effective superselection of possible quasiclassical rion or by extremizing an appropriate measure such as (and consistent) histories certainly falls into the latter purity or von Neumann entropy. In particular, the fact category. It is interesting to note that this approach has that macroscopic systems virtually always decohere into also led to a departure from the original “closed systems position eigenstates gives a physical explanation for why only” view to the study of local open quantum systems position is the ubiquitous determinate property of the and thus to the decomposition of the total Hilbert space world of our experience. into subsystems, within the consistent-histories formal- We have argued that, within the standard interpre- ism. Besides the fact that decoherence intrinsically re- tation of quantum mechanics, decoherence cannot solve quires the openness of systems, this move might also re- the problem of definite outcomes in quantum measure- flect the insight that the notion of classicality itself can ment: We are still left with a multitude of (albeit indi- be viewed as only arising from a conceptual division of vidually well-localized quasiclassical) components of the the universe into parts (see the discussion in Sec. III.A). wave function, and we need to supplement or other- Therefore environment-induced decoherence and su- wise to interpret this situation in order to explain why perselection have played a remarkable role in consistent- and how single outcomes are perceived. Accordingly, we histories interpretations: a practical one by suggesting a have discussed how environment-induced superselection physical selection mechanism for quasiclassical histories; of quasiclassical pointer states together with the local and a conceptual one by contributing to a shift in our suppression of interference terms can be put to great use view of originally rather fundamental concepts, such as in physically motivating, or potentially disproving, rules 38 and assumptions of alternative interpretive approaches Albert, D., and B. Loewer, 1996, in Perspectives on Quan- that change (or altogether abandon) the strict orthodox tum Reality, edited by R. Clifton (Kluwer, Dordrecht, The eigenvalue-eigenstate link and/or modify the unitary dy- Netherlands), pp. 81–91. namics to account for the perception of definite outcomes Albert, D. Z., and L. Vaidman, 1989, Phys. Lett. A 129, 1. Albrecht, A., 1992, Phys. Rev. D 46, 5504. and classicality in general. For example, to name just a 48 few applications, decoherence can provide a universal cri- Albrecht, A., 1993, Phys. Rev. D , 3768. Allori, V., 2001, Decoherence and the Classical Limit of Quan- terion for the selection of the branches in relative-state tum Mechanics, Ph.D. thesis, Physics Department, Univer- interpretations and a physical argument for the noninter- sity of Genova. ference between these branches from the point of view of Allori, V., D. D¨urr, S. Goldstein, and N. Zangh`ı, 2002, J. Opt. an observer; in modal interpretations, it can be used to B: Quantum Semiclass. Opt. 4, S482. specify empirically adequate sets of properties that can Allori, V., and N. Zangh`ı, 2001, biannual IQSA Meeting, Ce- be ascribed to systems; in collapse models, the free pa- sena, Italy, 2001, eprint quant-ph/0112009. rameters (and possibly even the nature of the reduction Anastopoulos, C., 1996, Phys. Rev. E 53, 4711. mechanism itself) might be derivable from environmental Anderson, P. 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