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# RMP 76, (2004) - Decoherence, the measurement problem, and interpretations of quantum mechanics

# RMP 76, (2004) - Decoherence, the measurement problem, and interpretations of quantum mechanics
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REVIEWS OF MODERN PHYSICS, VOLUME 76, OCTOBER 2004 , Decoherence, the measurement problem, and interpretations of quantum mechanics by Maximilian Schlosshauer
Department of Physics, University of Washington, Seattle, Washington 98195, USA
~Published 23 February 2005! Personal file link :: file : RMP 76, 1267 (2004), Maximilian Schlosshauer, Decoherence, the measurement problem, and interpretations of quantum mechanics.pdf Environment-induced decoherence and superselection have been a subject of intensive research over the past two decades, yet their implications for the foundational problems of quantum mechanics, most notably the quantum measurement problem, have remained a matter of great controversy. This paper is intended to clarify key features of the decoherence program, including its more recent results, and to investigate their application and consequences in the context of the main interpretive approaches of quantum mechanics. ## TOC ##[#sec-I] Introduction The implications of the decoherence program for the foundations of quantum mechanics have been the subject of an ongoing debate since the first precise formulation of the program in the early 1980s. The key idea promoted by decoherence is the insight that realistic quantum systems are never isolated, but are immersed in the surrounding environment and interact continuously with it. The decoherence program then studies, entirely within the standard quantum formalism (i.e., without adding any new elements to the mathematical theory or its interpretation), the resulting formation of quantum correlations between the states of the system and its environment and the often surprising effects of these system-environment interactions. In short, decoherence brings about a local suppression of interference between preferred states selected by the interaction with the environment. Bub (1997) termed decoherence part of the “new orthodoxy” of understanding quantum mechanics—as the working physicist’s way of motivating the postulates of quantum mechanics from physical principles. Proponents of decoherence called it an “historical accident” (Joos, 2000, p. 13 Joos, E., 2000, in Decoherence: Theoretical, Experimental, and Conceptual Problems, Lecture Notes in Physics No. 538, edited by P. Blanchard, D. Giulini, E. Joos, C. Kiefer, and I-O. Stamatescu (Springer, New York), p. 1.) that the implications for quantum mechanics and for the associated foundational problems were overlooked for so long. Zurek (2003a, p. 717 Zurek, W. H., 2003a, Rev. Mod. Phys. 75, 715 or arXiv:quant-ph/0105127. Decoherence, einselection, and the quantum origins of the classical.) suggests
The idea that the “openness” of quantum systems might have anything to do with the transition from quantum to classical was ignored for a very long time, probably because in classical physics problems of fundamental importance were always settled in isolated systems.
When the concept of decoherence was first introduced to the broader scientific community by Zurek’s (1991 Zurek, W. H., 1991, Phys. Today 44 (10), 36. Decoherence and the transition from quantum to classical. :: An updated version is available as arxiv.org - quant-ph/0306072.) article in Physics Today, it elicited a series of contentious comments from the readership (see the April 1993 issue of Physics Today). In response to his critics, Zurek (2003a, p. 718 ) states
In a field where controversy has reigned for so long this resistance to a new paradigm [namely, to decoherence] is no surprise.
Omnes (2002, p. 2 Omnès, R., 2002, Phys. Rev. A 65, 052119. Decoherence, irreversibility, and selection by decoherence of exclusive quantum states with definite probabilities) had this assessment:
The discovery of decoherence has already much improved our understanding of quantum mechanics. (…) [B]ut its foundation, the range of its validity and its full meaning are still rather obscure. This is due most probably to the fact that it deals with deep aspects of physics, not yet fully investigated.
In particular, the question whether decoherence provides, or at least suggests, a solution to the measurement problem of quantum mechanics has been discussed for several years. For example, Anderson (2001, p. 492 Anderson, P. W., 2001, Stud. Hist. Philos. Mod. Phys. 32, 487.) writes in an essay review
The last chapter (…) deals with the quantum measurement problem (…). My main test, allowing me to bypass the extensive discussion, was a quick, unsuccessful search in the index for the word “decoherence” which describes the process that used to be called “collapse of the wave function.”
Zurek speaks in various places of the “apparent” or “effective” collapse of the wave function induced by the interaction with environment (when embedded into a minimal additional interpretive framework) and concludes (Zurek, 1998, p. 1793 Zurek, W. H., 1998, Philos. Trans. R. Soc. London, Ser. A 356, 1793.)
A “collapse” in the traditional sense is no longer necessary. (…) [The] emergence of “objective existence” [from decoherence] (…) significantly reduces and perhaps even eliminates the role of the “collapse” of the state vector.
d’Espagnat, who considers the explanation of our experiences (i.e., of “appearances”) as the only “sure” requirement of a physical theory, states (d’Espagnat, 2000, p. 136 d’Espagnat, B., 2000, Phys. Lett. A 282, 133.)
For macroscopic systems, the appearances are those of a classical world (no interferences etc.), even in circumstances, such as those occurring in quantum measurements, where quantum effects take place and quantum probabilities intervene (…). Decoherence explains the just mentioned appearances and this is a most important result. (…) As long as we remain within the realm of mere predictions concerning what we shall observe (i.e., what will appear to us)—and refrain from stating anything concerning “things as they must be before we observe them”—no break in the linearity of quantum dynamics is necessary.
In his monumental book on the foundations of quantum mechanics (QM), Auletta (2000, p. 791 Auletta, G., 2000, Foundations and Interpretation of Quantum Mechanics in the Light of a Critical-Historical Analysis of the Problems and of a Synthesis of the Results (World Scientific, Singapore).) concludes that
the Measurement theory could be part of the interpretation of QM only to the extent that it would still be an open problem, and we think that this is largely no longer the case.
This is mainly so because, according to Auletta (2000, p. 289 ),
decoherence is able to solve practically all the problems of Measurement which have been discussed in the previous chapters.
On the other hand, even leading adherents of decoherence have expressed caution or even doubt that decoherence has solved the measurement problem. Joos (2000, p. 14 Joos, E., 2000, in Decoherence: Theoretical, Experimental, and Conceptual Problems, Lecture Notes in Physics No. 538, edited by P. Blanchard, D. Giulini, E. Joos, C. Kiefer, and I-O. Stamatescu (Springer, New York), p. 1.) writes
Does decoherence solve the measurement problem? Clearly not. What decoherence tells us, is that certain objects appear classical when they are observed. But what is an observation? At some stage, we still have to apply the usual probability rules of quantum theory.
Along these lines, Kiefer and Joos (1999, p. 5 Kiefer, C., and E. Joos, 1999, in Quantum Future: From Volta and Como to the Present and Beyond, edited by P. Blanchard and A. Jadczyk (Springer, Berlin), p. 105.) warn that
One often finds explicit or implicit statements to the effect that the above processes are equivalent to the collapse of the wave function (or even solve the measurement problem). Such statements are certainly unfounded.
In a response to Anderson’s (2001, p. 492 ) comment, Adler (2003, p. 136 Adler, S. L., 2003, Stud. Hist. Philos. Mod. Phys. 34 (1), 135.) states
I do not believe that either detailed theoretical calculations or recent experimental results show that decoherence has resolved the difficulties associated with quantum measurement theory.
Similarly, Bacciagaluppi (2003b, p. 3 Bacciagaluppi, G., 2003b, talk given at the workshop, Quantum Mechanics on a Large Scale, Vancouver, 23 April 2003. http://www.physics.ubc.ca/˜berciu/PHILIP/CONFERENCES/PWI03/FILES/baccia.ps) writes
Claims that simultaneously the measurement problem is real [and] decoherence solves it are confused at best.
Zeh asserts (Joos et al., 2003, Chap. 2 Joos, E., H. D. Zeh, C. Kiefer, D. Giulini, J. Kupsch, and I.-O. Stamatescu, 2003, Decoherence and the Appearance of a Classical World in Quantum Theory, 2nd ed. (Springer, New York).)
Decoherence by itself does not yet solve the 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 Schrödinger dynamics (…) is supplemented by a nonunitary collapse (…).
The key achievements of the decoherence program, apart from their implications for conceptual problems, do not seem to be universally understood either. Zurek (1998, p. 1800 ) remarks
[The] eventual diagonality of the density matrix (…) is a byproduct (…) but not the essence of decoherence. I emphasize this because diagonality of [the density matrix] in some basis has been occasionally (mis-)interpreted as a key accomplishment of decoherence. This is misleading. Any density matrix is diagonal in some basis. This has little bearing on the interpretation.
These remarks show that a balanced discussion of the key features of decoherence and their implications for the foundations of quantum mechanics is overdue. The decoherence program has made great progress over the past decade, and it would be inappropriate to ignore its relevance in tackling conceptual problems. However, it is equally important to realize the limitations of decoherence in providing consistent and noncircular answers to foundational questions. An excellent review of the decoherence program has recently been given by Zurek (2003a ). It deals primarily with the technicalities of decoherence, although it contains some discussion on how decoherence can be employed in the context of a relative-state interpretation to motivate basic postulates of quantum mechanics. A helpful first orientation and overview, the entry by Bacciagaluppi (2003a Bacciagaluppi, G., 2003a, in The Stanford Encyclopedia of Philosophy, Winter 2003 Edition, edited by E. N. Zalta, http://plato.stanford.edu/archives/win2003/entries/qm-decoherence/) in the Stanford Encyclopedia of Philosophy, features a relatively short (in comparison to the present paper) introduction to the role of decoherence in the foundations of quantum mechanics, including comments on the relationship between decoherence and several popular interpretations of quantum theory. In spite of these valuable recent contributions to the literature, a detailed and self-contained discussion of the role of decoherence in the foundations of quantum mechanics seems still to be lacking. This review article is intended to fill the gap. To set the stage, we shall first, in Sec. , review the measurement problem, which illustrates the key difficulties that are associated with describing quantum measurement within the quantum formalism and that are all in some form addressed by the decoherence program. In Sec. , we then introduce and discuss the main features of the theory of decoherence, with a particular emphasis on their foundational implications. Finally, in Sec. , we investigate the role of decoherence in various interpretive approaches of quantum mechanics, in particular with respect to the ability to motivate and support (or disprove) possible solutions to the measurement problem. ##/ I. Introduction ##[#sec-II] The Measurement Problem One of the most revolutionary elements introduced into physical theory by quantum mechanics is the superposition principle, mathematically founded in the linearity of the Hilbert-state space. If $|1\rangle$ and $|2\rangle$ are two states, then quantum mechanics tells us that any linear combination $\alpha |1\rangle+\beta |2\rangle$ also corresponds to a possible state. Whereas such superpositions of states have been experimentally extensively verified for microscopic systems (for instance, through the observation of interference effects), the application of the formalism to macroscopic systems appears to lead immediately to severe clashes with our experience of the everyday world. A book has never been observed to be in a state of being both “here” and “there” (i.e., to be in a superposition of macroscopically distinguishable positions), nor does a Schrödinger cat that is a superposition of being alive and dead bear much resemblence to reality as we perceive it. The problem is, then, how to reconcile the vastness of the Hilbert space of possible states with the observation of a comparatively few “classical” macrosopic states, defined by having a small number of determinate and robust properties such as position and momentum. Why does the world appear classical to us, in spite of its supposed underlying quantum nature, which would, in principle, allow for arbitrary superpositions? ###[#sec-II-A] Quantum measurement scheme This question is usually illustrated in the context of quantum measurement where microscopic superpositions are, via quantum entanglement, amplified into the macroscopic realm and thus lead to very “nonclassical” states that do not seem to correspond to what is actually perceived at the end of the measurement. In the ideal measurement scheme devised by von Neumann (1932 von Neumann, J., 1932, Mathematische Grundlagen der Quantenmechanik (Springer, Berlin).), a (typically microscopic) system $S$, represented by basis vectors ${ | s_n \rangle }$ in a Hilbert space $H_S$, interacts with a measurement apparatus $A$, described by basis vectors ${ | a_n \rangle }$ spanning a Hilbert space $H_A$, where the $| a_n \rangle$ are assumed to correspond to macroscopically distinguishable “pointer” positions that correspond to the outcome of a measurement if $S$ is in the state $| s_n \rangle$ .
1 Note that von Neumann’s scheme is in sharp contrast to the Copenhagen interpretation, where measurement is not treated as a system-apparatus interaction described by the usual quantum-mechanical formalism, but instead as an independent component of the theory, to be represented entirely in fundamentally classical terms.
Now, if $S$ is in a (microscopically “unproblematic”) superposition $\sum_n c_n | s_n \rangle$, and $A$ is in the initial “ready” state $| a_r \rangle$, the linearity of the Schrödinger equation entails that the total system $S~A$, assumed to be represented by the Hilbert product space $H_S \otimes H_A$, evolves according to \Big( \sum_n c_n | s_n \rangle \Big) | a_r \rangle \overset{t}{\rightarrow} \sum_n c_n | s_n \rangle | a_n \rangle This dynamical evolution is often referred to as a premeasurement in order to emphasize that the process described by Eq. does not suffice to directly conclude that a measurement has actually been completed. This is so for two reasons. First, the right-hand side is a superposition of system-apparatus states. Thus, without supplying an additional physical process (say, some collapse mechanism) or giving a suitable interpretation of such a superposition, it is not clear how to account, given the final composite state, for the definite pointer positions that are perceived as the result of an actual measurement—i.e., why do we seem to perceive the pointer to be in one position $| a_n \rangle$ but not in a superposition of positions? This is the problem of definite outcomes. Second, the expansion of the final composite state is in general not unique, and therefore the measured observable is not uniquely defined either. This is the problem of the preferred basis. In the literature, the first difficulty is typically referred to as the measurement problem, but the preferred-basis problem is at least equally important, since it does not make sense even to inquire about specific outcomes if the set of possible outcomes is not clearly defined. We shall therefore regard the measurement problem 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. ###/ II-A. Quantum measurement scheme ###[#sec-II-B] The problem of definite outcomes ####[#sec-II-B-1] Superpositions and ensembles The right-hand side of Eq. implies that after the premeasurement the combined system $SA$ is left in a pure state that represents a linear superposition of system-pointer states. It is a well-known and important property of quantum mechanics that a superposition of states is fundamentally different from a classical ensemble of states, where the system actually is in only one of the states but we simply do not know in which (this is often referred to as an “ignorance-interpretable,” or “proper” ensemble). This can be shown explicitly, especially on microscopic scales, by performing experiments that lead to the direct observation of interference patterns instead of the realization of one of the terms in the superposed pure state, for example, in a setup where electrons pass individually (one at a time) through a double slit. As is well known, this experiment clearly shows that, within the standard quantum-mechanical formalism, the electron must not be described by either one of the wave functions describing the passage through a particular slit ($\psi_1$ or $\psi_1$), but only by the superposition of these wave functions ($\psi_1 + \psi_1$). This is so because the correct density distribution $\rho$ of the pattern on the screen is not given by the sum of the squared wave functions describing the addition of individual passages through a single slit ($\rho = |\psi_1|^2 + |\psi_2|^2$), but only by the square of the sum of the individual wave functions ($\rho = |\psi_1 + \psi_2|^2$). Put differently, if an ensemble interpretation could be attached to a superposition, the latter would simply represent an ensemble of more fundamentally determined states, and based on the additional knowledge brought about by the results of measurements, we could simply choose a subensemble consisting of the definite pointer state obtained in the measurement. But then, since the time evolution has been strictly deterministic according to the Schrödinger equation, we could backtrack this subensemble in time and thus also specify the initial state more completely (“postselection”), and therefore this state necessarily could not be physically identical to the initially prepared state on the left-hand side of Eq. . ####/ II-B-1. Superpositions and ensembles ####[#sec-II-B-2] Superpositions and outcome attribution In the standard (“orthodox”) interpretation of quantum mechanics, an observable corresponding to a physical quantity has a definite value if and only if the system is in an eigenstate of the observable; if the system is, however, in a superposition of such eigenstates, as in Eq. , it is, according to the orthodox interpretation, meaningless to speak of the state of the system as having any definite value of the observable at all. (This is frequently referred to as the so-called eigenvalue-eigenstate link, or “e-e link” for short.) The e-e link, however, is by no means forced upon us by the structure of quantum mechanics or by empirical constraints (Bub, 1997 Bub, J., 1997, Interpreting the Quantum World, 1st ed. (Cambridge University, Cambridge, England).). The concept of (classical) “values” that can be ascribed through the e-e link based on observables and the existence of exact eigenstates of these observables has therefore frequently been either weakened or altogether abandonded. For instance, outcomes of measurements are typically registered in position space (pointer positions, etc.), but there exist no exact eigenstates of the position operator, and the pointer states are never exactly mutually orthogonal. One might then (explicitly or implicitly) promote a “fuzzy” e-e link, or give up the concept of observables and values entirely and directly interpret the time-evolved wave functions (working in the Schrödinger picture) and the corresponding density matrices. Also, if it is regarded as sufficient to explain our perceptions rather than describe the “absolute” state of the entire universe (see the argument below), one might only require that the (exact or fuzzy) e-e link hold in a “relative” sense, i.e., for the state of the rest of the universe relative to the state of the observer. Then, to solve the problem of definite outcomes, some interpretations (for example, modal interpretations and relative-state interpretations) interpret the final composite state in such a way as to explain the existence, or at least the subjective perception, of “outcomes” even if this state has the form of a superposition. Other interpretations attempt to solve the measurement problem by modifying the strictly unitary Schrödinger dynamics. Most prominently, the orthodox interpretation postulates a collapse mechanism that transforms a pure-state density matrix into an ignorance-interpretable ensemble of individual states (a “proper mixture”). Wave-function collapse theories add stochastic terms to the Schrödinger equation that induce an effective (albeit only approximate) collapse for states of macroscopic systems (Pearle, 1979, 1999 Pearle, P. M., 1979, Int. J. Theor. Phys. 48, 489.Pearle, P. M., 1999, in Open Systems and Measurement in Relativistic Quantum Theory, edited by H.-P. Breuer and F. Petruccioni (Springer, Berlin), p. 31.; Gisin, 1984 Gisin, N., 1984, Phys. Rev. Lett. 52 (19), 1657.; Ghirardi et al., 1986 Ghirardi, G. C., A. Rimini, and T. Weber, 1986, Phys. Rev. D 34, 470.), while other authors have suggested that collapse occurs at the level of the mind of a conscious observer (Wigner, 1963 Wigner, E. P., 1963, Am. J. Phys. 31, 6.; Stapp, 1993 Stapp, H. P., 1993, Mind, Matter, and Quantum Mechanics, 1st ed. (Springer, New York).). Bohmian mechanics, on the other hand, upholds a unitary time evolution of the wave function, but introduces an additional dynamical law that explicitly governs the always-determinate positions of all particles in the system. ####/ II-B-2. Superpositions and outcome attribution ####[#sec-II-B-3] Objective vs subjective definiteness In general, (macroscopic) definiteness—and thus a solution to the problem of outcomes in the theory of quantum measurement—can be achieved either on an ontological (objective) or an observational (subjective) level. Objective definiteness aims at ensuring “actual” definiteness in the macroscopic realm, whereas subjective definiteness only attempts to explain why the macroscopic world appears to be definite—and thus does not make any claims about definiteness of the underlying physical reality (whatever this reality might be). This raises the question of the significance of this distinction with respect to the formation of a satisfactory theory of the physical world. It might appear that a solution to the measurement problem based on ensuring subjective, but not objective, definiteness is merely good “for all practical purposes”—abbreviated, rather disparagingly, as “FAPP” by Bell (1990 Bell, J. S., 1990, in Sixty-Two Years of Uncertainty, edited by A. I. Miller (Plenum, New York), p. 17.)—and thus not capable of solving the “fundamental” problem that would seem relevant to the construction of the “precise theory” that Bell demanded so vehemently. It seems to the author, however, that this criticism is not justified, and that subjective definiteness should be viewed on a par with objective definiteness with respect to a satisfactory solution to the measurement problem. We demand objective definiteness because we experience definiteness on the subjective level of observation, and it should not be viewed as an a priori requirement for a physical theory. If we knew independently of our experience that definiteness existed in nature, subjective definiteness would presumably follow as soon as we had employed a simple model that connected the “external” physical phenomena with our “internal” perceptual and cognitive apparatus, where the expected simplicity of such a model can be justified by referring to the presumed identity of the physical laws governing external and internal processes. But since knowledge is based on experience, that is, on observation, the existence of objective definiteness could only be derived from the observation of definiteness. And, moreover, observation tells us that definiteness is in fact not a universal property of nature, but rather a property of macroscopic objects, where the borderline to the macroscopic realm is difficult to draw precisely; mesoscopic interference experiments have demonstrated clearly the blurriness of the boundary. Given the lack of a precise definition of the boundary, any demand for fundamental definiteness on the objective level should be based on a much deeper and more general commitment to a definiteness that applies to every physical entity (or system) across the board, regardless of spatial size, physical property, and the like. Therefore, if we realize that the often deeply felt commitment to a general objective definiteness is only based on our experience of macroscopic systems, and that this definiteness in fact fails in an observable manner for microscopic and even certain mesoscopic systems, the author sees no compelling grounds on which objective definiteness must be demanded as part of a satisfactory physical theory, provided that the theory can account for subjective, observational definiteness in agreement with our experience. Thus the author suggests that the same legitimacy be attributed to proposals for a solution of the measurement problem that achieve “only” subjective but not objective definiteness—after all, the measurement problem arises solely from a clash of our experience with certain implications of the quantum formalism. d’Espagnat (2000, pp. 134 and 135 ) has advocated a similar viewpoint:
The fact that we perceive such “things” as macroscopic objects lying at distinct places is due, partly at least, to the structure of our sensory and intellectual equipment. We should not, therefore, take it as being part of the body of sure knowledge that we have to take into account for defining a quantum state. (…) In fact, scientists most rightly claim that the purpose of science is to describe human experience, not to describe “what really is”; and as long as we only want to describe human experience, that is, as long as we are content with being able to predict what will be observed in all possible circumstances (…) we need not postulate the existence—in some absolute sense—of unobserved (i.e., not yet observed) objects lying at definite places in ordinary 3-dimensional space.
####/ II-B-3. Objective vs subjective definiteness ###/ II-B. The problem of definite outcomes ###[#sec-II-C] The preferred-basis problem The second difficulty associated with quantum measurement is known as the preferred-basis problem, which demonstrates that the measured observable is in general not uniquely defined by Eq. . For any choice of system states $\{ | s_n \rangle \}$, we can find corresponding apparatus states $\{ | a_n \rangle \}$, and vice versa, to equivalently rewrite the final state emerging from the premeasurement interaction, i.e., the right-hand side of Eq. . In general, however, for some choice of apparatus states the corresponding new system states will not be mutually orthogonal, so that the observable associated with these states will not be Hermitian, which is usually not desired (however, not forbidden—see the discussion by Zurek, 2003a ). Conversely, to ensure distinguishable outcomes, we must, in general, require the (at least approximate) orthogonality of the apparatus (pointer) states, and it then follows from the biorthogonal decomposition theorem that the expansion of the final premeasurement system-apparatus state of Eq. , | \psi \rangle = \sum_n c_n | s_n \rangle | a_n \rangle , is unique, but only if all coefficients $c_n$ are distinct. Otherwise, we can in general rewrite the state in terms of different state vectors, | \psi \rangle = \sum_n c'_n | s'_n \rangle | a'_n \rangle , such that the same postmeasurement state seems to correspond to two different measurements, that is, of the observables $\hat{A}=\sum_n \lambda_n | s_n \rangle \langle s_n |$ and $\hat{B}=\sum_n \lambda'_n | s'_n \rangle \langle s'_n |$ of the system, respectively, although in general $\hat{A}$ and $\hat{B}$ do not commute. As an example, consider a Hilbert space $H=H_1 \otimes H_2$ where $H_1$ and $H_2$ are two-dimensional spin spaces with states corresponding to spin up or spin down along a given axis. Suppose we are given an entangled spin state of the Einstein-Podolsky-Rosen form (Einstein et al., 1935 Einstein, A., B. Podolsky, and N. Rosen, 1935, Phys. Rev. 47, 777.) | \psi \rangle = \frac{1}{\sqrt{2}} \big( | z + \rangle_1 | z - \rangle_2 - | z - \rangle_1 | z + \rangle_2 \big) , where $| z \pm \rangle_{1,2}$ represents the eigenstates of the observable $\sigma_z$ corresponding to spin up or spin down along the $z$ axis of the two systems 1 and 2. The state $| \psi \rangle$ can, however, equivalently be expressed in the spin basis corresponding to any other orientation in space. For example, when using the eigenstates $| x \pm \rangle_{1,2}$ of the observable $\sigma_x$ (which represents a measurement of the spin orientation along the x axis) as basis vectors, we get | \psi \rangle = \frac{1}{\sqrt{2}} \big( | x + \rangle_1 | x - \rangle_2 - | x - \rangle_1 | x + \rangle_2 \big) , Now suppose that system 2 acts as a measuring device for the spin of system 1. Then Eqs. and imply that the measuring device has established a correlation with both the $z$ and the $x$ spin of system 1. This means that, if we interpret the formation of such a correlation as a measurement in the spirit of the von Neumann scheme (without assuming a collapse), our apparatus (system 2) could be considered as having measured also the $x$ spin once it has measured the $z$ spin, and vice versa—in spite of the noncommutativity of the corresponding spin observables $\sigma_z$ and $\sigma_x$. Moreover, since we can rewrite Eq. in infinitely many ways, it appears that once the apparatus has measured the spin of system 1 along one direction, it can also be regarded as having measured the spin along any other direction, again in apparent contradiction with quantum mechanics due to the noncommutativity of the spin observables corresponding to different spatial orientations. It thus seems that quantum mechanics has nothing to say about which observable(s) of the system is (are) being recorded, via the formation of quantum correlations, by the apparatus. This can be stated in a general theorem (Zurek, 1981 Zurek, W. H., 1981, Phys. Rev. D 24, 1516.; Auletta, 2000 ): When quantum mechanics is applied to an isolated composite object consisting of a system $S$ and an apparatus $A$, it cannot determine which observable of the system has been measured—in obvious contrast to our experience of the workings of measuring devices that seem to be “designed” to measure certain quantities. ###/ II-C. The preferred-basis problem ###[#sec-II-D] The quantum-to-classical transition and decoherence In essence, as we have seen above, the measurement problem deals with the transition from a quantum world, described by essentially arbitrary linear superpositions of state vectors, to our perception of “classical” states in the macroscopic world, that is, a comparatively small subset of the states allowed by the quantum-mechanical superposition principle, having only a few, but determinate and robust, properties, such as position, momentum, etc. The question of why and how our experience of a “classical” world emerges from quantum mechanics thus lies at the heart of the foundational problems of quantum theory. Decoherence has been claimed to provide an explanation for this quantum-to-classical transition by appealing to the ubiquitous immersion of virtually all physical systems in their environment (“environmental monitoring”). This trend can also be read off nicely from the titles of some papers and books on decoherence, for example, “The emergence of classical properties through interaction with the environment” (Joos and Zeh, 1985 Joos, E., and H. D. Zeh, 1985, Z. Phys. B: Condens. Matter 59, 223.), “Decoherence and the transition from quantum to classical” (Zurek, 1991 ), and “Decoherence and the appearance of a classical world in quantum theory” (Joos et al., 2003 ). We shall critically investigate in this paper to what extent the appeal to decoherence for an explanation of the quantum-to-classical transition is justified. ###/ II-D. The quantum-to-classical transition and decoherence ##[#sec-III] The Decoherence Program As remarked earlier, the theory of decoherence is based on a study of the effects brought about by the interaction of physical systems with their environment. In classical physics, the environment is usually viewed as a kind of disturbance, or noise, that perturbs the system under consideration in such a way as to negatively influence the study of its “objective” properties. Therefore science has established the idealization of isolated systems, with experimental physics aiming at eliminating any outer sources of disturbance as much as possible in order to discover the “true” underlying nature of the system under study. The distinctly nonclassical phenomenon of quantum entanglement, however, has demonstrated that the correlations between two systems can be of fundamental importance and can lead to properties that are not present in the individual systems . The earlier view of phenomena arising from quantum entanglement as “paradoxa” has generally been replaced by the recognition of entanglement as a fundamental property of nature.
2 Broadly speaking, this means that the (quantum-mechanical) whole is different from the sum of its parts.
The decoherence program is based on the idea that such quantum correlations are ubiquitous; that nearly every physical system must interact in some way with its environment (for example, with the surrounding photons that then create the visual experience within the observer), which typically consists of a large number of degrees of freedom that are hardly ever fully controlled. Only in very special cases of typically microscopic (atomic) phenomena, so goes the claim of the decoherence program, is the idealization of isolated systems applicable so that the predictions of linear quantum mechanics (i.e., a large class of superpositions of states) can actually be observationally confirmed. In the majority of the cases accessible to our experience, however, interaction with the environment is so dominant as to preclude the observation of the “pure” quantum world, imposing effective superselection rules (Wick et al., 1952, 1970; Galindo et al., 1962; Wightman, 1995; Cisnerosy et al., 1998; Giulini, 2000) onto the space of observable states that lead to states corresponding to the “classical” properties of our experience. Interference between such states gets locally suppressed and is thus claimed to become inaccessible to the observer.
3 For key ideas and concepts, see Zeh (1970 , 1973 , 1995 , 1997 , 2000 ); Zurek (1981 , 1982 , 1991 , 1993 , 2003a ); Kübler and Zeh (1973 ); Joos and Zeh (1985 ); Joos et al. (2003 ).
Probably the most surprising aspect of decoherence is the effectiveness of the system-environment interactions. Decoherence typically takes place on extremely 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 environment, it is usually very difficult to undo system-environment entanglement, which has been claimed as a source of our impression of irreversibility in nature (see, for example, Zurek, 1982, 2003a; Zurek and Paz, 1994; Kiefer and Joos, 1999; Zeh, 2001). In general, the effect of decoherence increases with the size of the system (from microscopic to macroscopic scales), but it is important to note that there exist, admittedly somewhat exotic, examples for which the decohering influence of the environment can be sufficiently shielded to lead to mesoscopic and even macroscopic superpositions. One such example would be the case of superconducting quantum interference devices (SQUID’s), in which superpositions of macroscopic currents become observable. Conversely, some microscopic systems (for instance, certain chiral molecules that exist in different distinct spatial configurations) can be subject to remarkably strong decoherence. The decoherence program has dealt with the following two main consequences of environmental interaction:
1. Environment-induced decoherence: The fast local suppression of interference between different states of the system. However, since only unitary time evolution is employed, global phase coherence is not actually destroyed—it becomes absent from the local density matrix that describes the system alone, but remains fully present in the total systemenvironment composition . We shall discuss environment-induced local decoherence in more detail in Sec. .
2. Environment-induced superselection: The selection of preferred sets of states, often referred to as “pointer states,” that are robust (in the sense of retaining correlations over time) in spite of their immersion in the environment. These states are determined by the form of the interaction between the system and its environment and are suggested to correspond to the “classical” states of our experience. We shall consider this mechanism in Sec. .
4 Note that the persistence of coherence in the total state is important to ensure the possibility of describing special cases in which mesoscopic or macrosopic superpositions have been experimentally realized.
Another, more recent aspect of the decoherence program, termed enviroment-assisted invariance or “envariance,” was introduced by Zurek (2003a, 2003b, 2004b) and further developed in Zurek (2004a). In particular, Zurek used envariance to explain the emergence of probabilities in quantum mechanics and to derive Born’s rule based on certain assumptions. We shall review envariance and Zurek’s derivation of the Born rule in Sec. . Finally, let us emphasize that decoherence arises from a direct application of the quantum-mechanical formalism to a description of the interaction of a physical system with its environment. By itself, decoherence is therefore neither an interpretation nor a modification of quantum mechanics. Yet the implications of decoherence need to be interpreted in the context of the different interpretations of quantum mechanics. Also, since decoherence effects have been studied extensively in both theoretical models and experiments (for a survey, see, for example, Joos et al., 2003; Zurek, 2003a), their existence can be taken as a well-confirmed fact. ###[#sec-III-A] Resolution into subsystems Note that decoherence derives from the presupposition of the existence and the possibility of a division of the world into “system(s)” and “environment.” In the decoherence program, the term “environment” is usually understood as the “remainder” of the system, in the sense that its degrees of freedom are typically not (cannot be, do not need to be) controlled and are not directly relevant to the observation under consideration (for example, the many microsopic degrees of freedom of the system), but that nonetheless the environment includes “all those degrees of freedom which contribute significantly to the evolution of the state” of the system (Zurek, 1981, p. 1520). This system-environment dualism is generally associated with quantum entanglement, which always describes a correlation between parts of the universe. As long as the universe is not resolved into individual subsystems, there is no measurement problem: the state vector $|\Psi\rangle$ of the entire universe evolves deterministically according to the Schrödinger equation $i\hbar (\partial / \partial t) |\Psi\rangle = \hat{H} |\Psi\rangle$, which poses no interpretive difficulty. Only when we decompose the total Hilbert-state space $H$ of the universe into a product of two spaces $H_1 \otimes H_2$, and accordingly form the joint-state vector $|\Psi\rangle = |\Psi_1\rangle |\Psi_2\rangle$, and want to ascribe an individual state (besides the joint state that describes a correlation) to one of the two systems (say, the apparatus), does the measurement problem arise. Zurek (2003a, p. 718) puts it like this:
In the absence of systems, the problem of interpretation seems to disappear. There is simply no need for “collapse” in a universe with no systems. Our experience of the classical reality does not apply to the universe as a whole, seen from the outside, but to the systems within it.
5 If we dare to postulate this total state—see counterarguments by Auletta (2000).
Moreover, terms like “observation,” “correlation,” and “interaction” will naturally make little sense without a division into systems. Zeh has suggested that the locality of the observer defines an observation in the sense that any observation arises from the ignorance of a part of the universe; and that this also defines the “facts” that can occur in a quantum system. Landsman (1995, pp. 45 and 46) argues similarly:
The essence of a “measurement,” “fact” or “event” in quantum mechanics lies in the nonobservation, or irrelevance, of a certain part of the system in question. (…) A world without parts declared or forced to be irrelevant is a world without facts.
However, the assumption of a decomposition of the universe into subsystems—as necessary as it appears to be for the emergence of the measurement problem and for the definition of the decoherence program—is definitely nontrivial. By definition, the universe as a whole is a closed system, and therefore there are no “unobserved degrees of freedom” of an external environment which would allow for the application of the theory of decoherence to determine the space of quasiclassical observables of the universe in its entirety. Also, there exists no general criterion for how the total Hilbert space is to be divided into subsystems, while at the same time much of what is called a property of the system will depend on its correlation with other systems. This problem becomes particularly acute if one would like decoherence not only to motivate explanations for the subjective perception of classicality (as in Zurek’s “existential interpretation”; see Zurek, 1993, 1998, 2003a, and Sec. below), but moreover to allow for the definition of quasiclassical “macrofacts.” Zurek (1998, p. 1820) admits this severe conceptual difficulty:
In particular, one issue which has been often taken for granted is looming big, as a foundation of the whole decoherence program. It is the question of what are the “systems” which play such a crucial role in all the discussions of the emergent classicality. (…) [A] compelling explanation of what are the systems—how to define them given, say, the overall Hamiltonian in some suitably large Hilbert space—would be undoubtedly most useful.
A frequently proposed idea is to abandon the notion of an “absolute” resolution and instead postulate the intrinsic relativity of the distinct state spaces and properties that emerge through the correlation between these relatively defined spaces (see, for example, the proposals, unrelated to decoherence, of Everett, 1957, Mermin, 1998a, 1998b; and Rovelli, 1996). This relative view of systems and correlations has counterintuitive, in the sense of nonclassical, implications. However, as in the case of quantum entanglement, these implications need not be taken as paradoxa that demand further resolution. Accepting some properties of nature as counterintuitive is indeed a satisfactory path to take in order to arrive at a description of nature that is as complete and objective as is allowed by the range of our experience (which is based on inherently local observations). ###/ III-A. Resolution into subsystems ###[#sec-III-B] The concept of reduced density matrices Since reduced density matrices are a key tool of decoherence, it will be worthwile to briefly review their basic properties and interpretation in the following. The concept of reduced density matrices emerged in the earliest days of quantum mechanics (Landau, 1927; von Neumann, 1932; Furry, 1936; for some historical remarks, see Pessoa, 1998). In the context of a system of two entangled systems in a pure state of the Einstein-PodolskyRosen-type, |\psi\rangle = \frac{1}{\sqrt{2}} \big( |+\rangle_1 |-\rangle_2 - |-\rangle_1 |+\rangle_2 \big) , it had been realized early that for an observable $\hat{O}$ that pertains only to system 1, $\hat{O} = \hat{O}_1 \otimes \hat{I}_2$, the pure-state density matrix $\rho = |\psi\rangle\langle\psi |$ yields, according to the trace rule $\langle\hat{O}\rangle = \text{Tr} (\rho\hat{O})$ and given the usual Born rule for calculating probabilities, exactly the same statistics as the reduced density matrix $\rho_1$ obtained by tracing over the degrees of freedom of system 2 (i.e., the states $|+\rangle_2$ and $|-\rangle_2$, \rho_1 = \text{Tr}_2 |\psi\rangle\langle\psi | = _2\hspace{-.35em}\langle +|\psi\rangle\langle\psi |+\rangle_2 + _2\hspace{-.35em}\langle -|\psi\rangle\langle\psi |-\rangle_2 since it is easy to show that, for this observable $\hat{O}$, \langle\hat{O}\rangle_{\psi} = \text{Tr} (\rho\hat{O}) = \text{Tr}_1 (\rho_1 \hat{O}_1) . This result holds in general for any pure state $|\psi\rangle = \sum_i \alpha_i |\phi_i \rangle_1 |\phi_i \rangle_2 \cdots |\phi_i \rangle_N$ of a resolution of a system into $N$ subsystems, where the $\{ |\phi_i \rangle_j \}$ are assumed to form orthonormal basis sets in their respective Hilbert spaces $H_j$, $j=1 \cdots N$. For any observable $\hat{O}$ that pertains only to system $j$, $\hat{O}$ = $\hat{I}_1$ $\otimes$ $\hat{I}_2$ $\otimes$ $\cdots$ $\otimes$ $\hat{I}_{j-1}$ $\otimes$ $\hat{O}_j$ $\otimes$ $\hat{I}_{j+1}$ $\otimes$ $\cdots$ $\otimes$ $\hat{I}_N$, the statistics of $\hat{O}$ generated by applying the trace rule will be identical regardless of whether we use the pure-state density matrix $\rho = |\psi\rangle\langle\psi |$ or the reduced density matrix $\rho_j = \text{Tr}_{~ 1,\dots ,~ j-1,~ j+1,\dots ,~ N} |\psi\rangle\langle\psi |$, since again $\langle\hat{O}\rangle = \text{Tr} (\rho\hat{O})$. The typical situation in which the reduced density matrix arises is this: Before a premeasurement-type interaction, the observer knows that each individual system is in some (unknown) pure state. After the interaction, i.e., after the correlation between the systems is established, the observer has access to only one of the systems, say, system 1; everything that can be known about the state of the composite system must therefore be derived from measurements on system 1, which will yield the possible outcomes of system 1 and their probability distribution. All information that can be extracted by the observer is then, exhaustively and correctly, contained in the reduced density matrix of system 1, assuming that the Born rule for quantum probabilities holds. Let us return to the Einstein-Podolsky-Rosen-type example, Eqs. and . If we assume that the states of system 2 are orthogonal, $_2\hspace{-.1em}\langle +|-\rangle_2 = 0$, $\rho_1$ becomes diagonal, \rho_1 = \text{Tr}_2 |\psi\rangle\langle\psi | = \frac{1}{2} \big( |+\rangle\langle +| \big)_1 + \frac{1}{2} \big( |-\rangle\langle -| \big)_1 . But this density matrix is formally identical to the density matrix that would be obtained if system 1 were in a mixed state, i.e., in either one of the two states $|+\rangle_1$ and $|-\rangle_1$ with equal probabilties—as opposed to the superposition $|\psi\rangle$, in which both terms are considered present, which could in principle be confirmed by suitable interference experiments. This implies that a measurement of an observable that only pertains to system 1 cannot discriminate between the two cases, pure vs mixed state .
6 As discussed by Bub (1997, pp. 208–210), this result also holds for any observable of the composite system that factorizes into the form $\hat{O} = \hat{O}_1 \otimes \hat{O}_2$, where $\hat{O}_1$ and $\hat{O}_2$ do not commute with the projection operators $(|\pm\rangle\langle\pm |)_1$ and $(|\pm\rangle\langle\pm |)_2$, respectively.
However, note that the formal identification of the reduced density matrix with a mixed-state density matrix is easily misinterpreted as implying that the state of the system can be viewed as mixed too (see also the discussion by d’Espagnat, 1988). Density matrices are only a calculational tool for computing the probability distribution of a set of possible outcomes of measurements; they do not specify the state of the system . Since the two systems are entangled and the total composite system is still described by a superposition, it follows from the standard rules of quantum mechanics that no individual definite state can be attributed to one of the systems. The reduced density matrix looks like a mixed-state density matrix because, if one actually measured an observable of the system, one would expect to get a definite outcome with a certain probability; in terms of measurement statistics, this is equivalent to the situation in which the system is in one of the states from the set of possible outcomes from the beginning, that is, before the measurement. As Pessoa (1998, p. 432) puts it, “taking a partial trace amounts to the statistical version of the projection postulate.”
7 In this context we note that any nonpure density matrix can be written in many different ways, demonstrating that any partition in a particular ensemble of quantum states is arbitrary.
###/ III-B. The concept of reduced density matrices ###[#sec-III-C] A modified von Neumann measurement scheme Let us reconsider the von Neumann model for ideal quantum-mechanical measurement, Eq. , but now with the environment included. We shall denote the environment by $E$ and represent its state before the measurement interaction by the initial state vector $|e_0 \rangle$ in a Hilbert space $H_E$. As usual, let us assume that the state space of the composite object system-apparatus-environment is given by the tensor product of the individual Hilbert spaces, $H_S \otimes H_A \otimes H_E$. The linearity of the Schrödinger equation then yields the following time evolution of the entire system $SAE$, \begin{align*} \Big( \sum_n c_n |s_n \rangle \Big) |a_r \rangle |e_0 \rangle &\overset{(1)}{\rightarrow} \Big( \sum_n c_n |s_n \rangle |a_n \rangle \Big) |e_0 \rangle \\ &\overset{(2)}{\rightarrow} \sum_n c_n |s_n \rangle |a_n \rangle |e_n \rangle \end{align*} where the $|e_n \rangle$ are the states of the environment associated with the different pointer states $|a_n \rangle$ of the measuring apparatus. Note that while for two subsystems, say, $S$ and $A$, there always exists a diagonal (“Schmidt”) decomposition of the final state of the form $\sum_n c_n |s_n \rangle |a_n \rangle$, for three subsystems (for example, $S$, $A$, and $E$), a decomposition of the form $\sum_n c_n |s_n \rangle |a_n \rangle |e_n \rangle$ is not always possible. This implies that the total Hamiltonian that induces a time evolution of the above kind, Eq. , must be of a special form .
8 For an example of such a Hamiltonian, see the model of Zurek (1981, 1982) and its outline in Sec. below. For a critical comment regarding limitations on the form of the evolution operator and the possibility of a resulting disagreement with experimental evidence, see Pessoa (1998).
Typically, the $|e_n \rangle$ will be product states of many microsopic subsystem states $|\varepsilon_n \rangle_i$ corresponding to the individual parts that form the environment, i.e., $|e_n \rangle = |\varepsilon_n \rangle_1 |\varepsilon_n \rangle_2 |\varepsilon_n \rangle_3 \cdots$. We see that a nonseparable and in most cases, for all practical purposes, irreversible (due to the enormous number of degrees of freedom of the environment) correlation has been established between the states of the system-apparatus combination $SA$ and the different states of the environment $E$. Note that Eq. also implies that the environment has recorded the state of the system—and, equivalently, the state of the system-apparatus composition. The environment, composed of many subsystems, thus acts as an amplifying, higher-order measuring device. ###/ III-C. A modified von Neumann measurement scheme ###[#sec-III-D] Decoherence and local suppression of interference Interaction with the environment typically leads to a rapid vanishing of the diagonal terms in the local density matrix describing the probability distribution for the outcomes of measurements on the system. This effect has become known as environment-induced decoherence, and it has also frequently been claimed to imply at least a partial solution to the measurement problem. ####[#sec-III-D-1] General formalism In Sec. , we have already introduced the concept of local (or reduced) density matrices and pointed out some caveats on their interpretation. In the context of the decoherence program, reduced density matrices arise as follows. Any observation will typically be restricted to the system-apparatus component, $SA$, while the many degrees of freedom of the environment $E$ remain unobserved. Of course, typically some degrees of freedom of the environment will always be included in our observation (e.g., some of the photons scattered off the apparatus) and we shall accordingly include them in the “observed part $SA$ of the universe.” The crucial point is that there still remains a comparatively large number of environmental degrees of freedom that will not be observed directly. Suppose then that the operator $\hat{O}_{SA}$ represents an observable of $SA$ only. Its expectation value $\langle\hat{O}_{SA}\rangle$ is given by \langle\hat{O}_{SA}\rangle =\text{Tr} \big( \hat{\rho}_{SAE} [ \hat{O}_{SA} \otimes \hat{I}_E ] \big) =\text{Tr}_{SA}\big(\hat{\rho}_{SA}\hat{O}_{SA}\big) , where the density matrix $\hat{\rho}_{SAE}$ of the total $SAE$ combination, \hat{\rho}_{SAE} =\sum_{mn} c_m c_n^* |s_m\rangle |a_m\rangle |e_m\rangle \langle s_n | \langle a_n | \langle e_n | , has, for all purposes of statistical prediction, been replaced by the local (or reduced) density matrix $\hat{\rho}_{SA}$, obtained by “tracing out the unobserved degrees of the environment,” that is, \hat{\rho}_{SA} =\text{Tr}_E \big( \hat{\rho}_{SAE} \big) =\sum_{mn} c_m c_n^* | s_m \rangle | a_m \rangle \langle s_n | \langle a_n | \langle e_n | | e_m \rangle . So far, $\hat{\rho}_{SA}$ contains characteristic interference terms $| s_m \rangle | a_m \rangle \langle s_n | \langle a_n |$, $m \neq n$, since we cannot assume from the outset that the basis vectors $| e_m \rangle$ of the environment are necessarily mutually orthogonal, i.e., that $\langle e_n | e_m \rangle = 0$ if $m \neq n$. Many explicit physical models for the interaction of a system with the environment (see Sec. below for a simple example), however, have shown that due to the large number of subsystems that compose the environment, the pointer states $| e_n \rangle$ of the environment rapidly approach orthogonality, $\langle e_n | e_m \rangle (t) \rightarrow \delta_{n,m}$, such that the reduced density matrix $\hat{\rho}_{SA}$ becomes approximately orthogonal in the “pointer basis” $\{ | a_n \rangle \}$; that is, \begin{align*} \hat{\rho}_{SA} \overset{t}{\rightarrow} \hat{\rho}_{SA}^d &\approx \sum_n | c_n |^2 | s_n \rangle | a_n \rangle \langle s_n | \langle a_n | \\ &= \sum_n | c_n |^2 \hat{P}_n^{(S)} \otimes \hat{P}_n^{(A)} . \end{align*} Here, $\hat{P}_n^{(S)}$ and $\hat{P}_n^{(A)}$ are the projection operators onto the eigenstates of $S$ and $A$, respectively. Therefore the interference terms have vanished in this local representation, i.e., phase coherence has been locally lost. This is precisely the effect referred to as environment-induced decoherence. The decohered local density matrices describing the probability distribution of the outcomes of a measurement on the system-apparatus combination are formally (approximately) identical to the corresponding mixed-state density matrix. But as we pointed out in Sec. , we must be careful in interpreting this state of affairs, since full coherence is retained in the total density matrix $\rho_{SAE}$. ####/ III-D-1. General formalism ####[#sec-III-D-2] An exactly solvable two-state model for decoherence To see how the approximate mutual orthogonality of the environmental state vectors arises, let us discuss a simple model first introduced by Zurek (1982). Consider a system $S$ with two spin states $\{ | \Uparrow \rangle , | \Downarrow \rangle \}$ that interacts with an environment $E$ described by a collection of $N$ other two-state spins represented by $\{ | \uparrow_k \rangle , | \downarrow_k \rangle \}$, $k=1 \cdots N$. The self-Hamiltonians $\hat{H}_S$ and $\hat{H}_E$ and the self-interaction Hamiltonian $\hat{H}_{EE}$ of the environment are taken to be equal to zero. Only the interaction Hamiltonian $\hat{H}_{SE}$ that describes the coupling of the spin of the system to the spins of the environment is assumed to be nonzero and of the form \begin{align*} \hat{H}_{SE} =&\Big( | \Uparrow \rangle \langle \Uparrow | - | \Downarrow \rangle \langle \Downarrow | \Big) \otimes \sum_k g_k \Big( | \uparrow_k \rangle \langle \uparrow_k | - | \downarrow_k \rangle \langle \downarrow_k | \Big) \\ &\underset{k' \neq k}{\otimes} \hat{I}_{k'} , \end{align*} where the $g_k$ are coupling constants and $\hat{I}_{k} = \Big( | \uparrow_k \rangle \langle \uparrow_k | + | \downarrow_k \rangle \langle \downarrow_k | \Big)$ is the identity operator for the $k$th environmental spin. Applied to the initial state before the interaction is turned on, | \psi (0) \rangle = \Big( a | \Uparrow \rangle + b | \Downarrow \rangle \Big) \overset{N}{\underset{k=1}{\otimes}} \Big( \alpha_k | \uparrow_k \rangle + \beta_k | \downarrow_k \rangle \Big) , this Hamiltonian yields a time evolution of the state given by | \psi (t) \rangle = a | \Uparrow \rangle | E_{\Uparrow} (t) \rangle + b | \Downarrow \rangle | E_{\Downarrow} (t) \rangle , where the two environmental states $| E_{\Uparrow} (t) \rangle$ and $| E_{\Downarrow} (t) \rangle$ are | E_{\Uparrow} (t) \rangle = | E_{\Downarrow} (-t) \rangle = \overset{N}{\underset{k=1}{\otimes}} \Big( \alpha_k e^{i g_k t} | \uparrow_k \rangle + \beta_k e^{-i g_k t} | \downarrow_k \rangle \Big) . The reduced density matrix $\rho_{S} (t) = \text{Tr} \big( | \psi (t) \rangle \langle \psi (t) | \big)$ is then \begin{align*} \rho_S (t) =&|a|^2 |\Uparrow\rangle\langle\Uparrow | + |b|^2 |\Downarrow\rangle\langle\Downarrow | + z(t) ab^* |\Uparrow\rangle\langle\Downarrow | \\ &+ z^* (t) a^* b \Downarrow\rangle\langle\Uparrow | , \end{align*} where the interference coefficient $z(t)$ which determines the weight of the off-diagonal elements in the reduced density matrix is given by z(t) = \langle E_{\Uparrow} (t) | E_{\Downarrow} (t) \rangle = \prod_{k=1}^{N} \Big( |\alpha_k|^2 e^{i g_k t} + |\beta_k|^2 e^{-i g_k t} \Big) , and thus | z(t) |^2 = \prod_{k=1}^{N} \Big\{ 1+[ ( |\alpha_k |^2 - |\beta_k |^2 )^2 - 1 ] \sin^2 2 g_k t \Big\} . At $t=0$, $z(t)=1$, i.e., the interference terms are fully present, as expected. If $|\alpha_k|^2 = 0$ or $1$ for each $k$, i.e., if the environment is in an eigenstate of the interaction Hamiltonian $\hat{H}_{SE}$ of the type $|\uparrow_1 \rangle |\uparrow_2 \rangle |\downarrow_2 \rangle \cdots |\uparrow_N \rangle$, and/or if $2 g_k t = m \pi ~ (m=0,1,\dots)$, then $z(t)^2 \equiv 1$, so coherence is retained over time. However, under realistic circumstances, we can typically assume a random distribution of the initial states of the environment (i.e., of coefficients $\alpha_k$, $\beta_k$) and of the coupling coefficients $g_k$. Then, in the long-time average, \langle |z(t)|^2 \rangle_{t \rightarrow \infty} \approx 2^{-N} \prod_{k=1}^{N} \big[ 1+(|\alpha_k |^2 - |\beta_k |^2 )^2 \big] \overset{N\rightarrow\infty}{\rightarrow} 0 , so the off-diagonal terms in the reduced density matrix become strongly damped for large $N$. It can also be shown directly that, given very general assumptions about the distribution of the couplings $g_k$ (namely, requiring their initial distribution to have finite variance), $z(t)$ exhibits a Gaussian time dependence of the form $z(t) \propto e^{iAt} e^{-B^2 t^2 /2}$, where $A$ and $B$ are real constants (Zurek et al., 2003). For the special case in which $\alpha_k = \alpha$ and $g_k = g$ for all $k$, this behavior of $z(t)$ can be immediately seen by first rewriting $z(t)$ as the binomial expansion z(t) = \big( |\alpha|^2 e^{igt} + |\beta|^2 e^{-igt} \big)^N = \sum_{l=0}^{N} \binom{N}{l} |\alpha|^{2l} |\beta|^{2(N-l)} e^{ig (2l-N)t} . For large $N$, the binomial distribution can then be approximated by a Gaussian, \binom{N}{l} |\alpha|^{2l} |\beta|^{2(N-l)} \approx \frac{e^{-(l-N|\alpha|^2)^2 / (2N |\alpha|^2 |\beta|^2)}}{\sqrt{2\pi N |\alpha|^2 |\beta|^2}} , in which case $z(t)$ becomes z(t) = \sum_{l=0}^{N} \frac{e^{-(l-N|\alpha|^2)^2 / (2N |\alpha|^2 |\beta|^2)}}{\sqrt{2\pi N |\alpha|^2 |\beta|^2}} e^{ig(2l-N)t} , that is, $z(t)$ is the Fourier transform of an (approximately) Gaussian distribution and is therefore itself (approximately) Gaussian. Detailed model calculations, in which the environment is typically represented by a more sophisticated model consisting of a collection of harmonic oscillators (Caldeira and Leggett, 1983; Unruh and Zurek, 1989; Hu et al., 1992; Zurek et al., 1993; Joos et al., 2003; Zurek, 2003a), have shown that the damping occurs on extremely short decoherence time scales $\tau_D$, which are typically many orders of magnitude shorter than the thermal relaxation. Even microscopic systems such as large molecules are rapidly decohered by the interaction with thermal radiation on a time scale that is much shorter than any practical observation could resolve; for mesoscopic systems such as dust particles, the 3K cosmic microwave background radiation is sufficient to yield strong and immediate decoherence (Joos and Zeh, 1985; Zurek, 1991). Within $\tau_D$, $|z(t)|$ approaches zero and remains close to zero, fluctuating with an average standard deviation of the random-walk-type $\sigma \sim \sqrt{N}$ (Zurek, 1982). However, the multiple periodicity of $z(t)$ implies that coherence, and thus the purity of the reduced density matrix, will reappear after a certain time $\tau_r$, which can be shown to be very long and of the Poincaré-type with $\tau_r \sim N!$. For macroscopic environments of realistic but finite sizes, $\tau_r$ can exceed the lifetime of the universe (Zurek, 1982), but nevertheless always remains finite. From a conceptual point of view, recurrence of coherence is of little relevance. The recurrence time could only be infinitely long in the hypothetical case of an infinitely large environment. In this situation, off-diagonal terms in the reduced density matrix would be irreversibly damped and lost in the limit $t\rightarrow\infty$, which has sometimes been regarded as describing a physical collapse of the state vector (Hepp, 1972). But the assumption of infinite sizes and times is never realized in nature (Bell, 1975), nor can information ever be truly lost (as achieved by a “true” state vector collapse) through unitary time evolution—full coherence is always retained at all times in the total density matrix $\rho_{SAE} (t) = |\psi(t)\rangle\langle\psi(t)|$. We can therefore state the general conclusion that, except for exceptionally well-isolated and carefully prepared microscopic and mesoscopic systems, the interaction 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 distribution of the possible outcomes of a measurement on the system, to become extremely small in a very short period of time, and this process is irreversible for all practical purposes. ####/ III-D-2. An exactly solvable two-state model for decoherence ###/ III-D. Decoherence and local suppression of interference ###[#sec-III-E] Environment-induced superselection Let us now turn to the second main consequence of the interaction with the environment, namely, the environment-induced selection of stable preferred-basis states. We discussed in Sec. the fact that the quantum-mechanical measurement scheme as represented by Eq. does not uniquely define the expansion of the postmeasurement state and thereby leaves open the question of which observable can be considered as having been measured by the apparatus. This situation is changed by the inclusion of the environment states in Eq. for the following two reasons:
1. Environment-induced superselection of a preferred basis. The interaction between the apparatus and the environment singles out a set of mutually commuting observables.
2. The existence of a tridecompositional uniqueness theorem (Elby and Bub, 1994; Clifton, 1995; Bub, 1997). If a state $|\psi\rangle$ in a Hilbert space $H_1 \otimes H_2 \otimes H_3$ can be decomposed into the diagonal (“Schmidt”) form $|\psi\rangle = \sum_i \alpha_i |\phi_i\rangle_1 |\phi_i\rangle_2 |\phi_i\rangle_3$, the expansion is unique provided that the $\{|\phi_i\rangle_1\}$ and $\{|\phi_i\rangle_2\}$ are sets of linearly independent, normalized vectors in $H_1$ and $H_2$, respectively, and that $\{|\phi_i\rangle_3\}$ is a set of mutually noncollinear normalized vectors in $H_3$. This can be generalized to an N-decompositional uniqueness theorem, in which $N\geq3$. Note that it is not always possible to decompose an arbitrary pure state of more than two systems ($N\geq3$) into the Schmidt form $|\psi\rangle = \sum_i \alpha_i |\phi_i\rangle_1 |\phi_i\rangle_2 \cdots |\phi_i\rangle_N$, but if the decomposition exists, its uniqueness is guaranteed.
The tridecompositional uniqueness theorem ensures that the expansion of the final state in Eq. is unique, which fixes the ambiguity in the choice of the set of possible outcomes. It demonstrates that the inclusion of (at least) a third “system” (here referred to as the environment) is necessary to remove the basis ambiguity. Of course, given any pure state in the composite Hilbert space $H_1 \otimes H_2 \otimes H_3$, the tridecompositional uniqueness theorem neither tells us whether a Schmidt decomposition exists nor specifies the unique expansion itself (provided the decomposition is possible), and since the precise states of the environment are generally not known, an additional criterion is needed that determines what the preferred states will be. ####[#sec-III-E-1] Stability criterion and pointer basis 12 page Editting... ####/ III-E-1. Stability criterion and pointer basis ####[#sec-III-E-2] Selection of quasiclassical properties ####/ III-E-2. Selection of quasiclassical properties ####[#sec-III-E-3] Implications for the preferred-basis problem ####/ III-E-3. Implications for the preferred-basis problem ####[#sec-III-E-4] Pointer basis vs instantaneous Schmidt states ####/ III-E-4. Pointer basis vs instantaneous Schmidt states ###/ III-E. Environment-induced superselection ###[#sec-III-F] Envariance, quantum probabilities, and the Born rule ####[#sec-III-F-1] Environment-assisted invariance ####/ III-F-1. Environment-assisted invariance ####[#sec-III-F-2] Deducing the Born rule ####/ III-F-2. Deducing the Born rule ####[#sec-III-F-3] Summary and outlook ####/ III-F-3. Summary and outlook ###/ III-F. Envariance, quantum probabilities, and the Born rule ##[#sec-IV] The Role of Decoherence in Interpretations of Quantum Mechanics ###[#sec-IV-A] General implications of decoherence for interpretations ###/ IV-A. General implications of decoherence for interpretations ###[#sec-IV-B] The standard and the Copenhagen interpretations ####[#sec-IV-B-1] The problem of definite outcomes ####/ IV-B-1. The problem of definite outcomes ####[#sec-IV-B-2] Observables, measurements, and environment-induced superselection ####/ IV-B-2. Observables, measurements, and environment-induced superselection ####[#sec-IV-B-3] The concept of classicality in the Copenhagen interpretation ####/ IV-B-3. The concept of classicality in the Copenhagen interpretation ###/ IV-B. The standard and the Copenhagen interpretations ###[#sec-IV-C] Relative-state interpretations ####[#sec-IV-C-1] Everett branches and the preferred-basis problem ####/ IV-C-1. Everett branches and the preferred-basis problem ####[#sec-IV-C-2] Probabilities in Everett interpretations ####/ IV-C-2. Probabilities in Everett interpretations ####[#sec-IV-C-3] The “existential interpretation” ####/ IV-C-3. The “existential interpretation” ###/ IV-C. Relative-state interpretations ###[#sec-IV-D] Modal interpretations ####[#sec-IV-D-1] Property assignment based on environment-induced superselection ####/ IV-D-1. Property assignment based on environment-induced superselection ####[#sec-IV-D-2] Property assignment based on instantaneous Schmidt decompositions ####/ IV-D-2. Property assignment based on instantaneous Schmidt decompositions ####[#sec-IV-D-3] Property assignment based on decompositions of the decohered density matrix ####/ IV-D-3. Property assignment based on decompositions of the decohered density matrix ####[#sec-IV-D-4] Concluding remarks ####/ IV-D-4. Concluding remarks ###/ IV-D. Modal interpretations ###[#sec-IV-E] Physical collapse theories ####[#sec-IV-E-1] The preferred-basis problem ####/ IV-E-1. The preferred-basis problem ####[#sec-IV-E-2] Simultaneous presence of decoherence and spontaneous localization ####/ IV-E-2. Simultaneous presence of decoherence and spontaneous localization ####[#sec-IV-E-3] The tails problem ####/ IV-E-3. The tails problem ####[#sec-IV-E-4] Connecting decoherence and collapse models ####/ IV-E-4. Connecting decoherence and collapse models ####[#sec-IV-E-5] Summary and outlook ####/ IV-E-5. Summary and outlook ###/ IV-E. Physical collapse theories ###[#sec-IV-F] Bohmian mechanics ####[#sec-IV-F-1] Particles as fundamental entities ####[#sec-IV-F-2] Bohmian trajectories and decoherence ###[#sec-IV-G] Consistent-histories interpretations ####[#sec-IV-G-1] Definition of histories ####[#sec-IV-G-2] Probabilities and consistency ####[#sec-IV-G-3] Selection of histories and classicality ####[#sec-IV-G-4] Consistent histories of open systems ####[#sec-IV-G-5] Schmidt states vs pointer basis as projectors ####[#sec-IV-G-6] Exact vs approximate consistency ####[#sec-IV-G-7] Consistency and environment-induced superselection ####[#sec-IV-G-8] Summary and discussion ##[#sec-V] Concluding Remarks We have presented an extensive discussion of the role of decoherence in the foundations of quantum mechanics, with a particular focus on the implications of decoherence for the measurement problem in the context of various interpretations of quantum mechanics. A key achievement of the decoherence program is the recognition that openness in quantum systems is important for their realistic description. The well-known phenomenon of quantum entanglement had already, early in the history of quantum mechanics, demonstrated that correlations between systems can lead to “paradoxical” properties of the composite system that cannot be composed from the properties of the individual systems. Nonetheless, the viewpoint of classical physics that the idealization of isolated systems is necessary to arrive at an “exact description” of physical systems has influenced quantum theory for a long time. It is the great merit of the decoherence program to have emphasized the ubiquity and essential inescapability of system-environment correlations and to have established the important role of such correlations as factors in the emergence of “classicality” from quantum systems. Decoherence also provides a realistic physical modeling and a generalization of the quantum measurement process, thus enhancing the “black-box” view of measurements in the standard (“orthodox”) interpretation and challenging the postulate of fundamentally classical measuring devices in the Copenhagen interpretation. With respect to the preferred-basis problem of quantum measurement, decoherence provides a very promising definition of preferred pointer states via a physically meaningful requirement, namely, the robustness criterion, and it describes methods for selecting operationally such states, for example, via the commutativity criterion or by extremizing an appropriate measure such as purity or von Neumann entropy. In particular, the fact that macroscopic systems virtually always decohere into position eigenstates gives a physical explanation for why position is the ubiquitous determinate property of the world of our experience. We have argued that, within the standard interpretation of quantum mechanics, decoherence cannot solve the problem of definite outcomes in quantum measurement: We are still left with a multitude of salbeit individually well-localized quasiclassicald components of the wave function, and we need to supplement or otherwise to interpret this situation in order to explain why and how single outcomes are perceived. Accordingly, we have discussed how environment-induced superselection of quasiclassical pointer states together with the local suppression of interference terms can be put to great use in physically motivating, or potentially disproving, rules and assumptions of alternative interpretive approaches that change (or altogether abandon) the strict orthodox eigenvalue-eigenstate link and/or modify the unitary dynamics to account for the perception of definite outcomes and classicality in general. For example, to name just a few applications, decoherence can provide a universal criterion for the selection of the branches in relative-state interpretations and a physical argument for the noninterference between these branches from the point of view of an observer; in modal interpretations, it can be used to specify empirically adequate sets of properties that can be ascribed to systems; in collapse models, the free parameters (and possibly even the nature of the reduction mechanism itself) might be derivable from environmental interactions; decoherence can also assist in the selection of quasiclassical particle trajectories in Bohmian mechanics, and it can serve as an efficient mechanism for singling out quasiclassical histories in the consistent-histories approach. Moreover, it has become clear that decoherence can ensure the empirical adequacy and thus empirical equivalence of different interpretive approaches, which has led some to the claim that the choice, for example, between the orthodox and the Everett interpretation becomes “purely a matter of taste, roughly equivalent to whether one believes mathematical language or human language to be more fundamental” (Tegmark, 1998, p. 855). It is fair to say that the decoherence program sheds new light on many foundational aspects of quantum mechanics. It paves a physics-based path towards motivating solutions to the measurement problem; it imposes constraints on the strands of interpretations that seek such a solution and thus makes them also more and more similar to each other. Decoherence remains an ongoing field of intense research, in both the theoretical and experimental domain, and we can expect further implications for the foundations of quantum mechanics from such studies in the near future. ##/ V. Concluding Remarks ##[.no-sec-N] Acknowledgments The author would like to thank A. Fine for many valuable discussions and comments throughout the process of writing this article. He gratefully acknowledges thoughtful and extensive feedback on this manuscript from S. L. Adler, V. Chaloupka, H.-D. Zeh, and W. H. Zurek. ##/ Acknowledgments ## RRA
1. Reviews of Modern Physics, volume 76, 1267 (2005), or ArXiv quantph/0312059 - Decoherence, the measurement problem, and interpretations of quantum mechanics, by Maximilian Schlosshauer

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