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REVIEWS OF MODERN PHYSICS, VOLUME 76, OCTOBER 2004 , Decoherence, the measurement problem, and interpretations of quantum mechanics
by Maximilian SchlosshauerDepartment of Physics, University of Washington, Seattle, Washington 98195, USA

Electronic address: MAXL@u.washington.edu

~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.

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.
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
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.
2 Broadly speaking, this means that the (quantum-mechanical) whole is different from the sum of its parts.

The decoherence program
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:
- 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. . - 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.
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.
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,
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
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.
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. - Environment-induced superselection of a preferred basis. The interaction between the apparatus and the environment singles out a set of mutually commuting observables.
- 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.

- 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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