Decoherence, the Measurement Problem, and Interpretations of Quantum Mechanics 0312059.pdf

Decoherence, the Measurement Problem, and

Interpretations of Quantum Mechanics

Maximilian Schlosshauer ∗

Department of Physics, University of Washington, Seattle, Washington 98195

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.

Contents

3. Property ascription based on decompositions of the

decohered density matrix

I. Introduction

4. Concluding remarks

E. Physical collapse theories

II. The measurement problem 3

A. Quantum measurement scheme 3

B. The problem of denite outcomes 4

1. Superpositions and ensembles 4

2. Superpositions and outcome attribution 4

3. Objective vs. subjective deniteness 5

C. The preferred basis problem 6

D. The quantum–to–classical transition and decoherence 6

1. The preferred basis problem

2. Simultaneous presence of decoherence and

spontaneous localization

3. The tails problem

4. Connecting decoherence and collapse models

5. Summary and outlook

F. Bohmian Mechanics

1. Particles as fundamental entities

III. The decoherence program 7

A. Resolution into subsystems 8

B. The concept of reduced density matrices 9

C. A modied von Neumann measurement scheme 9

D. Decoherence and local suppression of interference 10

1. General formalism 10

2. An exactly solvable two-state model for decoherence 11

E. Environment-induced superselection 12

1. Stability criterion and pointer basis 12

2. Selection of quasiclassical properties 13

3. Implications for the preferred basis problem 14

4. Pointer basis vs. instantaneous Schmidt states 15

F. Envariance, quantum probabilities and the Born rule 16

1. Environment-assisted invariance

2. Bohmian trajectories and decoherence

G. Consistent histories interpretations

1. Denition of histories

2. Probabilities and consistency

3. Selection of histories and classicality

4. Consistent histories of open systems

5. Schmidt states vs. pointer basis as projectors

6. Exact vs. approximate consistency

7. Consistency and environment-induced

superselection

8. Summary and discussion

V. Concluding remarks

Acknowledgments

2. Deducing the Born rule

3. Summary and outlook

References

IV. The role of decoherence in interpretations of

quantum mechanics 18

A. General implications of decoherence for interpretations 19

B. The Standard and the Copenhagen interpretation

I. INTRODUCTION

1. The problem of denite outcomes

2. Observables, measurements, and

environment-induced superselection

The implications of the decoherence program for the

foundations of quantum mechanics have been subject of

an ongoing debate since the rst precise formulation of

the program in the early 1980s. The key idea promoted

by decoherence is based on the insight that realistic quan-

tum systems are never isolated, but are immersed into

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 into the mathematical theory

or its interpretation), the resulting formation of quan-

tum correlations between the states of the system and

its environment and the often surprising eects of these

system–environment interactions. In short, decoherence

brings about a local suppression of interference between

3. The concept of classicality in the Copenhagen

interpretation 21

C. Relative-state interpretations 22

1. Everett branches and the preferred basis problem 23

2. Probabilities in Everett interpretations

3. The “existential interpretation”

D. Modal interpretations

1. Property ascription based on environment-induced

superselection

2. Property ascription based on instantaneous Schmidt

decompositions

Electronic address: MAXL@u.washington.edu

preferred states selected by the interaction with the en-

vironment.

Bub ( 1997 ) termed decoherence part of the “new or-

thodoxy” 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 ,

1999 , p. 13) that the implications for quantum mechan-

ics and for the associated foundational problems were

overlooked for so long. Zurek ( 2003a , p. 717) suggests:

existence” [from decoherence] (. . . ) signicantly

reduces and perhaps even eliminates the role of

the “collapse” of the state vector.

D’Espagnat, who advocates a view that considers the

explanation of our experiences (i.e., the “appearances”)

as the only “sure” demand for a physical theory, states

( d’Espagnat , 2000 , p. 136):

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 eects

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 ob-

serve (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.

The idea that the “openness” of quantum sys-

tems might have anything to do with the transi-

tion from quantum to classical was ignored for

a very long time, probably because in classi-

cal physics problems of fundamental importance

were always settled in isolated systems.

When the concept of decoherence was rst introduced

to the broader scientic audience by Zurek’s ( 1991 ) ar-

ticle that appeared in Physics Today, it sparked a series

of controversial comments from the readership (see the

April 1993 issue of Physics Today). In response to crit-

ics, Zurek ( 2003a , p. 718) states:

In his monumental book on the foundations of quantum

mechanics, Auletta ( 2000 , p. 791) concludes that

In a eld where controversy has reigned for so

long this resistance to a new paradigm [namely,

to decoherence] is no surprise.

the Measurement theory could be part of the in-

terpretation of QM only to the extent that it

would still be an open problem, and we think

that this largely no longer the case.

Omnes ( 2003 , p. 2) assesses:

This is mainly so because, so Auletta (p. 289),

The discovery of decoherence has already much

improved our understanding of quantum mechan-

ics. (. . . ) [B]ut its foundation, the range of its

validity and its full meaning are still rather ob-

scure. This is due most probably to the fact that

it deals with deep aspects of physics, not yet fully

investigated.

decoherence is able to solve practically all the

problems of Measurement which have been dis-

cussed in the previous chapters.

On the other hand, even leading adherents of decoher-

ence expressed caution in expecting that decoherence has

solved the measurement problem. Joos ( 1999 , p. 14)

writes:

In particular, the question whether decoherence provides,

or at least suggests, a solution to the measurement prob-

lem of quantum mechanics has been discussed for several

years. For example, Anderson ( 2001 , p. 492) writes in an

essay review:

Does decoherence solve the measurement prob-

lem? 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 prob-

ability rules of quantum theory.

The last chapter (. . . ) deals with the quantum

measurement problem (. . . ). My main test, al-

lowing 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 func-

tion”.

Along these lines, Kiefer and Joos ( 1998 , p. 5) warn that:

One often nds explicit or implicit statements to

the eect that the above processes are equivalent

to the collapse of the wave function (or even solve

the measurement problem). Such statements are

certainly unfounded.

Zurek speaks in various places of the “apparent” or “ef-

fective” collapse of the wave function induced by the in-

teraction with environment (when embedded into a min-

imal additional interpretive framework), and concludes

( Zurek , 1998 , p. 1793):

In a response to Anderson’s ( 2001 , p. 492) comment,

Adler ( 2003 , p. 136) states:

I do not believe that either detailed theoretical

calculations or recent experimental results show

that decoherence has resolved the di culties as-

sociated with quantum measurement theory.

A “collapse” in the traditional sense is no longer

necessary. (. . . ) [The] emergence of “objective

Similarly, Bacciagaluppi ( 2003b , p. 3) writes:

in some form addressed by the decoherence program. In

Sec. III , we then introduce and discuss the main features

of the theory of decoherence, with a particular emphasis

on their foundational implications. Finally, in Sec. IV ,

we investigate the role of decoherence in various inter-

pretive approaches of quantum mechanics, in particular

with respect to the ability to motivate and support (or

falsify) possible solutions to the measurement problem.

Claims that simultaneously the measurement

problem is real [and] decoherence solves it are

confused at best.

Zeh asserts ( Joos et al. , 2003 , Ch. 2):

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 Schrodinger dynamics

(. . . ) is supplemented by a nonunitary collapse

(. . . ).

II. THE MEASUREMENT PROBLEM

One of the most revolutionary elements introduced into

physical theory by quantum mechanics is the superposi-

tion principle, mathematically founded in the linearity of

the Hilbert state space. If|1and|2are two states, then

quantum mechanics tells us that also any linear combina-

tion α|1+β|2corresponds to a possible state. Whereas

such superpositions of states have been experimentally

extensively veried for microscopic systems (for instance

through the observation of interference eects), the appli-

cation of the formalism to macroscopic systems appears

to lead immediately to severe clashes with our experience

of the everyday world. Neither has a book ever observed

to be in a state of being both “here” and “there” (i.e., to

be in a superposition of macroscopically distinguishable

positions), nor seems a Schrodinger cat that is a superpo-

sition of being alive and dead to 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 comparably few “classi-

cal” macrosopic states, dened 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 that

would in principle allow for arbitrary superpositions?

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 diago-

nality of [the density matrix] in some basis has

been occasionally (mis-) interpreted as a key ac-

complishment of decoherence. This is mislead-

ing. Any density matrix is diagonal in some ba-

sis. This has little bearing on the interpretation.

These controversial remarks show that a balanced discus-

sion of the key features of decoherence and their implica-

tions for the foundations of quantum mechanics is over-

due. The decoherence program has made great progress

over the past decade, and it would be inappropriate to ig-

nore its relevance in tackling conceptual problems. How-

ever, it is equally important to realize the limitations of

decoherence in providing consistent and noncircular an-

swers to foundational questions.

An excellent review of the decoherence program has re-

cently been given by Zurek ( 2003a ). It dominantly deals

with the technicalities of decoherence, although it con-

tains some discussion on how decoherence can be em-

ployed in the context of a relative-state interpretation to

motivate basic postulates of quantum mechanics. Use-

ful for a helpful rst orientation and overview, the entry

by Bacciagaluppi ( 2003a ) in the Stanford Encyclopedia of

Philosophy features an (in comparison to the present pa-

per relatively short) introduction to the role of decoher-

ence 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 litera-

ture, a detailed and self-contained discussion of the role

of decoherence in the foundations of quantum mechanics

seems still outstanding. This review article is intended

to ll the gap.

To set the stage, we shall rst, in Sec. II , review the

measurement problem, which illustrates the key di cul-

ties that are associated with describing quantum mea-

surement within the quantum formalism and that are all

A. Quantum measurement scheme

This question is usually illustrated in the context

of quantum measurement where microscopic superposi-

tions are, via quantum entanglement, amplied 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 ),

a (typically microscopic) systemS, represented by basis

vectors{|s n

are

assumed to correspond to macroscopically distinguish-

able “pointer” positions that correspond to the outcome

of a measurement ifSis in the state|s n . 1

A , where the|a n

1 Note that von Neumann’s scheme is in sharp contrast to the

Copenhagen interpretation, where measurement is not treated

S , interacts with

a measurement apparatusA, described by basis vectors

{|a n

}in a Hilbert state spaceH

}spanning a Hilbert spaceH

, andAis in the initial “ready”

state|a r , the linearity of the Schrodinger equation en-

tails that the total systemSA, assumed to be represented

by the Hilbert product spaceH S ⊗H A , evolves according

n c n

|s n

This can explicitely be shown especially on microscopic

scales by performing experiments that lead to the direct

observation of interference patterns instead of the real-

ization 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 it 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 (ψ 1 or ψ 2 ), but only

by the superposition of these wave functions (ψ 1 + ψ 2 ),

since the correct density distribution ̺ 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 (̺ = |ψ 1

c n |s n

|a r t −→

c n |s n |a n . (2.1)

This dynamical evolution is often referred to as a pre-

measurement in order to emphasize that the process de-

scribed by Eq. ( 2.1 ) does not su ce to directly conclude

that a measurement has actually been completed. This is

so for two reasons. First, the right-hand side is a super-

position of system–apparatus states. Thus, without sup-

plying 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 -

nal composite state, for the denite 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 but not in a superposition of positions

(problem of denite outcomes)? Second, the expansion

of the nal composite state is in general not unique, and

therefore the measured observable is not uniquely dened

either (problem of the preferred basis). The rst di culty

is in the literature typically referred to as the measure-

ment problem, but the preferred basis problem is at least

equally important, since it does not make sense to even

inquire about specic outcomes if the set of possible out-

comes is not clearly dened. We shall therefore regard

the measurement problem as composed of both the prob-

lem of denite outcomes and the problem of the preferred

basis, and discuss these components in more detail in the

following.

| 2 ), but only by

the square of the sum of the individual wave functions

(̺ =|ψ 1 + ψ 2

+|ψ 2

| 2 ).

Put dierently, if an ensemble interpretation could be

attached to a superposition, the latter would simply rep-

resent 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 denite pointer

state obtained in the measurement. But then, since the

time evolution has been strictly deterministic according

to the Schrodinger equation, we could backtrack this

subensemble in time und thus also specify the initial

state more completely (“post-selection”), and therefore

this state necessarily could not be physically identical

to the initially prepared state on the left-hand side of

Eq. ( 2.1 ) .

2. Superpositions and outcome attribution

In the Standard (“orthodox”) interpretation of quan-

tum mechanics, an observable corresponding to a physi-

cal quantity has a denite value if and only if the system

is in an eigenstate of the observable; if the system is how-

ever in a superposition of such eigenstates, as in Eq. ( 2.1 ) ,

it is, according to the orthodox interpretation, meaning-

less to speak of the state of the system as having any

denite 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 me-

chanics or by empirical constraints ( Bub , 1997 ) . The con-

cept 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 fre-

quently 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 opera-

tor, and the pointer states are never exactly mutually

orthogonal. One might then (explicitely or implicitely)

promote a “fuzzy” e–e link, or give up the concept of

observables and values entirely and directly interpret the

B. The problem of denite outcomes

1. Superpositions and ensembles

The right-hand side of Eq. ( 2.1 ) implies that after the

premeasurement the combined systemSAis left in a pure

state that represents a linear superposition of system–

pointer states. It is a well-known and important prop-

erty of quantum mechanics that a superposition of states

is fundamentally dierent 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).

as a system–apparatus interaction described by the usual quan-

tum mechanical formalism, but instead as an independent com-

ponent of the theory, to be represented entirely in fundamentally

classical terms.

Now, ifSis in a (microscopically “unproblematic”)

superposition

| 2

time-evolved wave functions (working in the Schrodinger

picture) and the corresponding density matrices. Also,

if it is regarded as su cient to explain our perceptions

rather than describe the “absolute” state of the entire

universe (see the argument below), one might only re-

quire that the (exact or fuzzy) e–e link holds in a “rela-

tive” 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 denite outcomes, some

interpretations (for example, modal interpretations and

relative-state interpretations) interpret the nal-state su-

perposition in such a way as to explain the existence, or

at least the subjective perception, of “outcomes” even if

the nal composite state has the form of a superposition.

Other interpretations attempt to solve the measurement

problem by modifying the strictly unitary Schrodinger

dynamics. Most prominently, the orthodox interpreta-

tion 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 into the

Schrodinger equation that induce an eective (albeit only

approximate) collapse for states of macroscopic systems

( Ghirardi et al. , 1986 ; Gisin , 1984 ; Pearle , 1979 , 1999 ) ,

while other authors suggested that collapse occurs at the

level of the mind of a conscious observer ( Stapp , 1993 ;

Wigner , 1963 ). Bohmian mechanics, on the other hand,

upholds a unitary time evolution of the wavefunction, but

introduces an additional dynamical law that explicitely

governs the always determinate positions of all particles

in the system.

to a satisfactory solution to the measurement problem.

We demand objective deniteness because we experience

deniteness on the subjective level of observation, and it

shall not be viewed as an a priori requirement for a phys-

ical theory. If we knew independently of our experience

that deniteness exists in nature, subjective deniteness

would presumably follow as soon as we have employed a

simple model that connects the “external” physical phe-

nomena with our “internal” perceptual and cognitive ap-

paratus, where the expected simplicity of such a model

can be justied by referring to the presumed identity of

the physical laws governing external and internal pro-

cesses. But since knowledge is based on experience, that

is, on observation, the existence of objective deniteness

could only be derived from the observation of denite-

ness. And moreover, observation tells us that deniteness

is in fact not a universal property of nature, but rather a

property of macroscopic objects, where the borderline to

the macroscopic realm is di cult to draw precisely; meso-

scopic interference experiments demonstrated clearly the

blurriness of the boundary. Given the lack of a precise

denition of the boundary, any demand for fundamen-

tal deniteness on the objective level should be based

on a much deeper and more general commitment to a

deniteness that applies to every physical entity (or sys-

tem) across the board, regardless of spatial size, physical

property, and the like.

Therefore, if we realize that the often deeply felt com-

mitment to a general objective deniteness is only based

on our experience of macroscopic systems, and that this

deniteness in fact fails in an observable manner for mi-

croscopic and even certain mesoscopic systems, the au-

thor sees no compelling grounds on which objective de-

niteness must be demanded as part of a satisfactory phys-

ical theory, provided that the theory can account for sub-

jective, observational deniteness in agreement with our

experience. Thus the author suggests to attribute the

same legitimacy to proposals for a solution of the mea-

surement problem that achieve “only” subjective but not

objective deniteness—after all the measurement prob-

lem arises solely from a clash of our experience with cer-

tain implications of the quantum formalism. D’Espagnat

( 2000 , pp. 134–135) has advocated a similar viewpoint:

3. Objective vs. subjective deniteness

In general, (macroscopic) deniteness—and thus a so-

lution to the problem of outcomes in the theory of quan-

tum measurement—can be achieved either on an onto-

logical (objective) or an observational (subjective) level.

Objective deniteness aims at ensuring “actual” denite-

ness in the macroscopic realm, whereas subjective de-

niteness only attempts to explain why the macroscopic

world appears to be denite—and thus does not make

any claims about deniteness of the underlying physi-

cal reality (whatever this reality might be). This raises

the question of the signicance 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, deniteness is merely good “for all prac-

tical purposes”—abbreviated, rather disparagingly, as

“FAPP” by Bell ( 1990 ) —, and thus not capable of solv-

ing the “fundamental” problem that would seem relevant

to the construction of the “precise theory” that Bell de-

manded so vehemently.

It seems to the author, however, that this critism is

not justied, and that subjective deniteness should be

viewed on a par with objective denitess with respect

The fact that we perceive such “things” as macro-

scopic objects lying at distinct places is due,

partly at least, to the structure of our sensory and

intellectual equipment. We should not, there-

fore, take it as being part of the body of sure

knowledge that we have to take into account for

dening a quantum state. (. . . ) In fact, scien-

tists most righly 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 ob-

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