Essays
by H.D. Zeh
by H.D. Zeh
by E. Joos
How decoherence can
solve
the measurement problem
Decoherence may be defined as the uncontrollable dislocalization of
quantum
mechanical
superpositions. It is an unavoidable
consequence of
the interaction
of all local systems with their environments
according to the
Schrödinger equation. Since the dislocalization
propagates in
general without
bounds, this concept of decoherence
does not depend on any precise
boundaries between subsystems. All
systems should
be
entangled
with their growing environments, and generically cannot
possess
quantum
states by their own. They may then formally be described by a
reduced
density matrix ρ
representing
a "mixed state", with a von Neumann entropy -trace(ρ
lnρ) that
in
general varies in time. This reduced density matrix
is
operationally
indistinguishable from that describing an ensemble
of states – as
though a subsystem state did exist but were only
incompletely
known.
For this reason, it is often erroneously
identified with an ensemble.
Since the
dynamical situation of increasing entanglement
applies in particular
to systems representing macroscopic outcomes of
quantum measurements
("pointer positions"), decoherence has
occasionally
been
claimed
to
explain
the
probabilistic nature of quantum mechanics (quantum
indeterminism).
However,
such a conclusion would evidently contradict
the determinism of
the thereby presumed unitary global dynamics.
(Note that
the claim – if correct – requires decoherence
to be
irreversible, as the
measurement could otherwise be
undone
or
"erased" – see Quantum
teleportation
and
other quantum
misnomers). Although the claim is
operationally
unassailable, it is
wrong. The very
concept of a
density
matrix is already based on local
operations (measurements) which
presume the probability
interpretation, while the
global quantum state
always remains pure and uniquely
determined under the exact
unitary dynamics.
Because
of this popular "naive" misinterpretation of decoherence, I
have
often emphasized that the latter does "not by itself
solve
the
measurement problem". This remark has in turn been quoted
to argue
that
decoherence be quite irrelevant for a solution of the
measurement
problem. The argument has mostly been used by
physicists
who insist on a
"conventional" solution: either by means of
a
stochastic novel dynamical law, or on the basis of an ensemble of
as
yet unknown
(hidden)
variables. Their hope can indeed not
be fulfilled by decoherence,
and it may forever remain wishful
thinking. In particular,
"epistemic" interpretations of the wave
function (as merely
representing incomplete knowledge) usually
remain silent about the
nature of what this knowledge is about in order
to avoid
contradictions.
A stochastic collapse of the wave function
as a real physical process,
on the other hand, would require
a fundamental
non-linear
modification of the Schrödinger
equation. (It would not make any
difference if this stochastic dynamics
were derived from the presumed
deterministic dynamics of some
hypothetical, but
in principle
unobservable variables.) Since, in Tegmark's words,
decoherence
"looks and
smells
like a
collapse", it is instructive first to ask in
what
sense such
collapse theories would solve the measurement problem
if
their prospective
non-linear
dynamics were ever
confirmed
empirically (for example, by studying systems that
are completely
shielded
against decoherence – a very difficult
task).
According to von Neumann's analysis of the
measurement process, a
collapse could indeed
solve the measurement
problem,
although many
physicists seem to prefer the questionable
formulation that
the Schrödinger equation is exact but applicable only between
the
"preparation" and "measurement" of a quantum state. The
wave
function would then only represent a tool to calculate
probabilities
for other (classical?) variables, whose values "enter
existence" only
in
measurements. However, it
appears absurd to assume
that the wave function exists only for
the purpose of experimental
physicists to make predictions for their
experiments. It would then
also
remain completely open how macroscopic objects,
including
preparation and
measurement devices themselves, could ever
be consistently described as
real
physical
systems consisting of
atoms.
It is well known that superpositions of two or more quantum
states
represent (new) individual
physical properties as long as the system
remains isolated, while they
seem to turn
into
statistical ensembles
when measured and hence subjected
to
decoherence. (As to my
knowledge, no "real", that is, irreversible,
measurement
has ever
been performed in the absense of decoherence.)
So what
would it mean if
appropriate
non-linear collapse terms in the dynamics were confirmed
to
exist?
These theories require that an assumed or prepared wave
function for
the different positions of a macroscopic pointer
(or any
other macroscopic variable) indeterministically evolves or
jumps
into
one of many possible
narrow wave
packet that may represent a real
pointer
position. These wave packets resemble
Schrödinger's coherent
states, which he once used to
describe
quasi-classical
oscillators,
and which he hoped to be representative for all
quasi-classical
objects (apparent particles, in particular). His hope
failed because
of the dynamical dispersion of the wave packet under
the
Schrödinger equation, while coherent states successfully
describe
time-dependent quasi-classical states of electromagnetic
field modes,
which interact very weakly with their environment. The
ensemble of all
possible
outcomes of the postulated collapse into such wave
packets of pointer positions, weighted by the empirical
Born probabilities,
would be described by essentially the same density
matrix as
that
arising from
decoherence. This collapse assumption would
mean that
no fundamental
classical
concepts are needed any more for an interpretation of
quantum
mechanics.
Since macroscopic pointer states are assumed to collapse
into wave
packets in their position
representation, there is no
eigenvalue-eigenfunction link problem that
might arise in epistemic
interpretations. General "observables" then
occur as a derivable
concept.
As an application, consider the particle track
arising
in a Wilson or bubble chamber, described by a succession
of
collapse events. All the
little droplets (or bubbles in a bubble
chamber) can be interpreted as
macroscopic
"pointers" (or documents).
They can themselves be observed without
being changed
by means of
"ideal measurements". In unitary description, the state
of
the
apparently observed
"particle" (its wave function) becomes
entangled with all these pointer
states in a way
that describes a
superposition of many
different tracks, each one consisting of a
number of droplets at
correlated positions. This entanglement
would
disappear according to the collapse, as it essentially
removes all
but one of the tracks (which are described by components of the
global wave
function, that approximately factorize with respect to the
particle,
sets of droplets, and their environment). The lowering of
(local)
entropy as a consequence of the collapse is often
underestimated. So
one assumes
that
the kinematical concept of a
wave function is complete,
and hence, for example, that there are no
particles
in reality. In
contrast,
many interpretations of
quantum theory, such as the Copenhagen
interpretation or those based
on Feynman
paths or Bohm
trajectories, are all
entertaining
the
prejudice that
classical concepts are fundamental
at some
level.
Decoherence
leads to the same local
density
matrix (for the combined system of
droplets and "particle", which
therefore seems to represent
an
ensemble of tracks. The correlations between the wave functions
of
different
droplets as forming tracks were
already known to Mott in
the
early days of quantum mechanics, but he did not yet take
into
account the subsequent and unavoidable process of decoherence of
the
droplet positions by their
environment. Mott did not see the need
to solve any measurement
problem, as he had accepted the probability
interpretation in terms of
classical variables. In a global unitary
quantum description, however,
there
is still just one
global
superposition of all
"potential" tracks consisting of droplets,
entangled with the
particle wave function and the environment: a
universal
Schrödinger cat. Since one does not
obtain an ensemble of potential states without a
collapse,
one cannot select
one of
its members by a mere increase of information. As such a
selection seems to occur,
it is this apparent increase
of
information that requires further
analysis.
Therefore, now add an observer of the Wilson
chamber to this picture.
According to the
Schrödinger equation,
he, too, would necessarily become part
of the entanglement with the
"particle", the device, and the
environment. Clearly, the phase
relations originating from the initial
superposition have now been
irreversibly dislocalized (become an
uncontrollable
property of
the
state of the whole universe). They can never be experienced any
more
by an observer
who is assumed to be local as a consequence of the
locality of
dynamics, but this dynamical locality also means that
certain
components
of the universal wave function become dynamically
autonomous by means
of
decoherence
(see Quantum
nonlocality vs. Einstein locality). The in this way arising
branches of the
global
wave function form entirely different "worlds", which may
contain
different states of
various
observers.
If
we intend to associate
consciousness
with states of local observers, we can do
this only separately to
their
thus
dynamically defined component states. The observed
quantum
indeterminism must then
be
attributed
to the indeterministic
history of these quasi-classical world branches
with their internal
observers. No
indeterminism is required for the global
quantum state.
This identification of observers with states existing
only in certain
branching components of the global wave function
is
the
only
novel element that
has to be added to the quantum formalism
for a solution of the
measurement problem.
Different observers of
the same measurementresult
living in the same "world"
are
consistently
correlated with one
another in a
similar way
as
the
positions of different droplets forming an individual track in
the
Wilson chamber. However, redefining the very concept of
reality
operationally as applying only to the subjectively observed
branch
would eliminate what we already knew for merely pragmatic
reasons
(Occam's razor applied to the facts rather than to the laws)!
The
picture of branching "worlds" perfectly
describes quantum
measurements – although in an unconventional way.
Decoherence
may thus be
regarded as a "collapse without a collapse".
(Note,
however, that decoherence occuring in quantum processes in the
brain
must be expected to lead
to further indeterministic branching even
after the information about a
measurement result has arrived at the
sensoric system in a
quasi-classical form.) Why should we object to
the
consequence that there must
be
myriads of (by us) unobserved quasi-classical worlds
according to the
Schrödinger equation, or why should we
insist
on the existence of fundamental classical objects that we seem
to observe, but that we don't
need for a consistent physical
description of
our
observations?
Collapse theories (formulated by means of
fundamental stochastic
quantum Langevin
equations)
would not only
have to postulate
the
indeterministic transition of
quantum states into certain component
states, but
also their relative
probabilities according to the Born
rules as part of this modified
dynamics.
While even without
a
collapse, the relevant components (or robust "branches" of the
wave
function) can be
dynamically justified by the
dislocalization of
superpositions (decoherence), as described above,
the
probabilities
themselves
can not. All
attempts to
derive empirical facts must be doomed to remain
circular
in some way. For example, Wojciech
Zurek's recent
attempts to
derive Born's rules by "going beyond decoherence" are
based on local
operations that presuppose
the
existence of subsystem states, which
he further assumes to "possess"
certain probabilities. Together they
would then define a formal state
of
(objective?) "information". In
this way, Zurek even claims to avoid
those many
Everett "worlds"
without postulating a collapse in what he calls his
"existential
interpretation" – evidently in contradiction to the
assumed
unitary dynamics. This approach seems to confirm Max
Tegmark's
alternative between Many
Worlds or Many
Words!
According to Graham, one may derive the
observed
relative frequencies of measurement outcomes
(their
statistical distribution) by merely
assuming that
our final
(the present)
branch of the universal wave function (in which "we"
happen to live)
does
not
have an
extremely small norm. Although the
choice of the norm is here
completely equivalent
to assuming the Born
probabilities for all individual branchings, it is
a
natural choice
for such a postulate, since the norm is
conserved under the
Schrödinger equation (just as phase space
is
conserved
in
classical theories, where it similarly serves as an
appropriate
probability measure). Nonetheless, most physicists
seem
to insist on a metaphysical (pre-Humean) concept of dynamical
probabilities, which
would explain
the observed
frequencies of measurement results in a "causal" manner.
However,
this assumption seems to represent a prejudice
resulting from our
causal classical experience.
There is now a wealth of
observed mesoscopic realizations of
"Schrödinger
cats", produced
according to a general Schrödinger equation. They
include
superpositions of different states of electromagnetic fields,
interference
between partial waves representing
biomolecules passing through
different slits of an appropriate device,
or
superpositions of
currents consisting of millions of electrons moving
collectively in
opposite directions. They can all be used to
demonstrate their
gradual decoherence by interaction with the
environment (in contrast
to previously assumed spontaneous quantum
jumps), while there is
so
far no
indication whatsoever for a
genuine collapse. However, complex
biological systems (living
beings)
can hardly ever be sufficently isolated, since they have to
permanently
get rid of entropy. Such systems depend essentially on
the arrow of
time that is manifest in the growing
correlations
(most
importantly in the form of
quantum entanglement, and hence
decoherence).
Only in a
Gedanken Experiment
may we
conceive of an isolated observer, who for some interval
of
time
interacts with an also isolated measurement device, or even
directly
with a microscopic system (by absorbing a single photon, for
example).
One may also imagine an observer who is himself passing
through an
interference device while being aware of the slit he
passes through.
What
would that mean according to a universal
Schrödinger
equation? Since the observer's internal state of
knowledge must be
entangled with the variables that he has observed,
or with his path of
which he is aware, the corresponding
"global"
superposition defines several
distinct and dynamically
independent states
for him as different factor states in all
these components. So he
would subjectively believe
to pass through one slit
only.
Could we confirm such a
prediction in principle? If we observed the
otherwise
isolated
observer from
outside, he should behave just as any
microscopic system – thus
allowing for recoherence.
Unfortunately, he would
thereby have to lose all his memory about
what he experienced. So can
we not ask him before recoherence occurs?
This would require him to
emit
information in some physical form,
thereby preventing recoherence and
interference. An observer in a
state that allows interference
could
never tell us which passage he
was aware of! This demonstrates that the
Everett
branching is
ultimately subjective,
although
we may always assume it to
happen objectively as
soon as decoherence has become irreversible for
all
practical
purposes. As this usually occurs in the apparatus of
measurement,
this
description justifies the pragmatic Copenhagen
interpretation
– albeit in a conceptually consistent manner and without
presuming classical terms.
(For more see "Roots and
Fruits of
Decoherence"
- in particular Sects. 3, 5 and
6.)