May 2005 - last revised October 2009

How decoherence can solve the measurement problem

H. D. Zeh

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 definition of (sub)systems. All systems should thus be entangled with their growing environments, and generically cannot possess pure quantum states by their own. They may then formally be described by a reduced density matrix \rho that represents a "mixed state", with a von Neumann entropy -trace(\rho ln\rho) that grows in time (unless there were "advanced entanglement"). This reduced density matrix is operationally indistinguishable (by means of local operations) from that describing an ensemble of states – as though some really existing pure state were just incompletely known. One of the states diagonalizing this density matrix could then be selected by a mere increase of knowledge. For this reason, the mixed state arising from entanglement is often erroneously identified with such 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 assumed global unitary dynamics. (Note that this claimed solution – even if correct – would require decoherence to be irreversible, as the measurement could otherwise be undone or "erased" – see Quantum teleportation and other quantum  misnomers). Although the claim would then be operationally unassailable, it is wrong. The very concept of a density matrix is already based on local operations (measurements), which presume the probability interpretation to replace global unitarity at some point.

Because of the popularity of this "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 is irrelevant for the understanding of quantum measurements. This argument has mainly been used by physicists who insist on a traditional solution: by means of a stochastic interpretation that has to complement unitary dynamics. Their hope can indeed not be fulfilled by decoherence.

In particular, "epistemic" interpretations of the wave function (as merely representing incomplete knowledge) usually remain silent about what this missing knowledge is about, in order to avoid inconsistencies. A stochastic collapse of the wave function, on the other hand, would require a fundamental non-linear modification of the Schrödinger equation. 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 condition to be achieved in practice).

Some physicists prefer the questionable alternative that the Schrödinger equation is exact but applicable only between the "preparation" and "measurement" of a quantum state. 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 remain completely open how macroscopic objects, including preparation and measurement devices themselves, could ever be consistently described as physical systems consisting of atoms. It is well known that superpositions of two or more possible states may 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 absence of decoherence.)

So what would it mean if appropriate non-linear collapse terms in the dynamics were confirmed to exist? These theories require that any superposition of different positions of a macroscopic pointer (or any other macroscopic variables) indeterministically evolves or jumps into one of many possible narrow wave packets that may represent pointer states with an uncertainty that is negligible but quite natural for wave packets. 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 not only for micrroscopic particles because of the dynamical dispersion of the wave packet under the Schrödinger equation, while coherent states do 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, weighted by the empirical Born probabilities, can be described by a density matrix that is essentially the same as the reduced density matrix arising from decoherence. The collapse assumption would mean, though, that no fundamental classical concepts are needed any more for an interpretation of quantum mechanics. Since macroscopic pointer states are assumed to collapse into narrow wave packets in their position representation, there is no eigenvalue-eigenfunction link problem that is claimed to arise in epistemic interpretations. Observables are no fundamental concepts any more, as they can be derived from the specific measurement interaction Hamiltonian.

As an example, 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 disturbed by means of "ideal measurements". According to Mott's 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 superposition would disappear according to the collapse, which is assumed to remove all but one of the tracks. Individual tracks are globally described by wave packets that approximately factorize into localized final states of the particle, droplet positions, and their environment. So one assumes that the kinematical concept of a wave function is complete, which means that there are no particles. 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.

As mentioned above, decoherence leads to the same reduced density matrix (for the combined system of droplets and "particle"), which therefore seems to represent an ensemble of tracks. This was all 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 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 a genuine ensemble of pointer states, one cannot select one of its members by a mere increase of information. Since such a selection seems to occur in a measurement, it is this apparent increase of information that requires further analysis.

For this purpose, one has to include an observer of the pointer or the Wilson tracks into the description. 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 for dynamical reasons. This dynamical locality also means that decohered components of the universal wave function are dynamically autonomous (see Quantum nonlocality vs. Einstein locality). The in this way arising branches of the global wave function form entirely independent "worlds", which may contain different states of all observers who are involved in the process.

If we intend to associate unique contents of consciousness with physical states of local observers, we can do this only separately with their thus dynamically defined component states. The observed quantum indeterminism must then be attributed to the indeterministic history of these quasi-classical branch wave functions 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 measurement result living in the same branch 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. Redefining the very concept of reality operationally (that is, applying it only to "our" branch) would eliminate from reality most of what we just concluded to exist according to the unitary dynamics! The picture of branching "worlds" perfectly describes quantum measurements – although in an unconventional manner. Decoherence may 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 sensorial system already in a quasi-classical form.) Why should we reject the consequence of the Schrödinger equation that there must be myriads of (by us) unobserved quasi-classical worlds, or why should we insist on the existence of fundamental classical objects that we seem to observe, but that we don't need at all for a consistent physical description our observations?

Collapse theories (when formulated by means of fundamental stochastic quantum Langevin equations) would not only have to postulate the indeterministic transition of quantum states into definite component states, but also their relative probabilities according to the Born rules. 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. Since all outcomes are assumed to exist in this picture, all attempts to derive the empirical probabilities are doomed to remain circular.

According to Graham, one may derive the observed relative frequencies of measurement outcomes (their statistical distribution) by merely assuming that our final (presently experienced) branch of the universal wave function (in which "we" happen to live) does not have an extremely small norm in comparison to the others. 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 likewise 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 metaphysics seems to represent a prejudice resulting from our causal experience of the classical world.

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 describing 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 of 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 similarly 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 become entangled with the variables that he has observed, or with his path through the slits, 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 interference when "erasing" his memory. So he would have to lose all his memory about what he experienced in order to restore the complete superposition locally. Can we then not ask him before this 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 rules – albeit in a conceptually consistent manner and without presuming classical terms.

See also "Quantum discreteness is an illusion" (in particular Sects. 3 and 4) or "Roots and Fruits of Decoherence" (in particular Sects. 3, 5 and 6).

Back to