The Essence of the Concept of Decoherence

by H.D. Zeh

How decoherence can solve the measurement problem

by H.D. Zeh

Quantum nonlocality and Einstein locality

by H.D. Zeh

Do we need observables?

by E. Joos


The Essence of the Concept of Decoherence

Decoherence is now often identified with certain phenomena that have either been experimentally observed or are relevant in practice for other reasons. This may be in accord with an operational approach to quantum physics, but it tells us nothing about the basis of this concept in the established formalism from that it had originally been derived.

The second kind of phenomena are best known from attempts to realize quantum computers, where decoherence is regarded as an unwanted "distortion" caused by the environment. This picture has led to various unsuccessful attempts to construct "error correction codes" by means of redundant information storage as in classical computers. They can hardly ever be successful, since (1) quantum states cannot be cloned, while (2) genuine decoherence is an irreversible process. Only "virtual decoherence", defined by means of an unrealistic microscopic environment, could be reversed in practice. A similar reversability is used in delayed choice measurements or so-called quantum erasers, where a virtual measurement is "undone" (cf. here). The key experiments which have confirmed the phenomenon of decoherence, on the other hand, demonstrate the disappearance of an otherwise observed interference pattern, usually for mesoscopic objects whose effective environment may be varied to exhibit decoherence or not. Although this statistical phenomenon is indeed a consequence of the corresponding decoherence process, it can be observed only for ensembles of measurement results (such as many spots on a screen), while the irreversible process of decoherence affects each individual measurement. In these experiments, decoherence affects the investigated mesoscopic systems twice: first while they pass the slits of an interferometer or while they live for a short time as "Schrödinger cats" in an isolated cavity, and for the second time during the final measurement that leads to the "spontaneous" appearance of individual spots, bubbles, or clicks. Only the first decoherence process is investigated in these experiments, while for the measurement proper experimentalists analyzing their results often forget what they just demonstrated, and thus return to a pragmatic statistical interpretation without referring to the corresponding decoherence process. Some of them even regard their own inconsistency as support of a wave-particle dualism or of complementarity.

The reduced density matrix, derived from a complete description of the state of a global system by tracing out the environment, is a useful tool to describe the decoherence of a system under consideration. It can be used to investigate in detail how certain phase relations disappear from the system, thus transforming a pure state for it into a "mixed state", for example. Given a realistic environment, this tells us (very successfully) which variables must appear classically (lacking any superpositions), or in which situation we have to expect almost sudden "quantum jumps" or other stochastic "events" to occur. So it seems that we neither need fundamental classical variables any more, nor any indeterministic dynamics. However, this success, which led to the early popularity of the decoherence concept, is partly based on an ambiguity of the concept of the density matrix for a "mixed state". (That for a pure state is unique, since it is equivalent to the state vector or wave function.)

The reason for this ambiguity is that the density matrix is defined only to describe the correct probabilities for all measurements that can be performed on this quantum system according to Born's statistical interpretation. This means, first of all, that it cannot be used to derive this statistical interpretation in terms of stochastic quantum jumps itself. (The concept of "systems" is again an arbitrary tool - similar to the choice of coordinates. A system need not even be local, for example, although such an assumption appears usually natural.) A density matrix may represent a pure state (if it is a projector onto a state vector), or a mixed state. The latter may then either represent a statistical ensemble of pure states of the system with given probabilities, or form the reduced density matrix with respect to the state of some global system that includes all other systems with which the subsystem is entangled. In the first case one could simply "select" a pure state from the ensemble by an increase of information (just as for a classical probability distribution), in the second case one would first have to apply a stochastic interpretation to the global state in order to obtain an ensemble to select from. (A mixed state for the whole universe can always be interpreted as representing lacking information about a pure state.) Although the difference between entanglement and lacking information should by now be well known to all quantum physicists, this confusion is still responsible for many misunderstandings of decoherence. The concept of decoherence did in fact arise from the insight that entanglement describes a fundamental nonlocality rather than mere statistical correlations.

The complete situation can be described conceptually only by means of the wave function for the required global system. If either a measurement or an uncontrollable interaction with the environment happens to some system, the latter becomes entangled with whatever it interacted with. A pure system state (possibly one of the states diagonalizing its mixed density matrix) would thereby be transformed into a pure but entangled global state. An initial superposition is thus "dislocalized": it is neither in the system nor in the environment thereafter - something that can happen only in a nonlocal world. It has always been known that the quantum formalism is nonlocal - actually long before John Bell published his arguments which demonstrated once and for all that this nonlocality cannot be a statistical artifact due to incomplete knowledge about some as yet hidden local variables. (Note that in the literature one finds several popular but insufficient "measures of entanglement" which measure only that entanglement which can somehow be used while they neglect precisely all the uncontrollable entanglement that leads to decoherence.)

As I pointed out above, the reduced density matrix contains complete information about everything that can be observed at a local system. So, decoherence describes an irreversible transition of the "system" state into an apparent ensemble for all practical purposes. This irreversibility is induced by the time arrow characterizing the environment.  If a measurement apparatus could be treated as a (controllable) microscopic system, the measurement would be reversable (it could be "undone"). However, a macroscopic pointer must unavoidably interact with its uncontrollable environment in each individual measurement. Therefore, it appears quite unmotivated to invent any fundamental irreversible process, such as a collapse of the wave function, or to assume fundamental classical concepts to apply, precisely where and when the observable or irreversible phenomena occur. In particular, classical concepts (often defining the pointer basis of a measurement device) emerge according to the objective irreversible process of decoherence, while there remain various possibilities to explain why we observe individual measurement outcomes. If no new physics will be found to apply somewhere between apparatus and observer, we may have to accept the "many worlds" interpretation.

The essence of decoherence is thus given by the permanent increase of entanglement between all systems. It describes a situation very far from equilibrium, and it leads to the permanent dislocalization of superpositions. Its time arrow is formally analogous to the creation of "irrelevant" statistical correlations by Boltzmann collisions. Neglegting these classical correlations, for example by using a µ-space distribution, leads to an increase of ensemble entropy. This consequence remains true as well in quantum theory (in the sense of an "apparent ensemble entropy") if one neglects entanglement by relying on reduced density matrices of subsystems. However, one should keep in mind that entanglement represents individual properties of the combined systems (such as total angular momentum) - hence not just incomplete information. Certain entangled states, such as Bell states, are even considered as potential individual measurement outcomes in some experiments. In spite of the analogy with statistical correlations, the neglect of entanglement describes a change of the physical states. The arrow of time defined by the decoherence process requires a special initial condition for the universal wave function (namely: little or no initial entanglement). Evidently, this must be a physical condition - it cannot just be a condition for initial "human knowledge" or some kind of "information".


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.)


Quantum nonlocality vs. Einstein locality

H. D. Zeh

Quantum theory is kinematically nonlocal, while the theory of relativity (including relativistic quantum field theory) requires dynamical locality ("Einstein locality"). How can these two elements of the theory (well based on experimental results) be simultaneously meaningful and compatible? How can dynamical locality even be defined in terms of kinematically nonlocal concepts?

Dynamical locality in conventional terms means that there is no action at a distance: states "here" cannot directly influence states "there". Relativistically this has the consequence that dynamical effects can only arise within the forward light cones of their causes. However, generic quantum states are "neither here nor there", nor are they simply composed of "states here and states there" (with a logical "and" that would in the quantum formalism be represented as a direct product). Quantum systems at different places are usually entangled, and thus do not possess any states of their own. Therefore, quantum dynamics must in general describe the dynamics of global states. It may thus appear to be necessarily nonlocal.

This discrepancy is often muddled by insisting that reality is made up of local events or phenomena only. However, quantum entanglement does not merely represent statistical correlations that would represent incomplete information about a local reality. Individually observable quantities, such as the total angular momentum of composed systems, or the binding energy of the He atom, can not be defined in terms of local quantities. This nonlocality has been directly confirmed by the violation of Bell's inequalities or the existence of Greenberger-Horne-Zeilinger relations. If there were kinematically local concepts completely describing reality, they would indeed require some superluminal "spooky action at a distance" (in Einstein's words). Otherwise, however, such a picture may become meaningless, and nothing is teleported in so-called quantum teleportation experiments, for example. Instead, one has to carefully prepare an appropriate entangled state that contains, among its components, all states to be possibly teleported (or their dynamical predecessors) already at their final destination – similar to the hedgehog's wife in the Grimm brothers' story of Der Hase und der Igel (see Quantum teleportation and other quantum misnomers).

These kinematical aspects characterize quantum nonlocality. But what about Einstein locality in this picture? Why does the change of a global quantum state not allow superluminal signals, for example? The concept of locality in quantum theory requires more than a formal Hilbert space structure (relativistically as well as non-relativistically). It presumes a local Hilbert space basis (for example consisting of spatial fields and/or particles). Dynamical locality then means that the Hamiltonian is a sum over local terms, or an integral over a local Hamiltonian density in space, while all dynamical propagators for these local elements must relativistically obey the light cone structure.

This framework is most successfully represented by quantum field theory. It may be characterized by the following program:
(1) Define an underlying set of local "classical" fields (including a spatial metric) on a three-dimensional (or more general) manifold.
(2) Define quantum states as wave functionals of these fields (that is, nonlocal superpositions of different spatial fields).
(3) Assume that the Hamiltonian operator H (acting on wave functionals) is defined as an integral over a Hamiltonian density, written in terms of these fields at each space point.
(4) Using this Hamiltonian, write down a time-dependent Schrödinger equation for the wave functionals, or, in order to allow the inclusion of quantum gravity, a Wheeler-DeWitt equation: . The dynamics is then local (in the classical sense) for all local components, which, according to this construction, must span every nonlocal state. This concept defines the quantum version of Einstein locality. (I have here not discussed complications resulting from nonlocal gauge degrees of freedom.)

The local (additive) form of the Hamiltonian has an important dynamical consequence for nonlocal states. If two distant systems and are entangled, in the Schmidt decomposition, all matrix elements of H between components with different n must vanish, since the individual, local terms of H can only act on or . Such "dislocalized superpositions" arise unavoidably by means of decoherence, while their relocalization ("recoherence") would require an improbable accident in a causal universe (see The Physical Basis of the Direction of Time). The factorizing Schmidt components thus describe dynamically autonomous "worlds", which must contain separate observers, and which permanently branch by means of measurement-like processes.  This dynamical argument, based on nothing else but the Schrödinger equation with its local Hamiltonian, justifies Everett's collapse-free interpretation of quantum theory. (Note that the linearity of dynamics by itself would not be sufficient for this purpose, since it would not correctly describe quantum measurements and related phenomena.)

If, in the case of a Wheeler-DeWitt equation, a WKB approximation (based on a Born-Oppenheimer expansion in terms of the Planck mass) applies, orbit-like "wave tubes" in the "superspace" of spatial geometries (the configuration space of general relativity) may define quasi-classical spacetimes (such as solutions of the Einstein equations). The corresponding matter states obey a derived time-dependent Schrödinger equation with respect to a "WKB time" parameter along these quasi-classical orbits of spatial geometries (see C. Kiefer: Quantum Gravity, Cambridge UP, 2007). Wave tubes on the configuration space of geometry are decohered from one another by the matter states (which are thereby regarded as an environment to quantum geometry) according to the Wheeler-DeWitt equation. This decoherence along quasi-trajectories in superspace may lead to further quasi-classical fields, and possibly other quasi-local variables, which are robust in the sense that their different values define dynamically autonomous components ("branches"). Einstein locality then holds up to remaining quantum uncertainties of the spacetime metric (resulting from the non-vanishing widths of the wave packets in superspace).

In "effective" (phenomenological) quantum field theories, dynamical locality is often formulated by means of a condition of microcausality. It requires that commutators between field operators at spatially different spacetime points vanish. This condition is partially kinematic (as it presumes a local reference basis of quantum states), partially dynamical (as it uses the Heisenberg picture for field operators), and partially a matter of definition (as it requires a decomposition of the field operators in terms of "particles and antiparticles", which may depend on the effective vacuum, for example). The dynamical consistency of this microcausality condition is highly nontrivial. In principle, the properties of (anti-)commutators of (effective) field operators at different times should be derivable from those on an arbitrary simultaneity, t = t', by means of the given relativistic dynamics (Hamiltonian). They cannot be independently postulated for all times.

In his foundation of quantum field theory, Steven Weinberg derived microcausality and the locality of the Hamiltonian from his cluster decomposition principle. This is a phenomenological constraint to the S-matrix, which requires that "distant experiments give uncorrelated results". However, such a principle cannot form a fundamental element of the quantum theory, since (a) observable correlations may exist or controllably be prepared either as statistical correlations or as entanglement between distant systems, and (b) the concept of an S-matrix is (approximately) applicable only to sufficiently isolated (microscopic) systems. Macroscopic systems never cease to interact uncontrollably with their environment – thus giving rise to decoherence, and hence to their classical behavior or the appearance of "quantum events" (see How decoherence may solve the measurement problem). Only the latter justify the probability interpretation of the S-matrix – even for microscopic objects. So I feel that instead of going beyond the empirically founded effective theories when searching for mathematical consistency of hypothetical theories (in the hope for finding the final universal theory), physicsts should first analyze the physical consistency and meaning of effective field theories (see also Chap. 6 of The Physical Basis of the Direction of Time).

The original version of these essays can be found on the author's homepage