The Essence of the Concept of Decoherence  

H. Dieter Zeh (    -   June 2010 (last revised June 2011)

Decoherence is now often identified with certain phenomena that have either been experimentally observed or are relevant in practice for some 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" of he qubits 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 reversibility is used in delayed choice measurements or so-called quantum erasers, where a virtual measurement is "undone" (cf. here). The key experiments that 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 a decoherence process that affects the individual objects, it can be observed only for ensembles of measurement results (such as many spots on a screen). In these experiments, decoherence does in fact affect the investigated mesoscopic objects 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 discussed in these experiments, while for the measurement proper experimentalists analyzing their results often forget what they have just demonstrated, and thus return to a pragmatic statistical interpretation without referring to the second 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 for describing the decoherence of a system under consideration. It can be used to investigate in detail how certain phase relations disappear from this 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 do 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 of a "mixed state". (That for a pure state is unique, since it is just the dynadic product of the state vector or wave function with itself.)

The reason for this ambiguity is that the density matrix is defined only to describe the correct formal 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. An arbitrarily chosen system need not even be defined by spatial boundaries, for example, although such an assumption appears usually natural.) A density matrix may represent a pure state  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 or process 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 its 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 generic nonlocality of quantum states rather than  statistical correlations based on incomplete information.

The situation can be conceptually correctly and completely described by means of the wave function or Hilbert space vector for the required global system. If either a measurement or an uncontrollable interaction with the environment occurs to some system, the latter becomes entangled with whatever it interacted with. A factorizing pure global state 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 kinematically nonlocal world. It has always been known that the quantum formalism is nonlocal in this way - actually long before John Bell conceived of his arguments that demonstrated convincingly 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 questionable "measures of entanglement" which measure only "controllable" entanglement that can somehow be used, while they neglect precisely all the uncontrollable entanglement that leads to decoherence. One has to keep in mind that entanglement is defined as a property of pure quantum states, and based on their phase relations. The latter are simply averaged over in a mixed state - whatever its origin - thus apparently reducing the entanglement. However, an entangled state cannot lose this property just because it is considered in an ensemble of other states because of incomplete information.) The concept of information should therefore better be avoided when describing objective physical processes.

As I pointed out above, the reduced density matrix contains complete information about everything that can be observed at a certain system according to the Born rules. 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 some 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 decoherence phenomena occur according to the unitary quantum formalism. In particular, classical concepts (often describing 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 ever be found to apply somewhere between apparatus and observer, we may have to accept the "many worlds" interpretation. Before decoherence was understood as a unitary process that includes the environment, its occurrence was usually interpreted as a break-down of quantum mechanics that seemed to require the application of independently postulated classical concepts. However, while a collapse of the wave function would have to proceed with superluminal speed, the mere possibility of an interpretation in terms of branching observers based on unitary decoherence demonstrates a forteriori that it cannot be used to send superluminal signals.

Severeral authors have claimed that the concept of decoherence has failed to explain the measurement process. They are all either (wrongly) assuming that decoherence has to be based on the mentioned confusion between different interpretations of the density matrix, or they are (tacitly) criticizing only that the solution is not of the kind they had expected. One has to realize that the success of decoherence is not only challenged by traditionalists who still believe in complementarity and similar non-concepts, such as "quantum information", but perhaps even more passionately by those "dissidents" who are justifying their search for a quite novel theory precisely by their dissatisfaction with this pragmatic approach.

The essence of decoherence is thus given by the permanent and uncontrollable increase of entanglement between all systems. It describes the realistic situation of our world, which is very far from equilibrium, and it thus 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, would lead 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 for subsystems. However, one should keep in mind that entanglement represents individual properties of combined systems (such as total angular momentum) - hence not just incomplete information. Certain entangled states, such as Bell states, are even used 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 individual physical states. (In Everett's description, however, this can be understood as a change of the "relative state" with respect to the branching observer.) 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".