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arXiv:gr-qc/9304031 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Environment-induced decoherence, classicality, and consistency of quantum histories

Juan Pablo Paz, Wojciech Hubert Zurek

Published Sep 15, 1993• 160 citations
arXivHTMLPDF

Abstract

We prove that for an open system, in the Markovian regime, it is always possible to construct an infinite number of nontrivial sets of histories that exactly satisfy the probability sum rules. In spite of being perfectly consistent, these sets manifest a very nonclassical behavior: they are quite unstable under the addition of an extra instant to the list of times defining the history. To eliminate this feature---the implications of which we discuss---and to achieve the stability that characterizes the quasiclassical domain, it is necessary to separate the instants which define the history by time intervals significantly larger than the typical decoherence time. In this case environment induced superselection is very effective. The quasiclassical domain is defined by predictably evolving perferred states, ``pointer projectors,'' which give rise to consistent preferred sets of histories.

DOI
10.1103/physrevd.48.2728

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We prove that for an open system, in the Markovian regime, it is always possible to construct an infinite number of nontrivial sets of histories that exactly satisfy the probability sum rules. In spite of being perfectly consistent, these sets manifest a very nonclassical behavior: they are quite unstable under the addition of an extra instant to the list of times defining the history. To eliminate this feature---the implications of which we discuss---and to achieve the stability that characterizes the quasiclassical domain, it is necessary to separate the instants which define the history by time intervals significantly larger than the typical decoherence time. In this case environment induced superselection is very effective. The quasiclassical domain is defined by predictably evolving perferred states, ``pointer projectors,'' which give rise to consistent preferred sets of histories.
There are good motivations for considering some type of quantum histories formalism. Several possible formalisms are known, defined by different definitions of event and by different selection criteria for sets of histories. These formalisms have a natural interpretation, according to which nature somehow chooses one set of histories from among those allowed, and then randomly chooses to realise one history from that set; other interpretations are possible, but their scientific implications are essentially the same. The selection criteria proposed to date are reasonably natural, and certainly raise new questions. For example, the validity of ordering inferences which we normally take for granted – such as that a particle in one region is necessarily in a larger region containing it – depends on whether or not our history respects the criterion of ordered consistency, or merely consistency. However, the known selection criteria, including consistency and medium decoherence, are very weak. It is not possible to derive the predictions of classical mechanics or Copenhagen quantum mechanics from the theories they define, even given observational data in an extended time interval. Attempts to refine the consistent histories approach so as to solve this problem by finding a definition of quasiclassicality have so far not succeeded. On the other hand, it is shown that dynamical collapse models, of the type originally proposed by Ghirardi-Rimini-Weber, can be re-interpreted as set selection criteria within a quantum histories framework, in which context they appear as candidate solutions to the set selection problem. This suggests a new route to relativistic generalisation of these models, since covariant definitions of a quantum event are known.
We study the decoherence properties of a certain class of Markovian quantum open systems from both the Decohering Histories and Environment Induced Superselection paradigms. The class studied includes many familiar quantum optical cases. For this class, we show that there always exists a basis which leads to {\em exactly} consistent histories for any coarse graining {\em irrespective} of the initial conditions. The magnitude of the off--diagonal elements of the reduced density matrix ρρρ in this basis however, depends on the initial conditions. Necessary requirements for classicality as advanced by the two paradigms are thus in direct conflict in these systems.
In this universe, governed fundamentally by quantum mechanical laws, characterized by indeterminism and distributed probabilities, classical deterministic laws are applicable over a wide range of time, place, and scale. We review the origin of these laws in the context of the quantum mechanics of closed systems, most generally, the universe as a whole. There probabilities are predicted for members of decoherent sets of alternative histories of the universe, ie ones for which the interference between pairs in the set is negligible as measured by a decoherence functional. An expansion of the decoherence functional in the separation between histories allows the form of the deterministic equations of motion to be derived for suitable coarse grainings of a class of non-relativistic systems, including ones with general non-linear interactions. More coarse graining is needed to achieve classical predictability than naive arguments based on the uncertainty principle would suggest. Coarse graining is needed for decoherence, and coarse graining beyond that for the inertia necessary to resist the noise that mechanisms of decoherence produce. Sets of histories governed largely by deterministic laws constitute the quasiclassical realm of everyday experience which is an emergent feature of the closed system's initial condition and Hamiltonian. We analyse the sensitivity of the existence of a quasiclassical realm to the particular form of the initial condition. We find that almost any initial condition will exhibit a quasiclassical realm of some sort, but only a small fraction of the total number of possible initial states could reproduce the everyday quasiclassical realm of our universe. (Talk given at the Lanczos Centenary Conference, North Carolina State University, December 15, 1993.)
We examine the decoherence properties of a quantum open system as modeled by a quantum optical system in the Markov regime. We look for decoherence in both the environment-induced superselection (EIS) and consistent histories frameworks. We propose a general measure of the coherence of the reduced density matrix and find that EIS decoherence occurs in a number of bases for this model. The degree of "diagonality" achieved increases with bath temperature. We evaluate the decoherence functional of consistent histories for coarse-grained phase-space two-time projected histories. Using the measures proposed by Dowker and Halliwell we find that the consistency of the histories improves with increasing bath temperature, time, and final grain size and decreases with initial grain size. The peaking increases with increasing grain size and decreases with increasing bath temperature. Adopting the above-proposed measure of "coherence" to the decoherence functional gives similar results. The results agree in general with expectations while the anomalous dependence of the consistency on the initial grain size is discussed.
Selection of the preferred classical set of states in the process of decoherence is discussed with an emphasis on the role of correlations, information loss and entropy. Persistence of correlations between the observables of two systems (for instance, a record and a state of a system evolved from the initial conditions described by that record) in the presence of the environment is used to define classical behavior. From the viewpoint of an observer (or any system capable of maintaining records and utilising information) predictability is a measure of such persistence. \texitPredictability sieve – a procedure which employs both the statistical and algorithmic entropies to systematically explore all of the Hilbert space of an open system in order to eliminate the majority of the unpredictable and non-classical states, and to locate the islands of predictability including the preferred pointer basis, is proposed. The relation between the consistent histories approach and the preferred sets of states is considered. It is demonstrated that histories of sequences of events corresponding to projections onto the states of the pointer basis are consistent. The correspondence between the perceived reality and the evolving state of the quantum Universe based on a prolonged existence of states of individual systems and correlations between them is proposed. This emphasis on predictable existence of states of classical systems and records of classical observers suggests existential interpretation of quantum theory. Existential interpretation addresses the long-standing problem of the “collapse of the wave function” in a manner which goes beyond Everett's relative state and Bohr's Copenhagen interpretations, by identifying the collapse with the decay of non-classical superpositions of states (and of the observers records) into mixtures of preferred states chosen by the environment-induced superselection.
Selection of the preferred classical set of states in the process of decoherence -- so important for cosmological considerations -- is discussed with an emphasis on the role of information loss and entropy. {\it Persistence of correlations} between the observables of two systems (for instance, a record and a state of a system evolved from the initial conditions described by that record) in the presence of the environment is used to define classical behavior. From the viewpoint of an observer (or any system capable of maintaining records) {\it predictability} is a measure of such persistence. {\it Predictability sieve} -- a procedure which employs both the statistical and algorithmic entropies to systematically explore all of the Hilbert space of an open system in order to eliminate the majority of the unpredictable and non-classical states and to locate the islands of predictability including the preferred {\it pointer basis} is proposed. Predictably evolving states of decohering systems along with the time-ordered sequences of records of their evolution define the effectively classical branches of the universal wavefunction in the context of the ``Many Worlds Interpretation". The relation between the consistent histories approach and the preferred basis is considered. It is demonstrated that histories of sequences of events corresponding to projections onto the states of the pointer basis are consistent.