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arXiv:1412.2852

Is the Preferred Basis selected by the environment?

Tian Wang, David Hobill

Published Dec 9, 2014• 1 citation
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Abstract

We show that in a quantum measurement, the preferred basis is determined by the interaction between the apparatus and the quantum system, instead of by the environment. This interaction entangles three degrees of freedom, one system degree of freedom we are interested in and preserved by the interaction, one system degree of freedom that carries the change due to the interaction, and the apparatus degree of freedom which is always ignored. Considering all three degrees of freedom the composite state only has one decomposition, and this guarantees that the apparatus would end up in the expected preferred basis of our daily experiences. We also point out some problems with the environment-induced super-selection (Einselection) solution to the preferred basis problem, and clarifies a common misunderstanding of environmental decoherence and the preferred basis problem.

DOI
10.48550/arxiv.1412.2852

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We show that in a quantum measurement, the preferred basis is determined by the interaction between the apparatus and the quantum system, instead of by the environment. This interaction entangles three degrees of freedom, one system degree of freedom we are interested in and preserved by the interaction, one system degree of freedom that carries the change due to the interaction, and the apparatus degree of freedom which is always ignored. Considering all three degrees of freedom the composite state only has one decomposition, and this guarantees that the apparatus would end up in the expected preferred basis of our daily experiences. We also point out some problems with the environment-induced super-selection (Einselection) solution to the preferred basis problem, and clarifies a common misunderstanding of environmental decoherence and the preferred basis problem.
The preferred basis problem is mentioned in the literature in connection with the measurement problem and with the Many World Interpretation. It is argued that this problem actually corresponds to two inequivalent problems: (i) the preferred-decomposition problem, i.e., what singles out a preferred decomposition of a suitable state vector of a system as the sum of a finite or countable set of vectors?, and (ii) the preferred-representation problem, i.e., what singles out a preferred representation for the Hilbert space of a system? In this paper the preferred-decomposition problem is addressed and two processes, namely decoherence and permanent spatial decomposition (PSD), are examined and compared as possible solutions to this problem. It is shown that, perhaps contrary to common belief, in realistic situations decoherence is not sufficient to solve the preferred-decomposition problem. PSD is the (hypothesized) tendency of the wave function of the universe to decompose into permanently non-overlapping wave packets. Three phases can be roughly identified as composing PSD: Microscopic decomposition, amplification, and interaction with the environment. Decoherence theory considers only the interaction with the environment and ignores the first two phases. For this reason PSD is fundamentally different from decoherence and, unlike decoherence, provides a simple and non-elusive solution to the preferred-decomposition problem.
The effect of decoherence is analyzed for a free particle interacting with an environment via dissipative coupling. The interaction between the particle and the environment occurs by a coupling of the position operator of the particle with the environmental degrees of freedom. By examining the exact solution of the density-matrix equation, one finds that the density matrix becomes completely diagonal in momentum with time, while the position space density matrix remains nonlocal. This establishes the momentum basis as the emergent ``preferred basis'' selected by the environment, which is contrary to the general expectation that position should emerge as the preferred basis, since the coupling with the environment is via the position coordinate.
According to Zurek, decoherence is a process resulting from the interaction between a quantum system and its environment; this process singles out a preferred set of states, usually called pointer basis, that determines which observables will receive definite values. This means that decoherence leads to a sort of selection which precludes all except a small subset of the states in the Hilbert space of the system from behaving in a classical manner: environment-induced-superselection (einselection) is a consequence of the process of decoherence. The aim of this paper is to present a new approach to decoherence, different from the mainstream approach of Zurek and his collaborators. We will argue that this approach offers conceptual advantages over the traditional one when problems of foundations are considered; in particular, from the new perspective, decoherence in closed quantum systems becomes possible and the preferred basis acquires a well founded definition.
Environment induced decoherence, and other quantum processes, have been proposed in the literature to explain the apparent spontaneous selection—out of the many mathematically eligible bases—of a privileged measurement basis that corresponds to what we actually observe. This paper describes such processes, and demonstrates that—contrary to common belief—no such process can actually lead to a preferred basis in general. The key observation is that environment induced decoherence implicitly assumes a prior independence of the observed system, the observer and the environment. However, such independence cannot be guaranteed, and we show that environment induced decoherence does not succeed in establishing a preferred measurement basis in general. We conclude that the existence of the preferred basis must be postulated in quantum mechanics, and that changing the basis for a measurement is, and must be, described as an actual physical process.
The manner in which states of some quantum systems become effectively classical is of great significance for the foundations of quantum physics, as well as for problems of practical interest such as quantum engineering. In the past two decades it has become increasingly clear that many (perhaps all) of the symptoms of classicality can be induced in quantum systems by their environments. Thus decoherence is caused by the interaction in which the environment in effect monitors certain observables of the system, destroying coherence between the pointer states corresponding to their eigenvalues. This leads to environment-induced superselection or einselection, a quantum process associated with selective loss of information. Einselected pointer states are stable. They can retain correlations with the rest of the universe in spite of the environment. Einselection enforces classicality by imposing an effective ban on the vast majority of the Hilbert space, eliminating especially the flagrantly nonlocal ``Schr\"odinger-cat states.'' The classical structure of phase space emerges from the quantum Hilbert space in the appropriate macroscopic limit. Combination of einselection with dynamics leads to the idealizations of a point and of a classical trajectory. In measurements, einselection replaces quantum entanglement between the apparatus and the measured system with the classical correlation. Only the preferred pointer observable of the apparatus can store information that has predictive power. When the measured quantum system is microscopic and isolated, this restriction on the predictive utility of its correlations with the macroscopic apparatus results in the effective ``collapse of the wave packet.'' The existential interpretation implied by einselection regards observers as open quantum systems, distinguished only by their ability to acquire, store, and process information. Spreading of the correlations with the effectively classical pointer states throughout the environment allows one to understand ``classical reality'' as a property based on the relatively objective existence of the einselected states. Effectively classical pointer states can be ``found out'' without being re-prepared, e.g, by intercepting the information already present in the environment. The redundancy of the records of pointer states in the environment (which can be thought of as their ``fitness'' in the Darwinian sense) is a measure of their classicality. A new symmetry appears in this setting. Environment-assisted invariance or envariance sheds new light on the nature of ignorance of the state of the system due to quantum correlations with the environment and leads to Born's rules and to reduced density matrices, ultimately justifying basic principles of the program of decoherence and einselection.