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American Journal of Physics

Quantum Measurements Are Reversible

Asher Peres

Published Oct 1, 1974• 31 citations
Publisher

Abstract

Traditionally, quantum measurements have been associated with irreversible transformations of pure states into mixtures. In this paper we show that this old hypothesis is redundant. A simple model is constructed in which the measuring process is described by a continuous unitary transformation of the state vector of the observed system and the apparatus. The final state is pure, but appears to be a mixture if we ignore the degrees of freedom of the apparatus. This property is most strikingly illustrated by reformulating the problem in the Heisenberg picture: The state vector is fixed, and the dynamical variables are changed by the measurement.

DOI
10.1119/1.1987884

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Traditionally, quantum measurements have been associated with irreversible transformations of pure states into mixtures. In this paper we show that this old hypothesis is redundant. A simple model is constructed in which the measuring process is described by a continuous unitary transformation of the state vector of the observed system and the apparatus. The final state is pure, but appears to be a mixture if we ignore the degrees of freedom of the apparatus. This property is most strikingly illustrated by reformulating the problem in the Heisenberg picture: The state vector is fixed, and the dynamical variables are changed by the measurement.
A quantum measurement process, when non-trivial, is not a closed evolution: the appearance of classical outcomes is usually interpreted as the evidence of some decoherence-like mechanism causing quantum superpositions to degrade into classical mixtures. Such mechanism is due to a net flow of information from the input system (measurement object), through the physical apparatus interacting with the object (measurement probe), into some environment, the latter representing all those degrees of freedom which are not directly accessible by the experimenter. For this reason, the phenomenon of state reduction induced by the measurement process generally entails an irreversible state change. The aim of our contribution is to answer the following questions : how much information a measurement is able to extract? "How much" irreversible is the state reduction due to a particular measurement process? In which way information gain and irreversibility are related?
We will give a new model for measurements of a quantum system such that the measuring apparatuses are described by a unital separable non-type I nuclear simple C∗^*∗-algebra equipped with certain unital endomorphisms and pure states. An interaction between the quantum system and the apparatus is specified by a unitary associated with the combined system as before. Magnifying to the classical level some aspects of the quantum system so captured in the apparatus is explicitly done by applying the endomorphism; then the resulting state is the superposition of {\em phases} with weights. Nature will then choose each phase according to the probability prescribed by the weights just as does one when multiple phases appear as in phase transition. Thus in our model state-reduction (or collapse of the wave function) is a primary event; whether this corresponds to the measurement of an observable or which one if it does is another matter.
The Schr\"odinger equation cannot convert a pure state into a mixture (just as Newton's equations cannot display irreversibility). However, to observe phase relationships between macroscopically distinguishable states, one has to measure very peculiar operators. An example, constructed explicitly, shows that the classical analog of such an operator cannot be measured, because to do so would violate classical irreversibility. This result justifies von Neumann's measurement theory, without any hypothesis on the role of the observer.
Quantum mechanics is widely recognised as being incomplete. It is not consistent with the second law of thermodynamics and does not provide a scientifically credible physical account of the measurement process, the means by which coherence is broken and classically observable states are recorded. This has led to many ad hoc assumptions being used to account for various properties of quantum systems, among which is the coherence time of quantum devices that determines their ability to perform computations. Here, we show that all these properties can be accommodated naturally and consistently in the context of mixing quantum systems which exhibit continuous spectra, as arises in the thermodynamic limit in quantum statistical mechanics and quantum gravity. In particular, for isolated systems we show that the time-reversal symmetry associated with unitary time evolution of the quantum state gives rise to time-symmetry breaking and a semi-group evolution which attains thermodynamic equilibrium at long times. Moreover, the emergence of this non-unitary time-asymmetry leads to microcanonical equilibrium states in which all quantum coherence is lost and is accompanied by the transformation of pure states into mixtures, leading in turn to an increase in entropy. Inclusion of a macroscopic measurement apparatus shows how the outcome of a measurement corresponds to the von Neumann projection postulate, arising with probabilities in conformance with the Born rule. The mathematical structure of the theory which applies to quantum systems with continuous spectra is closely analogous to the classical ergodic theory of dynamical systems and the conditions under which they attain equilibrium states.
We study the extent to which the outcomes of a quantum measurement can be manipulated by changing the state of the measurement apparatus. The measurement process is modeled as decoherence induced by the experimenter, to gain knowledge about a particular system. The measurement apparatus is assumed to contain a large number of degrees of freedom, and the measurement outcomes are obtained by looking at the long interaction time limit. We study two cases which show contrasting behaviour. With a fixed axis coupling, the measurement is performed along the pointer basis with a high degree of robustness, for a wide variety of bath states. In a second model with Heisenberg interactions, the measurement outcomes can be altered considerably by changing the state of the bath.
We will give a new model for measurements of a quantum system, expressed as the compact operators on a Hilbert space, such that the measuring apparatuses are described by a unital separable non-type I nuclear simple C∗^*∗-algebra equipped with certain unital endomorphisms and pure states. An interaction between the quantum system and the apparatus is specified by a unitary associated with the combined system as before. Magnifying to the classical level some aspects of the quantum system so captured in the apparatus is explicitly done by applying the endomorphism; then the resulting state is the superposition of {\em phases} with weights. Nature will then choose each phase according to the probability prescribed by the weights just as does one when multiple phases appear as in phase transition. Thus in our model state-reduction (or collapse of the wave function) is a primary event; whether this corresponds to the measurement of an observable or which one if it does is another matter.