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arXiv:quant-ph/0207174 · Journal of Modern Optics

Quantum theory of preparation and measurement

DAVID PEGG, STEPHEN BARNETT, JOHN JEFFERS

Published Apr 1, 2002• 38 citations
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Abstract

The conventional postulate for the probabilistic interpretation of quantum mechanics is asymmetric in preparation and measurement, making retrodiction reliant on inference by use of Bayes' theorem. Here, a more fundamental symmetric postulate is presented, from which both predictive and retrodictive probabilities emerge immediately, even where measurement devices more general than those usually considered are involved. It is shown that the new postulate is perfectly consistent with the conventional postulate.

DOI
10.1080/09500340110109412

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The conventional postulate for the probabilistic interpretation of quantum mechanics is asymmetric in preparation and measurement, making retrodiction reliant on inference by use of Bayes' theorem. Here, a more fundamental symmetric postulate is presented, from which both predictive and retrodictive probabilities emerge immediately, even where measurement devices more general than those usually considered are involved. It is shown that the new postulate is perfectly consistent with the conventional postulate.
1 Abstract. The conventional postulate for the probabilistic interpretation of quantum mechanics is asymmetric in preparation and measurement, making retrodiction reliant on inference by use of Bayes ’ theorem. Here we present a more fundamental symmetric postulate from which both predictive and retrodictive probabilities emerge immediately, even where measurement devices more general than those usually considered are involved. We show that the new postulate is perfectly consistent with the conventional postulate. 2
Quantum theory can be viewed as a generalization of classical probability theory, but the analogy as it has been developed so far is not complete. Whereas the manner in which inferences are made in classical probability theory is independent of the causal relation that holds between the conditioned variable and the conditioning variable, in the conventional quantum formalism, there is a significant difference between how one treats experiments involving two systems at a single time and those involving a single system at two times. In this article, we develop the formalism of quantum conditional states, which provides a unified description of these two sorts of experiment. In addition, concepts that are distinct in the conventional formalism become unified: Channels, sets of states, and positive operator valued measures are all seen to be instances of conditional states; the action of a channel on a state, ensemble averaging, the Born rule, the composition of channels, and nonselective state-update rules are all seen to be instances of belief propagation. Using a quantum generalization of Bayes' theorem and the associated notion of Bayesian conditioning, we also show that the remote steering of quantum states can be described within our formalism as a mere updating of beliefs about one system given new information about another, and retrodictive inferences can be expressed using the same belief propagation rule as is used for predictive inferences. Finally, we show that previous arguments for interpreting the projection postulate as a quantum generalization of Bayesian conditioning are based on a misleading analogy and that it is best understood as a combination of belief propagation (corresponding to the nonselective state-update map) and conditioning on the measurement outcome.
Buschʼs theorem deriving the standard quantum probability rule can be regarded as a more general form of Gleasonʼs theorem. Here we show that a further generalization is possible by reducing the number of quantum postulates used by Busch. We do not assume that the positive measurement outcome operators are effects or that they form a probability operator measure. We derive a more general probability rule from which the standard rule can be obtained from the normal laws of probability when there is no measurement outcome information available, without the need for further quantum postulates. Our general probability rule has prediction–retrodiction symmetry and we show how it may be applied in quantum communications and in retrodictive quantum theory.
Busch's theorem deriving the standard quantum probability rule can be regarded as a more general form of Gleason's theorem. Here we show that a further generalisation is possible by reducing the number of quantum postulates used by Busch. We do not assume that the positive measurement outcome operators are effects or that they form a probability operator measure. We derive a more general probability rule from which the standard rule can be obtained from the normal laws of probability when there is no measurement outcome information available, without the need for further quantum postulates. Our general probability rule has prediction-retrodiction symmetry and we show how it may be applied in quantum communications and in retrodictive quantum theory.
Quantum theory can be viewed as a generalization of classical probability theory, but the analogy as it has been developed so far is not complete. Classical probability theory is independent of causal structure, whereas the conventional quantum formalism requires causal structure to be fixed in advance. In this paper, we develop the formalism of quantum conditional states, which unifies the description of experiments involving two systems at a single time with the description of those involving a single system at two times. The analogies between quantum theory and classical probability theory are expressed succinctly within the formalism and it unifies the mathematical description of distinct concepts, such as ensemble preparation procedures, measurements, and quantum dynamics. We introduce a quantum generalization of Bayes' theorem and the associated notion of Bayesian conditioning. Conditioning a quantum state on a classical variable is the correct rule for updating quantum states in light of classical data, regardless of the causal relationship between the classical variable and the quantum degrees of freedom, but it does not include the projection postulate as a special case. We show that previous arguments that projection is the quantum generalization of conditioning are based on misleading analogies. Since our formalism is causally neutral, conditioning provides a unification of the predictive and retrodictive formalisms for prepare-and-measure experiments and leads to an elegant derivation of the set of states that a system can be steered to by making measurements on a remote system.
We examine the assertion that the "reduction of the wave packet," implicit in the quantum theory of measurement introduces into the foundations of quantum physics a time-asymmetric element, which in turn leads to irreversibility. We argue that this time assymmetry is actually related to the manner in which statistical ensembles are constructed. If we construct an ensemble time symmetrically by using both initial and final states of the system to delimit the sample, then the resulting probability distribution turns out to be time symmetric as well. The conventional expressions for prediction as well as those for "retrodiction" may be recovered from the time-symmetric expressions formally by separating the final (or the initial) selection procedure from the measurements under consideration by sequences of "coherence destroying" manipulations. We can proceed from this situation, which resembles prediction, to true prediction (which does not involve any postselection) by adding to the time-symmetric theory a postulate which asserts that ensembles with unambiguous probability distributions may be constructed on the basis of preselection only. If, as we believe, the validity of this postulate and the falsity of its time reverse result from the macroscopic irreversibility of our universe as a whole, then the basic laws of quantum physics, including those referring to measurements, are as completely time symmetric as the laws of classical physics. As a by-product of our analysis, we also find that during the time interval between two noncommuting observations, we may assign to a system the quantum state corresponding to the observation that follows with as much justification as we assign, ordinarily, the state corresponding to the preceding measurement.