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arXiv:2103.06074 · Symmetry

Quantum Retrodiction: Foundations and Controversies

Stephen Mark Barnett, John Jeffers, David Thomas Pegg

Published Apr 2, 2021• 19 citations
arXivHTMLPDF

Abstract

Prediction is the making of statements, usually probabilistic, about future events based on current information. Retrodiction is the making of statements about past events based on current information. We present the foundations of quantum retrodiction and highlight its intimate connection with the Bayesian interpretation of probability. The close link with Bayesian methods enables us to explore controversies and misunderstandings about retrodiction that have appeared in the literature. To be clear, quantum retrodiction is universally applicable and draws its validity directly from conventional predictive quantum theory coupled with Bayes’ theorem.

DOI
10.3390/sym13040586

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Prediction is the making of statements, usually probabilistic, about future events based on current information. Retrodiction is the making of statements about past events based on current information. We present the foundations of quantum retrodiction and highlight its intimate connection with the Bayesian interpretation of probability. The close link with Bayesian methods enables us to explore controversies and misunderstandings about retrodiction that have appeared in the literature. To be clear, quantum retrodiction is universally applicable and draws its validity directly from conventional predictive quantum theory coupled with Bayes’ theorem.
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From data in the present we can predict the future and retrodict the past. These predictions and retrodictions are for histories – most simply time sequences of events. Quantum mechanics gives probabilities for individual histories in a decoherent set of alternative histories. This paper discusses several issues connected with the distinction between prediction and retrodiction in quantum cosmology: the difference between classical and quantum retrodiction, the permanence of the past, why we predict the future but remember the past, the nature and utility of reconstructing the past(s), and information theoretic measures of the utility of history.
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
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.
An attempt is made within the framework of the accepted quantum physics to achieve the maximum parallelism between prediction (inference of the future observational data from the present ones) and retrodiction (inference of the past observational data from the present ones). To implement this program, it is shown that the "retrodictive state function" (extrapolation of the present data to the past) can be just as useful as the ordinary "predictive state function" (extrapolation of the present data to the future). This leads to a formalism in which time-reversal becomes a linear transformation and double time-reversal becomes a ccc-number. In spite of this formal symmetry, it can be shown that the actual success of a retrodiction depends on the satisfaction of an additional condition which is not required in prediction, and which is not always fulfilled. From the same point of view, a logical loophole is pointed out in the indiscriminate application of the HHH-theorem to the past. The so-called irreversibility of observation is interpreted in terms of the decrease of "information" in the process of inference.
We further see the content and consistency of our claims on retrodictive aspects of quantum theory based on the previous papers, which deal with quantum mechanics of a single particle in one-dimensional space under successive mearsurements of particle position and momentum as a wave packet. The content of retrodiction can be best understood as statements on the past recorded in the present state. Conditional probabilites with the past and future states specified are consistently obtained through the ordinary predictive and/or our retrodictive probability formula in any combined way.