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Reviews of Modern Physics

Symmetry of Physical Laws. Part III. Prediction and Retrodiction

Satosi Watanabé

Published Apr 1, 1955• 127 citations
Publisher

Abstract

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 $c$-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 $H$-theorem to the past. The so-called irreversibility of observation is interpreted in terms of the decrease of "information" in the process of inference.

DOI
10.1103/revmodphys.27.179

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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.
The problem of retrodiction is associated with basic questions, but for this very reason it is, unfortunately, quite difficult, even in classical physics. In quantum physics it is even more subtle. In quantum mechanics, a consistent definition of what is called a "system" leads to problems that do not exist in classical physics, and correspondingly the answers to the questions raised are less straightforward. This chapter discusses the conditions under which it is possible to speak of an "independent time evolution" of a system S and to study what kinds of predictions can legitimately be made when these conditions are fulfilled. It describes retroactive computation to investigate whether or not the assumed knowledge in question is really consistent with the standard quantum mechanical formalism. The concept of retrodiction is related to the notions of time reversal and of irreversibility. Both these notions play a central role in physics.
Quantitative studies of irreversibility in statistical mechanics often involve the consideration of a reverse process, whose definition has been the object of many discussions, in particular for quantum mechanical systems. Here we show that the reverse channel very naturally arises as Bayesian retrodiction, both in classical and quantum theories. Fluctuation relations then become expressions of the statistical divergence between the forward (i.e., predictive) and the reverse (i.e., retrodictive) process. With the reverse process constructed on consistent logical inference, rather than on particular balancing mechanisms in microscopic physics, fluctuation relations acquire a much broader form and scope. Previous paradigmatic results like Jarzynski's and Crooks' fluctuation relations are recovered as special cases.
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.
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.
Prediction, of course, is the making of statements about the likelihood of future events. retrodiction (or postdiction) is the making of statement about past events. In these lectures we shall study the rules governing retrodictive probabilities both in the quantum and classical domains. In doing so we develop a powerful tool with which to interpret physical phenomena and both analyse and interpret experiments. We use retrodictive methods to analyze a number of phenomena in quantum optics including the familiar photon antibunching, entanglement and ghost imaging and the perhaps less familiar ideas of quantum scissors, quantum computation in the past a single-particle analysis of two-particle interference. Quantum retrodiction is a quantitative theory and the familiar techniques for evolving quantum states and modelling open systems are applicable to retrodictive problems, but only with the introduction of some novel features. We conclude with a few remarks on more philosophical issues related to the arrow of time and wavefunction "collapse" in the measurement process. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
The usual formalism of quantum mechanics is predictive with state vectors being assigned on the basis of past preparation procedures. The retrodictive formalism, in which some state vectors are assigned on the basis of future measurement results is less usual, although it has been known for some time. Although at first sight this formalism may appear to be anticausal and subject to paradoxes, if used carefully it still predicts the same measurable correlations as the more usual approach despite different assignments of state vectors to states in the interval between preparation and measurement of the system. Here we use a procedure which involves retrodiction to examine some quantum optical processes, including a recent state truncation process. The collapse of the state vector occurs at a different time from that for the usual approach underlining the difficulty in regarding the collapse as a real physical process.