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International Journal of Quantum Chemistry

Does quantum probability predict frequency?

William G. Faris

Published Jan 1, 1997• 0 citations

Abstract

In probability theory an event is characterized by a set of outcomes. The probability of an event is a prediction of the relative frequency of the outcomes that lie in the set, when the experiment is repeated independently. In quantum mechanics an event is identified with a closed subspace of Hilbert space. Do the probabilities of quantum events predict relative frequencies in the same way? The famous analysis of Bell shows that this cannot happen in general, for all quantum events. Nevertheless the probabilities must predict something, and if not relative frequencies then what? There are several answers to the question of how to interpret quantum probabilities as relative frequencies, but they all involve restrictions on the events that are considered. This article surveys three such interpretations, which correspond to three possible choices: measured events, macroscopic events, and position events. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 65: 389–398, 1997

DOI: 10.1002/(sici)1097-461x(1997)65:5<389::aid-qua3>3.0.co;2-s

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In probability theory an event is characterized by a set of outcomes. The probability of an event is a prediction of the relative frequency of the outcomes that lie in the set, when the experiment is repeated independently. In quantum mechanics an event is identified with a closed subspace of Hilbert space. Do the probabilities of quantum events predict relative frequencies in the same way? The famous analysis of Bell shows that this cannot happen in general, for all quantum events. Nevertheless the probabilities must predict something, and if not relative frequencies then what? There are several answers to the question of how to interpret quantum probabilities as relative frequencies, but they all involve restrictions on the events that are considered. This article surveys three such interpretations, which correspond to three possible choices: measured events, macroscopic events, and position events. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 65: 389–398, 1997

Definitions The concept of probability is of considerable importance in optics, as in any situation in which the outcome of a given trial or measurement is uncertain. Under these conditions it is desirable to be able to associate a measure with the likelihood of the outcome or the event in question; such a measure is called the probability of the event. Several different definitions of probability have been adopted at various times in the past. The classical definition is based on an exhaustive enumeration of the possible outcomes of an experiment or trial. If the trial has N distinguishable, mutually exclusive outcomes, which are equally likely to occur, and if n out of these N possible outcomes have an attribute or characteristic that we call ‘success’, then the probability of success in any one trial is given by the ratio n/N . For example, if we roll a die, and if each of the six digits is equally likely to be on top when the die comes to rest, there are N = 6 distinguishable outcomes. If we identify success with an even number, for example, then since there are three different ways in which success can be achieved, it follows that the probability of success when the die is rolled is given by 3/6 = 1/2. Unfortunately, an exhaustive enumeration of all possibilities is not always feasible. Another common definition of probability is based on the notion of relative frequency of success.

Quantum probability is very different from classical probability. Part of this difference is manifested in the generic inability of stochastic processes to describe the results of multi-time measurements of quantum mechanical systems and the fact that the complex-valued temporal correlation functions of quantum theory have no interpretation in terms of multi-time measurements. By analysing experiments involving measurements at more than one moments of time, we conclude that this inequivalence must be manifested either as a failure of the quantum logic or as the inability to define probabilities in multi-time measurements because the relative frequencies do not converge. These alternatives can be empirically distinguished as they correspond to different behaviours of the statistical data in multi-time measurements.

Quantum probability and quantum mechanics The formalism of quantum probability theory Discussions of quantum mechanics are often confused by a lack of clarity about what exactly constitutes ‘quantum mechanics’. It is therefore useful to try at the start to isolate a consistent mathematical core of quantum mechanics, and consider anything that goes beyond this core to be ‘interpretation’. For us, this core is quantum probability theory. Quantum probability is a generalization of classical probability, and therefore I begin with a brief review of the latter. I assume that the reader has some familiarity with the ideas of probability theory. What follows is just to provide a quick review, and to establish some notation and terminology. In modern classical probability theory, probabilities are defined over algebras of events. The motivation is straightforward: we begin with a set of ‘primitive’, or ‘simple’, events (the ‘sample space’), and form an algebra of events by taking all logical combinations of the simple events. For example, let us take the simple events to be the possible results of rolling a six-sided die one time, so that the sample space is the set {1,2,3,4,5,6}. We then form an algebra of events from the sample space by taking all possible logical combinations of the simple events. Logical combinations include, for example, ‘either 3 or 5’ and ‘not 3 and not 2’.