On the relation between quantum mechanical probabilities and event frequencies

Abstract

The probability `measure' for measurements at two consecutive moments of time is non-additive. These probabilities, on the other hand, may be determined by the limit of relative frequency of measured events, which are by nature additive. We demonstrate that there are only two ways to resolve this problem. The first solution places emphasis on the precise use of the concept of conditional probability for successive measurements. The physically correct conditional probabilities define additive probabilities for two-time measurements. These probabilities depend explicitly on the resolution of the physical device and do not, therefore, correspond to a function of the associated projection operators. It follows that quantum theory distinguishes between physical events and propositions about events, the latter are not represented by projection operators and that the outcomes of two-time experiments cannot be described by quantum logic. The alternative explanation is rather radical: it is conceivable that the relative frequencies for two-time measurements do not converge, unless a particular consistency condition is satisfied. If this is true, a strong revision of the quantum mechanical formalism may prove necessary. We stress that it is possible to perform experiments that will distinguish the two alternatives.

Introduction

Quantum mechanics is a probabilistic theory. It provides a set of rules that allows us to associate probabilities to specific physical events. There is little doubt that these rules have been proved remarkably successful in the description of any physical phenomenon that we have been able to study experimentally, but gravitational ones.

The reasoning in terms of probability in physical theories, however, is not entirely unproblematic. Most discussions about the interpretation of quantum theory—and the associated “paradoxes”—are related to the appropriate use of quantum probabilities. Some of the issues raised are not specifically quantum—they refer to the physical applicability of the general concepts of probability theory and date at least back to Boltzmann.

One may ask, for instance, whether the probabilities are subjective or objective—namely whether they refer to our knowledge about a physical system or to the physical system itself. In the latter case, one may further ask whether probabilities refer to an individual system—denoting perhaps its propensity to manifest one behavior or another- or to statistical ensembles. One may also question whether there exists a sample space for quantum phenomena, or all predictions have to make reference to a concrete measurement set-up.

There exists a common denominator in all interpretations of probability, either classical or quantum. We may not agree whether probabilities refer to the properties of the things themselves or not, but we do accept that probabilities refer to the statistics of measurement outcomes. Probabilities may or may not be physically meaningful a priori (before an experiment), but they can definitely be determined a posteriori, namely after a large number of experimental runs.

The probability of an event is defined as the limit of the relative frequency of this event as the number of trials goes to infinity. It may be argued that this is not the only way we employ probability in physics—after all statistical arguments enter into the design and preparation of any experiment. Still, whenever we want to compare the theoretical probabilities with concrete empirical data, we invariably employ the relation of probability to event frequencies.

In this paper, we analyse the basic properties of quantum mechanical probability for two-time measurements in two consecutive moments of time (two-time measurements). The key point of our argumentation is the empirical determination of probabilities as limits of relative frequencies. We employ this relation without committing to a frequency interpretation of probability [1]—we need not assume that probability, as a concept, is defined as a limit of relative frequencies. Neither do we commit to a specific interpretation of quantum theory. We only assume that the outcomes of measurements (that have actually been performed) are described by the probabilities obtained from the rules of quantum theory. Quantum mechanical probabilities may refer to other aspects of physical reality, but we need not make such an assumption. This thesis can hardly be rejected by any interpretation of quantum theory.

In a two-time measurement one determines specific properties of a physical system at two successive moments of time. The measurement outcomes may be sampled in the same manner they are sampled in the single-time measurements. Probabilities are then still determined by the limits of relative frequencies. We may still employ the rules of quantum theory to associate a probability to each possible measurement outcome. The problem is that the quantum mechanical probability `measure' for two-time histories does not satisfy the additivity property of probabilities. On the other hand, relative frequencies are always additive, since they are constructed by counting specific and indivisible physical events.

We next proceed to resolve this conflict. A physical theory must explain the observed phenomena—namely the frequencies of measured events. If these frequencies define probabilities, we have to accept that the conventional rule for probabilities of two-time measurements fails. We show that the derivation of this rule employs the concept of conditional probability in a rather ambiguous way. We address this problem and define thereby additive probabilities for two-time measurements. But the new probability assignment depends explicitly on the resolution of the physical device. The probabilities assigned to a specific sample set of measurements depend, therefore, on the physical characteristics of the apparatus and they are not a function of the associated projection operators. Projection operators cannot represent events universally. It follows that the YES–NO experiments [2] cannot reconstruct all probabilistic aspects of a physical system, and for this reason the outcomes of two-time experiments cannot be represented by any form of quantum logic.

The alternative is rather radical, but cannot be a priori rejected. It is conceivable that the relative frequencies for the two-time measurements do not converge (see [3] for a relevant interpretation of quantum probability). In that case, probabilities can only be defined for two-time events that satisfy a consistency condition—the same condition that appears in the consistent histories interpretation of quantum theory [4], [5], [6], [7], [8]. The failure of the frequencies to converge is eventually due to the interference between the two alternatives. This alternative explanation implies, of course, that one would need an all-new reformulation of quantum theory.

It is important to emphasise that the two possible resolutions of our problem may be empirically distinguished. It is possible—in theory and we believe in practice too—to design experiments that will determine whether the relative frequencies of two-time events converge or not. Either way, such experiments would shed much light in many counter-intuitive aspects of quantum probability.