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Quantum Probability: Conditioning and Inference

Romano Scozzafava

Published Jan 1, 2008• 0 citations
OpenAlex

Abstract

Looking at conditional probability from the right perspective allows one to avoid classic paradoxical aspects of quantum mechanics. We discuss, as an elementary analogue of quan-tum states, a classical example (an urn of unknown composition) with similar (putative) paradoxical situ-ations, despite the fact that there is no quantum effect to which they could be ascribed. A crucial feature is that we look at them from an epis-temic point of view and not from the viewpoint of a (seemingly) ontologi-cal reality.

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Looking at conditional probability from the right perspective allows one to avoid classic paradoxical aspects of quantum mechanics. We discuss, as an elementary analogue of quan-tum states, a classical example (an urn of unknown composition) with similar (putative) paradoxical situ-ations, despite the fact that there is no quantum effect to which they could be ascribed. A crucial feature is that we look at them from an epis-temic point of view and not from the viewpoint of a (seemingly) ontologi-cal reality.
In this note we discuss a few simple classical (as opposed to quantum) prediction problems. The thrust of this work is to examine the predictive role of probability theory and we shall see that some situations are not re-ally paradoxical. In particular we want to separate the role of probability as a predictive tool from one of the basic sources of strangeness in quan-tum mechanics, namely the principle of superposition of states. For that we shall re-examine some of the quantum paradoxes, recast as a problem about making a prediction about the result of a random experiment in classical physics when some information is known. 1
In this expository paper we discuss the role of probability in statistical physics: it should be that of a general tool guiding induction and must not be restricted only to the empirical interpretation coming from its evaluation by an observed frequency. It is true that in many cases the value of a probability can be sensibly expressed by a suitable frequency, but this does not entails that the latter can be taken as definition of probability. Moreover, in each situation in which one thinks that probabilities are physically real things and believes that something which is only “very probable” can be regarded as being “practically certain”, some “small” probabilities are actually ignored: this attitude of mind may render illegitimate also the probabilistic interpretation of the relevant physical laws. A “law” which is based on a probabilistic interpretation cannot aim at explaining why a fact that we forecast will occur, but rather at explaining why we forecast that this fact will occur. Finally, we emphasize the role of conditional probability, and we show that looking at it from the right perspective allows to avoids also classic paradoxical aspects of quantum mechanics, without the need to resort to the interpretation based on the dualism “particle-wave”.
In classical probability theory, the best predictor of a future observation of a random variable X,X,X, is its expected value EP[X]E_P[X]EP​[X] when no other information is available, whereas if information consisting in the observation of another random variable is available, then the best predictor is is another random variable EP[X∣Y].E_P[X|Y].EP​[X∣Y]. It is the purpose of this note to explore the analogue of this in the case of quantum mechanics. we shall see that when the result of an observation is taken into account by means of a non-commutative conditional expectation, exactly as in classical prediction theory, some of the usual paradoxes cease to be so. The moral of this note is that the mystery of the probabilistic interpretation of quantum mechanics lies in the superposition principle
After briefly recalling some of the main theses of quantum probability concerning the foundational aspects of quantum mechanics and of probability theory, we describe the various categories of axioms in classical probability theory and focus our attention on the classical conditioning axioms. Four classical conditioning axioms are proposed, their probabilistic and model-independent meaning is discussed and it is proven that: (i) the only mathematical realization of these axioms is the classical Bayes rule (or equivalently the classical theorem of composite probabilities) and (ii) von Neumann's projection postulate, which can be considered one of the first examples of quantum conditioning axioms, violates two of these axioms. Finally, we construct an example showing that, even in classical probability, an uncritical application of Bayes' rule can lead to non-optimal predictions. This article is part of the theme issue 'Quantum theory and topology in models of decision making (Part 2)'.
A number of books have appeared purporting to explain quantum mechanics to the general public. These books have put considerable emphasis on so-called quantum 'paradoxes', using them to portray the quantum theory as mysterious or 'spooky'. The author discusses these 'paradoxes' from a different viewpoint with the aim of demystifying the theory. Emphasis is shifted from an epistemological viewpoint (as exemplified by the 'probability interpretation') to an ontological one (the description of matter by physical fields). Difficulties in reconciling the probability interpretation with relativity are noted, and the importance of phase coherence, which is inadequately treated in the probability interpretation, is emphasised. In the light of such as approach, the 'paradoxes' are resolved.
Quantum mechanics (QM) has attracted a considerable amount of mysticism, in public opinion and even among academic researches, due to some of its conceptually puzzling features, such as the modification of reality by the observer and entanglement. We argue that many popular "quantum paradoxes" stem from a confusion between mathematical formalism and physics; We demonstrate this by explaining how the paradoxes go away once a different formalism, usually inconvenient to perform calculations, is used. we argue that some modern developments, well-studied in the research literature but generally overlooked by both popular science and teaching-level literature, make quantum mechanics (that is, "canonical" QM, not extensions of it) less conceptually problematic than it looks at first sight. When all this is looked at together, most "conceptual puzzles" of QM are not much different from the well-known paradoxes from probability theory. Consequently, "explanations of QM" involving physical action of consciouses or an infinity of universes are conceptually unnecessary