Maximum predictive power and the superposition principle

TL;DR

This paper explores the foundational principles behind quantum probability rules, emphasizing how the accumulation of information from observations leads to the superposition principle and the quantum rule for combining probabilities.

Contribution

It offers a reformulation of earlier work to show how the increase in information from observations naturally results in quantum probability rules and the superposition principle.

Findings

Information accumulation from observations leads to quantum probability structure.

The superposition principle emerges from the requirement of increasing predictive accuracy.

Quantum probability rules differ from classical probability in how they combine outcomes.

Abstract

Recently, there has been a discussion on the origin of the quantum probability rules (Deutsch quant-ph/9906015, Polley quant-ph/9906124, Barnum et al. quant-ph/9907024, Finkelstein quant-ph/9907004). This contribution, which is a slightly reformulated version of a paper published in Int.J.Theor.Phys. 33, 171 (1994), points out the follwoing: To an experimenter the world is a persistent stream of discrete data. All that is certain is that with each observation he/she knows more than before, simply because he/she can now answer the question "Which of the possible outcomes have you just registered?", while this was not possible before the observation. One can ask whether this relentless increase of information entails a specific structure. In particular, how must different observations be related in order to ensure that predictions become ever more accurate, the more past observations serve as input? This leads to the quantum rule for adding the complex square roots of probabilities, and not to adding the probabilities themselves, as classical probability would have it.