Derivation of quantum probability from measurement

TL;DR

This paper derives the quantum probability formula from measurement theory principles, showing it originates from initial states and the probability reproducibility condition, and emphasizes its physical validity in certain events.

Contribution

It provides a formal derivation of quantum probability from measurement theory, linking initial states and the probability reproducibility condition to the standard probability law.

Findings

Quantum probability formula originates from initial state expansion.

Probability reproducibility condition leads to the measurement result distribution.

Quantum probability law is physically valid in probability-one events.

Abstract

To begin with, it is pointed out that the form of the quantum probabil- ity formula originates in the very initial state of the object system as seen when the state is expanded with the eigen-projectors of the measured ob- servable. Making use of the probability reproducibility condition, which is a key concept in unitary measurement theory, one obtains the relevant coher- ent distribution of the complete-measurement results in the final unitary- measurement state in agreement with the mentioned probability formula. Treating the transition from the final unitary, or premeasurement, state, where all possible results are present, to one complete-measurement result sketchily in the usual way, the well-known probability formula is derived. In conclusion it is pointed out that the entire argument is only formal unless one makes it physical assuming that the quantum probability law is valid in the extreme case of probability-one (certain) events (projectors).