Quantum probability rule: a generalisation of the theorems of Gleason and Busch

TL;DR

This paper generalizes the quantum probability rule by reducing quantum postulates, deriving a more universal rule applicable in quantum communications and retrodictive quantum theory, extending Gleason's and Busch's theorems.

Contribution

It introduces a broader quantum probability rule that requires fewer assumptions, unifying and extending previous theorems by Busch and Gleason.

Findings

Derived a more general quantum probability rule without assuming effects or probability measures.

The rule reduces to the standard quantum probability rule under classical probability laws.

Demonstrated applications in quantum communications and retrodictive quantum theory.

Abstract

Busch's theorem deriving the standard quantum probability rule can be regarded as a more general form of Gleason's theorem. Here we show that a further generalisation is possible by reducing the number of quantum postulates used by Busch. We do not assume that the positive measurement outcome operators are effects or that they form a probability operator measure. We derive a more general probability rule from which the standard rule can be obtained from the normal laws of probability when there is no measurement outcome information available, without the need for further quantum postulates. Our general probability rule has prediction-retrodiction symmetry and we show how it may be applied in quantum communications and in retrodictive quantum theory.