Quantum Foundations in the Light of Quantum Information

TL;DR

This paper presents three simple theorems that deepen understanding of quantum probability, collapse, and entanglement, with implications for quantum foundations and interpretations.

Contribution

It offers new derivations of key quantum principles using Gleason-like theorems and classical probability concepts, broadening foundational insights.

Findings

Derived quantum probability law from POVMs, applicable to low-dimensional spaces.

Reformulated quantum collapse as a Bayesian update rule.

Derived tensor-product rule and entanglement from local measurements and classical communication.

Abstract

This paper reports three almost trivial theorems that nevertheless appear to have significant import for quantum foundations studies. 1) A Gleason-like derivation of the quantum probability law, but based on the positive operator-valued measures as the basic notion of measurement (see also Busch, quant-ph/9909073). Of note, this theorem also works for 2-dimensional vector spaces and for vector spaces over the rational numbers, where the standard Gleason theorem fails. 2) A way of rewriting the quantum collapse rule so that it looks almost precisely identical to Bayes rule for updating probabilities in classical probability theory. And 3) a derivation of the tensor-product rule for combining quantum systems (and with it the very notion of quantum entanglement) from Gleason-like considerations for local measurements on bipartite systems along with classical communication.