Topologies and Measurable Structures on the Projective Hilbert Space

TL;DR

This paper reviews different topologies and measurable structures on the projective Hilbert space, which represents pure quantum states, highlighting their natural properties and significance in quantum theory.

Contribution

It systematically analyzes and compares the natural topologies and measurable structures on the projective Hilbert space, clarifying their roles in quantum state representation.

Findings

P(H) has a natural topology suitable for quantum state analysis.

P(H) possesses a natural measurable structure compatible with quantum measurements.

The review clarifies the relationship between topological and measurable frameworks in quantum theory.

Abstract

A systematic review of the various topologies that can be defined on the projective Hilbert space P(H), i.e., on the set of the pure quantum states, is presented. It is shown that P(H) carries a natural topology as well as a natural measurable structure.