Quantum Probability Geometrically Realized in Projective Space

TL;DR

This paper geometrically reformulates quantum probability within projective space, offering a unified and visualizable framework that encompasses quantum events, collapse, and entanglement without relying on traditional zero-dimensional states.

Contribution

It introduces a geometric approach to quantum probability in projective space, extending the understanding of quantum events and entanglement within a purely geometric framework.

Findings

Quantum probability formulas are mapped to projective space.

Quantum theory is interpreted as probability theory of projective subspaces.

Framework is adaptable to general von Neumann algebras.

Abstract

The principal goal of this paper is to pass all quantum probability formulas to the projective space associated to the complex Hilbert space of a given quantum system, providing a more complete geometrization of quantum theory. Quantum events have consecutive and conditional probabilities, which have been used in the author's previous work to clarify `collapse' and to generalize the concept of entanglement by incorporating it into quantum probability theory. In this way all of standard textbook quantum theory can be understood as a geometric theory of projective subspaces without any special role for the zero-dimensional projective subspaces, which are also called pure states. The upshot is that quantum theory is the probability theory of projective subspaces, or equivalently, of quantum events. For the sake of simplicity the ideas are developed here in the context of a type I factor, but comments will be given about how to adopt this approach to more general von Neumann algebras.