Geometry and structure of quantum phase space

TL;DR

This paper develops a geometric framework for mixed quantum states using symplectic, complex, and Riemannian structures, offering new insights into quantum mechanics and information.

Contribution

It introduces a comprehensive geometric approach to quantum phase space for mixed states, unifying various structures to analyze quantum systems.

Findings

Provides a symplectic and complex structure on quantum state space

Enables geometric analysis of quantum systems and states

Discusses applications in quantum information and foundations

Abstract

The application of geometry to physics has provided us with new insightful information about many physical theories such as classical mechanics, general relativity, and quantum geometry (quantum gravity). The geometry also plays an important role in foundations of quantum mechanics and quantum information. In this work we discuss a geometric framework for mixed quantum states represented by density matrices, where the quantum phase space of density matrices is equipped with a symplectic structure, an almost complex structure, and a compatible Riemannian metric. This compatible triple allow us to investigate arbitrary quantum systems. We will also discuss some applications of the geometric framework.