Quantum Information Dimension and Geometric Entropy

TL;DR

This paper introduces geometric quantum information dimension and geometric entropy as tools to analyze quantum states through a differential-geometric approach, bridging quantum and classical mechanics insights.

Contribution

It adapts classical information measures to quantum systems using geometric quantum mechanics, providing new ways to quantify quantum information and entropy.

Findings

Defined quantum information dimension and geometric entropy for quantum states.

Computed these measures for various quantum systems and classes.

Provided insights into the physical interpretation of quantum information measures.

Abstract

Geometric quantum mechanics, through its differential-geometric underpinning, provides additional tools of analysis and interpretation that bring quantum mechanics closer to classical mechanics: state spaces in both are equipped with symplectic geometry. This opens the door to revisiting foundational questions and issues, such as the nature of quantum entropy, from a geometric perspective. Central to this is the concept of geometric quantum state -- the probability measure on a system's space of pure states. This space's continuity leads us to introduce two analysis tools, inspired by Renyi's information theory, to characterize and quantify fundamental properties of geometric quantum states: the quantum information dimension that is the rate of geometric quantum state compression and the dimensional geometric entropy that monitors information stored in quantum states. We recount their classical definitions, information-theoretic meanings, and physical interpretations, and adapt them to quantum systems via the geometric approach. We then explicitly compute them in various examples and classes of quantum system. We conclude commenting on future directions for information in geometric quantum mechanics.