Quantum Information Dimension and Geometric Entropy

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.

arXiv comment19 pages, 11 figures; http://csc.ucdavis.edu/~cmg/compmech/pubs/qidge.htm

References in corpus (18)

• SciPy 1.0--Fundamental Algorithms for Scientific Computing in Python
• Geometric Quantum Mechanics
• Generating random density matrices
• Exact emergent quantum state designs from quantum chaotic dynamics
• Geometrization of Quantum Mechanics
• Identifying Functional Thermodynamics in Autonomous Maxwellian Ratchets
• Random pure states: quantifying bipartite entanglement beyond the linear statistics
• Demon Dynamics: Deterministic Chaos, the Szilard Map, and the Intelligence of Thermodynamic Systems
• The Quantum Canonical Ensemble
• Monte Carlo sampling from the quantum state space. I
• Quantum and classical ergotropy from relative entropies
• Geometrical Description of Quantum Mechanics - Transformations and Dynamics
• Beyond Density Matrices: Geometric Quantum States
• Thermodynamics of Quantum Heat Bath
• An elementary introduction to the geometry of quantum states with pictures
• Information Content for Quantum States
• A geometric Hamiltonian description of composite quantum systems and quantum entanglement
• On quantum microcanonical equilibrium

Cited by in corpus (6)

• Dynamical purification and the emergence of quantum state designs from the projected ensemble
• Solvable model of deep thermalization with distinct design times
• A Maximum Entropy Principle in Deep Thermalization and in Hilbert-Space Ergodicity
• Branching States as The Emergent Structure of a Quantum Universe
• Quantum entropy couples matter with geometry
• Observation of Quantum Darwinism and the Origin of Classicality with Superconducting Circuits