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 continuous and 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 a 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 approach to information theory of continuous variables, to characterize and quantify fundamental properties of geometric quantum states: the quantum information dimension, which 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 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.
- Received 12 November 2021
- Accepted 13 May 2022
DOI:https://doi.org/10.1103/PRXQuantum.3.020355
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.
Published by the American Physical Society
Physics Subject Headings (PhySH)
Popular Summary
Quantum mechanics is the best framework we have to understand and predict phenomena at and below the molecular scale. Notoriously, it differs significantly from the classical mechanics of systems we encounter in everyday life. Geometric quantum mechanics exploits the best of both as it frames quantum physics in a way that closely resembles classical mechanics. Relying on differential geometry, it opens the door to a number of foundational questions and leads to new tools of analysis for quantum phenomena. In particular, this work exploits two concepts from Renyi’s information theory of continuous variables: the geometric quantum entropy and the quantum information dimension. The work rigorously defines them and introduces a range of tools, both analytical and numerical, to compute them in physically relevant settings. Examples include a finite-dimensional quantum system interacting with a finite environment, an electron in a two-dimensional box, chaotic quantum dynamics and quantum fractals via Baker’s and Standard maps, and the thermodynamic limit. Altogether, geometric quantum mechanics gives a new point of contact between quantum mechanics and analog information theory, with a deep and rich phenomenology to explore.
