Quantum information in Riemannian spaces
Abstract
We present a diffeomorphism-invariant formulation of differential entropy for Riemannian spaces, providing a fine-grained, coordinate-independent notion of quantum information for continuous variables in physical space. To this end, we consider the generalization of the Wigner quasiprobability density function to arbitrary Riemannian manifolds and analytically continue Shannon's differential entropy to account for contributions from intermediate virtual quantum states. We illustrate the framework by computing the quantum phase-space entropy of harmonic oscillator energy eigenstates in both Minkowski and anti-de Sitter geometries. Furthermore, we derive a generalized entropic uncertainty relation, extending the Bialynicki-Birula and Mycielski inequality to curved backgrounds. By bridging concepts from information theory, differential geometry, and quantum physics, our work provides a systematic approach to studying continuous-variable quantum information in curved spaces.
Cite
@article{arxiv.2412.02979,
title = {Quantum information in Riemannian spaces},
author = {Pablo G. Camara},
journal= {arXiv preprint arXiv:2412.02979},
year = {2025}
}
Comments
15 pages, 4 figures. This is the Accepted Manuscript version of an article accepted for publication in J. Phys. A: Math. Theor. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. This Accepted Manuscript is published under a CC BY license. The Version of Record is available online at doi.org/10.1088/1751-8121/add820