Quantum information in Riemannian spaces

TL;DR

This paper develops a coordinate-independent way to measure quantum information in curved spaces using a generalized differential entropy and phase-space methods, extending quantum information theory to Riemannian geometries.

Contribution

It introduces a diffeomorphism-invariant differential entropy framework for Riemannian spaces, generalizes the Wigner function, and extends entropic uncertainty relations to curved backgrounds.

Findings

Computed quantum phase-space entropy for harmonic oscillators in Minkowski and AdS spaces

Established a generalized entropic uncertainty relation in curved geometries

Bridged concepts from information theory, geometry, and quantum physics

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.