Quantum information in Riemannian spaces

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.