Quantum information in Riemannian spaces

• URL: http://arxiv.org/abs/2412.02979v3
• Date: Tue, 07 Jan 2025 13:55:14 GMT
• Title: Quantum information in Riemannian spaces
• Authors: Pablo G. Camara,
• Abstract summary: This work bridges concepts from information theory, geometry, and quantum physics. It provides a systematic approach to studying quantum information in continuous and curved sample spaces.
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• Abstract: We develop a diffeomorphism-invariant formulation of differential entropy for Riemannian spaces, addressing the lack of a fine-grained, coordinate-independent notion of information for continuous variables in physical space. We extend this formulation to the quantum domain by generalizing Wigner's quasiprobability density function to arbitrary Riemannian spaces and analytically continuing Shannon's differential entropy to include contributions from intermediate virtual quantum states. To demonstrate this framework, we compute the quantum phase space entropy of the harmonic oscillator energy eigenstates in Minkowski and anti-de Sitter geometries. Additionally, we derive a generalized quantum entropic uncertainty relation, extending the Bialynicki-Birula and Mycielski inequality to curved backgrounds. This work bridges concepts from information theory, geometry, and quantum physics to provide a systematic approach to studying quantum information in continuous and curved sample spaces.

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