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.

arXiv comment15 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

References in corpus (25)

• Decoherence, einselection, and the quantum origins of the classical
• Quantum information with continuous variables
• Gaussian Quantum Information
• Continuous variable quantum information: Gaussian states and beyond
• Direct Measurement of the Quantum Wavefunction
• Quantum Information and Relativity Theory
• Entropic Uncertainty Relations and their Applications
• Quantum Computation as Geometry
• Long-Distance Continuous-Variable Quantum Key Distribution over 202.81 km of Fiber
• The Second Law of Quantum Complexity
• Quantum entropy and special relativity
• Phase-space formulation of quantum mechanics and quantum state reconstruction for physical systems with Lie-group symmetries
• Continuous-variable quantum key distribution system: past, present, and future
• Deformation Quantization: Quantum Mechanics Lives and Works in Phase-Space
• A Quantum Exactly Solvable Nonlinear Oscillator with quasi-Harmonic Behaviour
• One-dimensional model of a quantum nonlinear harmonic oscillator
• A General Theory of Phase-Space Quasiprobability Distributions
• Wigner-Weyl isomorphism for quantum mechanics on Lie groups
• Quantum Wigner entropy
• Wigner functions for curved spaces I: On hyperboloids
• Classical dynamical coarse-grained entropy and comparison with the quantum version
• Quantum phase-space representation for curved configuration spaces
• Wigner distributions and quantum mechanics on Lie groups: the case of the regular representation
• Entropy-power uncertainty relations : towards a tight inequality for all Gaussian pure states
• Complex-valued Wigner entropy of a quantum state