Uncertainty relations for characteristic functions

Abstract

We present the uncertainty relation for the characteristic functions (ChUR) of the quantum mechanical position and momentum probability distributions. This inequality is more general than the Heisenberg Uncertainty Relation, and is saturated in two extremal cases for wavefunctions described by periodic Dirac combs. We further discuss a broad spectrum of applications of the ChUR, in particular, we constrain quantum optical measurements involving general detection apertures and provide the uncertainty relation that is relevant for Loop Quantum Cosmology. A method to measure the characteristic function directly using an auxiliary qubit is also briefly discussed.

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• Strong unitary and overlap uncertainty relations: theory and experiment
• Entropic uncertainty relations for successive measurements of canonically conjugate observables
• Mutual Unbiasedness in Coarse-grained Continuous Variables
• Uncertainty Relations for coarse-grained measurements: an overview
• Testing for entanglement with periodic coarse-graining
• Experimental investigation of uncertainty relations for non-Hermitian operators
• Work and Fluctuations: Coherent vs. Incoherent Ergotropy Extraction
• Improved unitary uncertainty relations
• Phase-number uncertainty from Weyl commutation relations