Quantum uncertainty relation saturated by the eigenstates of the harmonic oscillator

Abstract

We re-derive the Schrödinger-Robertson uncertainty principle for the position and momentum of a quantum particle. Our derivation does not directly employ commutation relations, but works by reduction to an eigenvalue problem related to the harmonic oscillator, which can then be further exploited to find a larger class of constrained uncertainty relations. We derive an uncertainty relation under the constraint of a fixed degree of Gaussianity and prove that, remarkably, it is saturated by all eigenstates of the harmonic oscillator. This goes beyond the common knowledge that the (Gaussian) ground state of the harmonic oscillator saturates the uncertainty relation.

9 pages, 3 figures

References in corpus (7)

• Observation of strong coupling between a micromechanical resonator and an optical cavity field
• Generation of a superposition of odd photon number states for quantum information networks
• A No-Go Theorem for Gaussian Quantum Error Correction
• Generation of large-amplitude coherent-state superposition via ancilla-assisted photon-subtraction
• Quantifying the non-Gaussian character of a quantum state by quantum relative entropy
• Quantifying non-Gaussianity for quantum information
• A measure of the non-Gaussian character of a quantum state

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• The role of quantum non-Gaussian distance in entropic uncertainty relation
• Complex-valued Wigner entropy of a quantum state
• Entropic Uncertainty Relations via Direct-Sum Majorization Relation for Generalized Measurements
• Resource distillation in convex Gaussian resource theories
• Moments and saturation properties of eigenstates