Universality in Uncertainty Relations for a Quantum Particle

Abstract

A general theory of preparational uncertainty relations for a quantum particle in one spatial dimension is developed. We derive conditions which determine whether a given smooth function of the particle's variances and its covariance is bounded from below. Whenever a global minimum exists, an uncertainty relation has been obtained. The squeezed number states of a harmonic oscillator are found to be universal: no other pure or mixed states will saturate any such relation. Geometrically, we identify a convex uncertainty region in the space of second moments which is bounded by the inequality derived by Robertson and Schrödinger. Our approach not only unifies existing uncertainty relations but also leads to new inequalities for second moments.

22 pages, 4 figures. Material rearranged to match published version

References in corpus (6)

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• Uncertainty Relations for Angular Momentum
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• Detecting entanglement of continuous variables with three mutually unbiased bases
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Cited by in corpus (6)

• Variance uncertainty relations without covariances for three and four observables
• State-independent preparation uncertainty relations
• Uncertainties of genuinely incompatible triple measurements based on statistical distance
• Geometry of Uncertainty Relations for Linear Combinations of Position and Momentum
• Preparational Uncertainty Relations for $N$ Continuous Variables
• Optimizing incompatible triple quantum measurements