Preparational Uncertainty Relations for $N$ Continuous Variables

Abstract

A smooth function of the second moments of $N$ continuous variables gives rise to an uncertainty relation if it is bounded from below. We present a method to systematically derive such bounds by generalizing an approach applied previously to a single continuous variable. New uncertainty relations are obtained for multi-partite systems which allow one to distinguish entangled from separable states. We also investigate the geometry of the "uncertainty region" in the $N(2N+1)$-dimensional space of moments. It is shown to be a convex set for any number continuous variables, and the points on its boundary found to be in one-to-one correspondence with pure Gaussian states of minimal uncertainty. For a single degree of freedom, the boundary can be visualized as one sheet of a "Lorentz-invariant" hyperboloid in the three-dimensional pace of second moments.

19 pages, 1 figure. Material rearranged to match published version

References in corpus (4)

• Uncertainty Relations for Angular Momentum
• Detection of transverse entanglement in phase space
• Detecting entanglement of continuous variables with three mutually unbiased bases
• Universality in Uncertainty Relations for a Quantum Particle

Cited by in corpus (5)

• State-independent Uncertainty Relations and Entanglement Detection in Noisy Systems
• Variance uncertainty relations without covariances for three and four observables
• State-independent preparation uncertainty relations
• Majorization uncertainty relations for mixed quantum states
• Universality in Uncertainty Relations for a Quantum Particle