Preparational Uncertainty Relations for $N$ Continuous Variables

TL;DR

This paper develops a systematic method to derive uncertainty relations for multiple continuous variables, enabling the distinction between entangled and separable states and exploring the geometry of the uncertainty region.

Contribution

It generalizes an approach to derive bounds on second moments for multiple variables, introducing new uncertainty relations and analyzing the geometry of the uncertainty region.

Findings

Uncertainty region is convex for any number of variables.

Boundary points correspond to pure Gaussian states of minimal uncertainty.

New criteria to distinguish entangled from separable states.

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.