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arXiv · 1606.09148

Preparational Uncertainty Relations for $N$ Continuous Variables

Spiros Kechrimparis·Stefan Weigert
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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.

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Spiros Kechrimparis, Stefan Weigert. 2016-10-16. Preparational Uncertainty Relations for $N$ Continuous Variables. https://doi.org/10.3390/math4030049

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