Preparational Uncertainty Relations for N Continuous Variables

Spyridon Kechrimparis; Stefan Weigert

doi:10.3390/math4030049

Preparational Uncertainty Relations for N Continuous Variables

Spyridon Kechrimparis, Stefan Weigert

Mathematics

Research output: Contribution to journal › Article › peer-review

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.

Original language	English
Pages (from-to)	1-17
Number of pages	17
Journal	Mathematics
DOIs	https://doi.org/10.3390/math4030049
Publication status	Published - 19 Jul 2016

Bibliographical note

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10.3390/math4030049Licence: CC BY

1606.09148v1
© 2016, The Author(s).
Accepted author manuscript, 316 KBLicence: CC BY

Cite this

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title = "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. ",

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