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Preparational Uncertainty Relations for N Continuous Variables

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JournalMathematics
DateAccepted/In press - 9 Jul 2016
DatePublished (current) - 19 Jul 2016
Number of pages17
Pages (from-to)1-17
Original languageEnglish

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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© 2016, The Author(s).

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