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Mathematics 2016, 4(3), 49; doi:10.3390/math4030049

Preparational Uncertainty Relations for N Continuous Variables

Department of Mathematics, University of York, York YO10 5DD, UK
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Academic Editors: Takayuki Miyadera and Teiko Heinosaari
Received: 31 March 2016 / Revised: 28 June 2016 / Accepted: 9 July 2016 / Published: 19 July 2016
(This article belongs to the Special Issue Mathematics of Quantum Uncertainty)
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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 that allow one to distinguish entangled from separable states. We also investigate the geometry of the “uncertainty region” in the N ( 2 N + 1 ) -dimensional space of moments. It is shown to be a convex set, and the points on its boundary are 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 space of second moments. View Full-Text
Keywords: quantum uncertainty; convexity; entanglement detection quantum uncertainty; convexity; entanglement detection
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Kechrimparis, S.; Weigert, S. Preparational Uncertainty Relations for N Continuous Variables. Mathematics 2016, 4, 49.

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