Preparational Uncertainty Relations for N Continuous Variables

Spiros Kechrimparis,Stefan Weigert

doi:10.3390/math4030049

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.

Highlights

Uncertainty relations express limitations on the precision with which one can measure specific properties of a quantum system, such as position and momentum of a quantum particle
We have presented a method to systematically determine lower bounds of uncertainty functionals, defined in terms of second moments of quantum systems with two or more continuous variables
In analogy to the one-dimensional case discussed in [13], we find that the states which extremize an uncertainty functional of N degrees of freedom must satisfy a eigenvalue equation which is quadratic in the 2N position and momentum operators

Summary

Introduction

Uncertainty relations express limitations on the precision with which one can measure specific properties of a quantum system, such as position and momentum of a quantum particle. Error-disturbance uncertainty relations refer to the constraints encountered when attempting to extract precise values through measurements on a single system. Both cases point to the uncertainty inherent in the quantum description of the world.

Lower bounds of uncertainty functionals

Inequalities for two or more continuous variables

The uncertainty region

Summary