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
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.
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