Universal quantum uncertainty relations: Minimumuncertainty wave packet depends on measure of spread
Abstract
Based on the statistical concept of the median, we propose a quantum uncertainty relation between semiinterquartile ranges of the position and momentum distributions of arbitrary quantum states. The relation is universal, unlike that based on the mean and standard deviation, as the latter may become nonexistent or ineffective in certain cases. We show that the medianbased one is not saturated for Gaussian distributions in position. Instead, the CauchyLorentz distributions in position turn out to be the one with the minimal uncertainty, among the states inspected, implying that the minimumuncertainty state is not unique but depends on the measure of spread used. Even the ordering of the states with respect to the distance from the minimum uncertainty state is altered by a change in the measure. We invoke the completeness of Hermite polynomials in the space of all quantum states to probe the medianbased relation. The results have potential applications in a variety of studies including those on the quantumtoclassical boundary and on quantum cryptography.
 Publication:

Physics Letters A
 Pub Date:
 June 2019
 DOI:
 10.1016/j.physleta.2019.03.012
 arXiv:
 arXiv:1706.00720
 Bibcode:
 2019PhLA..383.1850B
 Keywords:

 Quantum uncertainty relation;
 Quantum information;
 Quantum mechanics;
 Quantum Physics
 EPrint:
 9 pages