On some states minimizing uncertainty relations: A new look at these relations

TL;DR

This paper investigates states in quantum systems where the traditional uncertainty relations have zero lower bounds, revealing new states and insights into the nature of quantum uncertainty.

Contribution

The study identifies a large set of non-eigenstates with zero lower bounds in uncertainty relations, challenging conventional understanding.

Findings

States with zero lower bounds are not eigenstates of A or B.

Sum uncertainty relations do not constrain these states.

Uncertainty principle has dual aspects: lower bounds and correlation bounds.

Abstract

Analyzing Heisenberg--Robertson (HR) and Schr\"{o}dinger uncertainty relations we found, that there can exist a large set of states of the quantum system under considerations, for which the lower bound of the product of the standard deviations of a pair of non--commuting observables, $A$ and $B$, is zero, and which differ from those described in the literature. These states are not eigenstates of either the observable $A$ or $B$. The correlation function for these observables in such states is equal to zero. We have also shown that the so--called "sum uncertainty relations" also do not provide any information about lower bounds on the standard deviations calculated for these states. We additionally show that the uncertainty principle in its most general form has two faces: one is that it is a lower bound on the product of standard deviations, and the other is that the product of standard deviations is an upper bound on the modulus of the correlation function of a pair of the non--commuting observables in the state under consideration.