Uncertainty relations for incompatible observables: Newer results versus modified strategies

TL;DR

This paper critically examines the limitations of standard uncertainty relations for incompatible observables, proposes modified strategies to address these issues, and simplifies recent alternatives for broader applicability, including experimental considerations.

Contribution

It introduces new modified strategies to overcome limitations of existing uncertainty relations and simplifies recent approaches for multiple observables, with practical experimental and theoretical implications.

Findings

Modified uncertainty relations address trivialization issues.

Alternative strategies work without auxiliary states.

Simplified methods accommodate multiple observables.

Abstract

Two special situations where the standard uncertainty product inequality appears to be useless are modified. One such case is noted to also trivialize the recently-introduced alternatives [Phys. Rev. Lett. 113, 260401 (2014); Sci. Rep. 6, 23201 (2016)] involving sums of variances. A careful discussion is presented on the experimental justifications of some of the relations [Phys. Rev. A 93, 052108 (2016)] using qutrit and qubit states. Alternative bypass routes are put forward to tackle this situation, with and without involving any auxiliary state. This latter strategy is noted to be vital in an entirely different context concerned with the quality of approximate stationary states. The other case is more frustrating, but an effective method is advanced. En route, the recent alternatives are also simplified to easily accommodate even the cases of more than two observables. In favorable circumstances, an easy option in function space is obtained by virtue of symmetry that does not involve any auxiliary state. Pilot calculations reveal the advantages of our endeavor.