Improvement of Uncertainty Relations for Mixed States

TL;DR

This paper proposes improved uncertainty relations that account for the mixedness of quantum states, providing tighter bounds especially for thermal states of harmonic oscillators, with equalities achieved in certain cases.

Contribution

The authors introduce new terms into uncertainty relations that explicitly measure state mixedness, enhancing the traditional bounds for mixed quantum states.

Findings

Improved uncertainty relations incorporate state mixedness.

Equalities hold for thermal states of quantum harmonic oscillators.

Enhanced bounds are tighter for mixed states than traditional relations.

Abstract

We study a possible improvement of uncertainty relations. The Heisenberg uncertainty relation employs commutator of a pair of conjugate observables to set the limit of quantum measurement of the observables. The Schroedinger uncertainty relation improves the Heisenberg uncertainty relation by adding the correlation in terms of anti-commutator. However both relations are insensitive whether the state used is pure or mixed. We improve the uncertainty relations by introducing additional terms which measure the mixtureness of the state. For the momentum and position operators as conjugate observables and for the thermal state of quantum harmonic oscillator, it turns out that the equalities in the improved uncertainty relations hold.