Uncertainty relations based on state-dependent norm of commutator

TL;DR

This paper develops new state-dependent uncertainty relations based on the norm of commutators, which outperform traditional bounds especially for mixed states and mutually unbiased observables, revealing new quantum complementarity.

Contribution

Introduces two novel uncertainty relations based on state-dependent commutator norms, with one mathematically proven and the other supported by numerical evidence, surpassing existing bounds.

Findings

New uncertainty bounds outperform Robertson and Schrödinger relations.

Bounds are tighter for mixed states and mutually unbiased observables.

Numerical evidence supports the superiority of the second relation.

Abstract

We introduce two uncertainty relations based on the state-dependent norm of commutators, utilizing generalizations of the B\"ottcher-Wenzel inequality. The first relation is mathematically proven, while the second, tighter relation is strongly supported by numerical evidence. Both relations surpass the conventional Robertson and Schr\"odinger bounds, particularly as the quantum state becomes increasingly mixed. This reveals a previously undetected complementarity of quantum uncertainty, stemming from the non-commutativity of observables. We also compare our results with the Luo-Park uncertainty relation, demonstrating that our bounds can outperform especially for mutually unbiased observables.