Uncertainty relations based on state-dependent norm of commutator

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.