Tight Generalization of Robertson-Type Uncertainty Relations

We establish the tightest possible Robertson-type preparation uncertainty relation, which explicitly depends on the eigenvalue spectrum of the quantum state. The conventional constant 1/4 is replaced by a state-dependent coefficient with the largest and smallest eigenvalues of the density operator. This coefficient is optimal among all Robertson-type generalizations and does not admit further improvement. Our relation becomes more pronounced as the quantum state becomes more mixed, capturing a trade-off in quantum uncertainty that the conventional Robertson relation fails to detect. In addition, our result provides a strict generalization of the Schroedinger uncertainty relation, showing that the uncertainty trade-off is governed by the sum of the covariance term and a state-dependent improvement over the Robertson bound. As applications, we also refine error-disturbance trade-offs by incorporating spectral information of both the system and the measuring apparatus, thereby generalizing the Arthurs-Goodman and Ozawa inequalities.