Tight Generalization of Robertson-Type Uncertainty Relations

We establish the tightest possible Robertson-type preparation uncertainty relation, which explicitly depends on the eigenvalues of the quantum state. The conventional constant $\tfrac{1}{4}$ is replaced by a state-dependent coefficient $\frac{(\lambda_{\max} + \lambda_{\min})^2}{4(\lambda_{\max} - \lambda_{\min})^2}$, where $\lambda_{\max}$ and $\lambda_{\min}$ denote the largest and smallest eigenvalues of the density operator $\rho$, respectively. 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's relation fails to detect. In addition, our result also provides a strict generalization of the Schr\"oedinger's 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.