Tight Generalization of Robertson-Type Uncertainty Relations

Abstract

We establish the tightest possible Robertson-type preparation uncertainty relation, which explicitly depends on the eigenvalues of the quantum state. The conventional constant $\frac{1}{4}$ is replaced by a state-dependent coefficient $\frac{(\hat{I}{»}_{\max }+\hat{I}{»}_{\min }{)}^2}{4(\hat{I}{»}_{\max }-\hat{I}{»}_{\min }{)}^2}$, where $\hat{I}{»}_{\max }$ and $\hat{I}{»}_{\min }$ denote the largest and smallest eigenvalues of the density operator $\ddot{I}$, 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 Schröedinger'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.

7 pages, 1 figure