2025/05/26 by Gen Kimura, Kimura, Gen, Aina Mayumi +2 · 1 citation
Decision Sciences · #Risk and Safety Analysis
paper · pdf · doi:10.48550/arxiv.2505.19861
We establish the tightest possible Robertson-type preparation uncertainty relation, which explicitly depends on the eigenvalues of the quantum state. The conventional constant \tfrac14 is replaced by a state-dependent coefficient \frac(λmax + λmin)24(λmax - λmin)2, where λmax and λmin denote the largest and smallest eigenvalues of the density operator ρ, 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.