Tight Generalization of Robertson-Type Uncertainty Relations

TL;DR

This paper derives the tightest Robertson-type uncertainty relation that explicitly depends on a quantum state's eigenvalues, improving the understanding of uncertainty trade-offs especially for mixed states and generalizing several foundational inequalities.

Contribution

It introduces a state-dependent Robertson-type uncertainty relation with an optimal coefficient based on eigenvalues, refining previous bounds and generalizing Schrödinger's relation.

Findings

The new relation is tight and optimal among Robertson-type bounds.

It captures increased uncertainty for more mixed states.

It refines error-disturbance trade-offs by incorporating spectral information.

Abstract

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.