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Tight Generalization of Robertson-Type Uncertainty Relations

Gen Kimura, Aina Mayumi, Haruki Yamashita

TL;DR

This work derives a tight Robertson-type preparation uncertainty bound whose coefficient depends on the extreme eigenvalues of the state ρ, explicitly $c'_{opt}(ρ)=\frac{(λ_{max}+λ_{min})^2}{4(λ_{max}-λ_{min})^2}$, thereby sharpening the classic Robertson bound for mixed states. It further strengthens Schrödinger's relation with a state-dependent extra Covariance term and proves the bound's optimality. The authors also compare the generalized bounds to traditional ones in the qubit case, showing improvements that grow with state mixedness, and extend the framework to measurement-disturbance relations, yielding spectral-information–aware Arthurs–Goodman and Ozawa inequalities. Collectively, the results provide sharper, spectrum-informed uncertainty relations applicable to preparation and measurement scenarios in finite-dimensional quantum systems. The work highlights the practical impact of eigenvalue information on fundamental quantum limits and suggests avenues for more accurate error-disturbance analyses.

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 $\frac{(λ_{\max} + λ_{\min})^2}{4(λ_{\max} - λ_{\min})^2}$, where $ λ_{\max} $ and $ λ_{\min}$ 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.

Tight Generalization of Robertson-Type Uncertainty Relations

TL;DR

This work derives a tight Robertson-type preparation uncertainty bound whose coefficient depends on the extreme eigenvalues of the state ρ, explicitly , thereby sharpening the classic Robertson bound for mixed states. It further strengthens Schrödinger's relation with a state-dependent extra Covariance term and proves the bound's optimality. The authors also compare the generalized bounds to traditional ones in the qubit case, showing improvements that grow with state mixedness, and extend the framework to measurement-disturbance relations, yielding spectral-information–aware Arthurs–Goodman and Ozawa inequalities. Collectively, the results provide sharper, spectrum-informed uncertainty relations applicable to preparation and measurement scenarios in finite-dimensional quantum systems. The work highlights the practical impact of eigenvalue information on fundamental quantum limits and suggests avenues for more accurate error-disturbance analyses.

Abstract

We establish the tightest possible Robertson-type preparation uncertainty relation, which explicitly depends on the eigenvalues of the quantum state. The conventional constant is replaced by a state-dependent coefficient , where and denote the largest and smallest eigenvalues of the density operator , 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.

Paper Structure

This paper contains 6 sections, 2 theorems, 49 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Generalization of Robertson's relation
  3. Generalization of Schrödinger's relation
  4. Comparison with Robertson and Schrödinger bounds
  5. Applications to measurement type uncertainty relations
  6. Conclusions and Outlook

Key Result

Theorem 1

Let $A$ and $B$ be arbitrary quantum observables, and let $\rho$ be a quantum state that is not maximally mixed. Then the following uncertainty relation holds: where $\lambda_{\rm min}$ and $\lambda_{\rm max}$ denote the smallest and largest eigenvalues of $\rho$, respectively. Moreover, the bound is tight in the sense that there exist nonzero observables $A$ and $B$ for which the equality is att

Figures (1)

  • Figure 1: Comparison of the averaged bounds of Robertson's relation \ref{['rob2']} (blue, long-dashed), Schrödinger's relation \ref{['sch2']} (orange, short-dashed), the generalized Robertson's relation \ref{['grob2']} (green, dotted), and the generalized Schrödinger's relation \ref{['gsch2']} (red, dot-dashed).

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2