Tight Generalization of Robertson-Type Uncertainty Relations

• URL: http://arxiv.org/abs/2505.19861v1
• Date: Mon, 26 May 2025 11:46:54 GMT
• Title: Tight Generalization of Robertson-Type Uncertainty Relations
• Authors: Gen Kimura, Aina Mayumi, Haruki Yamashita,
• Abstract summary: We establish the tightest possible Robertson-type preparation uncertainty relation, which explicitly depends on the eigenvalue spectrum of the quantum state.<n>Our relation becomes more pronounced as the quantum state becomes more mixed, capturing a trade-off in quantum uncertainty.<n>As applications, we also refine error-disturbance trade-offs by incorporating spectral information of both the system and the measuring apparatus.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We establish the tightest possible Robertson-type preparation uncertainty relation, which explicitly depends on the eigenvalue spectrum of the quantum state. The conventional constant 1/4 is replaced by a state-dependent coefficient with the largest and smallest eigenvalues of the density operator. 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 relation fails to detect. In addition, our result provides a strict generalization of the Schroedinger 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.

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