Near-Optimal Variance-Based Uncertainty Relations

Abstract

Learning physical properties of a quantum system is essential for the developments of quantum technologies. However, Heisenberg's uncertainty principle constrains the potential knowledge one can simultaneously have about a system in quantum theory. Aside from its fundamental significance, the mathematical characterization of this restriction, known as `uncertainty relation', plays important roles in a wide range of applications, stimulating the formation of tighter uncertainty relations. In this work, we investigate the fundamental limitations of variance-based uncertainty relations, and introduce several `near optimal' bounds for incompatible observables. Our results consist of two morphologically distinct phases: lower bounds that illustrate the uncertainties about measurement outcomes, and the upper bound that indicates the potential knowledge we can gain. Combining them together leads to an \emph{uncertainty interval}, which captures the essence of uncertainties in quantum theory. Finally, we have detailed how to formulate lower bounds for product-form variance-based uncertainty relations by employing entropic uncertainty relations, and hence built a link between different forms of uncertainty relations.

7 pages, 4 figures; Final version with new title

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Cited by in corpus (5)

• Uncertainty relations for metric adjusted skew information and Cauchy-Schwarz inequality
• Wigner-Yanase skew information-based uncertainty relations for quantum channels
• Uncertainty from the Aharonov-Vaidman Identity
• Uncertainty of quantum channels based on symmetrized \r{ho}-absolute variance and modified Wigner-Yanase skew information
• Uncertainty relations based on the $\rho$-absolute variance for quantum channels