Near-optimal variance-based uncertainty relations

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.Ministry of Education (MOE)National Research Foundation (NRF)Published versionYX is supported by the Natural Sciences, the National Research Foundation (NRF). Singapore, under its NRFF Fellow programme (Grant No. NRF-NRFF2016-02), Singapore Ministry of Education Tier 1 Grants RG162/19 (S), the Quantum Engineering Program QEP-SF3, and No FQXi-RFP-1809 (The Role of Quantum Effects in Simplifying Quantum Agents) from the Foundational Questions Institute and Fetzer Franklin Fund (a donor-advised fund of Silicon Valley Community Foundation). BY acknowledges the support of Startup Funding of Guangdong Polytechnic Normal University No. 2021SDKYA178, and Guangdong Basic and Applied Basic Research Foundation No. 2020A1515111007. S-MF acknowledges the support of National Natural Science Foundation of China (NSFC) under Grant Nos. 12075159 and 12171044; Beijing Natural Science Foundation (Grant No. Z190005); the Academician Innovation Platform of Hainan Province. The work is supported by National Natural Science Foundation of China (grant Nos. 12126351, 12126314 and 11531004), Natural Science Foundation of Hubei Province grant No. 2020CFB538, China Scholarship Council and Simons Foundation grant No. 523868