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Abstract

Heisenberg's uncertainty principle introduced in 1927 usually states that non-commuting observables are not simultaneously measurable. However, the precise meaning of this statement has not been clarified for long. A more rigorous version states that nowhere commuting observables are not simultaneously measurable in any state, where "nowhere commuting" means "having no common eigenstates." Unfortunately, even this version is not true, as Heisenberg in 1930 mentioned an exception that in an eigenstate of one observable any two can be simultaneously measured. Since 1930, Heisenberg has not explicitly mentioned the uncertainty principle as the limitation to simultaneous measurements. In this talk, we discuss the uncertainty principle as the universal limitation to simultaneous measurements under a modern rigorous treatments of generalized quantum measurements, in particular, its quantitative expression, the characterization of simultaneous measurements of nowhere commuting observables, and some applications to problems in precision measurements and quantum information.