The original formulation of the uncertainty relation by Heisenberg, which is based on a thought experiment with an emphasis on measurement processes, lacked a simple mathematical basis compared to the widely accepted relations of Kennard and Robertson. In fact, a commonly assumed form of the Heisenberg-type error-disturbance relation has been recently invalidated by spin-measurements at Vienna [J. Erhart, et al., Nature Phys. 8, 185 (2012)]. On the other hand, the analysis of measurement processes is missing in the relations of Kennard and Robertson. Here we suggest to reformulate the Heisenberg uncertainty relation in such a manner that it incorporates both the intrinsic quantum fluctuations and the effects of measurement, and yet with the same mathematical rigor as the relations of Kennard and Robertson and thus universally valid. This relation, which assumes the form δxδp ∼ ℏ(instead of ℏ/2) when written in a popular notation, is regarded as a combination of the past works on the uncertainty relation by Arthurs and Kelly, who emphasized the role of measurement apparatus, and by Ozawa who clarified the mathematical structure.

Ref. K. Fujikawa. Phys. Rev. A85, 062117 (2012).