Geometric Formulation of Universally Valid Uncertainty Relation for Error

Abstract

We present a new geometric formulation of uncertainty relation valid for any quantum measurements of statistical nature. Owing to its simplicity and tangibility, our relation is universally valid and experimentally viable. Although our relation violates the na{\"i}ve non-commutativity bound $\hbar/2$ for the measurement of position and momentum, the spirit of the uncertainty principle still stands strong. Our relation entails, among others, the Ozawa relation as a corollary, and also reduces seamlessly to the standard Kennard-Robertson relation when the measurement is non-informative.