The Generalized Uncertainty Principle

The uncertainty principle lies at the heart of quantum physics, and is widely thought of as a fundamental limit of the measurement precision of incompatible observables. Here it is shown that the traditional uncertainty relation in fact belongs to the leading order approximation of a generalized uncertainty relation. That is, the leading order linear dependence of observables gives the Heisenberg type of uncertainty relations, while higher order nonlinear dependence may reveal more different and interesting correlation properties. Applications of the generalized uncertainty relation and the high order nonlinear dependence between observables in quantum information science are also discussed. A generalized uncertainty principle is proposed, by which the complementarity between observables is interpreted as a dependence rather than a constraint on the precision of measurements. In the new formalism of generalized uncertainty relation, the high order nonlinear dependences are indispensable and the conventional Heisenberg type uncertainty relation turns out to be merely the lowest order linear dependence.