A Universal Formulation of Uncertainty Relation for Error-Disturbance and Local Representability of Quantum Observables

Abstract

A universal formulation of the quantum uncertainty regarding quantum indeterminacy, quantum measurement, and its inevitable observer effect is presented with additional focus on the representability of quantum observables over a given state. The operational tangibility of the framework assures that the resultant general relations admit natural operational interpretations and characterisations, and thereby perhaps most importantly, their experimental verifiability. In view of the universal formulation, Heisenberg's original philosophy of the uncertainty principle, most typically exemplified in his famous gamma-ray microscope Gedankenexperiment, is revisited; it is reformulated and restated as a refined no-go theorem, albeit perhaps in a weaker form than was originally intended. The relations entail, in essence as corollaries to their special cases, several previously known relations, including most notably the standard Kennard-Robertson relation, the Arthurs-Kelly-Goodman relations for joint measurement as well as the Ozawa and the Watanabe-Sagawa-Ueda relations for error and error-disturbance.