A Universal Formulation of Uncertainty Relation for Errors under Local Representability

TL;DR

This paper introduces a universal framework for quantum measurement uncertainty relations based on local representability, providing operationally meaningful, experimentally verifiable bounds that unify and extend existing uncertainty principles.

Contribution

It presents a new universal formulation of uncertainty relations for quantum errors, emphasizing local representability and unifying several known relations under a common framework.

Findings

Reformulates Heisenberg's uncertainty principle as a refined no-go theorem.

Derives known relations like Arthurs-Kelly-Goodman, Ozawa, and Watanabe-Sagawa-Ueda as special cases.

Shows the Schrödinger relation as a special case when measurements are non-informative.

Abstract

A universal formulation of uncertainty relations for quantum measurements is presented with additional focus on the representability of quantum observables by classical observables over a given state. Owing to the simplicity and operational tangibility of the framework, the resultant general relations admit natural operational interpretations and characterisations, and are thus also experimentally verifiable. In view of the universal formulation, Heisenberg's philosophy of the uncertainty principle is also revisited; it is reformulated and restated as a refined no-go theorem, albeit perhaps in a weaker form than was originally intended. In fact, the relations entail, in essence as corollaries to their special cases, several previously known relations, including most notably the Arthurs-Kelly-Goodman, Ozawa, and Watanabe-Sagawa-Ueda relations for quantum measurements. The Schr{\"o}dinger relation (hence the standard Kennard-Robertson relation as its trivial corollary as well) is also shown to be a special case when the measurement is non-informative.