A Universal Formulation of Uncertainty Relation for Error and Disturbance

TL;DR

This paper introduces a universal, experimentally verifiable uncertainty relation applicable to any quantum measurement, unifying error, disturbance, and traditional quantum indeterminacy within a single framework.

Contribution

It presents a new, simple formulation of the uncertainty principle that applies universally, including error and disturbance, and aligns with Heisenberg's original ideas.

Findings

Violates the naive $rac{ ext{hbar}}{2}$ bound for position-momentum measurement

Recovers the Kennard--Robertson relation as a special case

Provides an experimentally verifiable uncertainty relation

Abstract

We present a universal formulation of uncertainty relation valid for any conceivable quantum measurement and the resultant observation (observer) effect of statistical nature. Owing to its simplicity and operational tangibility, our general relation is also experimentally verifiable. Our relation violates the traditional na{\"i}ve bound $\hbar/2$ for the position-momentum measurement while respecting Heisenberg's original philosophy of the uncertainty principle. Our error-disturbance relation admits a parallel formulation to our relation for errors, which also embraces the standard Kennard--Robertson (Schr{\"o}dinger) relation as a special case; this attains a unified picture of the three orthodox realms of uncertainty regarding quantum indeterminacy, measurement, and observation effect within a single framework.