Universal Uncertainty Principle, Simultaneous Measurability, and Weak Values

TL;DR

This paper proposes a new state-dependent theory allowing simultaneous measurement of non-commuting observables, challenging traditional views, and explores the relationship between weak values and measurement outcomes.

Contribution

It introduces a novel framework for simultaneous measurements based on state dependence, expanding the understanding of measurement in quantum mechanics.

Findings

Nowhere commuting observables can be measured simultaneously in certain states.

Weak values are related to the output distributions of these measurements.

The conventional equivalence of commutativity and measurability is challenged.

Abstract

In the conventional formulation, it is broadly accepted that simultaneous measurability and commutativity of observables are equivalent. However, several objections have been claimed that there are cases in which even nowhere commuting observables can be measured simultaneously. Here, we outline a new theory of simultaneous measurements based on a state-dependent formulation, in which nowhere commuting observables are shown to have simultaneous measurements in some states, so that the known objections to the conventional theory are theoretically justified. We also discuss new results on the relation between weak values and output probability distributions of simultaneous measurements.