Coexistence does not imply joint measurability

TL;DR

This paper clarifies that in quantum mechanics, coexistence of measurements does not necessarily imply they are jointly measurable, highlighting a nuanced distinction in measurement compatibility.

Contribution

It provides a definitive negative answer to whether coexistence implies joint measurability in general, refining the understanding of measurement compatibility.

Findings

Coexistence does not imply joint measurability in general.

Both notions are robust against small measurement imperfections.

The result clarifies the relationship between measurement compatibility concepts.

Abstract

One of the hallmarks of quantum theory is the realization that distinct measurements cannot in general be performed simultaneously, in stark contrast to classical physics. In this context the notions of coexistence and joint measurability are employed to analyze the possibility of measuring together two general quantum observables, characterizing different degrees of compatibility between measurements. It is known that two jointly measurable observables are always coexistent, and that the converse holds for various classes of observables, including the case of observables with two outcomes. Here we resolve, in the negative, the open question whether this equivalence holds in general. Our resolution strengthens the notions of coexistence and joint measurability by showing that both are robust against small imperfections in the measurement setups.