Notes on Joint Measurability of Quantum Observables

TL;DR

This paper investigates the properties of joint measurability of sharp quantum observables, demonstrating that certain intuitive properties do not extend to general cases and discussing potential generalizations.

Contribution

It clarifies which properties of joint measurability for sharp observables fail in the general case and explores possible generalizations.

Findings

Properties (i), (ii), (iii) do not hold in general for non-sharp observables

Joint measurability does not imply a unique joint observable in general

Partitioning properties do not guarantee joint measurability for all observables

Abstract

For sharp quantum observables the following facts hold: (i) if we have a collection of sharp observables and each pair of them is jointly measurable, then they are jointly measurable all together; (ii) if two sharp observables are jointly measurable, then their joint observable is unique and it gives the greatest lower bound for the effects corresponding to the observables; (iii) if we have two sharp observables and their every possible two outcome partitionings are jointly measurable, then the observables themselves are jointly measurable. We show that, in general, these properties do not hold. Also some possible candidates which would accompany joint measurability and generalize these apparently useful properties are discussed.