Non-disturbing quantum measurements

TL;DR

This paper investigates the relationships between non-disturbance, joint measurability, and commutativity of quantum observables, providing conditions for their equivalence or differences and methods to quantify disturbance.

Contribution

It clarifies when non-disturbance, joint measurability, and commutativity coincide or differ, and introduces a semidefinite program to decide and quantify disturbance.

Findings

Non-disturbance is generally not symmetric.

Conditions depend on outcomes, Hilbert space dimension, and algebraic properties.

Disturbance can be decided and quantified via semidefinite programming.

Abstract

We consider pairs of quantum observables (POVMs) and analyze the relation between the notions of non-disturbance, joint measurability and commutativity. We specify conditions under which these properties coincide or differ---depending for instance on the interplay between the number of outcomes and the Hilbert space dimension or on algebraic properties of the effect operators. We also show that (non-)disturbance is in general not a symmetric relation and that it can be decided and quantified by means of a semidefinite program.