Incompatibility of unbiased qubit observables and Pauli channels

Abstract

A quantum observable and a channel are considered compatible if they form parts of the same measurement device, otherwise they are incompatible. Constrains on compatibility between observables and channels can be quantified via relations highlighting the necessary tradeoffs between noise and disturbance within quantum measurements. In this paper we shall discuss the general properties of these compatibility relations and then fully characterize the compatibility conditions for an unbiased qubit observable and a Pauli channel. The implications of the characterization are demonstrated on some concrete examples.