  We develop a technique to find simultaneous measurements for noisy quantum
observables in finite-dimensional Hilbert spaces. We use the method to derive
lower bounds for the noise needed to make incompatible measurements jointly
measurable. Using our strategy together with recent devel- opments in the field
of one-sided quantum information processing we show that the attained lower
bounds are tight for various symmetric sets of quantum measurements. We use
this characterisation to prove the existence of so called 4-Specker sets in the
qubit case.
