  We combine traditional pointer-based simultaneous measurements of conjugate
observables with the concept of quantum Brownian motion of multipartite systems
to phenomenologically model simultaneous measurements of conjugate observables
in a thermal environment. This approach provides us with a formal solution of
the complete measurement dynamics for quadratic Hamiltonians and we can
therefore discuss the measurement uncertainty and optimal measurement times. As
a main result, we obtain a lower bound for the uncertainty of a noisy
measurement, which is an extension of a previously known uncertainty relation
and in which the squeezing of the system state to be measured plays an
important role. This also allows us to classify minimal uncertainty states in
more detail.
