  Heisenberg's uncertainty principle has recently led to general measurement
uncertainty relations for quantum systems: incompatible observables can be
measured jointly or in sequence only with some unavoidable approximation, which
can be quantified in various ways. The relative entropy is the natural
theoretical quantifier of the information loss when a `true' probability
distribution is replaced by an approximating one. In this paper, we provide a
lower bound for the amount of information that is lost by replacing the
distributions of the sharp position and momentum observables, as they could be
obtained with two separate experiments, by the marginals of any smeared joint
measurement. The bound is obtained by introducing an entropic error function,
and optimizing it over a suitable class of covariant approximate joint
measurements. We fully exploit two cases of target observables: (1)
$n$-dimensional position and momentum vectors; (2) two components of position
and momentum along different directions. In (1), we connect the quantum bound
to the dimension $n$; in (2), going from parallel to orthogonal directions, we
show the transition from highly incompatible observables to compatible ones.
For simplicity, we develop the theory only for Gaussian states and
measurements.
