  Heisenberg's uncertainty principle is one of the main tenets of quantum
theory. Nevertheless, and despite its fundamental importance for our
understanding of quantum foundations, there has been some confusion in its
interpretation: although Heisenberg's first argument was that the measurement
of one observable on a quantum state necessarily disturbs another incompatible
observable, standard uncertainty relations typically bound the indeterminacy of
the outcomes when either one or the other observable is measured. In this
paper, we quantify precisely Heisenberg's intuition. Even if two incompatible
observables cannot be measured together, one can still approximate their joint
measurement, at the price of introducing some errors with respect to the ideal
measurement of each of them. We present a new, tight relation characterizing
the optimal trade-off between the error on one observable versus the error on
the other. As a particular case, our approach allows us to characterize the
disturbance of an observable induced by the approximate measurement of another
one; we also derive a stronger error-disturbance relation for this scenario.
