How well can one jointly measure two incompatible observables on a given quantum state?
Heisenberg’s uncertainty principle is one of the main tenet s of quantum theory. Nevertheless, and despite its fundamental importance for our understanding of quantum fo undations, there has been some confusion in its interpretation: although Heisenberg’s first argument was t h t the measurement of one observable on a quantum state necessarily disturbs another incompatible observab l , standard uncertainty relations typically bound the indeterminacy of the outcomes when either one or the other ob servable is measured. In this paper, we quantify precisely Heisenberg’s intuition. Even if two incompatibl e observables cannot be measured together, one can still approximate their joint measurement, at the price of i ntroducing 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 th e approximate measurement of another one; we also derive a stronger error-disturbance relation for this scen ario.