How well can one jointly measure two incompatible observables on a given quantum state?

Understand

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.