Zero uncertainty states in the presence of quantum memory

The uncertainty principle imposes a fundamental limit on predicting the measurement outcomes of incompatible observables even if complete classical information system state is known. situation different one can build quantum memory entangled with system. Zero states (in contrast minimum states) are peculiar that eliminate uncertainties von Neumann once assisted by suitable measurements memory. Here we determine all zero any given set nondegenerate and entanglement required. It turns out maximally in generic case, vice versa, these only weakly incompatible. Our work establishes simple precise connection between maximum entanglement, which interest to foundational studies practical applications, including certification verification.