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