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The uncertainty principle imposes a fundamental limit on predicting themeasurement outcomes of incompatible observables even if complete classicalinformation of the system state is known. The situation is different if one canbuild a quantum memory entangled with the system. Zero uncertainty states (incontrast with minimum uncertainty states) are peculiar quantum states that caneliminate uncertainties of incompatible von Neumann observables once assistedby suitable measurements on the memory. Here we determine all zero uncertaintystates of any given set of nondegenerate observables and determine the minimumentanglement required. It turns out all zero uncertainty states are maximallyentangled in a generic case, and vice versa, even if these observables are onlyweakly incompatible. Our work establishes a simple and precise connectionbetween zero uncertainty and maximum entanglement, which is of interest tofoundational studies and practical applications, including quantumcertification and verification.

Zero Uncertainty States in the Presence of Quantum Memory