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.

即使已知系统状态的完整经典信息，不确定性原则也对预测不兼容可观测值的测量结果施加了基本限制。如果可以构建与系统纠缠在一起的量子存储器，情况就不同了。零不确定状态（与最小不确定状态相反）是特殊的量子状态，一旦在存储器上进行了适当的测量，便可以消除不相容的冯·诺伊曼可观测物的不确定性。在这里，我们确定任何给定的非退化可观量的所有零不确定状态，并确定所需的最小纠缠度。事实证明，在一般情况下，所有零不确定性状态都将最大程度地纠缠，反之亦然，即使这些可观测值仅是弱不兼容的。