A universal quantum certainty relation for arbitrary number of observables

We derive by lattice theory a universal quantum certainty relation for arbitrary M observables in N-dimensional system, which provides a state-independent maximum lower bound on the direct sum of the probability vectors in terms of majorization relation. While the utmost lower bound coincides with \((1/N,\ldots ,1/N)\) for any two observables with orthogonal bases, the majorization certainty relation for \(M\geqslant 3\) is shown to be nontrivial. The universal majorization bounds for three mutually complementary observables and a more general set of observables in dimension-2 are achieved. It is found that one cannot prepare a quantum state with probability vectors of incompatible observables spreading out arbitrarily. Moreover, we also explore the connections between quantum uncertainty and quantum coherence, and obtain a complementary relation for the quantum coherence as well, which characterizes a trade-off relation of quantum coherence with different bases and is illustrated by an explicit example.