Stronger uncertainty relations of mixed states

The Heisenberg-Robertson uncertainty relation bounds the product of the variances in the two possible measurement outcomes in terms of the expectation of the commutator of the observables. Notably, it does not capture the concept of incompatible observables because it can be trivial, i.e., the lower bound can be null even for two noncompatible observables. Here, we give two stronger uncertainty relations, relating to the sum of variances with respect to density matrix, whose lower bounds are guaranteed to be nontrivial whenever the two observables are incompatible on the state of the system; moreover, two stronger uncertainty relations in terms of the product of the variances of two observables are established. Also, several stronger uncertainty relations for three observables are established, relating to the sum and product of variances with respect to density matrix, respectively.