relation, where we have introduced the anticommutator, Mixed states The Robertson–Schrödinger uncertainty relation may be generalized in a straightforward way to describe mixed states. The Maccone–Pati uncertainty relations The Robertson–Schrödinger uncertainty relation can be trivial if the state of the system is chosen to be eigenstate of one of the observable. The stronger uncertainty relations proved by Maccone and Pati give non-trivial bounds on the sum of the variances for two incompatible observables. (Earlier works on uncertainty relations formulated as the sum of variances include, e.g., Ref. due to Huang.) For two non-commuting observables and the first stronger uncertainty relation is given by where , , is a normalized vector that is orthogonal to the state of the system and one should choose the sign of to make this real quantity a positive number. The second stronger uncertainty relation is given by where is a state orthogonal to . The form of implies that the right-hand side of the new uncertainty relation is nonzero unless is an eigenstate of . One may note that can be an eigenstate of without being an eigenstate of either or . However, when is an eigenstate of one of the two observables the Heisenberg–Schrödinger uncertainty relation becomes trivial. But the lower bound in the new relation is nonzero unless is an eigenstate of both. Improving the Robertson–Schrödinger uncertainty relation based on decompositions of the density matrix The Robertson–Schrödinger uncertainty can be improved noting that it must hold for all components in any decomposition of the density matrix given as Here, for the probailities and hold. Then, using the relation for , it follows that where the function in the bound is defined The above relation very often has a bound larger than that of the original Robertson–Schrödinger uncertainty relation. Thus, we need to calculate the bound of the Robertson–Schrödinger uncertainty for the mixed components of the quantum state rather than for the quantum state, and compute an average of their square roots. The following expression is stronger than the Robertson–Schrödinger uncertainty relation where on the right-hand side there is a concave roof over the decompositions of the density matrix. The improved relation above is saturated by all single-qubit quantum states. With similar arguments, one can derive a relation with a convex roof on the right-hand side where denotes the quantum Fisher information and the density matrix is decomposed to pure states as The derivation