Real-Valued Observables and Quantum Uncertainty

We first present a generalization of the Robertson-Heisenberg uncertainty principle. This generalization applies to mixed states and contains a covariance term. For faithful states, we characterize when the uncertainty inequality is an equality. We next present an uncertainty principle version for real-valued observables. Sharp versions and conjugates of real-valued observables are considered. The theory is illustrated with examples of dichotomic observables. We close with a discussion of real-valued coarse graining.