Average of uncertainty-product for observables

The goal of this paper is to calculate exactly the average of uncertainty-product of two bounded observables and to establish its typicality over the whole set of finite dimensional quantum pure states. Here we use the uniform ensembles of pure and isospectral states as well as the states distributed uniformly according to the measure induced by the Hilbert-Schmidt norm. Firstly, we investigate the average uncertainty of an observable over isospectral density matrices. By letting the isospectral density matrices be of rank-one, we get the average uncertainty of an observable restricted to pure quantum states. These results can help us check how large the gap is between the uncertainty-product and any obtained lower bounds about the uncertainty-product. Although our method in the present paper cannot give a tighter lower bound of uncertainty-product for bounded observables, it can help us drop any one that is not tighter than the known one substantially.