  We derive an optimal bound on the sum of entropic uncertainties of two or
more observables when they are sequentially measured on the same ensemble of
systems. This optimal bound is shown to be greater than or equal to the bounds
derived in the literature on the entropic uncertainties of two observables
which are measured on distinct but identically prepared ensembles of systems.
In the case of a two-dimensional Hilbert Space, the optimal bound for
successive measurements of two spin components, is seen to be strictly greater
than the optimal bound for the case when they are measured on distinct
ensembles, except when the spin components are mutually parallel or
perpendicular.
