Optimized entropic uncertainty for successive projective measurements

We focus here on the uncertainty of an observable Y caused by a precise measurement of X. We illustrate the effect by analyzing the general scenario of two successive measurements of spin components X and Y. We derive an optimized entropic uncertainty limit that quantifies the necessary amount of uncertainty observed in a subsequent measurement of Y. We compare this bound to recently derived error-disturbance relations and discuss how the bound quantifies the information of successive quantum measurements.