  We demonstrate a fundamental principle of disturbance tradeoff for quantum
measurements, along the lines of the celebrated uncertainty principle: The
disturbances associated with measurements performed on distinct yet identically
prepared ensembles of systems in a pure state cannot all be made arbitrarily
small. Indeed, we show that the average of the disturbances associated with a
set of projective measurements is strictly greater than zero whenever the
associated observables do not have a common eigenvector. For such measurements,
we show an equivalence between disturbance tradeoff measured in terms of
fidelity and the entropic uncertainty tradeoff formulated in terms of the
Tsallis entropy ($T_{2}$). We also investigate the disturbances associated with
the class of non-projective measurements, where the difference between the
disturbance tradeoff and the uncertainty tradeoff manifests quite clearly.
