A disturbance tradeoff principle for quantum measurements

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.