Universality and Optimality in the Information-Disturbance Tradeoff

We investigate the tradeoff between the quality of an approximate version of a given measurement and the disturbance it induces in the measured quantum system. We prove that if the target measurement is a non-degenerate von Neumann measurement, then the optimal tradeoff can always be achieved within a two-parameter family of quantum devices that is independent of the chosen distance measures. This form of almost universal optimality holds under mild assumptions on the distance measures such as convexity and basis independence, which are satisfied for all the usual cases that are based on norms, transport cost functions, relative entropies, fidelities, etc., for both worst-case and average-case analyses. We analyze the case of the cb-norm (or diamond norm) more generally for which we show dimension independence of the derived optimal tradeoff for general von Neumann measurements. A SDP solution is provided for general POVMs and shown to exist for arbitrary convex semialgebraic distance measures.