Uncertainty relations for approximation and estimation

• Several inequalities interpreted as uncertainty relations for approximation/estimation are derived from a single ‘versatile inequality’.
• The ‘versatile inequality’ sets a limit on the approximation of an observable and/or the estimation of a parameter by another observable.
• The ‘versatile inequality’ turns into an elaboration of the Robertson–Kennard (Schrödinger) inequality and the Cramér–Rao inequality.
• Both the position–momentum and the time–energy relation are treated in one framework.
• In every case, Aharonov's weak value arises as a key geometrical ingredient, deciding the optimal choice for the proxy functions.

We present a versatile inequality of uncertainty relations which are useful when one approximates an observable and/or estimates a physical parameter based on the measurement of another observable. It is shown that the optimal choice for proxy functions used for the approximation is given by Aharonov's weak value, which also determines the classical Fisher information in parameter estimation, turning our inequality into the genuine Cramér–Rao inequality. Since the standard form of the uncertainty relation arises as a special case of our inequality, and since the parameter estimation is available as well, our inequality can treat both the position–momentum and the time–energy relations in one framework albeit handled differently.