$\pi$-corrected Heisenberg limit

We show that for the most general adaptive noiseless estimation protocol, where $U_\varphi = e^{ i \varphi \Lambda}$ is a unitary map describing the elementary evolution of the probe system, the asymptotically saturable bound on the precision of estimating $\varphi$ takes the form $\Delta \varphi \geq \frac{\pi}{n (\lambda_+ - \lambda_-)}$, where $n$ is the number of applications of $U_\varphi$, while $\lambda_+$, $\lambda_-$ are the extreme eigenvalues of the generator $\Lambda$. This differs by a factor of $\pi$ from the conventional bound, which is derived directly from the properties of the quantum Fisher information. That is, the conventional bound is never saturable. This result applies both to idealized noiseless situations as well as to cases where noise can be effectively canceled out using variants of quantum-error correcting protocols.