$\pi$-Corrected Heisenberg Limit

We consider the precision $\Delta \varphi$ with which the parameter $\varphi$, appearing in the unitary map $U_\varphi = e^{ i \varphi \Lambda}$ acting on some type of probe system, can be estimated when there is a finite amount of prior information about $\varphi$. We show that, if $U_\varphi$ acts $n$ times in total, then, asymptotically in $n$, there is a tight lower bound $\Delta \varphi \geq \frac{\pi}{n (\lambda_+ - \lambda_-)}$, where $\lambda_+$, $\lambda_-$ are the extreme eigenvalues of the generator $\Lambda$. This is greater by a factor of $\pi$ than the conventional Heisenberg limit, derived from the properties of the quantum Fisher information. That is, the conventional bound is never saturable. Our result makes no assumptions on the measurement protocol, and is relevant not only in the noiseless case but also if noise can be eliminated using quantum error correction techniques.