# $\pi$-Corrected Heisenberg Limit

## TL;DR

This paper establishes a fundamental lower bound on parameter estimation precision in quantum systems, revealing a correction factor of pi to the traditional Heisenberg limit, applicable even with quantum error correction.

## Contribution

It introduces a pi-corrected Heisenberg limit that is tighter and universally applicable, correcting the conventional bound based on quantum Fisher information.

## Key findings

- The new bound is asymptotically tight for total uses of the unitary.
- The bound applies regardless of measurement protocol or noise elimination techniques.
- Conventional Heisenberg limit is never saturable in practice.

## Abstract

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.