Universality of the Heisenberg limit for estimates of random phase shifts

TL;DR

This paper rigorously proves the universal validity of the Heisenberg limit for phase estimation, closing previous loopholes and applying to all measurement schemes, including complex and nonlinear setups.

Contribution

It provides the first general, constraint-free, non-asymptotic proof of the Heisenberg limit applicable to all phase estimation methods.

Findings

Heisenberg limit holds universally for all phase estimation schemes.

The proof closes previous loopholes and applies to nonlinear and multimode probes.

The result is non-asymptotic and constraint-free, ensuring broad applicability.

Abstract

The Heisenberg limit traditionally provides a lower bound on the phase uncertainty scaling as 1/<N>, where <N> is the mean number of photons in the probe. However, this limit has a number of loopholes which potentially might be exploited, to achieve measurements with even greater accuracy. Here we close these loopholes by proving a completely rigorous form of the Heisenberg limit for the average error over all phase shifts. Our result gives the first completely general, constraint-free and non-asymptotic statement of the Heisenberg limit. It holds for all phase estimation schemes, including multiple passes, nonlinear phase shifts, multimode probes, and arbitrary measurements.