### 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, applicable to any estimate of a completely unknown phase shift. Our result gives a completely general, constraint-free, and nonasymptotic statement of the Heisenberg limit. It holds for all phase estimation schemes, including multiple passes, nonlinear phase shifts, multimode probes, and arbitrary measurements.

Original language | English |
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Article number | 041802 |

Pages (from-to) | 1-4 |

Number of pages | 4 |

Journal | Physical Review A - Atomic, Molecular, and Optical Physics |

Volume | 85 |

Issue number | 4 |

DOIs | |

Publication status | Published - 13 Apr 2012 |

### Bibliographical note

Hall, MJW, Berry, DW, Zwierz, M. and Wiseman, HM. Physical review A. Atomic, molecular, and optical physics, 85(4), 041802(R), 2012. Copyright (2012) by the American Physical Society. The original article can be found at http://dx.doi.org/10.1103/PhysRevA.85.041802## Fingerprint Dive into the research topics of 'Universality of the Heisenberg limit for estimates of random phase shifts'. Together they form a unique fingerprint.

## Cite this

*Physical Review A - Atomic, Molecular, and Optical Physics*,

*85*(4), 1-4. [041802]. https://doi.org/10.1103/PhysRevA.85.041802