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# Distributed quantum phase estimation with entangled photons

## Abstract

Distributed quantum metrology can enhance the sensitivity for sensing spatially distributed parameters beyond the classical limits. Here we demonstrate distributed quantum phase estimation with discrete variables to achieve Heisenberg limit phase measurements. Based on parallel entanglement in modes and particles, we demonstrate distributed quantum sensing for both individual phase shifts and an averaged phase shift, with an error reduction up to 1.4 dB and 2.7 dB below the shot-noise limit. Furthermore, we demonstrate a combined strategy with parallel mode entanglement and multiple passes of the phase shifter in each mode. In particular, our experiment uses six entangled photons with each photon passing the phase shifter up to six times, and achieves a total number of photon passes N = 21 at an error reduction up to 4.7 dB below the shot-noise limit. Our research provides a faithful verification of the benefit of entanglement and coherence for distributed quantum sensing in general quantum networks.

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## Data availability

The data that support the plots within this paper and other findings of this study are available from the corresponding authors upon reasonable request.

## Code availability

The code used for modelling the data is available from F.X. on reasonable request.

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## Acknowledgements

This work was supported by the National Key Research and Development (R&D) Plan of China (2018YFB0504300 and 2018YFA0306501), the National Natural Science Foundation of China (11425417, 61771443 and U1738140), the Shanghai Municipal Science and Technology Major Project (2019SHZDZX01), the Anhui Initiative in Quantum Information Technologies, the Chinese Academy of Sciences and the Shanghai Science and Technology Development Funds (18JC1414700).

## Author information

Authors

### Contributions

F.X., Y.-A.C. and J.-W.P. conceived the research and designed the experiments. L.-Z.L., F.X. and Y.-A.C. designed and characterized the multiphoton optical circuits. L.-Z.L., Z.-D.L., R.Z., X.-F.Y., Y.-Y.F., L.L. and N.-L.L carried out the experiments. L.-Z.L., R.Z. and Y.-A.C. analysed the data. Y.-Z.Z. and F.X. performed the theory calculations. L.-Z.L., Y.-Z.Z., F.X., Y.-A.C. and J.-W.P. wrote the manuscript, with input from all authors.

### Corresponding authors

Correspondence to Feihu Xu, Yu-Ao Chen or Jian-Wei Pan.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

### Peer review information

Nature Photonics thanks Xueshi Guo and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Extended data

### Extended Data Fig. 1 Experimental results for the parallel strategies of MsPe for mode 2 and mode 3.

a,c, The average outcome probability in the measurement basis σx2, for two-photon entangled states (2′5′ and 4′6 respectively). Blue (orange) lines represent the average outcome probability P+2′5′ and P+4′6 (P+2′5′ and P+4′6) for mode 2 and mode 3. b,d, The fisher information per trial, fitted from $${P}_{2^{\prime} 5^{\prime} }$$ and $${P}_{4^{\prime} 6}$$ for MsPs, respectively. The shaded areas correspond to the 90% confidence region, derived from uncertainty in the fitting parameters. Error bars are calculated from measurement statistics and too small to be visible. Red dot-dashed line: the theoretical limit for MePs. Blue dashed line: the theoretical limit of MsPe. Black dotted line: the theoretical value of the SNL for MsPs.

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Liu, LZ., Zhang, YZ., Li, ZD. et al. Distributed quantum phase estimation with entangled photons. Nat. Photonics 15, 137–142 (2021). https://doi.org/10.1038/s41566-020-00718-2

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• DOI: https://doi.org/10.1038/s41566-020-00718-2

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