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# Distributed quantum sensing in a continuous-variable entangled network

## Abstract

Networking is integral to quantum communications1 and has significant potential for upscaling quantum computer technologies2. Recently, it was realized that the sensing performances of multiple spatially distributed parameters may also be enhanced through the use of an entangled quantum network3,4,5,6,7,8,9,10. Here, we experimentally demonstrate how sensing of an averaged phase shift among four distributed nodes benefits from an entangled quantum network. Using a four-mode entangled continuous-variable state, we demonstrate deterministic quantum phase sensing with a precision beyond what is attainable with separable probes. The techniques behind this result can have direct applications in a number of areas ranging from molecular tracking to quantum networks of atomic clocks.

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## Data availability

The data represented in Figs. 2 and 3 and Supplementary Fig. 6b are available as Source Data or Supplementary Data. Raw oscilloscope data and data analysis scripts are available at https://doi.org/10.11583/DTU.9988805. All other data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

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## Acknowledgements

M.C. and J.B. acknowledge support from Villum Fonden via the QMATH Centre of Excellence (grant no. 10059), the European Research Council (ERC grant agreement no. 337603) and from the QuantERA ERA-NET Cofund in Quantum Technologies implemented within the European Union’s Horizon 2020 Programme (QuantAlgo project) via the Innovation Fund Denmark. X.G., C.R.B., S.I., M.V.L., T.G., J.S.N.-N. and U.L.A. acknowledge support from the Center for Macroscopic Quantum States (bigQ DNRF142). X.G., S.I. and J.S.N.-N. acknowledge support from Villum Fonden via the Young Investigator Programme (grant no. 10119).

## Author information

Authors

### Contributions

J.B., U.L.A., J.S.N.-N., T.G., X.G. and C.R.B. conceived the experiment. X.G., C.R.B. and M.V.L. performed the experiment and analysed the data. J.B., X.G., S.I., M.C. and J.S.N.-N. worked on the theoretical analysis. X.G. wrote the paper with contributions from J.B., C.R.B., S.I., J.S.N.-N. and U.L.A. J.S.N.-N. and U.L.A. supervised the project.

### Corresponding authors

Correspondence to Xueshi Guo, Jonas S. Neergaard-Nielsen or Ulrik L. Andersen.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

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 Setup.

See Supplementary Sec. II for details. (a) Single mode displaced squeezed state generation at the 3 MHz side band; (b) A proof-of-principle experimental setup for distributed phase sensing with entangled probes. The local oscillator (LO) is used as external phase reference, and the phase shift is introduced by wave plates; (c) The balanced homodyne detection setup corresponding to HD1 to HD4 in (b).

### Extended Data Fig. 2 Optimal sensitivities and quantum Cramér–Rao bounds (QCRB) for different scenarios.

All calculated with a total efficiency of η = 0.735 as in our experiment. The optimal sensitivity of our separable scheme $$\sigma _s^{opt}$$ and entangled scheme $$\sigma _e^{opt}$$ are plotted in solid blue and red, respectively. These are derived in the Supplementary Material Section I. The remaining four curves show the QCRBs derived in the Methods section: The optimal QCRB for ϕavg sensing with coherent probes ($$\sigma _{coh}^{CR}$$, dashed black), the separable scheme with squeezed probes ($$\sigma _s^{CR}$$, dashed blue), and the entangled scheme ($$\sigma _e^{CR}$$, dashed red), as well as the QCRB for single parameter phase sensing with a squeezed probe ($$\sigma _{sm}^{CR}$$, solid green).

## Supplementary information

### Supplementary Information

Supplementary sections I–IV, Figs. 1–10 and a table.

### Supplementary Data 1

The PSDs plotted in Supplementary Fig. 6b.

## Source data

### Source Data Fig. 2

The PSD plotted in Fig. 2.

### Source Data Fig. 3

The experimental result plotted in Fig. 3, including the error bars, which represent standard deviation.

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Reprints and Permissions

Guo, X., Breum, C.R., Borregaard, J. et al. Distributed quantum sensing in a continuous-variable entangled network. Nat. Phys. 16, 281–284 (2020). https://doi.org/10.1038/s41567-019-0743-x

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• DOI: https://doi.org/10.1038/s41567-019-0743-x

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