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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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Fig. 1: A sketch for estimating distributed multiparameters.
Fig. 2: Experimental set-up.
Fig. 3: Experimental results for estimating single parameters for mode 1, mode 2 and mode 3.
Fig. 4: Experimental results for the parallel strategies of MePe, MePs and MsPe.
Fig. 5: Experimental results for the combined strategy.

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.

References

  1. Giovannetti, V., Lloyd, S. & Maccone, L. Quantum-enhanced measurements: beating the standard quantum limit. Science 306, 1330–1336 (2004).

    ADS  Article  Google Scholar 

  2. Giovannetti, V., Lloyd, S. & Maccone, L. Advances in quantum metrology. Nat. Photon. 5, 222–229 (2011).

    ADS  Article  Google Scholar 

  3. Degen, C. L., Reinhard, F. & Cappellaro, P. Quantum sensing. Rev. Mod. Phys. 89, 035002 (2017).

    ADS  MathSciNet  Article  Google Scholar 

  4. Braun, D. et al. Quantum-enhanced measurements without entanglement. Rev. Mod. Phys. 90, 035006 (2018).

    ADS  MathSciNet  Article  Google Scholar 

  5. Kok, P., Lee, H. & Dowling, J. P. Creation of large-photon-number path entanglement conditioned on photodetection. Phys. Rev. A 65, 052104 (2002).

    ADS  Article  Google Scholar 

  6. Walther, P. et al. De Broglie wavelength of a non-local four-photon state. Nature 429, 158–161 (2004).

    ADS  Article  Google Scholar 

  7. Mitchell, M. W., Lundeen, J. S. & Steinberg, A. M. Super-resolving phase measurements with a multiphoton entangled state. Nature 429, 161–164 (2004).

    ADS  Article  Google Scholar 

  8. Nagata, T., Okamoto, R., O’Brien, J. L., Sasaki, K. & Takeuchi, S. Beating the standard quantum limit with four-entangled photons. Science 316, 726–729 (2007).

    ADS  Article  Google Scholar 

  9. Resch, K. J. et al. Time-reversal and super-resolving phase measurements. Phys. Rev. Lett. 98, 223601 (2007).

    ADS  Article  Google Scholar 

  10. Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free Heisenberg-limited phase estimation. Nature 450, 393–396 (2007).

    ADS  Article  Google Scholar 

  11. Gao, W.-B. et al. Experimental demonstration of a hyper-entangled ten-qubit Schrödinger cat state. Nat. Phys. 6, 331–335 (2010).

    Article  Google Scholar 

  12. Bell, B. et al. Multicolor quantum metrology with entangled photons. Phys. Rev. Lett. 111, 093603 (2013).

    ADS  Article  Google Scholar 

  13. Slussarenko, S. et al. Unconditional violation of the shot-noise limit in photonic quantum metrology. Nat. Photon. 11, 700–703 (2017).

    ADS  Article  Google Scholar 

  14. Schnabel, R., Mavalvala, N., McClelland, D. E. & Lam, P. K. Quantum metrology for gravitational wave astronomy. Nat. Commun. 1, 121 (2010).

    ADS  Article  Google Scholar 

  15. Aasi, J. et al. Enhanced sensitivity of the LIGO gravitational wave detector by using squeezed states of light. Nat. Photon. 7, 613–619 (2013).

    ADS  Article  Google Scholar 

  16. Humphreys, P. C., Barbieri, M., Datta, A. & Walmsley, I. A. Quantum enhanced multiple phase estimation. Phys. Rev. Lett. 111, 070403 (2013).

    ADS  Article  Google Scholar 

  17. Pérez-Delgado, C. A., Pearce, M. E. & Kok, P. Fundamental limits of classical and quantum imaging. Phys. Rev. Lett. 109, 123601 (2012).

    ADS  Article  Google Scholar 

  18. Polino, E., Valeri, M., Spagnolo, N. & Sciarrino, F. Photonic quantum metrology. AVS Quant. Sci. 2, 024703 (2020).

    ADS  Article  Google Scholar 

  19. Komar, P. et al. A quantum network of clocks. Nat. Phys. 10, 582–587 (2014).

    Article  Google Scholar 

  20. Chen, J.-Y., Pandurangan, G. & Xu, D. Robust computation of aggregates in wireless sensor networks: Distributed randomized algorithms and analysis. IEEE Trans. Parallel Distrib. Syst. 17, 987–1000 (2006).

    Article  Google Scholar 

  21. Dimakis, A. D. G., Sarwate, A. D. & Wainwright, M. J. Geographic gossip: efficient averaging for sensor networks. IEEE Trans. Signal Process. 56, 1205–1216 (2008).

    ADS  MathSciNet  Article  Google Scholar 

  22. Zhuang, Q., Zhang, Z. & Shapiro, J. H. Distributed quantum sensing using continuous-variable multipartite entanglement. Phys. Rev. A 97, 032329 (2018).

    ADS  Article  Google Scholar 

  23. Guo, X. et al. Distributed quantum sensing in a continuous-variable entangled network. Nat. Phys. 16, 281–284 (2020).

    Article  Google Scholar 

  24. Xia, Y. et al. Demonstration of a reconfigurable entangled radio-frequency photonic sensor network. Phys. Rev. Lett. 124, 150502 (2020).

    ADS  Article  Google Scholar 

  25. Ge, W., Jacobs, K., Eldredge, Z., Gorshkov, A. V. & Foss-Feig, M. Distributed quantum metrology with linear networks and separable inputs. Phys. Rev. Lett. 121, 043604 (2018).

    ADS  Article  Google Scholar 

  26. Proctor, T. J., Knott, P. A. & Dunningham, J. A. Multiparameter estimation in networked quantum sensors. Phys. Rev. Lett. 120, 080501 (2018).

    ADS  Article  Google Scholar 

  27. Gessner, M., Pezzè, L. & Smerzi, A. Sensitivity bounds for multiparameter quantum metrology. Phys. Rev. Lett. 121, 130503 (2018).

    ADS  Article  Google Scholar 

  28. Oh, C., Lee, C., Lie, S. H. & Jeong, H. Optimal distributed quantum sensing using Gaussian states. Phys. Rev. Res. 2, 023030 (2020).

    Article  Google Scholar 

  29. Juffmann, T., Klopfer, B. B., Frankort, T. L., Haslinger, P. & Kasevich, M. A. Multi-pass microscopy. Nat. Commun. 7, 12858 (2016).

    ADS  Article  Google Scholar 

  30. Hou, Z. et al. Control-enhanced sequential scheme for general quantum parameter estimation at the Heisenberg limit. Phys. Rev. Lett. 123, 040501 (2019).

    ADS  Article  Google Scholar 

  31. Xiang, G.-Y., Higgins, B. L., Berry, D., Wiseman, H. M. & Pryde, G. Entanglement-enhanced measurement of a completely unknown optical phase. Nat. Photon. 5, 43–47 (2011).

    ADS  Article  Google Scholar 

  32. Berni, A. A. et al. Ab initio quantum-enhanced optical phase estimation using real-time feedback control. Nat. Photon. 9, 577–581 (2015).

    ADS  Article  Google Scholar 

  33. Daryanoosh, S., Slussarenko, S., Berry, D. W., Wiseman, H. M. & Pryde, G. J. Experimental optical phase measurement approaching the exact Heisenberg limit. Nat. Commun. 9, 4606 (2018).

    ADS  Article  Google Scholar 

  34. Helstrom, C. W. Quantum detection and estimation theory. J. Stat. Phys. 1, 231–252 (1969).

    ADS  MathSciNet  Article  Google Scholar 

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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).

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Authors and Affiliations

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.

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The authors declare no competing interests.

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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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