- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Letter
  • Open Access

Efficient verification of entangled continuous-variable quantum states with local measurements

Ye-Chao Liu, Jiangwei Shang*, and Xiangdong Zhang†

  • Key Laboratory of Advanced Optoelectronic Quantum Architecture and Measurement of Ministry of Education, School of Physics, Beijing Institute of Technology, Beijing 100081, China

  • *jiangwei.shang@bit.edu.cn
  • †zhangxd@bit.edu.cn
PDF

Phys. Rev. Research 3, L042004 – Published 11 October, 2021

DOI: https://doi.org/10.1103/PhysRevResearch.3.L042004

Abstract

Continuous-variable quantum states are of particular importance in various quantum information processing tasks including quantum communication and quantum sensing. However, a bottleneck has emerged with the fast increasing in size of the quantum systems which severely hinders their efficient characterization. In this work, we establish a systematic framework for verifying entangled continuous-variable quantum states by employing local measurements only. Our protocol is able to achieve the unconditionally high verification efficiency which is quadratically better than quantum tomography as well as other nontomographic methods. Specifically, we demonstrate the power of our protocol by showing the efficient verification of entangled two-mode and multimode coherent states with local measurements.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (72)

  1. S. L. Braunstein and P. van Loock, Quantum information with continuous variables, Rev. Mod. Phys. 77, 513 (2005).
  2. C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, Gaussian quantum information, Rev. Mod. Phys. 84, 621 (2012).
  3. G. M. D'Ariano, C. Macchiavello, and M. G. A. Paris, Detection of the density matrix through optical homodyne tomography without filtered back projection, Phys. Rev. A 50, 4298 (1994).
  4. U. Leonhardt, Measuring the Quantum State of Light (Cambridge University Press, Cambridge, 1997), Vol. 22.
  5. G. M. D'Ariano, M. G. A. Paris, and M. F. Sacchi, Quantum tomography, Adv. Imag. Elect. Phys. 128, 205 (2003).
  6. M. Guţă and L. Artiles, Minimax estimation of the Wigner function in quantum homodyne tomography with ideal detectors, Math. Meth. Stat. 16, 1 (2007).
  7. S. Glancy and H. M. de Vasconcelos, Methods for producing optical coherent state superpositions, J. Opt. Soc. Am. B 25, 712 (2008).
  8. A. I. Lvovsky and M. G. Raymer, Continuous-variable optical quantum-state tomography, Rev. Mod. Phys. 81, 299 (2009).
  9. R. J. Donaldson, R. J. Collins, E. Eleftheriadou, S. M. Barnett, J. Jeffers, and G. S. Buller, Experimental Implementation of a Quantum Optical State Comparison Amplifier, Phys. Rev. Lett. 114, 120505 (2015).
  10. L. Aolita, C. Gogolin, M. Kliesch, and J. Eisert, Reliable quantum certification of photonic state preparations, Nat. Commun. 6, 8498 (2015).
  11. N. Liu, T. F. Demarie, S.-H. Tan, L. Aolita, and J. F. Fitzsimons, Client-friendly continuous-variable blind and verifiable quantum computing, Phys. Rev. A 100, 062309 (2019).
  12. U. Chabaud, T. Douce, F. Grosshans, E. Kashefi, and D. Markham, Building trust for continuous variable quantum states, arXiv:1905.12700.
  13. U. Chabaud, F. Grosshans, E. Kashefi, and D. Markham, Efficient verification of Boson sampling, arXiv:2006.03520.
  14. U. Chabaud, G. Roeland, M. Walschaers, F. Grosshans, V. Parigi, D. Markham, and N. Treps, Certification of non-Gaussian states with operational measurements, PRX Quantum 2, 020333 (2021).
  15. Y. Takeuchi, A. Mantri, T. Morimae, A. Mizutani, and J. F. Fitzsimons, Resource-efficient verification of quantum computing using Serflings bound, npj Quantum Inf. 5, 27 (2019).
  16. M. Hayashi, K. Matsumoto, and Y. Tsuda, A study of LOCC-detection of a maximally entangled state using hypothesis testing, J. Phys. A: Math. Gen. 39, 14427 (2006).
  17. S. Pallister, N. Linden, and A. Montanaro, Optimal Verification of Entangled States with Local Measurements, Phys. Rev. Lett. 120, 170502 (2018).
  18. T. Morimae, Y. Takeuchi, and M. Hayashi, Verification of hypergraph states, Phys. Rev. A 96, 062321 (2017).
  19. Y. Takeuchi and T. Morimae, Verification of Many-Qubit States, Phys. Rev. X 8, 021060 (2018).
  20. X.-D. Yu, J. Shang, and O. Gühne, Optimal verification of general bipartite pure states, npj Quantum Inf. 5, 112 (2019).
  21. Z. Li, Y.-G. Han, and H. Zhu, Efficient verification of bipartite pure states, Phys. Rev. A 100, 032316 (2019).
  22. K. Wang and M. Hayashi, Optimal verification of two-qubit pure states, Phys. Rev. A 100, 032315 (2019).
  23. H. Zhu and M. Hayashi, Optimal verification and fidelity estimation of maximally entangled states, Phys. Rev. A 99, 052346 (2019).
  24. H. Zhu and M. Hayashi, Efficient Verification of Hypergraph States, Phys. Rev. Appl. 12, 054047 (2019).
  25. H. Zhu and M. Hayashi, Efficient Verification of Pure Quantum States in the Adversarial Scenario, Phys. Rev. Lett. 123, 260504 (2019).
  26. H. Zhu and M. Hayashi, General framework for verifying pure quantum states in the adversarial scenario, Phys. Rev. A 100, 062335 (2019).
  27. Y.-C. Liu, X.-D. Yu, J. Shang, H. Zhu, and X. Zhang, Efficient Verification of Dicke States, Phys. Rev. Appl. 12, 044020 (2019).
  28. Z. Li, Y.-G. Han, and H. Zhu, Optimal Verification of Greenberger-Horne-Zeilinger States, Phys. Rev. Appl. 13, 054002 (2020).
  29. N. Dangniam, Y.-G. Han, and H. Zhu, Optimal verification of stabilizer states, Phys. Rev. Res. 2, 043323 (2020).
  30. W.-H. Zhang, C. Zhang, Z. Chen, X.-X. Peng, X.-Y. Xu, P. Yin, S. Yu, X.-J. Ye, Y.-J. Han, J.-S. Xu, G. Chen, C.-F. Li, and G.-C. Guo, Experimental Optimal Verification of Entangled States Using Local Measurements, Phys. Rev. Lett. 125, 030506 (2020).
  31. X. Jiang, K. Wang, K. Qian, Z. Chen, Z. Chen, L. Lu, L. Xia, F. Song, S. Zhu, and X. Ma, Towards the standardization of quantum state verification using optimal strategies, npj Quantum Inf. 6, 90 (2020).
  32. W.-H. Zhang, X. Liu, P. Yin, X.-X. Peng, G.-C. Li, X.-Y. Xu, S. Yu, Z.-B. Hou, Y.-J. Han, J.-S. Xu, Z.-Q. Zhou, G. Chen, C.-F. Li, and G.-C. Guo, Classical communication enhanced quantum state verification, npj Quantum Inf. 6, 103 (2020).
  33. Z. Li, Y.-G. Han, H.-F. Sun, J. Shang, and H. Zhu, Verification of phased Dicke states, Phys. Rev. A 103, 022601 (2021).
  34. Y.-C. Liu, J. Shang, R. Han, and X. Zhang, Universally Optimal Verification of Entangled States with Nondemolition Measurements, Phys. Rev. Lett. 126, 090504 (2021).
  35. Y.-C. Liu, J. Shang, X.-D. Yu, and X. Zhang, Efficient verification of quantum processes, Phys. Rev. A 101, 042315 (2020).
  36. H. Zhu and H. Zhang, Efficient verification of quantum gates with local operations, Phys. Rev. A 101, 042316 (2020).
  37. P. Zeng, Y. Zhou, and Z. Liu, Quantum gate verification and its application in property testing, Phys. Rev. Res. 2, 023306 (2020).
  38. N. C. Menicucci, P. van Loock, M. Gu, C. Weedbrook, T. C. Ralph, and M. A. Nielsen, Universal Quantum Computation with Continuous-Variable Cluster States, Phys. Rev. Lett. 97, 110501 (2006).
  39. F. Mattioli, Z. Zhou, A. Gaggero, R. Gaudio, R. Leoni, and A. Fiore, Photon-counting and analog operation of a 24-pixel photon number resolving detector based on superconducting nanowires, Opt. Express 24, 9067 (2016).
  40. D. Fukuda, G. Fujii, T. Numata, K. Amemiya, A. Yoshizawa, H. Tsuchida, H. Fujino, H. Ishii, T. Itatani, S. Inoue, and T. Zama, Titanium-based transition-edge photon number resolving detector with 98% detection efficiency with index-matched small-gap fiber coupling, Opt. Express 19, 870 (2011).
  41. A. Divochiy, F. Marsili, D. Bitauld, A. Gaggero, R. Leoni, F. Mattioli, A. Korneev, V. Seleznev, N. Kaurova, O. Minaeva, G. Gol'tsman, K. G. Lagoudakis, M. Benkhaoul, F. Lévy, and A. Fiore, Superconducting nanowire photon-number-resolving detector at telecommunication wavelengths, Nat. Photon. 2, 302 (2008).
  42. B. E. Kardynał, Z. L. Yuan, and A. J. Shields, An avalanche-photodiode-based photon-number-resolving detector, Nat. Photon. 2, 425 (2008).
  43. L. A. Morais, T. Weinhold, M. P. de Almeida, A. Lita, T. Gerrits, S. W. Nam, A. G. White, and G. Gillett, Precisely determining photon-number in real-time, arXiv:2012.10158.
  44. Y.-D. Wu, G. Bai, G. Chiribella, and N. Liu, Efficient Verification of Continuous-Variable Quantum States and Devices without Assuming Identical and Independent Operations, Phys. Rev. Lett. 126, 240503 (2021).
  45. S. J. van Enk and O. Hirota, Entangled coherent states: Teleportation and decoherence, Phys. Rev. A 64, 022313 (2001).
  46. X. Wang, Quantum teleportation of entangled coherent states, Phys. Rev. A 64, 022302 (2001).
  47. P. T. Cochrane, G. J. Milburn, and W. J. Munro, Macroscopically distinct quantum-superposition states as a bosonic code for amplitude damping, Phys. Rev. A 59, 2631 (1999).
  48. H. Jeong and M. S. Kim, Efficient quantum computation using coherent states, Phys. Rev. A 65, 042305 (2002).
  49. T. C. Ralph, A. Gilchrist, G. J. Milburn, W. J. Munro, and S. Glancy, Quantum computation with optical coherent states, Phys. Rev. A 68, 042319 (2003).
  50. A. P. Lund, T. C. Ralph, and H. L. Haselgrove, Fault-Tolerant Linear Optical Quantum Computing with Small-Amplitude Coherent States, Phys. Rev. Lett. 100, 030503 (2008).
  51. J. Joo, W. J. Munro, and T. P. Spiller, Quantum Metrology with Entangled Coherent States, Phys. Rev. Lett. 107, 083601 (2011).
  52. J. Joo, K. Park, H. Jeong, W. J. Munro, K. Nemoto, and T. P. Spiller, Quantum metrology for nonlinear phase shifts with entangled coherent states, Phys. Rev. A 86, 043828 (2012).
  53. J. Liu, X.-M. Lu, Z. Sun, and X. Wang, Quantum multiparameter metrology with generalized entangled coherent state, J. Phys. A: Math. Theor. 49, 115302 (2016).
  54. See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevResearch.3.L042004 for the Appendices.
  55. C. Wittmann, M. Takeoka, K. N. Cassemiro, M. Sasaki, G. Leuchs, and U. L. Andersen, Demonstration of Near-Optimal Discrimination of Optical Coherent States, Phys. Rev. Lett. 101, 210501 (2008).
  56. M. T. DiMario and F. E. Becerra, Robust Measurement for the Discrimination of Binary Coherent States, Phys. Rev. Lett. 121, 023603 (2018).
  57. V. Bužek, A. Vidiella-Barranco, and P. L. Knight, Superpositions of coherent states: Squeezing and dissipation, Phys. Rev. A 45, 6570 (1992).
  58. H. Takahashi, K. Wakui, S. Suzuki, M. Takeoka, K. Hayasaka, A. Furusawa, and M. Sasaki, Generation of Large-Amplitude Coherent-State Superposition Via Ancilla-Assisted Photon Subtraction, Phys. Rev. Lett. 101, 233605 (2008).
  59. V. V. Dodonov, I. A. Malkin, and V. I. Man'ko, Even and odd coherent states and excitations of a singular oscillator, Physica 72, 597 (1974).
  60. A. Gilchrist, K. Nemoto, W. J. Munro, T. C. Ralph, S. Glancy, S. L. Braunstein, and G. J. Milburn, Schrödinger cats and their power for quantum information processing, J. Opt. B: Quantum Semiclass. Opt. 6, S828 (2004).
  61. L.-M. Kuang and J.-Y. Zhu, Even and odd phase coherent states for Hermitian phase operator theory, J. Phys. A: Math. Gen. 29, 895 (1996).
  62. J.-C. Besse, S. Gasparinetti, M. C. Collodo, T. Walter, A. Remm, J. Krause, C. Eichler, and A. Wallraff, Parity Detection of Propagating Microwave Fields, Phys. Rev. X 10, 011046 (2020).
  63. B. C. Sanders, Entangled coherent states, Phys. Rev. A 45, 6811 (1992).
  64. B. C. Sanders, Review of entangled coherent states, J. Phys. A: Math. Theor. 45, 244002 (2012).
  65. W. J. Munro, G. J. Milburn, and B. C. Sanders, Entangled coherent-state qubits in an ion trap, Phys. Rev. A 62, 052108 (2000).
  66. H. Jeong, M. S. Kim, and J. Lee, Quantum-information processing for a coherent superposition state via a mixedentangled coherent channel, Phys. Rev. A 64, 052308 (2001).
  67. Be reminded that the noisy states in Eq. (21) are different in form from those in Eq. (4), but direct calculations can establish the relation between its passing probability and the infidelity ε (see Supplemental Material Appendix E [54]).
  68. Y. Israel, L. Cohen, X.-B. Song, J. Joo, H. S. Eisenberg, and Y. Silberberg, Entangled coherent states created by mixing squeezed vacuum and coherent light, Optica 6, 753 (2019).
  69. H. Jeong and N. B. An, Greenberger-Horne-Zeilinger–type and W-type entangled coherent states: Generation and Bell-type inequality tests without photon counting, Phys. Rev. A 74, 022104 (2006).
  70. M. Hayashi and T. Morimae, Verifiable Measurement-Only Blind Quantum Computing with Stabilizer Testing, Phys. Rev. Lett. 115, 220502 (2015).
  71. W. J. Munro, K. Nemoto, R. G. Beausoleil, and T. P. Spiller, High-efficiency quantum-nondemolition single-photon-number-resolving detector, Phys. Rev. A 71, 033819 (2005).
  72. C. Guerlin, J. Bernu, S. Deleglise, C. Sayrin, S. Gleyzes, S. Kuhr, M. Brune, J.-M. Raimond, and S. Haroche, Progressive field-state collapse and quantum non-demolition photon counting, Nature (London) 448, 889 (2007).

Outline

Information

Phys. Rev. Research 3, L042004– Published 11 October, 2021
  • Received 7 April 2021
  • Accepted 29 September 2021
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation