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Experimental entanglement quantification for unknown quantum states in a semi-device-independent manner

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Abstract

Using the concept of non-degenerate Bell inequality, we show that quantum entanglement, the critical resource for various quantum information processing tasks, can be quantified for any unknown quantum state in a semi-device-independent manner, where the quantification is based on the experimentally obtained probability distributions and prior knowledge of the quantum dimension only. Specifically, as an application of our approach to multi-level systems, we experimentally quantify the entanglement of formation and the entanglement of distillation for qutrit-qutrit quantum systems. In addition, to demonstrate our approach for multi-partite systems, we further quantify the geometric measure of entanglement of three-qubit quantum systems. Our results supply a general way to reliably quantify entanglement in multi-level and multi-partite systems, thus paving the way to characterize many-body quantum systems by quantifying the involved entanglement.

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References

  1. Gisin N, Thew R. Quantum communication. Nat Photon, 2007, 1: 165–171

    Article  Google Scholar 

  2. Bennett C H, Brassard G. Quantum cryptography: public key distribution and coin tossing. Theoretical Comput Sci, 2014, 560: 7–11

    Article  MathSciNet  MATH  Google Scholar 

  3. Ekert A K. Quantum cryptography based on Bell’s theorem. Phys Rev Lett, 1991, 67: 661–663

    Article  MathSciNet  MATH  Google Scholar 

  4. Gisin N, Ribordy G, Tittel W, et al. Quantum cryptography. Rev Mod Phys, 2002, 74: 145–195

    Article  MATH  Google Scholar 

  5. Raussendorf R, Briegel H J. A one-way quantum computer. Phys Rev Lett, 2001, 86: 5188–5191

    Article  MATH  Google Scholar 

  6. Vidal G. Efficient classical simulation of slightly entangled quantum computations. Phys Rev Lett, 2003, 91: 147902

    Article  Google Scholar 

  7. Gühne O, Tóth G. Entanglement detection. Phys Rep, 2009, 474: 1–75

    Article  MathSciNet  Google Scholar 

  8. Rosset D, Ferretti-Schöbitz R, Bancal J D, et al. Imperfect measurement settings: implications for quantum state tomography and entanglement witnesses. Phys Rev A, 2012, 86: 062325

    Article  Google Scholar 

  9. Levin M, Wen X G. Detecting topological order in a ground state wave function. Phys Rev Lett, 2006, 96: 110405

    Article  Google Scholar 

  10. Kitaev A, Preskill J. Topological entanglement entropy. Phys Rev Lett, 2006, 96: 110404

    Article  MathSciNet  Google Scholar 

  11. Acín A, Brunner N, Gisin N, et al. Device-independent security of quantum cryptography against collective attacks. Phys Rev Lett, 2007, 98: 230501

    Article  Google Scholar 

  12. Mayers D, Yao A. Self testing quantum apparatus. Quantum Inf Comput, 2004, 4: 273–286

    MathSciNet  MATH  Google Scholar 

  13. Ahrens J, Badziąg P, Cabello A, et al. Experimental device-independent tests of classical and quantum dimensions. Nat Phys, 2012, 8: 592–595

    Article  Google Scholar 

  14. Hendrych M, Gallego R, Mičuda M, et al. Experimental estimation of the dimension of classical and quantum systems. Nat Phys, 2012, 8: 588–591

    Article  Google Scholar 

  15. Hensen B, Bernien H, Dréau A E, et al. Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres. Nature, 2015, 526: 682–686

    Article  Google Scholar 

  16. Liu Y, Zhao Q, Li M H, et al. Device-independent quantum random-number generation. Nature, 2018, 562: 548–551

    Article  Google Scholar 

  17. Zhang W H, Chen G, Peng X X, et al. Experimental realization of robust self-testing of bell state measurements. Phys Rev Lett, 2019, 122: 090402

    Article  Google Scholar 

  18. Lo H K, Curty M, Qi B. Measurement-device-independent quantum key distribution. Phys Rev Lett, 2012, 108: 130503

    Article  Google Scholar 

  19. Braunstein S L, Pirandola S. Side-channel-free quantum key distribution. Phys Rev Lett, 2012, 108: 130502

    Article  Google Scholar 

  20. Moroder T, Gittsovich O. Calibration-robust entanglement detection beyond Bell inequalities. Phys Rev A, 2012, 85: 032301

    Article  Google Scholar 

  21. Liang Y C, Vértesi T, Brunner N. Semi-device-independent bounds on entanglement. Phys Rev A, 2011, 83: 022108

    Article  Google Scholar 

  22. Bennett C H, Divincenzo D P, Smolin J A, et al. Mixed-state entanglement and quantum error correction. Phys Rev A, 1996, 54: 3824–3851

    Article  MathSciNet  MATH  Google Scholar 

  23. Vedral V, Plenio M B, Rippin M A, et al. Quantifying entanglement. Phys Rev Lett, 1997, 78: 2275–2279

    Article  MathSciNet  MATH  Google Scholar 

  24. Brody D C, Hughston L P. Geometric quantum mechanics. J Geometry Phys, 2001, 38: 19–53

    Article  MathSciNet  MATH  Google Scholar 

  25. Wei T C, Goldbart P M. Geometric measure of entanglement and applications to bipartite and multipartite quantum states. Phys Rev A, 2003, 68: 042307

    Article  Google Scholar 

  26. Vedral V, Plenio M B. Entanglement measures and purification procedures. Phys Rev A, 1998, 57: 1619–1633

    Article  Google Scholar 

  27. Collins D, Gisin N, Popescu S, et al. Bell-type inequalities to detect true-body nonseparability. Phys Rev Lett, 2002, 88: 170405

    Article  Google Scholar 

  28. Mermin N D. Extreme quantum entanglement in a superposition of macroscopically distinct states. Phys Rev Lett, 1990, 65: 1838–1840

    Article  MathSciNet  MATH  Google Scholar 

  29. Ardehali M. Bell inequalities with a magnitude of violation that grows exponentially with the number of particles. Phys Rev A, 1992, 46: 5375–5378

    Article  MathSciNet  Google Scholar 

  30. Belinskiĭ A V, Klyshko D N. Interference of light and Bell’s theorem. Physics-Uspekhi, 1993, 36: 653–693

    Article  Google Scholar 

  31. Dai Y, Dong Y, Xu Z, et al. Experimentally accessible lower bounds for genuine multipartite entanglement and coherence measures. Phys Rev Appl, 2020, 13: 054022

    Article  Google Scholar 

  32. Kolda T G, Mayo J R. Shifted power method for computing tensor eigenpairs. SIAM J Matrix Anal Appl, 2011, 32: 1095–1124

    Article  MathSciNet  MATH  Google Scholar 

  33. Lin L, Wei Z. Quantifying multipartite quantum entanglement in a semi-device-independent manner. Phys Rev A, 2021, 104: 062433

    Article  MathSciNet  Google Scholar 

  34. Wei Z, Lin L. Analytic semi-device-independent entanglement quantification for bipartite quantum states. Phys Rev A, 2021, 103: 032215

    Article  MathSciNet  Google Scholar 

  35. Clauser J F, Horne M A, Shimony A, et al. Proposed experiment to test local hidden variable theories. Phys Rev Lett, 1970, 24: 549

    Article  MATH  Google Scholar 

  36. Scarani V, Gisin N. Spectral decomposition of Bell’s operators for qubits. J Phys A-Math Gen, 2001, 34: 6043–6053

    Article  MathSciNet  MATH  Google Scholar 

  37. Smith G, Smolin J A, Yuan X, et al. Quantifying coherence and entanglement via simple measurements. 2017. ArXiv: 1707.09928

  38. Nielsen M A, Chuang I L. Quantum Computation and Quantum Information. Cambridge: Cambridge University Press, 2000

    MATH  Google Scholar 

  39. Wei T C. Relative entropy of entanglement for multipartite mixed states: Permutation-invariant states and Dür states. Phys Rev A, 2008, 78: 012327

    Article  Google Scholar 

  40. Schumacher B, Nielsen M A. Quantum data processing and error correction. Phys Rev A, 1996, 54: 2629–2635

    Article  MathSciNet  Google Scholar 

  41. Lloyd S. Capacity of the noisy quantum channel. Phys Rev A, 1997, 55: 1613–1622

    Article  MathSciNet  Google Scholar 

  42. Cornelio M F, de Oliveira M C, Fanchini F F. Entanglement irreversibility from quantum discord and quantum deficit. Phys Rev Lett, 2011, 107: 020502

    Article  Google Scholar 

  43. Hu X M, Guo Y, Liu B H, et al. Beating the channel capacity limit for superdense coding with entangled ququarts. Sci Adv, 2018, 4: eaat9304

    Article  Google Scholar 

  44. Zhang C, Huang Y F, Zhang C J, et al. Generation and applications of an ultrahigh-fidelity four-photon Greenberger-Horne-Zeilinger state. Opt Express, 2016, 24: 27059

    Article  Google Scholar 

  45. Salavrakos A, Augusiak R, Tura J, et al. Bell inequalities tailored to maximally entangled states. Phys Rev Lett, 2017, 119: 040402

    Article  MathSciNet  Google Scholar 

  46. Ragnarsson S, van Loan C F. Block tensors and symmetric embeddings. Linear Algebra its Appl, 2013, 438: 853–874

    Article  MathSciNet  MATH  Google Scholar 

  47. Zohren S, Gill R D. Maximal violation of the Collins-Gisin-Linden-Massar-Popescu inequality for infinite dimensional states. Phys Rev Lett, 2008, 100: 120406

    Article  Google Scholar 

  48. Guo Y, Cheng S, Hu X, et al. Experimental measurement-device-independent quantum steering and randomness generation beyond qubits. Phys Rev Lett, 2019, 123: 170402

    Article  Google Scholar 

  49. Guo Y, Yu B C, Hu X M, et al. Measurement-device-independent quantification of irreducible high-dimensional entanglement. npj Quantum Inf, 2020, 6: 52

    Article  Google Scholar 

  50. Srednicki M. Chaos and quantum thermalization. Phys Rev E, 1994, 50: 888–901

    Article  Google Scholar 

  51. Abanin D A, Altman E, Bloch I, et al. Colloquium: many-body localization, thermalization, and entanglement. Rev Mod Phys, 2019, 91: 021001

    Article  MathSciNet  Google Scholar 

  52. Schreiber M, Hodgman S S, Bordia P, et al. Observation of many-body localization of interacting fermions in a quasirandom optical lattice. Science, 2015, 349: 842–845

    Article  MathSciNet  MATH  Google Scholar 

  53. Wen X G. Topological orders and edge excitations in fractional quantum Hall states. Adv Phys, 1995, 44: 405–473

    Article  Google Scholar 

Download references

Acknowledgements

This work was supported by National Key Research and Development Program of China (Grant No. 2021YFE0113100), National Natural Science Foundation of China (Grant Nos. 11821404, 11904357, 12204458, 61832015, 62272259), Fundamental Research Funds for the Central Universities, China Postdoctoral Science Foundation (Grant No. 2021M700138), China Postdoctoral for Innovative Talents (Grant No. BX2021289), USTC Tang Scholarship, and Science and Technological Fund of Anhui Province for Outstanding Youth (Grant No. 2008085J02). This work was partially carried out at the USTC Center for Micro and Nanoscale Research and Fabrication.

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Authors and Affiliations

  1. CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei, 230026, China

    Yu Guo, Huan Cao, Chao Zhang, Xiao-Min Hu, Bi-Heng Liu, Yun-Feng Huang, Yong-Jian Han, Chuan-Feng Li & Guang-Can Guo

  2. CAS Center For Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, 230026, China

    Yu Guo, Huan Cao, Chao Zhang, Xiao-Min Hu, Bi-Heng Liu, Yun-Feng Huang, Yong-Jian Han, Chuan-Feng Li & Guang-Can Guo

  3. School of Physics and Materials Engineering, Hefei Normal University, Hefei, 230601, China

    Yu Guo

  4. Yau Mathematical Sciences Center, Tsinghua University, Beijing, 100084, China

    Zhaohui Wei

  5. Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing, 100084, China

    Lijinzhi Lin, Xiaodie Lin & Zhaohui Wei

  6. Vienna Center for Quantum Science and Technology, Faculty of Physics, University of Vienna, Vienna, A-1090, Austria

    Huan Cao

Authors
  1. Yu Guo
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  2. Lijinzhi Lin
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  3. Huan Cao
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  4. Chao Zhang
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  5. Xiaodie Lin
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  7. Bi-Heng Liu
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  8. Yun-Feng Huang
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  9. Zhaohui Wei
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  11. Chuan-Feng Li
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  12. Guang-Can Guo
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Corresponding authors

Correspondence to Bi-Heng Liu, Zhaohui Wei, Yong-Jian Han or Chuan-Feng Li.

Additional information

Supporting information

See Supplemental Material for details about the quantities a1, \(\hat F\), and the Bell-type inequalities used in this work, including Refs. [46,47]. The supporting information is available online at info.scichina.com and link.springer.com. The supporting materials are published as submitted, without typesetting or editing. The responsibility for scientific accuracy and content remains entirely with the authors.

Supplementary File

11432_2022_3681_MOESM1_ESM.pdf

Supplementary Information: Experimental Entanglement Quantication for Unknown Quantum States in a Semi-Device-Independent Manner

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Guo, Y., Lin, L., Cao, H. et al. Experimental entanglement quantification for unknown quantum states in a semi-device-independent manner. Sci. China Inf. Sci. 66, 180506 (2023). https://doi.org/10.1007/s11432-022-3681-2

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  • DOI: https://doi.org/10.1007/s11432-022-3681-2

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