- 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

Covariant correlation-disturbance relation and its experimental realization with spin-1/2 particles

Ali Asadian1,*, Florian Gams2, and Stephan Sponar2,†

  • *Contact author: ali.asadian668@gmail.com
  • †Contact author: stephan.sponar@tuwien.ac.at
PDF

Phys. Rev. Research 8, L012011 – Published 13 January, 2026

DOI: https://doi.org/10.1103/llgb-gql9

Abstract

We formulate a precise trade-off relation between correlation and disturbance for sequential n-outcome quantum measurements in Hilbert spaces of arbitrary dimension. This relation highlights key symmetry properties useful for robust estimation and characterization of the measurement parameters against unitary noise, or in scenarios where shared reference frames are unavailable. In addition, we report on the experimental implementation of the proposal for the qubit case, more precisely in a neutron optical experiment, which is particularly valuable for calibrating and optimizing measurement devices, as confirmed by the theoretical results. Finally, we exploit the optimal trade-off relation for direct estimation of the characteristic noise of single-photon detectors, dark counts, and the finite detection efficiency.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (74)

  1. W. Heisenberg, Über den anschaulichen inhalt der quantentheoretischen Kinematik und Mechanik, Z. Phys. 43, 172 (1927).
  2. P. Busch, P. Lahti, and R. F. Werner, Colloquium: Quantum root-mean-square error and measurement uncertainty relations, Rev. Mod. Phys. 86, 1261 (2014).
  3. N. Bohr, The quantum postulate and the recent development of atomic theory 1, Nature (London) 121, 580 (1928).
  4. J. S. Bell, On the Einstein-Podolsky-Rosen paradox, Phys. Physique Fiz. 1, 195 (1964).
  5. J. S. Bell, On the problem of hidden variables in quantum mechanics, Rev. Mod. Phys. 38, 447 (1966).
  6. S. Kochen and E. P. Specker, The problem of hidden variables in quantum mechanics, J. Math. Mech. 17, 59 (1967).
  7. M. M. Wolf, D. Perez-Garcia, and C. Fernandez, Measurements incompatible in quantum theory cannot be measured jointly in any other no-signaling theory, Phys. Rev. Lett. 103, 230402 (2009).
  8. O. Gühne, E. Haapasalo, T. Kraft, J.-P. Pellonpää, and R. Uola, Colloquium: Incompatible measurements in quantum information science, Rev. Mod. Phys. 95, 011003 (2023).
  9. Y. Li, Y. Xiang, J. Tura, and Q. He, Necessary and sufficient condition for randomness certification from incompatibility, Phys. Rev. Lett. 135, 060201 (2025).
  10. P. Skrzypczyk, I. Šupić, and D. Cavalcanti, All sets of incompatible measurements give an advantage in quantum state discrimination, Phys. Rev. Lett. 122, 130403 (2019).
  11. F. Buscemi, E. Chitambar, and W. Zhou, Complete resource theory of quantum incompatibility as quantum programmability, Phys. Rev. Lett. 124, 120401 (2020).
  12. K. Sen, S. Halder, and U. Sen, Incompatibility of local measurements providing an advantage in local quantum state discrimination, Phys. Rev. A 109, 012415 (2024).
  13. D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, Measurement of qubits, Phys. Rev. A 64, 052312 (2001).
  14. F. B. Maciejewski, Z. Zimborás, and M. Oszmaniec, Mitigation of readout noise in near-term quantum devices by classical post-processing based on detector tomography, Quantum 4, 257 (2020).
  15. A. Elben, S. T. Flammia, H.-Y. Huang, R. Kueng, J. Preskill, B. Vermersch, and P. Zoller, The randomized measurement toolbox, Nat. Rev. Phys. 5, 9 (2023).
  16. A. J. Leggett and A. Garg, Quantum mechanics versus macroscopic realism: Is the flux there when nobody looks?, Phys. Rev. Lett. 54, 857 (1985).
  17. C. Emary, N. Lambert, and F. Nori, Leggett–Garg inequalities, Rep. Prog. Phys. 77, 016001 (2014).
  18. G. C. Knee, K. Kakuyanagi, M.-C. Yeh, Y. Matsuzaki, H. Toida, H. Yamaguchi, S. Saito, A. J. Leggett, and W. J. Munro, A strict experimental test of macroscopic realism in a superconducting flux qubit, Nat. Commun. 7, 13253 (2016).
  19. E. Kreuzgruber, R. Wagner, N. Geerits, H. Lemmel, and S. Sponar, Violation of a Leggett-Garg inequality using ideal negative measurements in neutron interferometry, Phys. Rev. Lett. 132, 260201 (2024).
  20. G. Vitagliano and C. Budroni, Leggett-Garg macrorealism and temporal correlations, Phys. Rev. A 107, 040101 (2023).
  21. C. A. Fuchs and A. Peres, Quantum-state disturbance versus information gain: Uncertainty relations for quantum information, Phys. Rev. A 53, 2038 (1996).
  22. I. I. Hirschman, A note on entropy, Am. J. Math. 79, 152 (1957).
  23. W. Beckner, Inequalities in Fourier analysis, Ann. Math. 102, 159 (1975).
  24. I. Białynicki-Birula and J. Mycielski, Uncertainty relations for information entropy in wave mechanics, Commun. Math. Phys. 44, 129 (1975).
  25. D. Deutsch, Uncertainty in quantum measurements, Phys. Rev. Lett. 50, 631 (1983).
  26. H. Maassen and J. B. M. Uffink, Generalized entropic uncertainty relations, Phys. Rev. Lett. 60, 1103 (1988).
  27. M. Berta, M. Christandl, R. Colbeck, J. M. Renes, and R. Renner, The uncertainty principle in the presence of quantum memory, Nat. Phys. 6, 659 (2010).
  28. P. J. Coles, J. Kaniewski, and S. Wehner, Equivalence of wave-particle duality to entropic uncertainty, Nat. Commun. 5, 5814 (2014).
  29. M. A. Nielsen and I. Chuang, in Quantum Computation and Quantum Information, edited by M. A. Nielsen and I. Chuang (Cambridge University Press, Cambridge, 2000).
  30. M. Ozawa, Universally valid reformulation of the Heisenberg uncertainty principle on noise and disturbance in measurement, Phys. Rev. A 67, 042105 (2003).
  31. M. Ozawa, Uncertainty relations for noise and disturbance in generalized quantum measurements, Ann. Phys. 311, 350 (2004).
  32. R. F. Werner, The uncertainty relation for joint measurement of position and momentum, Quant. Inf. Comput. 4, 546 (2004).
  33. F. Buscemi, M. J. W. Hall, M. Ozawa, and M. M. Wilde, Noise and disturbance in quantum measurements: An information-theoretic approach, Phys. Rev. Lett. 112, 050401 (2014).
  34. P. J. Coles and F. Furrer, State-dependent approach to entropic measurement–disturbance relations, Phys. Lett. A 379, 105 (2015).
  35. K. Baek and W. Son, Entropic uncertainty relations for successive generalized measurements, Mathematics 4, 41 (2016).
  36. R. Schwonnek, D. Reeb, and R. F. Werner, Measurement uncertainty for finite quantum observables, Mathematics 4, 38 (2016).
  37. A. Barchielli, M. Gregoratti, and A. Toigo, Measurement uncertainty relations for discrete observables: Relative entropy formulation, Commun. Math. Phys. 357, 1253 (2018).
  38. N. Saberian, S. J. Akhtarshenas, and F. Shahbeigi, Measurement sharpness and disturbance tradeoff, Phys. Rev. A 109, 012201 (2024).
  39. K. Mohan, A. Tavakoli, and N. Brunner, Sequential random access codes and self-testing of quantum measurement instruments, New J. Phys. 21, 083034 (2019).
  40. N. Miklin, J. J. Borkała, and M. Pawłowski, Semi-device-independent self-testing of unsharp measurements, Phys. Rev. Res. 2, 033014 (2020).
  41. H. Anwer, S. Muhammad, W. Cherifi, N. Miklin, A. Tavakoli, and M. Bourennane, Experimental characterization of unsharp qubit observables and sequential measurement incompatibility via quantum random access codes, Phys. Rev. Lett. 125, 080403 (2020).
  42. P. Roy and A. K. Pan, Device-independent self-testing of unsharp measurements, New J. Phys. 25, 013040 (2023).
  43. R. Stricker, D. Vodola, A. Erhard, L. Postler, M. Meth, M. Ringbauer, P. Schindler, R. Blatt, M. Müller, and T. Monz, Characterizing quantum instruments: From nondemolition measurements to quantum error correction, PRX Quantum 3, 030318 (2022).
  44. S.-L. Chen and J. Eisert, Semi-device-independently characterizing quantum temporal correlations, Phys. Rev. Lett. 132, 220201 (2024).
  45. S. Gherardini and G. De Chiara, Quasiprobabilities in quantum thermodynamics and many-body systems, PRX Quantum 5, 030201 (2024).
  46. C. Raj, T. Prasad, A. Chaturvedi, L. Pollyceno, D. Spegel-Lexne, S. Gómez, J. Argillander, A. Alarcón, G. B. Xavier, M. Pawłowski, and P. R. Dieguez, Certifying semi-device-independent security via wave-particle duality experiments, npj Quantum Information (2025).
  47. T. Heinosaari and T. Miyadera, Universality of sequential quantum measurements, Phys. Rev. A 91, 022110 (2015).
  48. R. Uola, G. Vitagliano, and C. Budroni, Leggett-Garg macrorealism and the quantum nondisturbance conditions, Phys. Rev. A 100, 042117 (2019).
  49. G. Schild and C. Emary, Maximum violations of the quantum-witness equality, Phys. Rev. A 92, 032101 (2015).
  50. M. H. Mohammady, T. Miyadera, and L. Loveridge, Measurement disturbance and conservation laws in quantum mechanics, Quantum 7, 1033 (2023).
  51. See Supplemental Material at http://link.aps.org/supplemental/10.1103/llgb-gql9 for details.
  52. B.-G. Englert, Fringe visibility and which-way information: An inequality, Phys. Rev. Lett. 77, 2154 (1996).
  53. T. H. Yoon and M. Cho, Quantitative complementarity of wave-particle duality, Sci. Adv. 7, eabi9268 (2021).
  54. D. Spegel-Lexne, S. Gómez, J. Argillander, M. Pawłowski, P. R. Dieguez, A. Alarcón, and G. B. Xavier, Experimental demonstration of the equivalence of entropic uncertainty with wave-particle duality, Sci. Adv. 10, eadr2007 (2024).
  55. P. Busch, P. Lahti, and R. F. Werner, Proof of Heisenberg's error-disturbance relation, Phys. Rev. Lett. 111, 160405 (2013).
  56. P. Busch, P. Lahti, and R. F. Werner, Heisenberg uncertainty for qubit measurements, Phys. Rev. A 89, 012129 (2014).
  57. M. Ozawa, Universal uncertainty principle in the measurement operator formalism, J. Opt. B: Quantum Semiclassical Opt. 7, S672 (2005).
  58. M. J. W. Hall, Prior information: How to circumvent the standard joint-measurement uncertainty relation, Phys. Rev. A 69, 052113 (2004).
  59. Y.-L. Mao, Z.-H. Ma, R.-B. Jin, Q.-C. Sun, S.-M. Fei, Q. Zhang, J. Fan, and J.-W. Pan, Error-disturbance trade-off in sequential quantum measurements, Phys. Rev. Lett. 122, 090404 (2019).
  60. R. H. Hadfield, Single-photon detectors for optical quantum information applications, Nat. Photon. 3, 696 (2009).
  61. J. S. Lundeen, A. Feito, H. Coldenstrodt-Ronge, K. L. Pregnell, C. Silberhorn, T. C. Ralph, J. Eisert, M. B. Plenio, and I. A. Walmsley, Tomography of quantum detectors, Nat. Phys. 5, 27 (2009).
  62. M. D. Eisaman, J. Fan, A. Migdall, and S. V. Polyakov, Invited review article: Single-photon sources and detectors, Rev. Sci. Instrum. 82, 071101 (2011).
  63. A. Zhang, J. Xie, H. Xu, K. Zheng, H. Zhang, Y.-T. Poon, V. Vedral, and L. Zhang, Experimental self-characterization of quantum measurements, Phys. Rev. Lett. 124, 040402 (2020).
  64. D. Orsucci, J.-D. Bancal, N. Sangouard, and P. Sekatski, How post-selection affects device-independent claims under the fair sampling assumption, Quantum 4, 238 (2020).
  65. J. Hloušek, M. Ježek, and J. Fiurášek, Direct experimental certification of quantum non-Gaussian character and Wigner function negativity of single-photon detectors, Phys. Rev. Lett. 126, 043601 (2021).
  66. L. Catani, M. Leifer, G. Scala, D. Schmid, and R. W. Spekkens, What is nonclassical about uncertainty relations? Phys. Rev. Lett. 129, 240401 (2022).
  67. L. Catani, M. Leifer, G. Scala, D. Schmid, and R. W. Spekkens, Aspects of the phenomenology of interference that are genuinely nonclassical, Phys. Rev. A 108, 022207 (2023).
  68. J. J. Halliwell, Leggett-Garg inequalities and no-signaling in time: A quasiprobability approach, Phys. Rev. A 93, 022123 (2016).
  69. D. Schmid, A review and reformulation of macroscopic realism: Resolving its deficiencies using the framework of generalized probabilistic theories, Quantum 8, 1217 (2024).
  70. S. Wagner, J.-D. Bancal, N. Sangouard, and P. Sekatski, Device-independent characterization of quantum instruments, Quantum 4, 243 (2020).
  71. S. Khandelwal and A. Tavakoli, Simulating quantum instruments with projective measurements and quantum postprocessing, Phys. Rev. Lett. 135, 040202 (2025).
  72. J. Bowles, F. Baccari, and A. Salavrakos, Bounding sets of sequential quantum correlations and device-independent randomness certification, Quantum 4, 344 (2020).
  73. R. Gallego, L. E. Würflinger, R. Chaves, A. Acín, and M. Navascués, Nonlocality in sequential correlation scenarios, New J. Phys. 16, 033037 (2014).
  74. S. Cheng, L. Liu, T. J. Baker, and M. J. W. Hall, Limitations on sharing Bell nonlocality between sequential pairs of observers, Phys. Rev. A 104, L060201 (2021).

Outline

Information

Phys. Rev. Research 8, L012011– Published 13 January, 2026
  • Received 6 November 2024
  • Revised 19 May 2025
  • Accepted 5 December 2025
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