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  • Open Access

General Class of Functionals for Certifying Quantum Incompatibility

Kuan-Yi Lee1,2,*, Jhen-Dong Lin2,†, Adam Miranowicz3, and Yueh-Nan Chen2,4,‡

  • *Contact author: kuanyi.lee@sns.it
  • †Contact author: jhendonglin@gmail.com
  • ‡Contact author: yuehnan@mail.ncku.edu.tw
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Phys. Rev. Lett. 136, 120203 – Published 24 March, 2026

DOI: https://doi.org/10.1103/rsx1-zvbp

Abstract

Quantum steering, measurement incompatibility, and instrument incompatibility have recently been recognized as unified manifestations of quantum incompatibility. Building on this perspective, we develop a general framework for constructing optimization-free, nonlinear incompatibility witnesses based on convex functionals, valid in arbitrary dimensions. We prove that these witnesses are nontrivial precisely when the underlying functional is nonaffine on extremal points (e.g., pure states for ensembles). For pure bipartite states, the witnesses yield lower bounds on entanglement measures, thereby outperforming most linear steering inequalities in the pure-state regime. Moreover, the construction extends in full generality to certify measurement and instrument incompatibility, where the witnesses act as genuine incompatibility monotones. We demonstrate the versatility of our approach with two operationally relevant functionals: the Wigner-Yanase skew information and an ℓ2-type coherence functional.

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References (71)

  1. T. Heinosaari, T. Miyadera, and M. Ziman, An invitation to quantum incompatibility, J. Phys. A 49, 123001 (2016).
  2. 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).
  3. R. Gallego and L. Aolita, Resource theory of steering, Phys. Rev. X 5, 041008 (2015).
  4. Y. Xiang, S. Cheng, Q. Gong, Z. Ficek, and Q. He, Quantum steering: Practical challenges and future directions, PRX Quantum 3, 030102 (2022).
  5. 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).
  6. F. Buscemi, All entangled quantum states are nonlocal, Phys. Rev. Lett. 108, 200401 (2012).
  7. N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, Bell nonlocality, Rev. Mod. Phys. 86, 419 (2014).
  8. N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, Quantum cryptography, Rev. Mod. Phys. 74, 145 (2002).
  9. C. Branciard, E. G. Cavalcanti, S. P. Walborn, V. Scarani, and H. M. Wiseman, One-sided device-independent quantum key distribution: Security, feasibility, and the connection with steering, Phys. Rev. A 85, 010301(R) (2012).
  10. M. Piani and J. Watrous, Necessary and sufficient quantum information characterization of Einstein-Podolsky-Rosen steering, Phys. Rev. Lett. 114, 060404 (2015).
  11. R. Takagi, B. Regula, K. Bu, Z.-W. Liu, and G. Adesso, Operational advantage of quantum resources in subchannel discrimination, Phys. Rev. Lett. 122, 140402 (2019).
  12. P. Skrzypczyk and N. Linden, Robustness of measurement, discrimination games, and accessible information, Phys. Rev. Lett. 122, 140403 (2019).
  13. 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).
  14. K. C. Tan, V. Narasimhachar, and B. Regula, Fisher information universally identifies quantum resources, Phys. Rev. Lett. 127, 200402 (2021).
  15. K. Flatt, H. Lee, C. R. I. Carceller, J. B. Brask, and J. Bae, Contextual advantages and certification for maximum-confidence discrimination, PRX Quantum 3, 030337 (2022).
  16. K.-Y. Lee, J.-D. Lin, A. Miranowicz, F. Nori, H.-Y. Ku, and Y.-N. Chen, Steering-enhanced quantum metrology using superpositions of noisy phase shifts, Phys. Rev. Res. 5, 013103 (2023).
  17. R. Uola, F. Lever, O. Gühne, and J.-P. Pellonpää, Unified picture for spatial, temporal, and channel steering, Phys. Rev. A 97, 032301 (2018).
  18. H. M. Wiseman, S. J. Jones, and A. C. Doherty, Steering, entanglement, nonlocality, and the Einstein-Podolsky-Rosen paradox, Phys. Rev. Lett. 98, 140402 (2007).
  19. M. T. Quintino, T. Vértesi, and N. Brunner, Joint measurability, Einstein-Podolsky-Rosen steering, and Bell nonlocality, Phys. Rev. Lett. 113, 160402 (2014).
  20. R. Uola, T. Moroder, and O. Gühne, Joint measurability of generalized measurements implies classicality, Phys. Rev. Lett. 113, 160403 (2014).
  21. R. Uola, C. Budroni, O. Gühne, and J.-P. Pellonpää, One-to-one mapping between steering and joint measurability problems, Phys. Rev. Lett. 115, 230402 (2015).
  22. H.-Y. Ku, C.-Y. Hsieh, S.-L. Chen, Y.-N. Chen, and C. Budroni, Complete classification of steerability under local filters and its relation with measurement incompatibility, Nat. Commun. 13, 4973 (2022).
  23. Y.-N. Chen, C.-M. Li, N. Lambert, S.-L. Chen, Y. Ota, G.-Y. Chen, and F. Nori, Temporal steering inequality, Phys. Rev. A 89, 032112 (2014).
  24. K. Bartkiewicz, A. Černoch, K. Lemr, A. Miranowicz, and F. Nori, Temporal steering and security of quantum key distribution with mutually unbiased bases against individual attacks, Phys. Rev. A 93, 062345 (2016).
  25. S.-L. Chen, N. Lambert, C.-M. Li, A. Miranowicz, Y.-N. Chen, and F. Nori, Quantifying non-Markovianity with temporal steering, Phys. Rev. Lett. 116, 020503 (2016).
  26. F. Buscemi, E. Chitambar, and W. Zhou, Complete resource theory of quantum incompatibility as quantum programmability, Phys. Rev. Lett. 124, 120401 (2020).
  27. L. Leppäjärvi and M. Sedlák, Incompatibility of quantum instruments, Quantum 8, 1246 (2024).
  28. K. Ji and E. Chitambar, Incompatibility as a resource for programmable quantum instruments, PRX Quantum 5, 010340 (2024).
  29. D. Cavalcanti and P. Skrzypczyk, Quantum steering: A review with focus on semidefinite programming, Rep. Prog. Phys. 80, 024001 (2016).
  30. S. Designolle, M. Farkas, and J. Kaniewski, Incompatibility robustness of quantum measurements: A unified framework, New J. Phys. 21, 113053 (2019).
  31. I. Kogias, P. Skrzypczyk, D. Cavalcanti, A. Acín, and G. Adesso, Hierarchy of steering criteria based on moments for all bipartite quantum systems, Phys. Rev. Lett. 115, 210401 (2015).
  32. S.-L. Chen, C. Budroni, Y.-C. Liang, and Y.-N. Chen, Natural framework for device-independent quantification of quantum steerability, measurement incompatibility, and self-testing, Phys. Rev. Lett. 116, 240401 (2016).
  33. T. Pramanik, M. Kaplan, and A. S. Majumdar, Fine-grained Einstein-Podolsky-Rosen–steering inequalities, Phys. Rev. A 90, 050305(R) (2014).
  34. T. Kriváchy, F. Fröwis, and N. Brunner, Tight steering inequalities from generalized entropic uncertainty relations, Phys. Rev. A 98, 062111 (2018).
  35. Ana C. S. Costa, R. Uola, and O. Gühne, Steering criteria from general entropic uncertainty relations, Phys. Rev. A 98, 050104(R) (2018).
  36. 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).
  37. E. G. Cavalcanti, C. J. Foster, M. Fuwa, and H. M. Wiseman, Analog of the Clauser–Horne–Shimony–Holt inequality for steering, J. Opt. Soc. Am. B 32, A74 (2015).
  38. E. P. Wigner and M. M. Yanase, Information contents of distributions, Proc. Natl. Acad. Sci. U.S.A. 49, 910 (1963).
  39. S. Luo, Wigner-Yanase skew information and uncertainty relations, Phys. Rev. Lett. 91, 180403 (2003).
  40. T. Baumgratz, M. Cramer, and M. B. Plenio, Quantifying coherence, Phys. Rev. Lett. 113, 140401 (2014).
  41. B. Yadin, M. Fadel, and M. Gessner, Metrological complementarity reveals the Einstein-Podolsky-Rosen paradox, Nat. Commun. 12, 2410 (2021).
  42. K.-Y. Lee, J.-D. Lin, K. Lemr, A. Černoch, A. Miranowicz, F. Nori, H.-Y. Ku, and Y.-N. Chen, Unveiling quantum steering by quantum-classical uncertainty complementarity, npj Quantum Inf. 11, 72 (2025).
  43. T. Heinosaari, D. Reitzner, and P. Stano, Notes on joint measurability of quantum observables, Found. Phys. 38, 1133–1147 (2008).
  44. B. Regula, Convex geometry of quantum resource quantification, J. Phys. A: Math. Theor. 51, 045303 (2018).
  45. See Supplemental Material at http://link.aps.org/supplemental/10.1103/rsx1-zvbp for additional proofs and examples.
  46. R. Alicki and M. Fannes, Entanglement boost for extractable work from ensembles of quantum batteries, Phys. Rev. E 87, 042123 (2013).
  47. D. Girolami, T. Tufarelli, and G. Adesso, Characterizing nonclassical correlations via local quantum uncertainty, Phys. Rev. Lett. 110, 240402 (2013).
  48. D. Girolami, Observable measure of quantum coherence in finite dimensional systems, Phys. Rev. Lett. 113, 170401 (2014).
  49. I. Marvian, R. W. Spekkens, and P. Zanardi, Quantum speed limits, coherence, and asymmetry, Phys. Rev. A 93, 052331 (2016).
  50. D. P. Pires, M. Cianciaruso, L. C. Céleri, G. Adesso, and D. O. Soares-Pinto, Generalized geometric quantum speed limits, Phys. Rev. X 6, 021031 (2016).
  51. R. Takagi, Skew informations from an operational view via resource theory of asymmetry, Sci. Rep. 9, 14562 (2019).
  52. K. Yamaguchi and H. Tajima, Beyond i.i.d. in the resource theory of asymmetry: An information-spectrum approach for quantum Fisher information, Phys. Rev. Lett. 131, 200203 (2023).
  53. A. Streltsov, G. Adesso, and M. B. Plenio, Colloquium: Quantum coherence as a resource, Rev. Mod. Phys. 89, 041003 (2017).
  54. F. Caravelli, B. Yan, L. P. García-Pintos, and A. Hamma, Energy storage and coherence in closed and open quantum batteries, Quantum 5, 505 (2021).
  55. G. Styliaris, N. Anand, L. Campos Venuti, and P. Zanardi, Quantum coherence and the localization transition, Phys. Rev. B 100, 224204 (2019).
  56. L. P. Hughston, R. Jozsa, and W. K. Wootters, A complete classification of quantum ensembles having a given density matrix, Phys. Lett. A 183, 14 (1993).
  57. R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Quantum entanglement, Rev. Mod. Phys. 81, 865 (2009).
  58. I. Šupić and J. Bowles, Self-testing of quantum systems: A review, Quantum 4, 337 (2020).
  59. S. Sarkar, D. Saha, J. Kaniewski, and R. Augusiak, Self-testing quantum systems of arbitrary local dimension with minimal number of measurements, npj Quantum Inf. 7, 151 (2021).
  60. J. Kaniewski, Weak form of self-testing, Phys. Rev. Res. 2, 033420 (2020).
  61. A. Tavakoli and R. Uola, Measurement incompatibility and steering are necessary and sufficient for operational contextuality, Phys. Rev. Res. 2, 013011 (2020).
  62. C. Budroni, A. Cabello, O. Gühne, M. Kleinmann, and J.-A. Larsson, Kochen-Specker contextuality, Rev. Mod. Phys. 94, 045007 (2022).
  63. Y.-L. Mao, H. Chen, C. Niu, Z.-D. Li, S. Yu, and J. Fan, Testing Heisenberg-type measurement uncertainty relations of three observables, Phys. Rev. Lett. 131, 150203 (2023).
  64. M. T. Quintino, C. Budroni, E. Woodhead, A. Cabello, and D. Cavalcanti, Device-independent tests of structures of measurement incompatibility, Phys. Rev. Lett. 123, 180401 (2019).
  65. C.-Y. Hsieh, H.-Y. Ku, and C. Budroni, Characterisation and fundamental limitations of irreversible stochastic steering distillation, arXiv:2309.06191.
  66. M. Piani, Channel steering, J. Opt. Soc. Am. B 32, A1 (2015).
  67. A. Mitra et al., Compatibility of quantum instruments, Phys. Rev. A 105, 052202 (2022).
  68. A. Mitra et al., Characterizing and quantifying the incompatibility of quantum instruments, Phys. Rev. A 107, 032217 (2023).
  69. J. Ghai et al., Instrument-based quantum resources: Quantification, hierarchies and towards constructing resource theories, arXiv:2508.09134.
  70. H.-Y. Ku, J. Kadlec, A. Černoch, M. T. Quintino, W. Zhou, K. Lemr, N. Lambert, A. Miranowicz, S.-L. Chen, F. Nori, and Y.-N. Chen, Quantifying quantumness of channels without entanglement, PRX Quantum 3, 020338 (2022).
  71. K.-Y. Lee, Supporting code for “General class of functionals for certifying quantum incompatibility,” 10.5281/zenodo.18761270.

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Phys. Rev. Lett. 136, 120203– Published 24 March, 2026
  • Received 16 October 2025
  • Revised 5 January 2026
  • Accepted 23 February 2026
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