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  • Open Access

Resource theory of interactive quantum instruments

Chung-Yun Hsieh1,*, Armin Tavakoli2, Huan-Yu Ku3, and Paul Skrzypczyk1

  • *Contact author: chung-yun.hsieh@bristol.ac.uk
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Phys. Rev. A 113, 062445 – Published 17 June, 2026

DOI: https://doi.org/10.1103/dx6d-4kjy

Abstract

Quantum instruments describe both the classical outcome and the updated quantum state in a measurement process. To do this in a nontrivial way, instruments must have the capability to interact coherently with the state that they measure. Here we develop a resource theory for instruments. We consider a relevant quantifier of the separation between interactive and noninteractive instruments and show that it admits three distinct operational interpretations in terms of quantum information tasks. These concern (i) the preservation of maximally entangled states after a local measurement, (ii) the average ability to preserve random states after measurement, and (iii) the ability to recover the classical information generated from measuring half of a maximally entangled state. We also introduce a natural set of allowed operations and show that the third task fully characterizes the resource content of instruments. Our general framework reproduces as special cases established resource theories for channels and measurements.

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References (38)

  1. E. Chitambar and G. Gour, Quantum resource theories, Rev. Mod. Phys. 91, 025001 (2019).
  2. S. Khandelwal and A. Tavakoli, Simulating quantum instruments with projective measurements and quantum postprocessing, Phys. Rev. Lett. 135, 040202 (2025).
  3. 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).
  4. C.-Y. Hsieh, R. Uola, and P. Skrzypczyk, Quantum complementarity: A novel resource for unambiguous exclusion and encryption, arXiv:2309.11968.
  5. 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).
  6. A. Tavakoli and R. Uola, Measurement incompatibility and steering are necessary and sufficient for operational contextuality, Phys. Rev. Res. 2, 013011 (2020).
  7. F. Buscemi, E. Chitambar, and W. Zhou, Complete resource theory of quantum incompatibility as quantum programmability, Phys. Rev. Lett. 124, 120401 (2020).
  8. K. Ji and E. Chitambar, Incompatibility as a resource for programmable quantum instruments, PRX Quantum 5, 010340 (2024).
  9. R. Takagi and B. Regula, General resource theories in quantum mechanics and beyond: Operational characterization via discrimination tasks, Phys. Rev. X 9, 031053 (2019).
  10. M. Oszmaniec and T. Biswas, Operational relevance of resource theories of quantum measurements, Quantum 3, 133 (2019).
  11. A. F. Ducuara and P. Skrzypczyk, Operational interpretation of weight-based resource quantifiers in convex quantum resource theories, Phys. Rev. Lett. 125, 110401 (2020).
  12. R. Uola, T. Bullock, T. Kraft, J.-P. Pellonpää, and N. Brunner, All quantum resources provide an advantage in exclusion tasks, Phys. Rev. Lett. 125, 110402 (2020).
  13. A. F. Ducuara and P. Skrzypczyk, Characterization of quantum betting tasks in terms of Arimoto mutual information, PRX Quantum 3, 020366 (2022).
  14. 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).
  15. 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).
  16. D. Cavalcanti and P. Skrzypczyk, Quantum steering: A review with focus on semidefinite programming, Rep. Prog. Phys. 80, 024001 (2017).
  17. P. Skrzypczyk and N. Linden, Robustness of measurement, discrimination games, and accessible information, Phys. Rev. Lett. 122, 140403 (2019).
  18. C.-Y. Hsieh and S.-L. Chen, Thermodynamic approach to quantifying incompatible instruments, Phys. Rev. Lett. 133, 170401 (2024).
  19. J. Ghai and A. Mitra, Instrument-based quantum resources: Quantification, hierarchies and towards constructing resource theories, arXiv:2508.09134.
  20. F. Buscemi, K. Kobayashi, S. Minagawa, P. Perinotti, and A. Tosini, Unifying different notions of quantum incompatibility into a strict hierarchy of resource theories of communication, Quantum 7, 1035 (2023).
  21. L. Leppäjärvi and M. Sedlák, Incompatibility of quantum instruments, Quantum 8, 1246 (2024).
  22. K. Kuroiwa, R. Takagi, G. Adesso, and H. Yamasaki, Robustness- and weight-based resource measures without convexity restriction: Multicopy witness and operational advantage in static and dynamical quantum resource theories, Phys. Rev. A 109, 042403 (2024).
  23. S. Boyd and L. Vandenberghe, Convex Optimization (Cambridge University Press, Cambridge, UK, 2004).
  24. P. Skrzypczyk and D. Cavalcanti, Semidefinite Programming in Quantum Information Science, 2053-2563 (IOP, Bristol, 2023).
  25. R. Takagi, K. Wang, and M. Hayashi, Application of the resource theory of channels to communication scenarios, Phys. Rev. Lett. 124, 120502 (2020).
  26. M. Horodecki, P. Horodecki, and R. Horodecki, General teleportation channel, singlet fraction, and quasidistillation, Phys. Rev. A 60, 1888 (1999).
  27. R. Konig, R. Renner, and C. Schaffner, The operational meaning of min- and max-entropy, IEEE Trans. Inf. Theory 55, 4337 (2009).
  28. N. Datta, Min- and max-relative entropies and a new entanglement monotone, IEEE Trans. Inf. Theory 55, 2816 (2009).
  29. M. A. Nielsen, A simple formula for the average gate fidelity of a quantum dynamical operation, Phys. Lett. A 303, 249 (2002).
  30. F. Buscemi and N. Datta, The quantum capacity of channels with arbitrarily correlated noise, IEEE Trans. Inf. Theory 56, 1447 (2010).
  31. M.-D. Choi, Completely positive linear maps on complex matrices, Linear Algebra Appl. 10, 285 (1975).
  32. A. Jamiołkowski, Linear transformations which preserve trace and positive semidefiniteness of operators, Rep. Math. Phys. 3, 275 (1972).
  33. M. Slater, Lagrange multipliers revisited, in Traces and Emergence of Nonlinear Programming, edited by G. Giorgi and T. H. Kjeldsen (Springer, Basel, 2014), pp. 293–306.
  34. J. Watrous, The Theory of Quantum Information (Cambridge University Press, Cambridge, UK, 2018).
  35. A. Tavakoli, A. Pozas-Kerstjens, P. Brown, and M. Araújo, Semidefinite programming relaxations for quantum correlations, Rev. Mod. Phys. 96, 045006 (2024).
  36. A. W. Harrow, The church of the symmetric subspace, arXiv:1308.6595.
  37. M. Sion, On general minimax theorems, Pac. J. Math. 8, 171 (1958).
  38. H. Komiya, Elementary proof for Sion's minimax theorem, Kodai Math. J. 11, 5 (1988).

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Phys. Rev. A 113, 062445– Published 17 June, 2026
  • Received 14 April 2026
  • Accepted 28 May 2026
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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.

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