Strict hierarchy between $n$-wise measurement simulability, compatibility structures, and multi-copy compatibility

Quantum 10, 2231 (2026). https://doi.org/10.22331/q-2026-10-08-2231 The incompatibility of quantum measurements, i.e. the fact that certain observable quantities cannot be measured jointly is widely regarded as a distinctive quantum feature with important implications for the foundations and the applications of quantum information theory. While the standard incompatibility of multiple measurements has been the focus of attention since the inception of quantum theory, its generalizations, such as measurement simulability, $n$-wise incompatibility, and mulit-copy incompatibility have only been proposed recently. Here, we point out that all these generalizations are differing notions of the question of how many measurements are genuinely contained in a measurement device. We then show, that all notions do differ not only in their operational meaning but also mathematically in the set of measurement assemblages they describe. We then fully resolve the relations between these different generalizations, by showing a strict hierarchy between these notions. Hence, we provide a general framework for generalized measurement incompatibility. Finally, we consider the implications our results have for recent works using these different notions.

The incompatibility of quantum measurements, i.e. the fact that certain observable quantities cannot be measured jointly is widely regarded as a distinctive quantum feature with important implications for the foundations and the applications of quantum information theory. While the standard incompatibility of multiple measurements has been the focus of attention since the inception of quantum theory, its generalizations, such as measurement simulability, $n$-wise incompatibility, and mulit-copy incompatibility have only been proposed recently. Here, we point out that all these generalizations are differing notions of the question of how many measurements are genuinely contained in a measurement device. We then show, that all notions do differ not only in their operational meaning but also mathematically in the set of measurement assemblages they describe. We then fully resolve the relations between these different generalizations, by showing a strict hierarchy between these notions. Hence, we provide a general framework for generalized measurement incompatibility. Finally, we consider the implications our results have for recent works using these different notions.

[1] J. A. Wheeler and W. H. Zurek, eds., Quantum Theory and Measurement (Princeton University Press, Princeton, 1983).

[2] P. Busch, P. Lahti, J.-P. Pellonpää, and K. Ylinen, Quantum Measurement (Springer International Publishing, 2016). https://doi.org/10.1007/978-3-319-43389-9

[3] M. Herrero-Collantes and J. C. Garcia-Escartin, Quantum random number generators, Rev. Mod. Phys. 89, 015004 (2017). https://doi.org/10.1103/RevModPhys.89.015004

[4] A. Acín and L. Masanes, Certified randomness in quantum physics, Nature 540, 213–219 (2016). https://doi.org/10.1038/nature20119

[5] W. C. Myrvold and J. Christian, Quantum Reality, Relativistic Causality, and Closing the Epistemic Circle: Essays in Honour of Abner Shimony (Springer Netherlands, 2009). https://doi.org/10.1007/978-1-4020-9107-0

[6] W. Heisenberg, Über quantentheoretische umdeutung kinematischer und mechanischer beziehungen, Zeitschrift für Physik 33, 879 (1925). https://doi.org/10.1007/BF01328377

[7] W. Heisenberg, Über den anschaulichen inhalt der quantentheoretischen kinematik und mechanik, Zeitschrift für Physik 43, 172 (1927). https://doi.org/10.1007/bf01397280

[8] H. P. Robertson, The uncertainty principle, Phys. Rev. 34, 163 (1929). https://doi.org/10.1103/PhysRev.34.163

[9] R. Uola, A. C. S. Costa, H. C. Nguyen, and O. Gühne, Quantum steering, Rev. Mod. Phys. 92, 015001 (2020a). https://doi.org/10.1103/RevModPhys.92.015001

[10] D. Cavalcanti and P. Skrzypczyk, Quantum steering: a review with focus on semidefinite programming, Reports on Progress in Physics 80, 024001 (2016). https://doi.org/10.1088/1361-6633/80/2/024001

[11] 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). https://doi.org/10.1103/PhysRevLett.98.140402

[12] M. T. Quintino, T. Vértesi, and N. Brunner, Joint Measurability, Einstein-Podolsky-Rosen Steering, and Bell Nonlocality, Phys. Rev. Lett. 113, 160402 (2014), arXiv:1406.6976 [quant-ph]. https://doi.org/10.1103/PhysRevLett.113.160402 arXiv:1406.6976

[13] R. Uola, T. Moroder, and O. Gühne, Joint Measurability of Generalized Measurements Implies Classicality, Phys. Rev. Lett. 113, 160403 (2014), arXiv:1407.2224 [quant-ph]. https://doi.org/10.1103/PhysRevLett.113.160403 arXiv:1407.2224

[14] 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). https://doi.org/10.1103/PhysRevLett.115.230402

[15] S. M. Barnett and S. Croke, Quantum state discrimination, Advances in Optics and Photonics 1, 238 (2009). https://doi.org/10.1364/aop.1.000238

[16] F. Buscemi, E. Chitambar, and W. Zhou, Complete resource theory of quantum incompatibility as quantum programmability, Phys. Rev. Lett. 124, 120401 (2020). https://doi.org/10.1103/PhysRevLett.124.120401

[17] C. Carmeli, T. Heinosaari, and A. Toigo, Quantum incompatibility witnesses, Phys. Rev. Lett. 122, 130402 (2019). https://doi.org/10.1103/PhysRevLett.122.130402

[18] R. Uola, T. Kraft, J. Shang, X.-D. Yu, and O. Gühne, Quantifying quantum resources with conic programming, Phys. Rev. Lett. 122, 130404 (2019). https://doi.org/10.1103/PhysRevLett.122.130404

[19] M. Oszmaniec and T. Biswas, Operational relevance of resource theories of quantum measurements, Quantum 3, 133 (2019). https://doi.org/10.22331/q-2019-04-26-133

[20] A. F. Ducuara and P. Skrzypczyk, Operational interpretation of weight-based resource quantifiers in convex quantum resource theories, Phys. Rev. Lett. 125, 110401 (2020). https://doi.org/10.1103/PhysRevLett.125.110401

[21] 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 (2020b). https://doi.org/10.1103/PhysRevLett.125.110402

[22] L. Guerini, M. T. Quintino, and L. Aolita, Distributed sampling, quantum communication witnesses, and measurement incompatibility, Phys. Rev. A 100, 042308 (2019), arXiv:1904.08435 [quant-ph]. https://doi.org/10.1103/PhysRevA.100.042308 arXiv:1904.08435

[23] A. Tavakoli and R. Uola, Measurement incompatibility and steering are necessary and sufficient for operational contextuality, Physical Review Research 2, 013011 (2020), arXiv:1905.03614 [quant-ph]. https://doi.org/10.1103/PhysRevResearch.2.013011 arXiv:1905.03614

[24] N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, Bell nonlocality, Rev. Mod. Phys. 86, 419 (2014). https://doi.org/10.1103/RevModPhys.86.419

[25] J. S. Bell, On the Einstein Podolsky Rosen paradox, Physics Physique Fizika 1, 195 (1964). https://doi.org/10.1103/PhysicsPhysiqueFizika.1.195

[26] M. T. Quintino, J. Bowles, F. Hirsch, and N. Brunner, Incompatible quantum measurements admitting a local-hidden-variable model, Phys. Rev. A 93, 10.1103/physreva.93.052115 (2016), 1510.06722 [quant-ph]. https://doi.org/10.1103/physreva.93.052115 arXiv:1510.06722

[27] F. Hirsch, M. T. Quintino, and N. Brunner, Quantum measurement incompatibility does not imply bell nonlocality, Phys. Rev. A 97, 012129 (2018). https://doi.org/10.1103/PhysRevA.97.012129

[28] E. Bene and T. Vértesi, Measurement incompatibility does not give rise to bell violation in general, New Journal of Physics 20, 013021 (2018). https://doi.org/10.1088/1367-2630/aa9ca3

[29] C. Budroni, A. Cabello, O. Gühne, M. Kleinmann, and J.-A. Larsson, Kochen-specker contextuality, Rev. Mod. Phys. 94, 045007 (2022). https://doi.org/10.1103/RevModPhys.94.045007

[30] R. Gallego, N. Brunner, C. Hadley, and A. Acín, Device-Independent Tests of Classical and Quantum Dimensions, Phys. Rev. Lett. 105, 230501 (2010), arXiv:1010.5064 [quant-ph]. https://doi.org/10.1103/PhysRevLett.105.230501 arXiv:1010.5064

[31] P. E. Frenkel and M. Weiner, Classical Information Storage in an n-Level Quantum System, Communications in Mathematical Physics 340, 563 (2015), arXiv:1304.5723 [cs.IT]. https://doi.org/10.1007/s00220-015-2463-0 arXiv:1304.5723

[32] S. Egelhaaf, J. Pauwels, M. T. Quintino, and R. Uola, Certifying measurement incompatibility in prepare-and-measure and Bell scenarios, Journal of Physics A Mathematical General 58, 095304 (2025), arXiv:2407.06787 [quant-ph]. https://doi.org/10.1088/1751-8121/adb3ff arXiv:2407.06787

[33] C. Carmeli, T. Heinosaari, and A. Toigo, Quantum random access codes and incompatibility of measurements, EPL (Europhysics Letters) 130, 50001 (2020). https://doi.org/10.1209/0295-5075/130/50001

[34] 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). https://doi.org/10.1103/PhysRevLett.125.080403

[35] T. Heinosaari, T. Miyadera, and M. Ziman, An invitation to quantum incompatibility, Journal of Physics A: Mathematical and Theoretical 49, 123001 (2016). https://doi.org/10.1088/1751-8113/49/12/123001

[36] 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). https://doi.org/10.1103/RevModPhys.95.011003

[37] L. Guerini, J. Bavaresco, M. T. Cunha, and A. Acín, Operational framework for quantum measurement simulability, Journal of Mathematical Physics 58, 092102 (2017). https://doi.org/10.1063/1.4994303

[38] C. Carmeli, T. Heinosaari, D. Reitzner, J. Schultz, and A. Toigo, Quantum incompatibility in collective measurements, Mathematics 4, 54 (2016). https://doi.org/10.3390/math4030054

[39] 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). https://doi.org/10.1103/PhysRevLett.123.180401

[40] R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Quantum entanglement, Rev. Mod. Phys. 81, 865 (2009). https://doi.org/10.1103/RevModPhys.81.865

[41] O. Gühne and G. Tóth, Entanglement detection, Physics Reports 474, 1–75 (2009). https://doi.org/10.1016/j.physrep.2009.02.004

[42] P. Horodecki, Łukasz Rudnicki, and K. Życzkowski, Multipartite entanglement (2024), arXiv:2409.04566 [quant-ph]. https://doi.org/10.48550/ARXIV.2409.04566 arXiv:2409.04566

[43] M. Luo, New genuinely multipartite entanglement, Advanced Quantum Technologies 4, 10.1002/qute.202000123 (2020). https://doi.org/10.1002/qute.202000123

[44] T. Kraft, S. Designolle, C. Ritz, N. Brunner, O. Gühne, and M. Huber, Quantum entanglement in the triangle network, Phys. Rev. A 103, L060401 (2021). https://doi.org/10.1103/PhysRevA.103.L060401

[45] L. Tendick, Quantum correlations cannot be reproduced with a finite number of measurements in any no-signaling theory, Phys. Rev. Lett. 135, 030201 (2025). https://doi.org/10.1103/7vjx-62ng

[46] L. Tendick, H. Kampermann, and D. Bruß, Distributed quantum incompatibility, Phys. Rev. Lett. 131, 120202 (2023a). https://doi.org/10.1103/PhysRevLett.131.120202

[47] I. Veeren, M. Plávala, L. Leppäjärvi, and R. Uola, Semi-device-independent certification of the number of measurements, Phys. Rev. A 109, 062203 (2024). https://doi.org/10.1103/PhysRevA.109.062203

[48] A. F. Ducuara, R. Takakura, F. J. Hernandez, and C. E. Susa, Multiobject operational tasks for measurement incompatibility, Phys. Rev. Res. 7, 033050 (2025). https://doi.org/10.1103/m7ln-tb1s

[49] 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). https://doi.org/10.1103/PhysRevLett.122.130403

[50] A. K. Das, S. Mukherjee, D. Saha, D. Das, and A. S. Majumdar, An operational approach to classifying measurement incompatibility (2025), arXiv:2401.01236 [quant-ph]. https://doi.org/10.1098/rspa.2025.0045 arXiv:2401.01236

[51] Y. Li, Y. Xiang, J. Tura, and Q. He, Necessary and sufficient condition for randomness certification from incompatibility, Phys. Rev. Lett. 135, 060201 (2025). https://doi.org/10.1103/zp36-2xg4

[52] R. F. Werner, Optimal cloning of pure states, Phys. Rev. A 58, 1827 (1998). https://doi.org/10.1103/PhysRevA.58.1827

[53] S. T. Ali, C. Carmeli, T. Heinosaari, and A. Toigo, Commutative povms and fuzzy observables, Foundations of Physics 39, 593–612 (2009). https://doi.org/10.1007/s10701-009-9292-y

[54] A. Fine, Hidden variables, joint probability, and the bell inequalities, Phys. Rev. Lett. 48, 291 (1982). https://doi.org/10.1103/PhysRevLett.48.291

[55] S. Boyd and L. Vandenberghe, Convex Optimization (Cambridge University Press, 2004). https://doi.org/10.1017/CBO9780511804441

[56] P. Busch, Unsharp reality and joint measurements for spin observables, Phys. Rev. D 33, 2253 (1986). https://doi.org/10.1103/PhysRevD.33.2253

[57] E. Haapasalo, T. Heinosaari, and J.-P. Pellonpää, Quantum measurements on finite dimensional systems: relabeling and mixing, Quantum Information Processing 11, 1751–1763 (2011). https://doi.org/10.1007/s11128-011-0330-2

[58] M. Oszmaniec, L. Guerini, P. Wittek, and A. Acín, Simulating positive-operator-valued measures with projective measurements, Phys. Rev. Lett. 119, 190501 (2017). https://doi.org/10.1103/PhysRevLett.119.190501

[59] L. Tendick, M. Kliesch, H. Kampermann, and D. Bruß, Distance-based resource quantification for sets of quantum measurements, Quantum 7, 1003 (2023b). https://doi.org/10.22331/q-2023-05-15-1003

[60] J. Bavaresco, M. Murao, and M. T. Quintino, Strict Hierarchy between Parallel, Sequential, and Indefinite-Causal-Order Strategies for Channel Discrimination, Phys. Rev. Lett. 127, 200504 (2021), arXiv:2011.08300 [quant-ph]. https://doi.org/10.1103/PhysRevLett.127.200504 arXiv:2011.08300

[61] S. N. Filippov, T. Heinosaari, and L. Leppäjärvi, Simulability of observables in general probabilistic theories, Phys. Rev. A 97, 062102 (2018). https://doi.org/10.1103/PhysRevA.97.062102

[62] 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). https://doi.org/10.1103/PhysRevLett.103.230402

[63] M. Plávala, General probabilistic theories: An introduction, Physics Reports 1033, 1–64 (2023). https://doi.org/10.1016/j.physrep.2023.09.001

[64] Online repository for code used in the proof of result 1, https://github.com/LucasTendick/measurement_simulability (2025). https://github.com/LucasTendick/measurement_simulability

[65] Online repository for code used in the proof of result 2, https://github.com/costantinobudroni/measurement_simulability (2024). https://github.com/costantinobudroni/measurement_simulability

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Strict hierarchy between $n$-wise measurement simulability, compatibility structures, and multi-copy compatibility

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