External quantum fluctuations select measurement contexts
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AbstractQuantum paradoxes show that the outcomes of different quantum measurements cannot be described by a single measurement-independent reality. Any theoretical description of a quantum measurement implies the selection of a specific measurement context. Here, we investigate generalised quantum measurements, in order to identify the mechanism by which this specific context is selected. We show that external quantum fluctuations, represented by the initial state of the measurement apparatus, play an essential role in the selection of the context. This has the non-trivial consequence that, when considering measurements other than just idealised projection-valued measures, different outcomes of a single measurement setup can represent different measurement contexts. We further show this result underpins recent claims that contextuality can occur in scenarios without measurement incompatibility.Popular summaryQuantum paradoxes show that the outcomes of different quantum measurements cannot be described by a single measurement-independent reality. Any theoretical description of a quantum measurement implies the selection of a specific measurement context – a set of incompatible options, like settings on a switch, or non-overlapping answers to a question (e.g., "yes" or "no"). In quantum mechanics, combining inferences about a system from multiple different contexts can lead to seemingly nonsensical results. Here, we investigate generalised quantum measurements, which don't require that the different measurement outcomes are non-overlapping (e.g., "yes", "maybe", and "no"), in order to identify the mechanism by which this specific context is selected. We show that external quantum fluctuations (from the immediate environment outside our system), represented by the initial state of the measurement apparatus, play an essential role in the selection of the context. This has the non-trivial consequence that, when considering measurements other than just idealised projection-valued measures (e.g., measurements where the measurement outcomes aren't incompatible, so can overlap), different outcomes of a single measurement setup can represent different measurement contexts. We further show this result underpins recent claims that contextuality can occur in scenarios without measurement incompatibility.► BibTeX data@article{Hance2026externalquantum, doi = {10.22331/q-2026-05-13-2106}, url = {https://doi.org/10.22331/q-2026-05-13-2106}, title = {External quantum fluctuations select measurement contexts}, author = {Hance, Jonte R. and Ji, Ming and Matsushita, Tomonori and Hofmann, Holger F.}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2106}, month = may, year = {2026} }► References [1] Simon Kochen and Ernst Specker. ``The problem of hidden variables in quantum mechanics''. Indiana Univ. Math. J. 17, 59–87 (1968). https://doi.org/10.1512/iumj.1968.17.17004 [2] Costantino Budroni, Adán Cabello, Otfried Gühne, Matthias Kleinmann, and Jan-Åke Larsson. ``Kochen-Specker contextuality''. Rev. Mod. Phys. 94, 045007 (2022). https://doi.org/10.1103/RevModPhys.94.045007 [3] R. W. Spekkens. ``Contextuality for preparations, transformations, and unsharp measurements''. Phys. Rev. A 71, 052108 (2005). https://doi.org/10.1103/PhysRevA.71.052108 [4] Robert W. Spekkens. ``Negativity and contextuality are equivalent notions of nonclassicality''. Phys. Rev. Lett. 101, 020401 (2008). https://doi.org/10.1103/PhysRevLett.101.020401 [5] Matthew F. Pusey. ``Anomalous weak values are proofs of contextuality''. Phys. Rev. Lett. 113, 200401 (2014). https://doi.org/10.1103/PhysRevLett.113.200401 [6] John H. Selby, David Schmid, Elie Wolfe, Ana Belén Sainz, Ravi Kunjwal, and Robert W. Spekkens. ``Contextuality without incompatibility''. Phys. Rev. Lett. 130, 230201 (2023). https://doi.org/10.1103/PhysRevLett.130.230201 [7] Ming Ji and Holger F. Hofmann. ``Quantitative relations between different measurement contexts''. Quantum 8, 1255 (2024). https://doi.org/10.22331/q-2024-02-14-1255 [8] John H. Selby, David Schmid, Elie Wolfe, Ana Belén Sainz, Ravi Kunjwal, and Robert W. Spekkens. ``Accessible fragments of generalized probabilistic theories, cone equivalence, and applications to witnessing nonclassicality''. Phys. Rev. A 107, 062203 (2023). https://doi.org/10.1103/PhysRevA.107.062203 [9] Holger F. Hofmann. ``Sequential propagation of a single photon through five measurement contexts in a three-path interferometer''. Optica Quantum 1, 63–70 (2023). https://doi.org/10.1364/OPTICAQ.502468 [10] Berthold-Georg Englert. ``Fringe visibility and which-way information: An inequality''. Phys. Rev. Lett. 77, 2154–2157 (1996). https://doi.org/10.1103/PhysRevLett.77.2154 [11] Tomonori Matsushita and Holger F. Hofmann. ``Uncertainty limits of the information exchange between a quantum system and an external meter''. Phys. Rev. A 104, 012219 (2021). https://doi.org/10.1103/PhysRevA.104.012219 [12] Israel Gelfand and Mark Neumark. ``On the imbedding of normed rings into the ring of operators in Hilbert space''. Recueil Mathématique (Nouvelle série) 12, 197–217 (1943). url: https://www.mathnet.ru/eng/sm/v54/i2/p197. https://www.mathnet.ru/eng/sm/v54/i2/p197 [13] Wojciech Hubert Zurek. ``Decoherence, einselection, and the quantum origins of the classical''. Rev. Mod. Phys. 75, 715–775 (2003). https://doi.org/10.1103/RevModPhys.75.715 [14] Teiko Heinosaari and Michael M. Wolf. ``Nondisturbing quantum measurements''. Journal of Mathematical Physics 51, 092201 (2010). https://doi.org/10.1063/1.3480658 [15] Teiko Heinosaari, Takayuki Miyadera, and Mário 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 [16] Martin J. Renner. ``Compatibility of generalized noisy qubit measurements''. Phys. Rev. Lett. 132, 250202 (2024). https://doi.org/10.1103/PhysRevLett.132.250202 [17] Kengo Matsuyama, Holger F Hofmann, and Masataka Iinuma. ``Experimental investigation of the relation between measurement uncertainties and non-local quantum correlations''. Journal of Physics Communications 5, 115012 (2021). https://doi.org/10.1088/2399-6528/ac3109 [18] Salvatore Virzì, Enrico Rebufello, Francesco Atzori, Alessio Avella, Fabrizio Piacentini, Rudi Lussana, Iris Cusini, Francesca Madonini, Federica Villa, Marco Gramegna, Eliahu Cohen, Ivo Pietro Degiovanni, and Marco Genovese. ``Entanglement-preserving measurement of the Bell parameter on a single entangled pair''. Quantum Science and Technology 9, 045027 (2024). https://doi.org/10.1088/2058-9565/ad6a37 [19] Holger F. Hofmann. ``Local measurement uncertainties impose a limit on nonlocal quantum correlations''. Phys. Rev. A 100, 012123 (2019). https://doi.org/10.1103/PhysRevA.100.012123 [20] Marco Genovese and Fabrizio Piacentini. ``Consequences of the single-pair measurement of the Bell parameter''. Phys. Rev. A 111, 022204 (2025). https://doi.org/10.1103/PhysRevA.111.022204 [21] Marian Kupczynski. ``Comment on ``Consequences of the single-pair measurement of the Bell parameter''''. Phys. Rev. A 112, 066201 (2025). https://doi.org/10.1103/hvm8-wtvz [22] Justo Pastor Lambare. ``Comment on ``Consequences of the single-pair measurement of the Bell parameter''''. Phys. Rev. A 112, 066202 (2025). https://doi.org/10.1103/gf3c-r5sv [23] Marco Genovese and Fabrizio Piacentini. ``Reply to ``Comments on `Consequences of the single-pair measurement of the Bell parameter' ''''. Phys. Rev. A 112, 066203 (2025). https://doi.org/10.1103/prqc-2jbw [24] Jonte R Hance. ``Counterfactual restrictions and Bell’s theorem''. Journal of Physics Communications 8, 122001 (2024). https://doi.org/10.1088/2399-6528/ad9b6d [25] Jonte R Hance, Tomonori Matsushita, and Holger F Hofmann. ``Counterfactuality, back-action, and information gain in multi-path interferometers''. Quantum Science and Technology 9, 045015 (2024). https://doi.org/10.1088/2058-9565/ad63c7 [26] Holger F. Hofmann. ``What does the operator algebra of quantum statistics tell us about the objective causes of observable effects?''. Entropy 22 (2020). https://doi.org/10.3390/e22060638 [27] John Von Neumann. ``Mathematical foundations of quantum mechanics: New edition''.
Princeton University Press. (2018). https://doi.org/10.2307/j.ctt1wq8zhp [28] Robert Golub and Steven K Lamoreaux. ``A retrospective review of von Neumann’s analysis of hidden variables in quantum mechanics''. Academia Quantum 1 (2024). https://doi.org/10.20935/AcadQuant7311Cited byCould not fetch Crossref cited-by data during last attempt 2026-05-13 09:13:56: Could not fetch cited-by data for 10.22331/q-2026-05-13-2106 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-05-13 09:13:57: No response from ADS or unable to decode the received json data when getting the list of citing works.This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions. AbstractQuantum paradoxes show that the outcomes of different quantum measurements cannot be described by a single measurement-independent reality. Any theoretical description of a quantum measurement implies the selection of a specific measurement context. Here, we investigate generalised quantum measurements, in order to identify the mechanism by which this specific context is selected. We show that external quantum fluctuations, represented by the initial state of the measurement apparatus, play an essential role in the selection of the context. This has the non-trivial consequence that, when considering measurements other than just idealised projection-valued measures, different outcomes of a single measurement setup can represent different measurement contexts. We further show this result underpins recent claims that contextuality can occur in scenarios without measurement incompatibility.Popular summaryQuantum paradoxes show that the outcomes of different quantum measurements cannot be described by a single measurement-independent reality. Any theoretical description of a quantum measurement implies the selection of a specific measurement context – a set of incompatible options, like settings on a switch, or non-overlapping answers to a question (e.g., "yes" or "no"). In quantum mechanics, combining inferences about a system from multiple different contexts can lead to seemingly nonsensical results. Here, we investigate generalised quantum measurements, which don't require that the different measurement outcomes are non-overlapping (e.g., "yes", "maybe", and "no"), in order to identify the mechanism by which this specific context is selected. We show that external quantum fluctuations (from the immediate environment outside our system), represented by the initial state of the measurement apparatus, play an essential role in the selection of the context. This has the non-trivial consequence that, when considering measurements other than just idealised projection-valued measures (e.g., measurements where the measurement outcomes aren't incompatible, so can overlap), different outcomes of a single measurement setup can represent different measurement contexts. We further show this result underpins recent claims that contextuality can occur in scenarios without measurement incompatibility.► BibTeX data@article{Hance2026externalquantum, doi = {10.22331/q-2026-05-13-2106}, url = {https://doi.org/10.22331/q-2026-05-13-2106}, title = {External quantum fluctuations select measurement contexts}, author = {Hance, Jonte R. and Ji, Ming and Matsushita, Tomonori and Hofmann, Holger F.}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2106}, month = may, year = {2026} }► References [1] Simon Kochen and Ernst Specker. ``The problem of hidden variables in quantum mechanics''. Indiana Univ. Math. J. 17, 59–87 (1968). https://doi.org/10.1512/iumj.1968.17.17004 [2] Costantino Budroni, Adán Cabello, Otfried Gühne, Matthias Kleinmann, and Jan-Åke Larsson. ``Kochen-Specker contextuality''. Rev. Mod. Phys. 94, 045007 (2022). https://doi.org/10.1103/RevModPhys.94.045007 [3] R. W. Spekkens. ``Contextuality for preparations, transformations, and unsharp measurements''. Phys. Rev. A 71, 052108 (2005). https://doi.org/10.1103/PhysRevA.71.052108 [4] Robert W. Spekkens. ``Negativity and contextuality are equivalent notions of nonclassicality''. Phys. Rev. Lett. 101, 020401 (2008). https://doi.org/10.1103/PhysRevLett.101.020401 [5] Matthew F. Pusey. ``Anomalous weak values are proofs of contextuality''. Phys. Rev. Lett. 113, 200401 (2014). https://doi.org/10.1103/PhysRevLett.113.200401 [6] John H. Selby, David Schmid, Elie Wolfe, Ana Belén Sainz, Ravi Kunjwal, and Robert W. Spekkens. ``Contextuality without incompatibility''. Phys. Rev. Lett. 130, 230201 (2023). https://doi.org/10.1103/PhysRevLett.130.230201 [7] Ming Ji and Holger F. Hofmann. ``Quantitative relations between different measurement contexts''. Quantum 8, 1255 (2024). https://doi.org/10.22331/q-2024-02-14-1255 [8] John H. Selby, David Schmid, Elie Wolfe, Ana Belén Sainz, Ravi Kunjwal, and Robert W. Spekkens. ``Accessible fragments of generalized probabilistic theories, cone equivalence, and applications to witnessing nonclassicality''. Phys. Rev. A 107, 062203 (2023). https://doi.org/10.1103/PhysRevA.107.062203 [9] Holger F. Hofmann. ``Sequential propagation of a single photon through five measurement contexts in a three-path interferometer''. Optica Quantum 1, 63–70 (2023). https://doi.org/10.1364/OPTICAQ.502468 [10] Berthold-Georg Englert. ``Fringe visibility and which-way information: An inequality''. Phys. Rev. Lett. 77, 2154–2157 (1996). https://doi.org/10.1103/PhysRevLett.77.2154 [11] Tomonori Matsushita and Holger F. Hofmann. ``Uncertainty limits of the information exchange between a quantum system and an external meter''. Phys. Rev. A 104, 012219 (2021). https://doi.org/10.1103/PhysRevA.104.012219 [12] Israel Gelfand and Mark Neumark. ``On the imbedding of normed rings into the ring of operators in Hilbert space''. Recueil Mathématique (Nouvelle série) 12, 197–217 (1943). url: https://www.mathnet.ru/eng/sm/v54/i2/p197. https://www.mathnet.ru/eng/sm/v54/i2/p197 [13] Wojciech Hubert Zurek. ``Decoherence, einselection, and the quantum origins of the classical''. Rev. Mod. Phys. 75, 715–775 (2003). https://doi.org/10.1103/RevModPhys.75.715 [14] Teiko Heinosaari and Michael M. Wolf. ``Nondisturbing quantum measurements''. Journal of Mathematical Physics 51, 092201 (2010). https://doi.org/10.1063/1.3480658 [15] Teiko Heinosaari, Takayuki Miyadera, and Mário 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 [16] Martin J. Renner. ``Compatibility of generalized noisy qubit measurements''. Phys. Rev. Lett. 132, 250202 (2024). https://doi.org/10.1103/PhysRevLett.132.250202 [17] Kengo Matsuyama, Holger F Hofmann, and Masataka Iinuma. ``Experimental investigation of the relation between measurement uncertainties and non-local quantum correlations''. Journal of Physics Communications 5, 115012 (2021). https://doi.org/10.1088/2399-6528/ac3109 [18] Salvatore Virzì, Enrico Rebufello, Francesco Atzori, Alessio Avella, Fabrizio Piacentini, Rudi Lussana, Iris Cusini, Francesca Madonini, Federica Villa, Marco Gramegna, Eliahu Cohen, Ivo Pietro Degiovanni, and Marco Genovese. ``Entanglement-preserving measurement of the Bell parameter on a single entangled pair''. Quantum Science and Technology 9, 045027 (2024). https://doi.org/10.1088/2058-9565/ad6a37 [19] Holger F. Hofmann. ``Local measurement uncertainties impose a limit on nonlocal quantum correlations''. Phys. Rev. A 100, 012123 (2019). https://doi.org/10.1103/PhysRevA.100.012123 [20] Marco Genovese and Fabrizio Piacentini. ``Consequences of the single-pair measurement of the Bell parameter''. Phys. Rev. A 111, 022204 (2025). https://doi.org/10.1103/PhysRevA.111.022204 [21] Marian Kupczynski. ``Comment on ``Consequences of the single-pair measurement of the Bell parameter''''. Phys. Rev. A 112, 066201 (2025). https://doi.org/10.1103/hvm8-wtvz [22] Justo Pastor Lambare. ``Comment on ``Consequences of the single-pair measurement of the Bell parameter''''. Phys. Rev. A 112, 066202 (2025). https://doi.org/10.1103/gf3c-r5sv [23] Marco Genovese and Fabrizio Piacentini. ``Reply to ``Comments on `Consequences of the single-pair measurement of the Bell parameter' ''''. Phys. Rev. A 112, 066203 (2025). https://doi.org/10.1103/prqc-2jbw [24] Jonte R Hance. ``Counterfactual restrictions and Bell’s theorem''. Journal of Physics Communications 8, 122001 (2024). https://doi.org/10.1088/2399-6528/ad9b6d [25] Jonte R Hance, Tomonori Matsushita, and Holger F Hofmann. ``Counterfactuality, back-action, and information gain in multi-path interferometers''. Quantum Science and Technology 9, 045015 (2024). https://doi.org/10.1088/2058-9565/ad63c7 [26] Holger F. Hofmann. ``What does the operator algebra of quantum statistics tell us about the objective causes of observable effects?''. Entropy 22 (2020). https://doi.org/10.3390/e22060638 [27] John Von Neumann. ``Mathematical foundations of quantum mechanics: New edition''.
Princeton University Press. (2018). https://doi.org/10.2307/j.ctt1wq8zhp [28] Robert Golub and Steven K Lamoreaux. ``A retrospective review of von Neumann’s analysis of hidden variables in quantum mechanics''. Academia Quantum 1 (2024). https://doi.org/10.20935/AcadQuant7311Cited byCould not fetch Crossref cited-by data during last attempt 2026-05-13 09:13:56: Could not fetch cited-by data for 10.22331/q-2026-05-13-2106 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-05-13 09:13:57: No response from ADS or unable to decode the received json data when getting the list of citing works.This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.