Solvable Criterion for the Contextuality of any Prepare-and-Measure Scenario

Victor Gitton and Mischa P. Woods

Institute for Theoretical Physics, ETH Zürich, Switzerland

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Abstract

Starting from arbitrary sets of quantum states and measurements, referred to as the prepare-and-measure scenario, an operationally noncontextual ontological model of the quantum statistics associated with the prepare-and-measure scenario is constructed. The operationally noncontextual ontological model coincides with standard Spekkens noncontextual ontological models for tomographically complete scenarios, while covering the non-tomographically complete case with a new notion of a reduced space, which we motivate following the guiding principles of noncontextuality. A mathematical criterion, called $\textit{unit separability}$, is formulated as the relevant classicality criterion – the name is inspired by the usual notion of quantum state separability. Using this criterion, we derive a new upper bound on the cardinality of the ontic space. Then, we recast the unit separability criterion as a (possibly infinite) set of linear constraints, from which we obtain two separate hierarchies of algorithmic tests to witness the non-classicality or certify the classicality of a scenario. Finally, we reformulate our results in the framework of generalized probabilistic theories and discuss the implications for simplex-embeddability in such theories.

Starting from arbitrary sets of quantum states and measurements, referred to as the prepare-and-measure scenario, an operationally noncontextual ontological model of the quantum statistics associated with the prepare-and-measure scenario is constructed. The operationally noncontextual ontological model coincides with standard Spekkens noncontextual ontological models for tomographically complete scenarios, while covering the non-tomographically complete case with a new notion of a reduced space, which we motivate following the guiding principles of noncontextuality. A mathematical criterion, called unit separability, is formulated as the relevant classicality criterion – the name is inspired by the usual notion of quantum state separability. Using this criterion, we derive a new upper bound on the cardinality of the ontic space. Then, we recast the unit separability criterion as a (possibly infinite) set of linear constraints, from which we obtain two separate hierarchies of algorithmic tests to witness the non-classicality or certify the classicality of a scenario. Finally, we reformulate our results in the framework of generalized probabilistic theories and discuss the implications for simplex-embeddability in such theories.

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[1] A. Einstein, B. Podolsky, and N. Rosen. ``Can quantum-mechanical description of physical reality be considered complete?''. Phys. Rev. 47, 777–780 (1935).
https:/​/​doi.org/​10.1103/​PhysRev.47.777

[2] J. S. Bell. ``On the Einstein Podolsky Rosen paradox''. Physics Physique Fizika 1, 195–200 (1964).
https:/​/​doi.org/​10.1103/​PhysicsPhysiqueFizika.1.195

[3] Simon Kochen and E. P. Specker. ``The problem of hidden variables in quantum mechanics''. Journal of Mathematics and Mechanics 17, 59–87 (1967). url: http:/​/​www.jstor.org/​stable/​24902153.
http:/​/​www.jstor.org/​stable/​24902153

[4] R. W. Spekkens. ``Contextuality for preparations, transformations, and unsharp measurements''. Phys. Rev. A 71, 052108 (2005).
https:/​/​doi.org/​10.1103/​PhysRevA.71.052108

[5] David Schmid and Robert W. Spekkens. ``Contextual advantage for state discrimination''. Phys. Rev. X 8, 011015 (2018).
https:/​/​doi.org/​10.1103/​PhysRevX.8.011015

[6] David R. M. Arvidsson-Shukur, Nicole Yunger Halpern, Hugo V. Lepage, Aleksander A. Lasek, Crispin H. W. Barnes, and Seth Lloyd. ``Quantum advantage in postselected metrology''. Nature Communications 11, 3775 (2020).
https:/​/​doi.org/​10.1038/​s41467-020-17559-w

[7] Mark Howard, Joel J. Wallman, Victor Veitch, and Joseph Emerson. ``Contextuality supplies the magic for quantum computation''. Nature 510, 351–355 (2014).
https:/​/​doi.org/​10.1038/​nature13460

[8] Juan Bermejo-Vega, Nicolas Delfosse, Dan E. Browne, Cihan Okay, and Robert Raussendorf. ``Contextuality as a resource for models of quantum computation on qubits''. Physical Review Letters 119, 120505 (2017).
https:/​/​doi.org/​10.1103/​PhysRevLett.119.120505

[9] David Schmid, John H. Selby, Elie Wolfe, Ravi Kunjwal, and Robert W. Spekkens. ``Characterization of noncontextuality in the framework of generalized probabilistic theories''. PRX Quantum 2, 010331 (2021).
https:/​/​doi.org/​10.1103/​PRXQuantum.2.010331

[10] 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

[11] David Schmid, Robert W. Spekkens, and Elie Wolfe. ``All the noncontextuality inequalities for arbitrary prepare-and-measure experiments with respect to any fixed set of operational equivalences''. Phys. Rev. A 97, 062103 (2018).
https:/​/​doi.org/​10.1103/​PhysRevA.97.062103

[12] Robert W. Spekkens. ``The status of determinism in proofs of the impossibility of a noncontextual model of quantum theory''. Foundations of Physics 44, 1125–1155 (2014).
https:/​/​doi.org/​10.1007/​s10701-014-9833-x

[13] Christopher Ferrie and Joseph Emerson. ``Frame representations of quantum mechanics and the necessity of negativity in quasi-probability representations''. Journal of Physics A: Mathematical and Theoretical 41, 352001 (2008).
https:/​/​doi.org/​10.1088/​1751-8113/​41/​35/​352001

[14] A. Jamiołkowski. ``Linear transformations which preserve trace and positive semidefiniteness of operators''. Reports on Mathematical Physics 3, 275 – 278 (1972).
https:/​/​doi.org/​10.1016/​0034-4877(72)90011-0

[15] Asher Peres. ``Separability criterion for density matrices''. Phys. Rev. Lett. 77, 1413–1415 (1996).
https:/​/​doi.org/​10.1103/​PhysRevLett.77.1413

[16] Farid Shahandeh. ``Contextuality of general probabilistic theories''. PRX Quantum 2, 010330 (2021).
https:/​/​doi.org/​10.1103/​PRXQuantum.2.010330

[17] David Schmid, John H. Selby, Matthew F. Pusey, and Robert W. Spekkens. ``A structure theorem for generalized-noncontextual ontological models'' (2020). url: arxiv.org/​abs/​2005.07161.
arXiv:2005.07161

[18] David Avis. ``A revised implementation of the reverse search vertex enumeration algorithm''. Pages 177–198. Birkhäuser Basel. (2000).
https:/​/​doi.org/​10.1007/​978-3-0348-8438-9_9

[19] Anirudh Krishna, Robert W. Spekkens, and Elie Wolfe. ``Deriving robust noncontextuality inequalities from algebraic proofs of the Kochen-Specker theorem: the Peres-Mermin square''. New Journal of Physics 19, 123031 (2017).
https:/​/​doi.org/​10.1088/​1367-2630/​aa9168

[20] Michael J. Panik. ``Fundamentals of convex analysis''. Theory and Decision Library. Springer, Dordrecht. (1993).
https:/​/​doi.org/​10.1007/​978-94-015-8124-0

[21] Aram W. Harrow, Anand Natarajan, and Xiaodi Wu. ``An improved semidefinite programming hierarchy for testing entanglement''. Communications in Mathematical Physics 352, 881–904 (2017).
https:/​/​doi.org/​10.1007/​s00220-017-2859-0

[22] Peter Janotta and Haye Hinrichsen. ``Generalized probability theories: what determines the structure of quantum theory?''. Journal of Physics A: Mathematical and Theoretical 47, 323001 (2014).
https:/​/​doi.org/​10.1088/​1751-8113/​47/​32/​323001

[23] M. Thamban Nair and Arindama Singh. ``Linear algebra''. Springer, Singapore. (2018).
https:/​/​doi.org/​10.1007/​978-981-13-0926-7

[24] Joseph Muscat. ``Functional analysis''. Springer, Cham. (2014).
https:/​/​doi.org/​10.1007/​978-3-319-06728-5

[25] Isaac Namioka and R. R. Phelps. ``Tensor products of compact convex sets.''. Pacific J. Math. 31, 469–480 (1969).
https:/​/​doi.org/​10.2140/​pjm.1969.31-2

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[4] David Schmid, John H. Selby, Matthew F. Pusey, and Robert W. Spekkens, "A structure theorem for generalized-noncontextual ontological models", arXiv:2005.07161, (2020).

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[6] David Schmid, John Selby, Elie Wolfe, Ravi Kunjwal, and Robert W. Spekkens, "The Characterization of Noncontextuality in the Framework of Generalized Probabilistic Theories", arXiv:1911.10386, (2019).

[7] Martin Plávala and Otfried Gühne, "Contextuality as a precondition for entanglement", arXiv:2209.09942, (2022).

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[9] Carlos de Gois, George Moreno, Ranieri Nery, Samuraí Brito, Rafael Chaves, and Rafael Rabelo, "General Method for Classicality Certification in the Prepare and Measure Scenario", PRX Quantum 2 3, 030311 (2021).

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