Solvable Criterion for the Contextuality of any Prepare-and-Measure Scenario
Institute for Theoretical Physics, ETH Zürich, Switzerland
| Published: | 2022-06-07, volume 6, page 732 |
| Eprint: | arXiv:2003.06426v4 |
| Doi: | https://doi.org/10.22331/q-2022-06-07-732 |
| Citation: | Quantum 6, 732 (2022). |
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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.
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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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[1] Martin Plávala and Otfried Gühne, "Contextuality as a Precondition for Quantum Entanglement", Physical Review Letters 132 10, 100201 (2024).
[2] Taira Giordani, Rafael Wagner, Chiara Esposito, Anita Camillini, Francesco Hoch, Gonzalo Carvacho, Ciro Pentangelo, Francesco Ceccarelli, Simone Piacentini, Andrea Crespi, Nicolò Spagnolo, Roberto Osellame, Ernesto F. Galvão, and Fabio Sciarrino, "Experimental certification of contextuality, coherence, and dimension in a programmable universal photonic processor", Science Advances 9 44, eadj4249 (2023).
[3] John H. Selby, Elie Wolfe, David Schmid, Ana Belén Sainz, and Vinicius P. Rossi, "Linear Program for Testing Nonclassicality and an Open-Source Implementation", Physical Review Letters 132 5, 050202 (2024).
[4] David Schmid, John H. Selby, Matthew F. Pusey, and Robert W. Spekkens, "A structure theorem for generalized-noncontextual ontological models", Quantum 8, 1283 (2024).
[5] Markus P. Müller and Andrew J. P. Garner, "Testing Quantum Theory by Generalizing Noncontextuality", Physical Review X 13 4, 041001 (2023).
[6] 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", Physical Review A 107 6, 062203 (2023).
[7] Roberto D. Baldijão, Rafael Wagner, Cristhiano Duarte, Bárbara Amaral, and Marcelo Terra Cunha, "Emergence of Noncontextuality under Quantum Darwinism", PRX Quantum 2 3, 030351 (2021).
[8] 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).
[9] Martin Plávala, "Incompatibility in restricted operational theories: connecting contextuality and steering", Journal of Physics A Mathematical General 55 17, 174001 (2022).
[10] 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).
[11] Sergii Strelchuk and Mischa P. Woods, "Measuring time with stationary quantum clocks", arXiv:2106.07684, (2021).
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