Characterising and bounding the set of quantum behaviours in contextuality scenarios

Anubhav Chaturvedi; Máté Farkas; Victoria J Wright

doi:10.22331/q-2021-06-29-484

Characterising and bounding the set of quantum behaviours in contextuality scenarios

Anubhav Chaturvedi^1,2, Máté Farkas³, and Victoria J Wright^2,3

¹Institute of Theoretical Physics and Astrophysics, National Quantum Information Centre, Faculty of Mathematics, Physics and Informatics, University of Gdansk, 80-952 Gdansk, Poland
²International Centre for Theory of Quantum Technologies, University of Gdansk, 80-308, Gdansk, Poland
³ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels, Spain

Published:	2021-06-29, volume 5, page 484
Eprint:	arXiv:2010.05853v3
Doi:	https://doi.org/10.22331/q-2021-06-29-484
Citation:	Quantum 5, 484 (2021).

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Abstract

The predictions of quantum theory resist generalised noncontextual explanations. In addition to the foundational relevance of this fact, the particular extent to which quantum theory violates noncontextuality limits available quantum advantage in communication and information processing. In the first part of this work, we formally define contextuality scenarios via prepare-and-measure experiments, along with the polytope of general contextual behaviours containing the set of quantum contextual behaviours. This framework allows us to recover several properties of set of quantum behaviours in these scenarios, including contextuality scenarios and associated noncontextuality inequalities that require for their violation the individual quantum preparation and measurement procedures to be mixed states and unsharp measurements. With the framework in place, we formulate novel semidefinite programming relaxations for bounding these sets of quantum contextual behaviours. Most significantly, to circumvent the inadequacy of pure states and projective measurements in contextuality scenarios, we present a novel unitary operator based semidefinite relaxation technique. We demonstrate the efficacy of these relaxations by obtaining tight upper bounds on the quantum violation of several noncontextuality inequalities and identifying novel maximally contextual quantum strategies. To further illustrate the versatility of these relaxations, we demonstrate $\textit{monogamy of preparation contextuality}$ in a tripartite setting, and present a secure semi-device independent quantum key distribution scheme powered by quantum advantage in parity oblivious random access codes.

Featured image: The landscape of contextual behaviours: The diagram illustrates the relationship between sets of behaviours achievable in different theories in a generic contextuality scenario. In direct analogy with the polytopes of no-signalling and local correlations in Bell experiments, the set of all valid contextual behaviours, $\mathcal{C}$, and the set of noncontextual behaviours, $\mathcal{NC}$, form polytopes. Continuing the analogy, the set of quantum behaviours, $\mathcal{Q}$, although convex, does not form a polytope, and satisfies $\mathcal{NC}\subsetneq\mathcal{Q}\subsetneq\mathcal{C}$. We find contextuality scenarios in which the subsets of behaviours obtained from pure quantum states $\mathcal{Q}^\Psi$ (possibly convex) and projective measurements $\mathcal{Q}^{\Pi}$ (convex) form strict subsets of the set of quantum contextual behaviours, thereby demonstrating that in such scenarios mixed quantum states and unsharp measurements can access greater contextuality. The latter observations constitute significant impediments in the formulation of a hierarchy of semidefinite programming (SDP) relaxations for bounding the set of quantum behaviours in generic contextuality scenarios. We circumvent these difficulties by employing a novel SDP technique entailing a set of moment matrices, each hinged on a preparation, indexed by unitary operators instead of measurement operators.

► BibTeX data

@article{Chaturvedi2021characterising,
  doi = {10.22331/q-2021-06-29-484},
  url = {https://doi.org/10.22331/q-2021-06-29-484},
  title = {Characterising and bounding the set of quantum behaviours in contextuality scenarios},
  author = {Chaturvedi, Anubhav and Farkas, M{\'{a}}t{\'{e}} and Wright, Victoria J},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {5},
  pages = {484},
  month = jun,
  year = {2021}
}

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Cited by

[1] Vaisakh M, Ram krishna Patra, Mukta Janpandit, Samrat Sen, Manik Banik, and Anubhav Chaturvedi, "Mutually unbiased balanced functions and generalized random access codes", Physical Review A 104 1, 012420 (2021).

[2] Jaskaran Singh, Rajendra Singh Bhati, and Arvind, "Revealing quantum contextuality using a single measurement device", Physical Review A 107 1, 012201 (2023).

[3] Armin Tavakoli, Alejandro Pozas-Kerstjens, Ming-Xing Luo, and Marc-Olivier Renou, "Bell nonlocality in networks", Reports on Progress in Physics 85 5, 056001 (2022).

[4] Oskari Kerppo, "Partial-ignorance communication tasks in quantum theory", Physical Review A 105 6, 062607 (2022).

[5] Rafael Santos, Chellasamy Jebarathinam, and Remigiusz Augusiak, "Scalable noncontextuality inequalities and certification of multiqubit quantum systems", Physical Review A 106 1, 012431 (2022).

[6] Hammad Anwer, Natalie Wilson, Ralph Silva, Sadiq Muhammad, Armin Tavakoli, and Mohamed Bourennane, "Noise-robust preparation contextuality shared between any number of observers via unsharp measurements", Quantum 5, 551 (2021).

[7] Costantino Budroni, Adán Cabello, Otfried Gühne, Matthias Kleinmann, and Jan-Åke Larsson, "Kochen-Specker contextuality", Reviews of Modern Physics 94 4, 045007 (2022).

[8] Armin Tavakoli, Emmanuel Zambrini Cruzeiro, Roope Uola, and Alastair A. Abbott, "Bounding and Simulating Contextual Correlations in Quantum Theory", PRX Quantum 2 2, 020334 (2021).

[9] Rafael Wagner, Roberto D. Baldijão, Alisson Tezzin, and Bárbara Amaral, "Using a resource theoretic perspective to witness and engineer quantum generalized contextuality for prepare-and-measure scenarios", arXiv:2102.10469, (2021).

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