Quantum violations in the Instrumental scenario and their relations to the Bell scenario

Thomas Van Himbeeck1,2, Jonatan Bohr Brask3, Stefano Pironio1, Ravishankar Ramanathan1, Ana Belén Sainz4,5, and Elie Wolfe4

1Laboratoire d'Information Quantique, Université Libre de Bruxelles, 1050 Bruxelles, Belgium
2Centre for Quantum Information & Communication, Université Libre de Bruxelles, Belgium
3Department of Applied Physics, University of Geneva, 1211 Geneva, Switzerland
4Perimeter Institute for Theoretical Physics, 31 Caroline St. N, Waterloo, Ontario, Canada, N2L 2Y5
5International Centre for Theory of Quantum Technologies, University of Gdańsk, 80-308 Gdańsk, Poland

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The causal structure of any experiment implies restrictions on the observable correlations between measurement outcomes, which are different for experiments exploiting classical, quantum, or post-quantum resources. In the study of Bell nonlocality, these differences have been explored in great detail for more and more involved causal structures. Here, we go in the opposite direction and identify the simplest causal structure which exhibits a separation between classical, quantum, and post-quantum correlations. It arises in the so-called Instrumental scenario, known from classical causal models. We derive inequalities for this scenario and show that they are closely related to well-known Bell inequalities, such as the Clauser-Horne-Shimony-Holt inequality, which enables us to easily identify their classical, quantum, and post-quantum bounds as well as strategies violating the first two. The relations that we uncover imply that the quantum or post-quantum advantages witnessed by the violation of our Instrumental inequalities are not fundamentally different from those witnessed by the violations of standard inequalities in the usual Bell scenario. However, non-classical tests in the Instrumental scenario require fewer input choices than their Bell scenario counterpart, which may have potential implications for device-independent protocols.

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