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

TL;DR

This paper identifies the simplest causal structure, the Instrumental scenario, that shows a separation between classical, quantum, and post-quantum correlations, and relates its inequalities to Bell inequalities, with implications for device-independent protocols.

Contribution

It derives inequalities for the Instrumental scenario, relates them to Bell inequalities, and highlights their potential for simpler non-classical tests in quantum information.

Findings

Instrumental inequalities are closely related to Bell inequalities.

Quantum and post-quantum bounds can be identified for the Instrumental scenario.

Fewer input choices are needed in the Instrumental scenario for non-classical tests.

Abstract

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.