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.
Highlights
Classical and quantum physics provide fundamentally different predictions about the correlation which can be observed in experiments with multiple parties
Depending on whether the experiment is modeled using classical random variables, quantum states and measurements, or post-quantum resources, these limitations may be different, leading to observable differences between classical models, quantum mechanics, and general probabilistic theories. This was first pointed out by Bell [1], who found that models which attempt to describe an experiment in terms of causal relations between classical random variables, and where the actions of one party cannot influence the local observations of separate parties, imply restrictions on the observable correlations, known as Bell inequalities
We show here that the Instrumental scenario does provide a separation between the sets of classical, quantum, and generalized probabilistic theories (GPT) correlations
Summary
Classical and quantum physics provide fundamentally different predictions about the correlation which can be observed in experiments with multiple parties. The causal structure, represented by the DAG, constrains the possible correlations between the observed variables, which further depends on whether the hidden variables are classical, quantum or GPT. Given a DAG, one can evaluate the HLP condition If this condition is not satisfied, the sets of classical, quantum, and GPT correlations are equal, i.e., the causal structure represented by the DAG is uninteresting as it does not lead to observable. HLP have applied their criterion to all possible DAGs with 7 nodes or less [15], identifying all DAGs that possibly admit a separation between classical, quantum, and post-quantum correlations They have found a single DAG that is simpler than the Bell DAG, where ‘simple’ means that it involves fewer nodes and edges. We start by describing the instrumental scenario in more detail and relating it to the Bell scenario
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