Towards an Understanding of Quantum and Post-Quantum Correlations in Three Causal Settings

Understanding quantum and post-quantum correlations in various causal settings, is fundamental to singling out the first set from the latter. We present progress in this study in three directions: correlation self-testing of quantum theory, exploration of post-quantum correlations within an indefinite causal order, and certification of nonlocal quantum correlations in the absence of freedom of choice.

Correlation self-testing refers to identifying a set of tasks optimal performance of which, can only be achieved using quantum theory. The adaptive CHSH game, which was proposed as a candidate task, requires a theory to allow for entanglement swapping, if a post-classical performance is to be achieved. Its performance, however, was not explicitly tested in theories that do. In fact, a theory in which this task can be executed better than quantum theory, has also been proposed. We show that this theory does not violate Chained Bell inequalities and therefore can be ruled out. We present adaptive GHZ and adaptive Chained Bell Games and analyse its performance in various theories. Finally, we show that the existing theories that allow for entanglement swapping can also be ruled out using the adaptive CHSH game.

In the second part, we introduce an operational definition of superposition, consistent with quantum theory. We then construct a toy theory that admits superposition, under this definition, and can generate all non-signalling correlations in the Bell setting. We show, that even in an indefinite causal order, in particular, the switch setting, this theory can generate post-quantum correlations. We certify this using theory-independent techniques.

In the final part, we consider a relaxation of parameter independence in the Bell setting by allowing the parties to communicate with each other over a binary symmetric channel. We manage to show, that unless this channel is perfect, all quantum correlations cannot be reproduced using classical strategies. In addition, one way signalling is just as effective as two way signalling, for this channel model.