Bipartite post-quantum steering in generalised scenarios

The study of stronger-than-quantum effects is a fruitful line of research that provides valuable insight into quantum theory. Unfortunately, traditional bipartite steering scenarios can always be explained by quantum theory. Here we show that, by relaxing this traditional setup, bipartite steering incompatible with quantum theory is possible. The two scenarios we describe, which still feature Alice remotely steering Bob's system, are: (i) one where Bob also has an input and operates on his subsystem, and (ii) the `instrumental steering' scenario. We show that such bipartite post-quantum steering is a genuinely new type of post-quantum nonlocality, which does not follow from post-quantum Bell nonlocality. In addition, we present a method to bound quantum violations of steering inequalities in these scenarios.
Comment: 21 pages, one figure. Comments welcome ! V2: We include a relaxation (characterised by a semidefinite program) of the set of quantum assemblages in the Bob-with-Input scenario