We investigate the trade-off between information gain and disturbance for a class of weak von Neumann measurements on spin-$frac{1}{2}$ particles, and derive the unusual measurement pointer state that saturates this trade-off. We then consider the fu
ndamental question of sharing the non-locality of a single particle of an entangled pair among multiple observers, and demonstrate that by exploiting the information gain disturbance trade-off, one can obtain an arbitrarily long sequence of consecutive and independent violations of the CHSH-Bell inequality.
There have been a number of attempts to derive the set of quantum non-local correlations from reasonable physical principles. Here we introduce $tilde{Q}$, a set of multipartite supra-quantum correlations that has appeared under different names in fi
elds as diverse as graph theory, quantum gravity and quantum information science. We argue that $tilde{Q}$ may correspond to the set of correlations of a reasonable physical theory, in which case the research program to reconstruct quantum theory from device-independent principles is met with strong obstacles. In support of this conjecture, we prove that $tilde{Q}$ is closed under classical operations and satisfies the physical principles of Non-Trivial Communication Complexity, No Advantage for Nonlocal Computation, Macroscopic Locality and Local Orthogonality. We also review numerical evidence that almost quantum correlations satisfy Information Causality.
