One of the main features of quantum mechanics (QM) is non-locality, that is the impossibility of describing the world using a local hidden variable (LHV) model. This aspect can be studied by using Bell's inequalities, which must be obeyed by any LHV model and are thus a fundamental tool for the investigation of non-locality in quantum mechanical systems.
An intriguing property of quantum non-locality is its monogamy: given three non-signaling observers (Alice, Bob1 and Bob2), it is impossible to have a simultaneous violation of Alice-Bob1 and Alice-Bob2 Bell's inequalities. However, a recent theoretical work  predicts that, if the non-signaling hypothesis is dropped, this restriction no longer holds. Indeed, by sending a two-qubit entangled state to Alice and the Bobs and allowing Bob1 to weakly measure the state before Bob2's measurement, it is possible to have non-local correlations between Alice and both Bobs. Even though there is implicit signaling, given by the fact that the state received by Bob2 depends on Bob1's measurement, the two Bobs are independent, because the don't need to agree on a common measurement strategy and can in principle be unaware of each other's presence.
Here , we experimentally test this hypothesis in the case of a photonic polarization entangled state, by varying the strength of Bob1's weak measurement before Bob2's strong measurement. We test the existence of non-local correlations by evaluating a particular Bell's inequality, proposed by Clauser-Horse-Shimony-Holt (CHSH), independently for Alice-Bob1 and Alice-Bob2. Our results show a large simultaneous violation of the CHSH inequality between Alice and both Bobs.
In addition to its relevance for fundamental physics, our demonstration may have impact also on Quantum Key Distribution or in the certification of Quantum Random Number Generators based on weak measurements.
 Silva et al., PRL 114, 250401 (2015)
 Schiavon et al., Quantum Sci. Technol. 2, 015010 (2017)