## 2 Various Questions on Quantum Non-Locality  
Quantum non-locality and entanglement exhibit a strict and deep interrelation, being at the same time two well distinct concepts, and the understanding of the differences in a quantitative terms deserves attention in order to clarify the balance between these two inequivalent resources. As a starting point, an interesting aspect to be considered in this chapter concerns whether QM violates relativistic locality, and on the consequence if a superluminal influence between entangled entities can be a suitable explanation for quantum non-locality. It is important to notice that a preferred reference frame is not in contrast to special relativity and in particular that a universal preferred reference frame has been already observed, being the cosmic microwave background. Ref. [45] posed a lower limit of more than ten thousand time the speed of light in the vacuum. This limit has been recently enhanced [46] to \(\beta>5\cdot 10^{6}\). In addition, the analysis reported in [47] shows that this hidden influence, although propagating to a subluminal speed in a preferred reference frame, does it allow always for superluminal communications: this peculiarity refers to four-partite systems, but it is not valid for bipartite states. On the other side, this is not true for all the other models [9] (including de Broglie-Bohm and GRW [48]) for which the propagation in space of the correlations is not continuous. Other interesting facets of the argument, from which it results evident that the exact relation between entanglement and nonlocality is still poorly understood, refers for example to the analytic demonstration [49] that quantum non-locality does not imply distillability of quantum correlations: this was deduced by considering a 3-qubit entangled state that is separable along any bipartition, but nevertheless violates a Bell inequality. Another case refers to the reasoning of [50], where it is demonstrated that despite a particular class of states can exhibit entanglement without violating Bell inequalities, furthermore it is possible to obtain more non-locality with less entanglement, not only at the qubit level [12] but also when there is no restriction on the size of the entanglement [51; 52; 53], a situation experimentally investigated in [54]. An argument related to states that exhibit on one side a maximal non-locality without possessing a maximum degree of entanglement has been investigated in [55]. This study put in evidence three possible scenarios plausible with this situation:  
1. the case [12] in which states that are non-maximally entangled violate Bell inequalities when using single-particle detectors with non-ideal quantum efficiency below 82%;
2. for non-maximally entangled states, the Kullback-Leibler distance with the closest local distribution [56] is larger;
3. the simulation of entanglement with non-local resources is more favorable when maximally entangled states are considered rather then the non-maximally ones. A peculiar property analyzed in [57] shows that the totality of non-fully-separable states with an arbitrary dimension in the Hilbert space and number of parties, violates a Bell inequality if mixed with another state that, if taken alone, could not act a violation of the same Bell inequality. From another study it appeared evident that an emerging quantum non-locality [58; 59] is always unveiled by mixing a \(\sigma_{AB}\) state not violating Bell inequalities with any entangled state \(\rho_{AB}\), a peculiar trait still to be seized in detail for potential exploitation in quantum communication protocols. In the work reported in [60], an interesting comparison is analyzed between the resource theories of non-locality (local operations and shared randomness (LOSR) paradigm) and entanglement (local operations and classical communication (LOCC) paradigm), and it is shown how a class of non-local games can witness quantum entanglement, however weak, and reveal non-locality in any entangled quantum state.