## 2 Various Questions on Quantum Non-Locality
This introduces another significant line of research, i.e., the quantification of non-locality, an important argument when this is considered as a resource for quantum technologies, but also for better understanding its fundamental meaning. A study related to the quantification of nonclassicality and focused on the global impact of local unitary evolutions [61] puts in evidence that only for those composite quantum systems exhibiting quantum correlations (evaluated in the context of quantum-versus-classical paradigm [62]) the global state can undergo a transformation operated by means of any nontrivial unitary evolution; this property holds also when states non violating the Bell inequality are considered. More recently it was demonstrated [63] that all entangled states allow implementing a quantum channel, which cannot be reproduced classically, in particular teleportation according to a specific \"teleportation witnesss\". A relevant way to quantify non-locality [64; 65] takes into account the minimal amount of information necessary to preserve quantum correlation in a communication exchange between two parties, which in the particular case of a 2-qubit maximally entangled system undergoing projective measurements this quantification is accounted to be exactly 1 bit (on the contrary the situation can be different when the system is non-maximal entangled, a matter still under debate). On a different perspective it is founded the argument detailed in [66], considering general hidden variable models that can be expressed in terms of both local and non-local parts, and demonstrating the existence of experimentally verifiable quantum correlations that are incompatible with any hidden variable model having a nontrivial local part. Following this reasoning, it is then conceivable that if the non-local HVs are not considered, measurement results rely exclusively on local HV parameters and experimental settings. In this sense, this is equivalent to saying that quantum mechanics cannot be mimicked by HV theories provided with a local component. The following reported approach started by considering that in a typical Bell experiment, correlations are observed between the measurement results of [PERSON] and Bob and averages are evaluated over measurements of many pairs of particles, but when non-locality appears from statistics, this does not directly means that all individual pairs behave non-locally. This inspired the reasoning [67] for which a set of the pairs behaves locally while another fraction in a non-local way. In other terms, it was argued that the relative quantum distribution probability \(P_{Q}(a,b|A,B,\rho)\) for a given quantum state \(\rho\) can be expressed in terms of local (\(P_{L}\)) and non-local (\(P_{NL}\)) parts, where respectively the \(P_{L}\) satisfies Bell inequalities and the other \(P_{NL}\) does not depend on outcomes or experimental settings:  
\[P_{Q}=p_{L}(\rho)\cdot P_{L}+[1-p_{L}(\rho)]\cdot P_{NL}. \tag{4}\]  
A tricky problem resides in determining the weight of the local part (that represents in some sense the measure of locality for \(P_{Q}\) ) and in finding a suitable way to minimize it. A step in this direction was the demonstration that the local part is identically null for maximally entangled 2-qubit systems, returning a complete non-local behaviour, while an investigation analyzing non-maximally pure 2-qubit and 3-qubit states can be found in [68; 69]. Instead, focusing on the non-local component of Equation (4) and making use of non-local resources uniquely, with no-signaling constraint, it is possible to simulate correlations for all pure 2-qubit entangled systems without communication [70], therefore exploiting no-signaling resources exclusively. A similar decomposition approach for contextuality has been validated in [71]. Taking advantage of the general way to express any q-qubit pure state in the form \(|\Psi_{\theta}\rangle=\cos(\theta)|00\rangle+\sin(\theta)|11\rangle\), Ref.