The results of measurements performed locally on entangled quantum systems shared among multiple parties can be correlated in ways which are inexplicable by any classical mechanism. This phenomenon, known as Bell nonlocality, is a fundamental and fascinating aspect of quantum theory .
What is meant by classically inexplicable? It means that there is no classical model (often called a local hidden variable model) with the same underlying causal structure that can reproduce the quantum predictions. The causal structure is the implicit geometry of the setup – for example the fact that each party’s local measurement result is independent of the other parties’ measurement choices (known as no-signaling), or that the measurement choices are themselves independent of everything else (known as measurement independence).
If we consider trying to reproduce quantum predictions using classical mechanisms with relaxed causal structure, then indeed they can often do so. For example, if there is communication among all the parties, then this classical mechanism can readily reproduce any quantum correlation, nonlocal or not. What is remarkable is that some seemingly powerful causal relaxations still cannot reproduce all of the predictions of quantum theory. For example, if the communication is restricted to all but one of the parties, then this is not able to reproduce everything that can be achieved by making local measurements on multipartite entangled states . The nonlocality of such ‘genuine multipartite nonlocal’ correlations is therefore shown to be very strong, as highlighted by the difficultly of classically reproducing them, even given much more freedom.
One barrier to having a systematic study of causal relaxations is that the number of relaxations grows exponentially in the number of parties. At least it seemed to naively. The main result of the work of Chaves, Cavalcanti and Aolita  is to identify that large classes of causal relaxations are in fact equivalent to each other, as far as the non-signaling correlations they can produce are concerned. What is really of interest then is the number of different inequivalent classes of causal relaxations, which they show is much smaller, and has a natural hierarchical structure, depending on the total number of relaxations introduced.
Focusing on the tripartite scenario, they demonstrate the power of their result by showing that there are in fact only 8 different inequivalent classes of causal relaxations which are interesting – ones which are not powerful enough to reproduce all non-signaling correlations. Previous results had shown that quantum correlations are inexplicable by 6 of these classes . Of the remaining two classes, the first, which sits at the top of the hierarchy, and is termed the star (since one party receives communication from all others), is shown, somewhat astonishingly, not to be able to reproduce all quantum nonlocal correlations, despite being the strongest possible causal relaxation. The second class, termed the circle (since each party communicates to their neighbour in a circular fashion), is left as the intriguing class – known neither to reproduce all quantum correlations nor whether it falls short.
The true power of the results comes in the unifying and simplified view they provide for studying relaxed causal structures. What was previously a vast forest is now a well organised playground, ready to be explored, and played in, as we continue to push forward our understanding of quantum non-locality.
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