Hi all,
Title:
Tsallis entropies and causal structures

Abstract:
Distinguishing between classical and non-classical correlations that could arise from a given causal structure is a central problem in the study of causality which has applications in device independent quantum cryptography. The entropy vector method is often employed for tackling this problem and so far, the Shannon entropy has been used in this method for analysing causal structures, but it has been found to have limitations in certifying non-classicality for certain causal structures.  

A natural question that arises is whether using other entropic measures can avoid these limitations. We discuss the use of Tsallis entropies for this task; at first they appear to be a good candidate for the task as they satisfy certain desirable properties for the entropy vector method and one might expect them to provide such advantages for other causal structures as well.

However, in the presence of a causal structure, we find that the Tsallis entropies do not satisfy the simple conditional independence constraint that the Shannon entropy does. We derive the corresponding causal constraints satisfied by the Tsallis entropies for some specific cases, conjecture the general causal condition for these entropies and provide various evidence supporting this conjecture. Even if we assume that these causal constraints hold, the computational task of projecting to the marginal polytope to obtain new entropic inequalities becomes too time consuming for relatively small causal structures such as the bipartite Bell scenario. We then analyse the post-selected bipartite Bell scenario with 2 inputs and 3 outputs per party, predicting the existence of a family of non-local correlations whose non-locality may be undetectable by both Shannon and Tsallis entropic inequalities. Contrary to expectation, our work indicates that Tsallis entropies may not be useful for analysing causal structures with present techniques and computational power.