Hi all,
today Vilasini is visiting from York and will tell us about causal
structures and Tsallis entropies. See the title and abstract below. Note
that we're starting a little later today and in a different room than
usual.
Best,
-joe
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.