My research focuses on the von Neumann entropy, and, in particular, on the inequalities which govern the entropies of different subsystems of a multipartite quantum state. These inequalities are important in quantum Shannon theory, and also have applications to quantum entanglement.
There is essentially only one inequality of this type which is known, that of strong subadditivity. The question is: are there any others?
I have also studied similar questions for entropy measures in the family of toy theories knowns as 'generalised probabilistic theories'.
"Infinitely many constrained inequalities for the von Neumann entropy"
"Measurement entropy in Generalized Non-Signalling Theory cannot detect bipartite non-locality"