The structure of Rényi entropic inequalities

We investigate the universal inequalities relating the -Rényi entropies of the marginals of a multipartite quantum state. This is in analogy to the same question for the Shannon and von Neumann entropies ( =1), which are known to satisfy several non-trivial inequalities such as strong subadditivity. Somewhat surprisingly, we find for 0< <1 that the only inequality is non-negativity: in other words, any collection of non-negative numbers assigned to the non-empty subsets of n parties can be arbitrarily well approximated by the -entropies of the 2n − 1 marginals of a quantum state. For >1, we show analogously that there are no non-trivial homogeneous (in particular, no linear) inequalities. On the other
hand, it is known that there are further, nonlinear and indeed non-homogeneous, inequalities delimiting the -entropies of a general quantum state. Finally,
we also treat the case of Rényi entropies restricted to classical states (i.e. probability distributions), which, in addition to non-negativity, are also subject to
monotonicity. For 6= 0, 1, we show that this is the only other homogeneous relation.