The structure of Renyi entropic inequalities

Understand

We investigate the universal inequalities relating the alpha-Renyi entropies of the marginals of a multi-partite quantum state.

• This is in analogy to the same question for the Shannon and von Neumann entropy (alpha=1) which are known to satisfy several non-trivial inequalities such as strong subadditivity.
• Somewhat surprisingly, we find for 0<alpha<1, that the only inequality is non-negativity: In other words, any collection of non-negative numbers assigned to the nonempty subsets of n parties can be arbitrarily well approximated by the alpha-entropies of the 2^n-1 marginals of a quantum state.
• For alpha>1 we show analogously that there are no non-trivial homogeneous (in particular no linear) inequalities.