Noah Linden, Milán Mosonyi, Andreas Winter
We investigate the universal inequalities relating the \alpha-R\'enyi 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. On the other hand, it is known that there are further, non-linear and indeed non-homogeneous, inequalities delimiting the \alpha-entropies of a general quantum state. Finally, we also treat the case of R\'enyi entropies restricted to classical states (i.e. probability distributions), which in addition to non-negativity are also subject to monotonicity. For $\alpha$ different from 0 and 1 we show that this is the only other homogeneous relation.
View original:
http://arxiv.org/abs/1212.0248
No comments:
Post a Comment