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arXiv · 1212.0248

The structure of Renyi entropic inequalities

Noah Linden·Milán Mosonyi·Andreas Winter
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Abstract

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 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 Renyi 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.

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Noah Linden, Milán Mosonyi, Andreas Winter. 2012-12-07. The structure of Renyi entropic inequalities. https://doi.org/10.1098/rspa.2012.0737

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