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Title: The structure of Rényi entropic inequalities
Authors: Linden, N.
Mosonyi, M.
Winter, A. 
Keywords: Entropy inequalities
Homogeneous inequalities
Multi-partite quantum states
Rényi entropies
Issue Date: 8-Oct-2013
Citation: Linden, N., Mosonyi, M., Winter, A. (2013-10-08). The structure of Rényi entropic inequalities. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 469 (2158) : -. ScholarBank@NUS Repository.
Abstract: We investigate the universal inequalities relating the α-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 (α =1), which are known to satisfy several non-trivial inequalities such as strong subadditivity. Somewhat surprisingly, we find for 0 < α < 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 α-entropies of the 2n α 1 marginals of a quantum state. For α >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 α-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 α ≠ 0, 1, we show that this is the only other homogeneous relation. © 2013 The Author(s).
Source Title: Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
ISSN: 13645021
DOI: 10.1098/rspa.2012.0737
Appears in Collections:Staff Publications

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