Checking the strict positivity of Kraus maps is NP-hard

Stéphane Gaubert 1, 2, 3 Zheng Qu 4
1 TROPICAL - TROPICAL
Inria Saclay - Ile de France
2 MAXPLUS - Max-plus algebras and mathematics of decision
CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique, Inria Saclay - Ile de France, X - École polytechnique, CNRS - Centre National de la Recherche Scientifique : UMR
Abstract : Basic properties in Perron-Frobenius theory are positivity, primitivity, and irreducibility. Whereas these properties can be checked in polynomial time for stochastic matrices, we show that for Kraus maps - the noncommutative generalization of stochastic matrices - checking positivity is NP-hard. This is in contrast with irreducibility and primitivity, which we show to be checkable in strongly polynomial time for completely positive maps - the noncommutative generalization of nonpositive matrices. As an intermediate result, we get that the bilinear feasibility problem over $\mathbb{Q}$ is NP-hard.
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Information Processing Letters, Elsevier, 2017, 118, pp.35--43. 〈10.1016/j.ipl.2016.09.008〉
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https://hal.inria.fr/hal-01097942
Contributeur : Stephane Gaubert <>
Soumis le : lundi 22 décembre 2014 - 14:45:02
Dernière modification le : jeudi 10 mai 2018 - 02:04:00

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Stéphane Gaubert, Zheng Qu. Checking the strict positivity of Kraus maps is NP-hard. Information Processing Letters, Elsevier, 2017, 118, pp.35--43. 〈10.1016/j.ipl.2016.09.008〉. 〈hal-01097942〉

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